(* 0 Function definition list: An(x1) = An(x1) Zero = Zero One = One x1 + x2 = x1 + x2 x1 * x2 = x1 * x2 Minus(x1) = Minus(x1) Inv(x1) = Inv(x1) Sup(x1) = Sup(x1) Nat = {x1|(Ax2. Zero E x2 & (Ax3. x3 E x2 -> x3 + One E x2) -> x1 E x2)} Reals = {x1|Real(x1)} x1 Closure x2 = {x3|(Ax4. x1 E x4 & (Ax5. p1(x5) E x4 & x5 E x2 -> p2(x5) E x4) -> x3 E x4)} x1 Union x2 = {x3|x3 E x1 v x3 E x2} Sing(x1) = {x2|x2 = x1} x1 Couple x2 = {x3|x3 = x1 v x3 = x2} x1 Kpair x2 = Sing(x1) Couple x1 Couple x2 Si(x1) = {x2|(Ax3. x3 E x1 -> x2 E x3)} Projtwo(x1) = {x2| (Ex3.x2 E x3 & x3 E x1) & (Ax3.(Ax4. x2 E x3 & x2 E x4 & x3 E x1 & x4 E x1 -> x3 = x4))} Three = One + One + One Two = One + One Half = Inv(Two)*) (* Sequent snapshot: 1: (Ax2. Zero < x2 & Real(x2) & x2 < Sup({x3|Zero < x3 & x3 * x3 < a1}) -> [a1 + Minus(Sup({x70| Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv( Three * Sup({x72|Zero < x72 & x72 * x72 < a1})) <= x2) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) *) (* 0 Predicate definition list: Real(x1) == Real(x1) x1 < x2 == x1 < x2 x1 Equal x2 == x1 + Zero = x2 + Zero x1 <= x2 == x1 Equal x2 v x1 < x2 x1 Function x2 == (Ax3. x3 E x1 -> p1(x3) E p1(x2) & p2(x3) E p2(x2)) & (Ax3.(Ax4.(Ax5. [x3,x5] E x1 & [x4,x5] E x1 -> x3 = x4))) Sequence(x1) == x1 Function [Nat,Reals]*) (* Sequent snapshot: 1: (Ax2. Zero < x2 & Real(x2) & x2 < Sup({x3|Zero < x3 & x3 * x3 < a1}) -> [a1 + Minus(Sup({x70| Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv( Three * Sup({x72|Zero < x72 & x72 * x72 < a1})) <= x2) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) *) (* 0 -------------------- Not Proved ------------------ Line 1: |- 1: (Ax1.Real(x1) & Zero < x1 -> (Ex2.x2 * x2 = x1)) By 2 -------------------- Not Proved ------------------ Line 2: |- 1: Real(a1) & Zero < a1 -> (Ex3.x3 * x3 = a1) By 3 -------------------- Not Proved ------------------ Line 3: 1: Real(a1) & Zero < a1 |- 1: (Ex3.x3 * x3 = a1) By 4 -------------------- Not Proved ------------------ Line 4: 1: Real(a1) 2: Zero < a1 |- 1: (Ex3.x3 * x3 = a1) By 5 -------------------- Not Proved ------------------ Line 5: 1: Real(a1) 2: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 7, 6 -------------------- Not Proved ------------------ Line 7: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) Equal a1 v Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1}) < a1 v a1 < Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1}) 2: Real(a1) 3: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 9, 8 -------------------- Not Proved ------------------ Line 9: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 v a1 < Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1}) 2: Real(a1) 3: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 15, 16 -------------------- Not Proved ------------------ Line 15: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 2: Real(a1) 3: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 18, 17 -------------------- Not Proved ------------------ Line 18: 1: (Ax5. x5 E {x7|Zero < x7 & x7 * x7 < a1} -> x5 <= a1 + One) -> (Ax8. x8 E {x10| Zero < x10 & x10 * x10 < a1} -> x8 <= Sup({x11|Zero < x11 & x11 * x11 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 20, 19 -------------------- Not Proved ------------------ Line 20: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) By 60, 59 -------------------- Not Proved ------------------ Line 60: 1: (Ax2. Zero < x2 & Real(x2) & x2 < Sup({x3|Zero < x3 & x3 * x3 < a1}) -> [a1 + Minus(Sup({x70| Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv( Three * Sup({x72|Zero < x72 & x72 * x72 < a1})) <= x2) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 2: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 59: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 |- 1: (Ax2. Zero < x2 & Real(x2) & x2 < Sup({x3|Zero < x3 & x3 * x3 < a1}) -> [a1 + Minus(Sup({x70| Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv( Three * Sup({x72|Zero < x72 & x72 * x72 < a1})) <= x2) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 61 -------------------- Proved ---------------------- Line 61: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 |- 1: Zero < a23 & Real(a23) & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> [a1 + Minus(Sup({x71| Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv( Three * Sup({x73|Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 62 -------------------- Proved ---------------------- Line 62: 1: Zero < a23 & Real(a23) & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 63 -------------------- Proved ---------------------- Line 63: 1: Real(a23) & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 64 -------------------- Proved ---------------------- Line 64: 1: Real(a23) 2: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 5: Real(a1) 6: Zero < a1 7: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 66, 65 -------------------- Proved ---------------------- Line 66: 1: Zero < a23 -> Zero + Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x3| Zero < x3 & x3 * x3 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 68, 67 -------------------- Proved ---------------------- Line 68: 1: Zero + Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x3| Zero < x3 & x3 * x3 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 70, 71, 69 -------------------- Proved ---------------------- Line 70: 1: Zero + Sup({x3|Zero < x3 & x3 * x3 < a1}) = Sup({x3| Zero < x3 & x3 * x3 < a1}) 2: Zero + Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x3| Zero < x3 & x3 * x3 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 72 -------------------- Proved ---------------------- Line 72: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 74, 75, 76, 77, 73 -------------------- Proved ---------------------- Line 74: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 97, 96 -------------------- Proved ---------------------- Line 97: 1: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] v a1 < [a23 + Sup({x3| Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] v [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 99, 98 -------------------- Proved ---------------------- Line 99: 1: a1 < [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] v [a23 + Sup({x3| Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 100, 101 -------------------- Proved ---------------------- Line 100: 1: a1 < [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 261, 260 -------------------- Proved ---------------------- Line 261: 1: [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: a1 < [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 262 -------------------- Proved ---------------------- Line 262: 1: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 264, 265, 266, 263 -------------------- Proved ---------------------- Line 264: 1: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 268, 267 -------------------- Proved ---------------------- Line 268: 1: a1 < a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) & a23 * a23 + Two * a23 * Sup({x9|Zero < x9 & x9 * x9 < a1}) + Sup({x10| Zero < x10 & x10 * x10 < a1}) * Sup({x11| Zero < x11 & x11 * x11 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) -> a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 3: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 270, 269 -------------------- Proved ---------------------- Line 270: 1: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 274, 273 -------------------- Proved ---------------------- Line 274: 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) < a1 -> Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 276, 275 -------------------- Proved ---------------------- Line 276: 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 278, 277 -------------------- Proved ---------------------- Line 278: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) = Zero 2: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 279 -------------------- Proved ---------------------- Line 279: 1: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 2: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 281, 280 -------------------- Proved ---------------------- Line 281: 1: a1 < a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1}) -> a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < [a23 * Sup({x10|Zero < x10 & x10 * x10 < a1}) + Two * a23 * Sup({x11| Zero < x11 & x11 * x11 < a1}) + Sup({x12|Zero < x12 & x12 * x12 < a1}) * Sup({x13|Zero < x13 & x13 * x13 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 283, 282 -------------------- Proved ---------------------- Line 283: 1: a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1})] + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 285, 284 -------------------- Proved ---------------------- Line 285: 1: [a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1})] + Minus(Sup({x11| Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1})) = a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + [Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x14| Zero < x14 & x14 * x14 < a1}) * Sup({x15| Zero < x15 & x15 * x15 < a1})] + Minus( Sup({x16|Zero < x16 & x16 * x16 < a1}) * Sup({x17|Zero < x17 & x17 * x17 < a1})) 2: a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1})] + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 286 -------------------- Proved ---------------------- Line 286: 1: a1 + Minus( Sup({x11|Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1})) < a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + [ Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x14| Zero < x14 & x14 * x14 < a1}) * Sup({x15| Zero < x15 & x15 * x15 < a1})] + Minus( Sup({x16|Zero < x16 & x16 * x16 < a1}) * Sup({x17|Zero < x17 & x17 * x17 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 288, 287 -------------------- Proved ---------------------- Line 288: 1: [Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x18|Zero < x18 & x18 * x18 < a1}) * Sup({x19|Zero < x19 & x19 * x19 < a1})) = Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x20| Zero < x20 & x20 * x20 < a1}) * Sup({x21| Zero < x21 & x21 * x21 < a1}) + Minus(Sup({x22| Zero < x22 & x22 * x22 < a1}) * Sup({x23| Zero < x23 & x23 * x23 < a1})) 2: a1 + Minus( Sup({x11|Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1})) < a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + [ Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x14| Zero < x14 & x14 * x14 < a1}) * Sup({x15| Zero < x15 & x15 * x15 < a1})] + Minus( Sup({x16|Zero < x16 & x16 * x16 < a1}) * Sup({x17|Zero < x17 & x17 * x17 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 289 -------------------- Proved ---------------------- Line 289: 1: a1 + Minus( Sup({x20|Zero < x20 & x20 * x20 < a1}) * Sup({x21| Zero < x21 & x21 * x21 < a1})) < a23 * Sup({x22|Zero < x22 & x22 * x22 < a1}) + Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x23| Zero < x23 & x23 * x23 < a1}) * Sup({x24| Zero < x24 & x24 * x24 < a1}) + Minus(Sup({x25| Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 291, 290 -------------------- Proved ---------------------- Line 291: 1: Sup({x25|Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1}) + Minus(Sup({x25| Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) = Zero 2: a1 + Minus( Sup({x20|Zero < x20 & x20 * x20 < a1}) * Sup({x21| Zero < x21 & x21 * x21 < a1})) < a23 * Sup({x22|Zero < x22 & x22 * x22 < a1}) + Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x23| Zero < x23 & x23 * x23 < a1}) * Sup({x24| Zero < x24 & x24 * x24 < a1}) + Minus(Sup({x25| Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 292 -------------------- Proved ---------------------- Line 292: 1: a1 + Minus( Sup({x21|Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1})) < a23 * Sup({x23|Zero < x23 & x23 * x23 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Zero 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 294, 295, 293 -------------------- Proved ---------------------- Line 294: 1: Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Zero = Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 2: a1 + Minus( Sup({x21|Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1})) < a23 * Sup({x23|Zero < x23 & x23 * x23 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Zero 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 296 -------------------- Proved ---------------------- Line 296: 1: a1 + Minus( Sup({x25|Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) < a23 * Sup({x27|Zero < x27 & x27 * x27 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 298, 299, 297 -------------------- Proved ---------------------- Line 298: 1: a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) = Three * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 2: a1 + Minus( Sup({x25|Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) < a23 * Sup({x27|Zero < x27 & x27 * x27 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 300 -------------------- Proved ---------------------- Line 300: 1: a1 + Minus( Sup({x27|Zero < x27 & x27 * x27 < a1}) * Sup({x28| Zero < x28 & x28 * x28 < a1})) < Three * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 302, 301 -------------------- Proved ---------------------- Line 302: 1: a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) = Sup({x26| Zero < x26 & x26 * x26 < a1}) * a23 2: a1 + Minus( Sup({x27|Zero < x27 & x27 * x27 < a1}) * Sup({x28| Zero < x28 & x28 * x28 < a1})) < Three * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 303 -------------------- Proved ---------------------- Line 303: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 305, 304 -------------------- Proved ---------------------- Line 305: 1: Zero < Inv(Three * Sup({x34|Zero < x34 & x34 * x34 < a1})) & a1 + Minus(Sup({x30| Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x35|Zero < x35 & x35 * x35 < a1}) * a23 -> [ a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1}))] * Inv(Three * Sup({x36| Zero < x36 & x36 * x36 < a1})) < [ Three * Sup({x37|Zero < x37 & x37 * x37 < a1}) * a23] * Inv( Three * Sup({x38|Zero < x38 & x38 * x38 < a1})) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 307, 306 -------------------- Proved ---------------------- Line 307: 1: [a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1}))] * Inv(Three * Sup({x34| Zero < x34 & x34 * x34 < a1})) < [ Three * Sup({x35|Zero < x35 & x35 * x35 < a1}) * a23] * Inv( Three * Sup({x36|Zero < x36 & x36 * x36 < a1})) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 323, 322 -------------------- Proved ---------------------- Line 323: 1: [Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23] * Inv( Three * Sup({x38|Zero < x38 & x38 * x38 < a1})) = Three * [ Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x40|Zero < x40 & x40 * x40 < a1})) 2: [a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1}))] * Inv(Three * Sup({x34| Zero < x34 & x34 * x34 < a1})) < [ Three * Sup({x35|Zero < x35 & x35 * x35 < a1}) * a23] * Inv( Three * Sup({x36|Zero < x36 & x36 * x36 < a1})) 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 324 -------------------- Proved ---------------------- Line 324: 1: [a1 + Minus( Sup({x39|Zero < x39 & x39 * x39 < a1}) * Sup({x40| Zero < x40 & x40 * x40 < a1}))] * Inv(Three * Sup({x41| Zero < x41 & x41 * x41 < a1})) < Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 326, 325 -------------------- Proved ---------------------- Line 326: 1: [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x45|Zero < x45 & x45 * x45 < a1})) = Sup({x42| Zero < x42 & x42 * x42 < a1}) * a23 * Inv(Three * Sup({x46|Zero < x46 & x46 * x46 < a1})) 2: [a1 + Minus( Sup({x39|Zero < x39 & x39 * x39 < a1}) * Sup({x40| Zero < x40 & x40 * x40 < a1}))] * Inv(Three * Sup({x41| Zero < x41 & x41 * x41 < a1})) < Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 327 -------------------- Proved ---------------------- Line 327: 1: [a1 + Minus( Sup({x46|Zero < x46 & x46 * x46 < a1}) * Sup({x47| Zero < x47 & x47 * x47 < a1}))] * Inv(Three * Sup({x48| Zero < x48 & x48 * x48 < a1})) < Three * Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23 * Inv( Three * Sup({x49|Zero < x49 & x49 * x49 < a1})) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 329, 328 -------------------- Proved ---------------------- Line 329: 1: a23 * Inv(Three * Sup({x50|Zero < x50 & x50 * x50 < a1})) = Inv( Three * Sup({x51|Zero < x51 & x51 * x51 < a1})) * a23 2: [a1 + Minus( Sup({x46|Zero < x46 & x46 * x46 < a1}) * Sup({x47| Zero < x47 & x47 * x47 < a1}))] * Inv(Three * Sup({x48| Zero < x48 & x48 * x48 < a1})) < Three * Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23 * Inv( Three * Sup({x49|Zero < x49 & x49 * x49 < a1})) 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 330 -------------------- Proved ---------------------- Line 330: 1: [a1 + Minus( Sup({x51|Zero < x51 & x51 * x51 < a1}) * Sup({x52| Zero < x52 & x52 * x52 < a1}))] * Inv(Three * Sup({x53| Zero < x53 & x53 * x53 < a1})) < Three * Sup({x54|Zero < x54 & x54 * x54 < a1}) * Inv(Three * Sup({x55| Zero < x55 & x55 * x55 < a1})) * a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 332, 331 -------------------- Proved ---------------------- Line 332: 1: [Three * Sup({x55|Zero < x55 & x55 * x55 < a1})] * Inv( Three * Sup({x57|Zero < x57 & x57 * x57 < a1})) * a23 = Three * Sup({x58|Zero < x58 & x58 * x58 < a1}) * Inv( Three * Sup({x59|Zero < x59 & x59 * x59 < a1})) * a23 2: [a1 + Minus( Sup({x51|Zero < x51 & x51 * x51 < a1}) * Sup({x52| Zero < x52 & x52 * x52 < a1}))] * Inv(Three * Sup({x53| Zero < x53 & x53 * x53 < a1})) < Three * Sup({x54|Zero < x54 & x54 * x54 < a1}) * Inv(Three * Sup({x55| Zero < x55 & x55 * x55 < a1})) * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 333 -------------------- Proved ---------------------- Line 333: 1: [a1 + Minus( Sup({x58|Zero < x58 & x58 * x58 < a1}) * Sup({x59| Zero < x59 & x59 * x59 < a1}))] * Inv(Three * Sup({x60| Zero < x60 & x60 * x60 < a1})) < [ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x62|Zero < x62 & x62 * x62 < a1})) * a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 335, 334 -------------------- Proved ---------------------- Line 335: 1: [ [Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x63|Zero < x63 & x63 * x63 < a1}))] * a23 = [Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv(Three * Sup({x64|Zero < x64 & x64 * x64 < a1})) * a23 2: [a1 + Minus( Sup({x58|Zero < x58 & x58 * x58 < a1}) * Sup({x59| Zero < x59 & x59 * x59 < a1}))] * Inv(Three * Sup({x60| Zero < x60 & x60 * x60 < a1})) < [ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x62|Zero < x62 & x62 * x62 < a1})) * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 336 -------------------- Proved ---------------------- Line 336: 1: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 338, 337 -------------------- Proved ---------------------- Line 338: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero v [ Three * Sup({x67|Zero < x67 & x67 * x67 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) = One 2: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 339, 340 -------------------- Proved ---------------------- Line 339: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero 2: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 353, 354, 355, 356, 352 -------------------- Proved ---------------------- Line 353: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) = Zero 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 358, 357 -------------------- Proved ---------------------- Line 358: 1: [Three * Sup({x33|Zero < x33 & x33 * x33 < a1})] * a23 = Three * Sup({x34|Zero < x34 & x34 * x34 < a1}) * a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 15: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) = Zero |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 359 -------------------- Proved ---------------------- Line 359: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) = Zero 2: a1 + Minus( Sup({x34|Zero < x34 & x34 * x34 < a1}) * Sup({x35| Zero < x35 & x35 * x35 < a1})) < [ Three * Sup({x36|Zero < x36 & x36 * x36 < a1})] * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 360 -------------------- Proved ---------------------- Line 360: 1: a1 + Minus( Sup({x35|Zero < x35 & x35 * x35 < a1}) * Sup({x36| Zero < x36 & x36 * x36 < a1})) < Zero * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 13: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 362, 361 -------------------- Proved ---------------------- Line 362: 1: Zero * a23 = Zero 2: a1 + Minus( Sup({x35|Zero < x35 & x35 * x35 < a1}) * Sup({x36| Zero < x36 & x36 * x36 < a1})) < Zero * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 363 -------------------- Proved ---------------------- Line 363: 1: a1 + Minus( Sup({x36|Zero < x36 & x36 * x36 < a1}) * Sup({x37| Zero < x37 & x37 * x37 < a1})) < Zero 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) |- By NOTBOTH Proof of NOTBOTH begins -------------------- Proved ---------------------- Line 46.3: 1: a2 < a1 2: a1 < a2 |- By 4, 5 -------------------- Proved ---------------------- Line 46.4: |- 1: a2 < a1 & a1 < a2 -> a2 < a2 By NOTBOTH.TRANS Proof of NOTBOTH.TRANS begins -------------------- Proved ---------------------- Line 19.19: |- 1: a1 < a2 & a2 < a3 -> a1 < a3 By NOTBOTH.TRANS.TRANSTheorem reference error Proof of NOTBOTH.TRANS ends -------------------- Proved ---------------------- Line 46.5: 1: a2 < a1 & a1 < a2 -> a2 < a2 2: a2 < a1 3: a1 < a2 |- By 6, 7 -------------------- Proved ---------------------- Line 46.6: 1: a2 < a1 2: a1 < a2 |- 1: a2 < a1 & a1 < a2 By 8, 9 -------------------- Trivial --------------------- Line 46.8: 1: a2 < a1 |- 1: a2 < a1 -------------------- Trivial --------------------- Line 46.9: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 46.7: 1: a2 < a2 |- By 10, 11 -------------------- Proved ---------------------- Line 46.10: |- 1: ~a2 < a2 By NOTBOTH.IRR Proof of NOTBOTH.IRR begins -------------------- Proved ---------------------- Line 17.17: |- 1: ~a1 < a1 By NOTBOTH.IRR.IRRTheorem reference error Proof of NOTBOTH.IRR ends -------------------- Proved ---------------------- Line 46.11: 1: ~a2 < a2 2: a2 < a2 |- By 12 -------------------- Trivial --------------------- Line 46.12: 1: a2 < a2 |- 1: a2 < a2 Proof of NOTBOTH ends -------------------- Proved ---------------------- Line 361: |- 1: Zero * a23 = Zero By MZERO2 Proof of MZERO2 begins -------------------- Proved ---------------------- Line 43.1: |- 1: Zero * a1 = Zero By 2, 3 -------------------- Proved ---------------------- Line 43.2: |- 1: Zero * a1 = a1 * Zero By MZERO2.CTIMES Proof of MZERO2.CTIMES begins -------------------- Proved ---------------------- Line 8.8: |- 1: a1 * a2 = a2 * a1 By MZERO2.CTIMES.CTIMESTheorem reference error Proof of MZERO2.CTIMES ends -------------------- Proved ---------------------- Line 43.3: 1: Zero * a1 = a1 * Zero |- 1: Zero * a1 = Zero By 4 -------------------- Proved ---------------------- Line 43.4: |- 1: a1 * Zero = Zero By MZERO2.MZERO Proof of MZERO2.MZERO begins -------------------- Proved ---------------------- Line 30.1: |- 1: a1 * Zero = Zero By 2, 3 -------------------- Proved ---------------------- Line 30.2: 1: a1 * Zero + a1 * Zero = a1 * Zero |- 1: a1 * Zero = Zero By 4, 5, 6, 7 -------------------- Proved ---------------------- Line 30.4: 1: Real(a1 * Zero) 2: Real(a1 * Zero) |- 1: a1 * Zero + a1 * Zero = a1 * Zero == a1 * Zero = Zero By MZERO2.MZERO.NULLADD Proof of MZERO2.MZERO.NULLADD begins -------------------- Proved ---------------------- Line 25.3: 1: Real(a1) 2: Real(a2) |- 1: a1 + a2 = a1 == a2 = Zero By 4 -------------------- Proved ---------------------- Line 25.4: 1: Real(a1) 2: Real(a2) |- 1: (a1 + a2 = a1 -> a2 = Zero) & (a2 = Zero -> a1 + a2 = a1) By 5, 6 -------------------- Proved ---------------------- Line 25.5: 1: Real(a2) |- 1: a1 + a2 = a1 -> a2 = Zero By 7 -------------------- Proved ---------------------- Line 25.7: 1: a1 + a2 = a1 2: Real(a2) |- 1: a2 = Zero By 8, 9 -------------------- Proved ---------------------- Line 25.8: 1: a1 + a2 = a1 |- 1: [a1 + a2] + Minus(a1) = a1 + Minus(a1) By 10 -------------------- Trivial --------------------- Line 25.10: |- 1: a1 + Minus(a1) = a1 + Minus(a1) -------------------- Proved ---------------------- Line 25.9: 1: [a1 + a2] + Minus(a1) = a1 + Minus(a1) 2: Real(a2) |- 1: a2 = Zero By 11, 12 -------------------- Proved ---------------------- Line 25.11: |- 1: a1 + Minus(a1) = Zero By MZERO2.MZERO.NULLADD.MINUS Proof of MZERO2.MZERO.NULLADD.MINUS begins -------------------- Proved ---------------------- Line 14.14: |- 1: a1 + Minus(a1) = Zero By MZERO2.MZERO.NULLADD.MINUS.MINUSTheorem reference error Proof of MZERO2.MZERO.NULLADD.MINUS ends -------------------- Proved ---------------------- Line 25.12: 1: [a1 + a2] + Minus(a1) = a1 + Minus(a1) 2: a1 + Minus(a1) = Zero 3: Real(a2) |- 1: a2 = Zero By 13 -------------------- Proved ---------------------- Line 25.13: 1: [a1 + a2] + Minus(a1) = Zero 2: a1 + Minus(a1) = Zero 3: Real(a2) |- 1: a2 = Zero By 14, 15 -------------------- Proved ---------------------- Line 25.14: 1: [a1 + a2] + Minus(a1) = Zero 2: [a1 + a2] + Minus(a1) = a2 |- 1: a2 = Zero By 16 -------------------- Trivial --------------------- Line 25.16: 1: a2 = Zero |- 1: a2 = Zero -------------------- Proved ---------------------- Line 25.15: 1: a1 + Minus(a1) = Zero 2: Real(a2) |- 1: [a1 + a2] + Minus(a1) = a2 By 17, 18 -------------------- Proved ---------------------- Line 25.17: |- 1: a1 + a2 = a2 + a1 By MZERO2.MZERO.NULLADD.CPLUS Proof of MZERO2.MZERO.NULLADD.CPLUS begins -------------------- Proved ---------------------- Line 7.7: |- 1: a1 + a2 = a2 + a1 By MZERO2.MZERO.NULLADD.CPLUS.CPLUSTheorem reference error Proof of MZERO2.MZERO.NULLADD.CPLUS ends -------------------- Proved ---------------------- Line 25.18: 1: a1 + a2 = a2 + a1 2: a1 + Minus(a1) = Zero 3: Real(a2) |- 1: [a1 + a2] + Minus(a1) = a2 By 19 -------------------- Proved ---------------------- Line 25.19: 1: a1 + a2 = a2 + a1 2: a1 + Minus(a1) = Zero 3: Real(a2) |- 1: [a1 + a2] + Minus(a1) = a2 By 20, 21 -------------------- Proved ---------------------- Line 25.20: |- 1: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) By MZERO2.MZERO.NULLADD.APLUS Proof of MZERO2.MZERO.NULLADD.APLUS begins -------------------- Proved ---------------------- Line 9.9: |- 1: [a1 + a2] + a3 = a1 + a2 + a3 By MZERO2.MZERO.NULLADD.APLUS.APLUSTheorem reference error Proof of MZERO2.MZERO.NULLADD.APLUS ends -------------------- Proved ---------------------- Line 25.21: 1: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) 2: a1 + Minus(a1) = Zero 3: Real(a2) 4: a1 + a2 = a2 + a1 |- 1: [a1 + a2] + Minus(a1) = a2 By 22 -------------------- Proved ---------------------- Line 25.22: 1: [a2 + a1] + Minus(a1) = a2 + Zero 2: Real(a2) 3: a1 + a2 = a2 + a1 |- 1: [a1 + a2] + Minus(a1) = a2 By 23, 24 -------------------- Proved ---------------------- Line 25.23: |- 1: Real(a2) -> a2 + Zero = a2 By MZERO2.MZERO.NULLADD.IPLUS Proof of MZERO2.MZERO.NULLADD.IPLUS begins -------------------- Proved ---------------------- Line 12.12: |- 1: Real(a1) -> a1 + Zero = a1 By MZERO2.MZERO.NULLADD.IPLUS.IPLUSTheorem reference error Proof of MZERO2.MZERO.NULLADD.IPLUS ends -------------------- Proved ---------------------- Line 25.24: 1: Real(a2) -> a2 + Zero = a2 2: [a2 + a1] + Minus(a1) = a2 + Zero 3: Real(a2) 4: a1 + a2 = a2 + a1 |- 1: [a1 + a2] + Minus(a1) = a2 By 25, 26 -------------------- Trivial --------------------- Line 25.25: 1: Real(a2) |- 1: Real(a2) -------------------- Proved ---------------------- Line 25.26: 1: [a2 + a1] + Minus(a1) = a2 + Zero 2: a2 + Zero = a2 3: a1 + a2 = a2 + a1 |- 1: [a1 + a2] + Minus(a1) = a2 By 27 -------------------- Proved ---------------------- Line 25.27: 1: [a2 + a1] + Minus(a1) = a2 + Zero 2: a2 + Zero = a2 3: a1 + a2 = a2 + a1 |- 1: [a1 + a2] + Minus(a1) = a2 By 28 -------------------- Proved ---------------------- Line 25.28: 1: a1 + a2 = a2 + a1 2: [a2 + a1] + Minus(a1) = a2 |- 1: [a1 + a2] + Minus(a1) = a2 By 29 -------------------- Trivial --------------------- Line 25.29: 1: [a2 + a1] + Minus(a1) = a2 |- 1: [a2 + a1] + Minus(a1) = a2 -------------------- Proved ---------------------- Line 25.6: 1: Real(a1) |- 1: a2 = Zero -> a1 + a2 = a1 By 30 -------------------- Proved ---------------------- Line 25.30: 1: a2 = Zero 2: Real(a1) |- 1: a1 + a2 = a1 By 31 -------------------- Proved ---------------------- Line 25.31: 1: Real(a1) |- 1: a1 + Zero = a1 By 32, 33 -------------------- Proved ---------------------- Line 25.32: |- 1: Real(a1) -> a1 + Zero = a1 By MZERO2.MZERO.NULLADD.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 25.33: 1: Real(a1) -> a1 + Zero = a1 2: Real(a1) |- 1: a1 + Zero = a1 By 35, 36 -------------------- Trivial --------------------- Line 25.35: 1: Real(a1) |- 1: Real(a1) -------------------- Trivial --------------------- Line 25.36: 1: a1 + Zero = a1 |- 1: a1 + Zero = a1 Proof of MZERO2.MZERO.NULLADD ends -------------------- Proved ---------------------- Line 30.5: 1: a1 * Zero + a1 * Zero = a1 * Zero == a1 * Zero = Zero 2: a1 * Zero + a1 * Zero = a1 * Zero |- 1: a1 * Zero = Zero By 8 -------------------- Proved ---------------------- Line 30.8: 1: ( a1 * Zero + a1 * Zero = a1 * Zero -> a1 * Zero = Zero) & ( a1 * Zero = Zero -> a1 * Zero + a1 * Zero = a1 * Zero) 2: a1 * Zero + a1 * Zero = a1 * Zero |- 1: a1 * Zero = Zero By 9 -------------------- Proved ---------------------- Line 30.9: 1: a1 * Zero + a1 * Zero = a1 * Zero -> a1 * Zero = Zero 2: a1 * Zero + a1 * Zero = a1 * Zero |- 1: a1 * Zero = Zero By MZERO2.MZERO.MP Proof of MZERO2.MZERO.MP begins -------------------- Proved ---------------------- Line 28.1: 1: P1 -> P2 2: P1 |- 1: P2 By 2, 3 -------------------- Trivial --------------------- Line 28.2: 1: P1 |- 1: P1 -------------------- Trivial --------------------- Line 28.3: 1: P2 |- 1: P2 Proof of MZERO2.MZERO.MP ends -------------------- Proved ---------------------- Line 30.6: |- 1: Real(a1 * Zero) By MZERO2.MZERO.RTIMES Proof of MZERO2.MZERO.RTIMES begins -------------------- Proved ---------------------- Line 3.3: |- 1: Real(a1 * a2) By MZERO2.MZERO.RTIMES.RTIMESTheorem reference error Proof of MZERO2.MZERO.RTIMES ends -------------------- Proved ---------------------- Line 30.7: |- 1: Real(a1 * Zero) By MZERO2.MZERO.RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 30.3: |- 1: a1 * Zero + a1 * Zero = a1 * Zero By 10, 11 -------------------- Proved ---------------------- Line 30.10: |- 1: a1 * [Zero + Zero] = a1 * Zero + a1 * Zero By MZERO2.MZERO.DIST Proof of MZERO2.MZERO.DIST begins -------------------- Proved ---------------------- Line 11.11: |- 1: a1 * [a2 + a3] = a1 * a2 + a1 * a3 By MZERO2.MZERO.DIST.DISTTheorem reference error Proof of MZERO2.MZERO.DIST ends -------------------- Proved ---------------------- Line 30.11: 1: a1 * [Zero + Zero] = a1 * Zero + a1 * Zero |- 1: a1 * Zero + a1 * Zero = a1 * Zero By 12 -------------------- Proved ---------------------- Line 30.12: |- 1: a1 * [Zero + Zero] = a1 * Zero By 13, 14 -------------------- Proved ---------------------- Line 30.13: |- 1: Real(Zero) -> Zero + Zero = Zero By MZERO2.MZERO.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 30.14: 1: Real(Zero) -> Zero + Zero = Zero |- 1: a1 * [Zero + Zero] = a1 * Zero By 15, 16 -------------------- Proved ---------------------- Line 30.15: |- 1: Real(Zero) By MZERO2.MZERO.RZERO Proof of MZERO2.MZERO.RZERO begins -------------------- Proved ---------------------- Line 29.1: |- 1: Real(Zero) By 2, 3 -------------------- Proved ---------------------- Line 29.2: |- 1: U1 + Minus(U1) = Zero By MZERO2.MZERO.RZERO.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 29.3: 1: U1 + Minus(U1) = Zero |- 1: Real(Zero) By 4 -------------------- Proved ---------------------- Line 29.4: |- 1: Real(U1 + Minus(U1)) By MZERO2.MZERO.RZERO.RPLUS Proof of MZERO2.MZERO.RZERO.RPLUS begins -------------------- Proved ---------------------- Line 2.2: |- 1: Real(a1 + a2) By MZERO2.MZERO.RZERO.RPLUS.RPLUSTheorem reference error Proof of MZERO2.MZERO.RZERO.RPLUS ends Proof of MZERO2.MZERO.RZERO ends -------------------- Proved ---------------------- Line 30.16: 1: Zero + Zero = Zero |- 1: a1 * [Zero + Zero] = a1 * Zero By 17 -------------------- Trivial --------------------- Line 30.17: |- 1: a1 * Zero = a1 * Zero Proof of MZERO2.MZERO ends Proof of MZERO2 ends -------------------- Proved ---------------------- Line 357: |- 1: [Three * Sup({x33|Zero < x33 & x33 * x33 < a1})] * a23 = Three * Sup({x34|Zero < x34 & x34 * x34 < a1}) * a23 By ATIMES Proof of ATIMES begins -------------------- Proved ---------------------- Line 10.10: |- 1: [a1 * a2] * a3 = a1 * a2 * a3 By ATIMES.ATIMESTheorem reference error Proof of ATIMES ends -------------------- Trivial --------------------- Line 354: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero 2: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 355: |- 1: Real(Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 356: |- 1: Real(Zero) By RZEROAlready shown (29) -------------------- Proved ---------------------- Line 352: 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero 2: Real(Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) 3: Real(Zero) |- 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) = Zero By EQUALITY Proof of EQUALITY begins -------------------- Proved ---------------------- Line 36.4: 1: a1 Equal a2 2: Real(a1) 3: Real(a2) |- 1: a1 = a2 By 5 -------------------- Proved ---------------------- Line 36.5: 1: a1 + Zero = a2 + Zero 2: Real(a1) 3: Real(a2) |- 1: a1 = a2 By 6, 7 -------------------- Proved ---------------------- Line 36.6: |- 1: Real(a1) -> a1 + Zero = a1 By EQUALITY.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 36.7: 1: Real(a1) -> a1 + Zero = a1 2: a1 + Zero = a2 + Zero 3: Real(a1) 4: Real(a2) |- 1: a1 = a2 By 8, 9 -------------------- Trivial --------------------- Line 36.8: 1: Real(a1) |- 1: Real(a1) -------------------- Proved ---------------------- Line 36.9: 1: a1 + Zero = a1 2: a1 + Zero = a2 + Zero 3: Real(a2) |- 1: a1 = a2 By 10 -------------------- Proved ---------------------- Line 36.10: 1: a1 + Zero = a1 2: a1 = a2 + Zero 3: Real(a2) |- 1: a1 = a2 By 11, 12 -------------------- Proved ---------------------- Line 36.11: |- 1: Real(a2) -> a2 + Zero = a2 By EQUALITY.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 36.12: 1: Real(a2) -> a2 + Zero = a2 2: a1 + Zero = a1 3: a1 = a2 + Zero 4: Real(a2) |- 1: a1 = a2 By 13, 14 -------------------- Trivial --------------------- Line 36.13: 1: Real(a2) |- 1: Real(a2) -------------------- Proved ---------------------- Line 36.14: 1: a2 + Zero = a2 2: a1 = a2 + Zero 3: a1 + Zero = a1 |- 1: a1 = a2 By 15 -------------------- Trivial --------------------- Line 36.15: 1: a1 = a2 2: a1 + Zero = a1 3: a2 + Zero = a2 |- 1: a1 = a2 Proof of EQUALITY ends -------------------- Proved ---------------------- Line 340: 1: [Three * Sup({x67|Zero < x67 & x67 * x67 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) = One 2: [a1 + Minus( Sup({x64|Zero < x64 & x64 * x64 < a1}) * Sup({x65| Zero < x65 & x65 * x65 < a1}))] * Inv(Three * Sup({x66| Zero < x66 & x66 * x66 < a1})) < [[ Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1}))] * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 341 -------------------- Proved ---------------------- Line 341: 1: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 343, 342 -------------------- Proved ---------------------- Line 343: 1: Real(a23) -> a23 * One = a23 2: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 345, 344 -------------------- Proved ---------------------- Line 345: 1: a23 * One = One * a23 2: Real(a23) -> a23 * One = a23 3: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 4: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 5: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 6: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 346 -------------------- Proved ---------------------- Line 346: 1: Real(a23) -> One * a23 = a23 2: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 15: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 348, 347 -------------------- Proved ---------------------- Line 348: 1: One * a23 = a23 2: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 3: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 15: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 349 -------------------- Proved ---------------------- Line 349: 1: [a1 + Minus( Sup({x66|Zero < x66 & x66 * x66 < a1}) * Sup({x67| Zero < x67 & x67 * x67 < a1}))] * Inv(Three * Sup({x68| Zero < x68 & x68 * x68 < a1})) < a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 350 -------------------- Proved ---------------------- Line 350: 1: [a1 + Minus( Sup({x66|Zero < x66 & x66 * x66 < a1}) * Sup({x67| Zero < x67 & x67 * x67 < a1}))] * Inv(Three * Sup({x68| Zero < x68 & x68 * x68 < a1})) < a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) Equal a23 v [a1 + Minus(Sup({x71| Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv( Three * Sup({x73|Zero < x73 & x73 * x73 < a1})) < a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 351 -------------------- Trivial --------------------- Line 351: 1: [a1 + Minus( Sup({x66|Zero < x66 & x66 * x66 < a1}) * Sup({x67| Zero < x67 & x67 * x67 < a1}))] * Inv(Three * Sup({x68| Zero < x68 & x68 * x68 < a1})) < a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) < a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) Equal a23 -------------------- Trivial --------------------- Line 347: 1: Real(a23) 2: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 5: Real(a1) 6: Zero < a1 7: Zero < a23 8: a23 * One = One * a23 9: [a1 + Minus( Sup({x65|Zero < x65 & x65 * x65 < a1}) * Sup({x66| Zero < x66 & x66 * x66 < a1}))] * Inv(Three * Sup({x67| Zero < x67 & x67 * x67 < a1})) < One * a23 10: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 11: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 12: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 14: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) |- 1: Real(a23) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 344: |- 1: a23 * One = One * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 342: |- 1: Real(a23) -> a23 * One = a23 By ITIMES Proof of ITIMES begins -------------------- Proved ---------------------- Line 13.13: |- 1: Real(a1) -> a1 * One = a1 By ITIMES.ITIMESTheorem reference error Proof of ITIMES ends -------------------- Proved ---------------------- Line 337: |- 1: Three * Sup({x67|Zero < x67 & x67 * x67 < a1}) Equal Zero v [ Three * Sup({x67|Zero < x67 & x67 * x67 < a1})] * Inv( Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) = One By INV Proof of INV begins -------------------- Proved ---------------------- Line 15.15: |- 1: a1 Equal Zero v a1 * Inv(a1) = One By INV.INVTheorem reference error Proof of INV ends -------------------- Proved ---------------------- Line 334: |- 1: [ [Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv( Three * Sup({x63|Zero < x63 & x63 * x63 < a1}))] * a23 = [Three * Sup({x61|Zero < x61 & x61 * x61 < a1})] * Inv(Three * Sup({x64|Zero < x64 & x64 * x64 < a1})) * a23 By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 331: |- 1: [Three * Sup({x55|Zero < x55 & x55 * x55 < a1})] * Inv( Three * Sup({x57|Zero < x57 & x57 * x57 < a1})) * a23 = Three * Sup({x58|Zero < x58 & x58 * x58 < a1}) * Inv( Three * Sup({x59|Zero < x59 & x59 * x59 < a1})) * a23 By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 328: |- 1: a23 * Inv(Three * Sup({x50|Zero < x50 & x50 * x50 < a1})) = Inv( Three * Sup({x51|Zero < x51 & x51 * x51 < a1})) * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 325: |- 1: [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x45|Zero < x45 & x45 * x45 < a1})) = Sup({x42| Zero < x42 & x42 * x42 < a1}) * a23 * Inv(Three * Sup({x46|Zero < x46 & x46 * x46 < a1})) By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 322: |- 1: [Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23] * Inv( Three * Sup({x38|Zero < x38 & x38 * x38 < a1})) = Three * [ Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x40|Zero < x40 & x40 * x40 < a1})) By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 306: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < Inv(Three * Sup({x34|Zero < x34 & x34 * x34 < a1})) & a1 + Minus(Sup({x30| Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x35|Zero < x35 & x35 * x35 < a1}) * a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 309, 308 -------------------- Trivial --------------------- Line 309: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 308: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 311, 310 -------------------- Proved ---------------------- Line 311: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 313, 312 -------------------- Proved ---------------------- Line 313: 1: Zero < Three 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 315, 314 -------------------- Proved ---------------------- Line 315: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 317, 316 -------------------- Proved ---------------------- Line 317: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 319, 318 -------------------- Trivial --------------------- Line 319: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 318: 1: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) By 321, 320 -------------------- Trivial --------------------- Line 321: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 8: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 9: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 12: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 320: 1: Zero < a23 2: a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x32|Zero < x32 & x32 * x32 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 |- 1: Zero < a23 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 316: |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 314: 1: Zero < Three 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) By TIMESPOS Proof of TIMESPOS begins -------------------- Proved ---------------------- Line 93.3: 1: Zero < a1 2: Zero < a2 |- 1: Zero < a1 * a2 By 4, 5 -------------------- Proved ---------------------- Line 93.4: |- 1: Zero < a1 & Zero < a2 -> Zero * a1 < a2 * a1 By TIMESPOS.MTIMES Proof of TIMESPOS.MTIMES begins -------------------- Proved ---------------------- Line 22.6: |- 1: Zero < a3 & a1 < a2 -> a1 * a3 < a2 * a3 By TIMESPOS.MTIMES.MTIMESTheorem reference error Proof of TIMESPOS.MTIMES ends -------------------- Proved ---------------------- Line 93.5: 1: Zero < a1 & Zero < a2 -> Zero * a1 < a2 * a1 2: Zero < a1 3: Zero < a2 |- 1: Zero < a1 * a2 By 6, 7 -------------------- Proved ---------------------- Line 93.6: 1: Zero < a1 2: Zero < a2 |- 1: Zero < a1 & Zero < a2 By 8, 9 -------------------- Trivial --------------------- Line 93.8: 1: Zero < a1 |- 1: Zero < a1 -------------------- Trivial --------------------- Line 93.9: 1: Zero < a2 |- 1: Zero < a2 -------------------- Proved ---------------------- Line 93.7: 1: Zero * a1 < a2 * a1 |- 1: Zero < a1 * a2 By 10, 11 -------------------- Proved ---------------------- Line 93.10: |- 1: Zero * a1 = Zero By TIMESPOS.MZERO2Already shown (43) -------------------- Proved ---------------------- Line 93.11: 1: Zero * a1 = Zero 2: Zero * a1 < a2 * a1 |- 1: Zero < a1 * a2 By 12 -------------------- Proved ---------------------- Line 93.12: 1: Zero * a1 = Zero 2: Zero < a2 * a1 |- 1: Zero < a1 * a2 By 13, 14 -------------------- Proved ---------------------- Line 93.13: |- 1: a2 * a1 = a1 * a2 By TIMESPOS.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 93.14: 1: a2 * a1 = a1 * a2 2: Zero < a2 * a1 3: Zero * a1 = Zero |- 1: Zero < a1 * a2 By 15 -------------------- Trivial --------------------- Line 93.15: 1: Zero < a1 * a2 2: Zero * a1 = Zero 3: a2 * a1 = a1 * a2 |- 1: Zero < a1 * a2 Proof of TIMESPOS ends -------------------- Proved ---------------------- Line 312: |- 1: Zero < Three By THREEPOS Proof of THREEPOS begins -------------------- Proved ---------------------- Line 92.1: |- 1: Zero < Three By 2, 3 -------------------- Proved ---------------------- Line 92.2: |- 1: Zero < One By THREEPOS.ZEROLESSONE Proof of THREEPOS.ZEROLESSONE begins -------------------- Proved ---------------------- Line 49.1: |- 1: Zero < One By 2, 3, 4 -------------------- Proved ---------------------- Line 49.2: 1: Real(One) |- 1: Zero <= One * One By THREEPOS.ZEROLESSONE.SQUARENONNEG Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG begins -------------------- Proved ---------------------- Line 42.2: 1: Real(a1) |- 1: Zero <= a1 * a1 By 3 -------------------- Proved ---------------------- Line 42.3: 1: Real(a1) |- 1: Zero Equal a1 * a1 v Zero < a1 * a1 By 4 -------------------- Proved ---------------------- Line 42.4: 1: Real(a1) |- 1: Zero Equal a1 * a1 2: Zero < a1 * a1 By 5, 6 -------------------- Proved ---------------------- Line 42.5: |- 1: a1 Equal Zero v a1 < Zero v Zero < a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.TRI Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.TRI begins -------------------- Proved ---------------------- Line 18.18: |- 1: a1 Equal a2 v a1 < a2 v a2 < a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.TRI.TRITheorem reference error Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.TRI ends -------------------- Proved ---------------------- Line 42.6: 1: a1 Equal Zero v a1 < Zero v Zero < a1 2: Real(a1) |- 1: Zero Equal a1 * a1 2: Zero < a1 * a1 By 7, 8 -------------------- Proved ---------------------- Line 42.7: 1: a1 Equal Zero 2: Real(a1) |- 1: Zero Equal a1 * a1 By 9, 10, 11, 12, 13 -------------------- Proved ---------------------- Line 42.9: 1: a1 Equal Zero 2: Real(a1) 3: Real(Zero) |- 1: a1 = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.EQUALITYAlready shown (36) -------------------- Proved ---------------------- Line 42.10: 1: a1 = Zero 2: a1 Equal Zero |- 1: Zero Equal a1 * a1 By 14 -------------------- Proved ---------------------- Line 42.14: 1: a1 = Zero 2: a1 Equal Zero |- 1: Zero Equal Zero * Zero By 15, 16 -------------------- Proved ---------------------- Line 42.15: |- 1: Zero * Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MZEROAlready shown (30) -------------------- Proved ---------------------- Line 42.16: 1: Zero * Zero = Zero 2: a1 = Zero 3: a1 Equal Zero |- 1: Zero Equal Zero * Zero By 17 -------------------- Proved ---------------------- Line 42.17: 1: a1 = Zero 2: a1 Equal Zero |- 1: Zero Equal Zero By 18 -------------------- Trivial --------------------- Line 42.18: 1: Zero Equal Zero 2: a1 = Zero |- 1: Zero Equal Zero -------------------- Trivial --------------------- Line 42.11: 1: a1 Equal Zero |- 1: a1 Equal Zero -------------------- Trivial --------------------- Line 42.12: 1: Real(a1) |- 1: Real(a1) -------------------- Proved ---------------------- Line 42.13: |- 1: Real(Zero) By THREEPOS.ZEROLESSONE.SQUARENONNEG.RZEROAlready shown (29) -------------------- Proved ---------------------- Line 42.8: 1: a1 < Zero v Zero < a1 |- 1: Zero < a1 * a1 By 19, 20 -------------------- Proved ---------------------- Line 42.19: 1: a1 < Zero |- 1: Zero < a1 * a1 By 21, 22, 23, 24 -------------------- Proved ---------------------- Line 42.21: 1: a1 < Zero 2: a1 < Zero |- 1: Zero * a1 < a1 * a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG begins -------------------- Proved ---------------------- Line 41.3: 1: a3 < Zero 2: a1 < a2 |- 1: a2 * a3 < a1 * a3 By 4, 5, 6 -------------------- Proved ---------------------- Line 41.4: 1: a3 < Zero |- 1: Zero < Minus(a3) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ begins -------------------- Proved ---------------------- Line 37.3: 1: a1 < Zero |- 1: Zero < Minus(a1) By 4, 5 -------------------- Proved ---------------------- Line 37.4: |- 1: a1 + Minus(a1) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 37.5: 1: a1 + Minus(a1) = Zero 2: a1 < Zero |- 1: Zero < Minus(a1) By 6, 7 -------------------- Proved ---------------------- Line 37.6: 1: a1 + Minus(a1) = Zero 2: a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) |- 1: Zero < Minus(a1) By 8 -------------------- Proved ---------------------- Line 37.8: 1: a1 + Minus(a1) = Zero 2: Zero < Zero + Minus(a1) |- 1: Zero < Minus(a1) By 9, 10 -------------------- Proved ---------------------- Line 37.9: |- 1: Zero + Minus(a1) = Minus(a1) + Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 37.10: 1: Zero + Minus(a1) = Minus(a1) + Zero 2: Zero < Zero + Minus(a1) 3: a1 + Minus(a1) = Zero |- 1: Zero < Minus(a1) By 11 -------------------- Proved ---------------------- Line 37.11: 1: Zero + Minus(a1) = Minus(a1) + Zero 2: Zero < Minus(a1) + Zero 3: a1 + Minus(a1) = Zero |- 1: Zero < Minus(a1) By 12, 13, 14 -------------------- Proved ---------------------- Line 37.12: 1: Real(Minus(a1)) |- 1: Minus(a1) + Zero = Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.IPLUSa Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.IPLUSa begins -------------------- Proved ---------------------- Line 26.2: 1: Real(a1) |- 1: a1 + Zero = a1 By 3, 4 -------------------- Proved ---------------------- Line 26.3: |- 1: Real(a1) -> a1 + Zero = a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.IPLUSa.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 26.4: 1: Real(a1) -> a1 + Zero = a1 2: Real(a1) |- 1: a1 + Zero = a1 By 5, 6 -------------------- Trivial --------------------- Line 26.5: 1: Real(a1) |- 1: Real(a1) -------------------- Trivial --------------------- Line 26.6: 1: a1 + Zero = a1 |- 1: a1 + Zero = a1 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.IPLUSa ends -------------------- Proved ---------------------- Line 37.13: 1: Minus(a1) + Zero = Minus(a1) 2: Zero < Minus(a1) + Zero 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a1) = Minus(a1) + Zero |- 1: Zero < Minus(a1) By 15 -------------------- Trivial --------------------- Line 37.15: 1: Zero < Minus(a1) 2: a1 + Minus(a1) = Zero 3: Zero + Minus(a1) = Minus(a1) + Zero 4: Minus(a1) + Zero = Minus(a1) |- 1: Zero < Minus(a1) -------------------- Proved ---------------------- Line 37.14: |- 1: Real(Minus(a1)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.RMINUS Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.RMINUS begins -------------------- Proved ---------------------- Line 4.4: |- 1: Real(Minus(a1)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.RMINUS.RMINUSTheorem reference error Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.RMINUS ends -------------------- Proved ---------------------- Line 37.7: 1: a1 + Minus(a1) = Zero 2: a1 < Zero |- 1: a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) By 16, 17 -------------------- Proved ---------------------- Line 37.16: |- 1: a1 < a1 + Minus(a1) -> a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS begins -------------------- Proved ---------------------- Line 21.1: |- 1: a1 < a2 -> a1 + a3 < a2 + a3 By 2, 3 -------------------- Proved ---------------------- Line 21.2: |- 1: a1 < a2 == a1 + a3 < a2 + a3 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS.MPLUS0 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS.MPLUS0 begins -------------------- Proved ---------------------- Line 20.20: |- 1: a1 < a2 == a1 + a3 < a2 + a3 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS.MPLUS0.MPLUS0Theorem reference error Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS.MPLUS0 ends -------------------- Proved ---------------------- Line 21.3: 1: a1 < a2 == a1 + a3 < a2 + a3 |- 1: a1 < a2 -> a1 + a3 < a2 + a3 By 4 -------------------- Proved ---------------------- Line 21.4: 1: (a1 < a2 -> a1 + a3 < a2 + a3) & ( a1 + a3 < a2 + a3 -> a1 < a2) |- 1: a1 < a2 -> a1 + a3 < a2 + a3 By 5 -------------------- Trivial --------------------- Line 21.5: 1: a1 < a2 -> a1 + a3 < a2 + a3 |- 1: a1 < a2 -> a1 + a3 < a2 + a3 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ.MPLUS ends -------------------- Proved ---------------------- Line 37.17: 1: a1 < a1 + Minus(a1) -> a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) 2: a1 + Minus(a1) = Zero 3: a1 < Zero |- 1: a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) By 18, 19 -------------------- Proved ---------------------- Line 37.18: 1: a1 + Minus(a1) = Zero 2: a1 < Zero |- 1: a1 < a1 + Minus(a1) By 20 -------------------- Trivial --------------------- Line 37.20: 1: a1 < Zero |- 1: a1 < Zero -------------------- Trivial --------------------- Line 37.19: 1: a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) |- 1: a1 + Minus(a1) < [a1 + Minus(a1)] + Minus(a1) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.NEGINEQ ends -------------------- Proved ---------------------- Line 41.5: 1: Zero < Minus(a3) 2: a1 < a2 |- 1: a2 * a3 < a1 * a3 By 7, 8 -------------------- Proved ---------------------- Line 41.7: |- 1: a1 < a2 == Minus(a2) < Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS begins -------------------- Proved ---------------------- Line 35.1: |- 1: a1 < a2 == Minus(a2) < Minus(a1) By 2 -------------------- Proved ---------------------- Line 35.2: |- 1: (a1 < a2 -> Minus(a2) < Minus(a1)) & ( Minus(a2) < Minus(a1) -> a1 < a2) By 3, 4 -------------------- Proved ---------------------- Line 35.3: |- 1: a1 < a2 -> Minus(a2) < Minus(a1) By 5, 6 -------------------- Proved ---------------------- Line 35.5: |- 1: a1 < a2 -> a1 + Minus(a1) + Minus(a2) < a2 + Minus(a1) + Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 35.6: 1: a1 < a2 -> a1 + Minus(a1) + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: a1 < a2 -> Minus(a2) < Minus(a1) By 7 -------------------- Proved ---------------------- Line 35.7: 1: a1 < a2 -> a1 + Minus(a1) + Minus(a2) < a2 + Minus(a1) + Minus(a2) 2: a1 < a2 |- 1: Minus(a2) < Minus(a1) By 8, 9 -------------------- Trivial --------------------- Line 35.8: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 35.9: 1: a1 + Minus(a1) + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: Minus(a2) < Minus(a1) By 10, 11 -------------------- Proved ---------------------- Line 35.10: |- 1: [a1 + Minus(a1)] + Minus(a2) = a1 + Minus(a1) + Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 35.11: 1: [a1 + Minus(a1)] + Minus(a2) = a1 + Minus(a1) + Minus(a2) 2: a1 + Minus(a1) + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: Minus(a2) < Minus(a1) By 12 -------------------- Proved ---------------------- Line 35.12: 1: [a1 + Minus(a1)] + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: Minus(a2) < Minus(a1) By 13, 14 -------------------- Proved ---------------------- Line 35.13: |- 1: a1 + Minus(a1) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 35.14: 1: a1 + Minus(a1) = Zero 2: [a1 + Minus(a1)] + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: Minus(a2) < Minus(a1) By 15 -------------------- Proved ---------------------- Line 35.15: 1: a1 + Minus(a1) = Zero 2: Zero + Minus(a2) < a2 + Minus(a1) + Minus(a2) |- 1: Minus(a2) < Minus(a1) By 16, 17, 18 -------------------- Proved ---------------------- Line 35.16: 1: Real(Minus(a2)) |- 1: Zero + Minus(a2) = Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb begins -------------------- Proved ---------------------- Line 33.2: 1: Real(a1) |- 1: Zero + a1 = a1 By 3, 4 -------------------- Proved ---------------------- Line 33.3: |- 1: Zero + a1 = a1 + Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 33.4: 1: Zero + a1 = a1 + Zero 2: Real(a1) |- 1: Zero + a1 = a1 By 5 -------------------- Proved ---------------------- Line 33.5: 1: Real(a1) |- 1: a1 + Zero = a1 By 6, 7, 8, 9 -------------------- Proved ---------------------- Line 33.6: 1: Real(a1) 2: Real(Zero) |- 1: a1 + Zero = a1 == Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb.NULLADDAlready shown (25) -------------------- Proved ---------------------- Line 33.7: 1: a1 + Zero = a1 == Zero = Zero |- 1: a1 + Zero = a1 By 10 -------------------- Proved ---------------------- Line 33.10: 1: (a1 + Zero = a1 -> Zero = Zero) & (Zero = Zero -> a1 + Zero = a1) |- 1: a1 + Zero = a1 By 11 -------------------- Proved ---------------------- Line 33.11: 1: Zero = Zero -> a1 + Zero = a1 |- 1: a1 + Zero = a1 By 12, 13 -------------------- Trivial --------------------- Line 33.12: |- 1: Zero = Zero -------------------- Proved ---------------------- Line 33.13: 1: Zero = Zero -> a1 + Zero = a1 2: Zero = Zero |- 1: a1 + Zero = a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb.MPAlready shown (28) -------------------- Trivial --------------------- Line 33.8: 1: Real(a1) |- 1: Real(a1) -------------------- Proved ---------------------- Line 33.9: |- 1: Real(Zero) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb.RZEROAlready shown (29) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSb ends -------------------- Proved ---------------------- Line 35.17: 1: Zero + Minus(a2) = Minus(a2) 2: Zero + Minus(a2) < a2 + Minus(a1) + Minus(a2) 3: a1 + Minus(a1) = Zero |- 1: Minus(a2) < Minus(a1) By 19 -------------------- Proved ---------------------- Line 35.19: 1: Minus(a2) < a2 + Minus(a1) + Minus(a2) 2: a1 + Minus(a1) = Zero 3: Zero + Minus(a2) = Minus(a2) |- 1: Minus(a2) < Minus(a1) By 20, 21 -------------------- Proved ---------------------- Line 35.20: |- 1: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 35.21: 1: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) 2: Minus(a2) < a2 + Minus(a1) + Minus(a2) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) |- 1: Minus(a2) < Minus(a1) By 22 -------------------- Proved ---------------------- Line 35.22: 1: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) 2: Minus(a2) < a2 + Minus(a2) + Minus(a1) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) |- 1: Minus(a2) < Minus(a1) By 23, 24 -------------------- Proved ---------------------- Line 35.23: |- 1: [a2 + Minus(a2)] + Minus(a1) = a2 + Minus(a2) + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 35.24: 1: [a2 + Minus(a2)] + Minus(a1) = a2 + Minus(a2) + Minus(a1) 2: Minus(a2) < a2 + Minus(a2) + Minus(a1) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) 5: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) |- 1: Minus(a2) < Minus(a1) By 25 -------------------- Proved ---------------------- Line 35.25: 1: Minus(a2) < [a2 + Minus(a2)] + Minus(a1) 2: a1 + Minus(a1) = Zero 3: Zero + Minus(a2) = Minus(a2) 4: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) |- 1: Minus(a2) < Minus(a1) By 26, 27 -------------------- Proved ---------------------- Line 35.26: |- 1: a2 + Minus(a2) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 35.27: 1: a2 + Minus(a2) = Zero 2: Minus(a2) < [a2 + Minus(a2)] + Minus(a1) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) 5: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) |- 1: Minus(a2) < Minus(a1) By 28 -------------------- Proved ---------------------- Line 35.28: 1: a2 + Minus(a2) = Zero 2: Minus(a2) < Zero + Minus(a1) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) 5: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) |- 1: Minus(a2) < Minus(a1) By 29, 30, 31 -------------------- Proved ---------------------- Line 35.29: 1: Real(Minus(a1)) |- 1: Zero + Minus(a1) = Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.IPLUSbAlready shown (33) -------------------- Proved ---------------------- Line 35.30: 1: Zero + Minus(a1) = Minus(a1) 2: Minus(a2) < Zero + Minus(a1) 3: a1 + Minus(a1) = Zero 4: Zero + Minus(a2) = Minus(a2) 5: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) 6: a2 + Minus(a2) = Zero |- 1: Minus(a2) < Minus(a1) By 32 -------------------- Trivial --------------------- Line 35.32: 1: Minus(a2) < Minus(a1) 2: a1 + Minus(a1) = Zero 3: Zero + Minus(a2) = Minus(a2) 4: Minus(a1) + Minus(a2) = Minus(a2) + Minus(a1) 5: a2 + Minus(a2) = Zero 6: Zero + Minus(a1) = Minus(a1) |- 1: Minus(a2) < Minus(a1) -------------------- Proved ---------------------- Line 35.31: |- 1: Real(Minus(a1)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.RMINUSAlready shown (4) -------------------- Proved ---------------------- Line 35.18: |- 1: Real(Minus(a2)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.RMINUSAlready shown (4) -------------------- Proved ---------------------- Line 35.4: |- 1: Minus(a2) < Minus(a1) -> a1 < a2 By 33 -------------------- Proved ---------------------- Line 35.33: 1: Minus(a2) < Minus(a1) |- 1: a1 < a2 By 34, 35 -------------------- Proved ---------------------- Line 35.34: |- 1: Minus(a2) < Minus(a1) -> Minus(a2) + a1 + a2 < Minus(a1) + a1 + a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 35.35: 1: Minus(a2) < Minus(a1) -> Minus(a2) + a1 + a2 < Minus(a1) + a1 + a2 2: Minus(a2) < Minus(a1) |- 1: a1 < a2 By 36, 37 -------------------- Trivial --------------------- Line 35.36: 1: Minus(a2) < Minus(a1) |- 1: Minus(a2) < Minus(a1) -------------------- Proved ---------------------- Line 35.37: 1: Minus(a2) + a1 + a2 < Minus(a1) + a1 + a2 |- 1: a1 < a2 By 38, 39 -------------------- Proved ---------------------- Line 35.38: 1: Minus(a2) + a1 + a2 = a1 + Zero 2: Minus(a2) + a1 + a2 < Minus(a1) + a1 + a2 |- 1: a1 < a2 By 40 -------------------- Proved ---------------------- Line 35.40: 1: a1 + Zero < Minus(a1) + a1 + a2 2: Minus(a2) + a1 + a2 = a1 + Zero |- 1: a1 < a2 By 41, 42 -------------------- Proved ---------------------- Line 35.41: 1: Minus(a1) + a1 + a2 = a2 + Zero 2: a1 + Zero < Minus(a1) + a1 + a2 3: Minus(a2) + a1 + a2 = a1 + Zero |- 1: a1 < a2 By 43 -------------------- Proved ---------------------- Line 35.43: 1: Minus(a1) + a1 + a2 = a2 + Zero 2: a1 + Zero < a2 + Zero 3: Minus(a2) + a1 + a2 = a1 + Zero |- 1: a1 < a2 By 44, 45 -------------------- Proved ---------------------- Line 35.44: |- 1: a1 < a2 == a1 + Zero < a2 + Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MPLUS0Already shown (20) -------------------- Proved ---------------------- Line 35.45: 1: a1 < a2 == a1 + Zero < a2 + Zero 2: Minus(a1) + a1 + a2 = a2 + Zero 3: a1 + Zero < a2 + Zero 4: Minus(a2) + a1 + a2 = a1 + Zero |- 1: a1 < a2 By 46 -------------------- Proved ---------------------- Line 35.46: 1: (a1 < a2 -> a1 + Zero < a2 + Zero) & (a1 + Zero < a2 + Zero -> a1 < a2) 2: Minus(a1) + a1 + a2 = a2 + Zero 3: a1 + Zero < a2 + Zero 4: Minus(a2) + a1 + a2 = a1 + Zero |- 1: a1 < a2 By 47 -------------------- Proved ---------------------- Line 35.47: 1: a1 + Zero < a2 + Zero -> a1 < a2 2: a1 + Zero < a2 + Zero |- 1: a1 < a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MPAlready shown (28) -------------------- Proved ---------------------- Line 35.42: |- 1: Minus(a1) + a1 + a2 = a2 + Zero By 48, 49 -------------------- Proved ---------------------- Line 35.48: |- 1: [Minus(a1) + a1] + a2 = Minus(a1) + a1 + a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 35.49: 1: [Minus(a1) + a1] + a2 = Minus(a1) + a1 + a2 |- 1: Minus(a1) + a1 + a2 = a2 + Zero By 50 -------------------- Proved ---------------------- Line 35.50: |- 1: [Minus(a1) + a1] + a2 = a2 + Zero By 51, 52 -------------------- Proved ---------------------- Line 35.51: |- 1: Minus(a1) + a1 = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUS2 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUS2 begins -------------------- Proved ---------------------- Line 34.1: |- 1: Minus(a1) + a1 = Zero By 2, 3 -------------------- Proved ---------------------- Line 34.2: |- 1: Minus(a1) + a1 = a1 + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUS2.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 34.3: 1: Minus(a1) + a1 = a1 + Minus(a1) |- 1: Minus(a1) + a1 = Zero By 4 -------------------- Proved ---------------------- Line 34.4: |- 1: a1 + Minus(a1) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUS2.MINUSAlready shown (14) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUS2 ends -------------------- Proved ---------------------- Line 35.52: 1: Minus(a1) + a1 = Zero |- 1: [Minus(a1) + a1] + a2 = a2 + Zero By 53 -------------------- Proved ---------------------- Line 35.53: |- 1: Zero + a2 = a2 + Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 35.39: |- 1: Minus(a2) + a1 + a2 = a1 + Zero By 54, 55 -------------------- Proved ---------------------- Line 35.54: |- 1: Minus(a2) + a1 + a2 = [a1 + a2] + Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 35.55: 1: Minus(a2) + a1 + a2 = [a1 + a2] + Minus(a2) |- 1: Minus(a2) + a1 + a2 = a1 + Zero By 56 -------------------- Proved ---------------------- Line 35.56: |- 1: [a1 + a2] + Minus(a2) = a1 + Zero By 57, 58 -------------------- Proved ---------------------- Line 35.57: |- 1: [a1 + a2] + Minus(a2) = a1 + a2 + Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 35.58: 1: [a1 + a2] + Minus(a2) = a1 + a2 + Minus(a2) |- 1: [a1 + a2] + Minus(a2) = a1 + Zero By 59 -------------------- Proved ---------------------- Line 35.59: |- 1: a1 + a2 + Minus(a2) = a1 + Zero By 60, 61 -------------------- Proved ---------------------- Line 35.60: |- 1: a2 + Minus(a2) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 35.61: 1: a2 + Minus(a2) = Zero |- 1: a1 + a2 + Minus(a2) = a1 + Zero By 62 -------------------- Trivial --------------------- Line 35.62: |- 1: a1 + Zero = a1 + Zero Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MINUSLESS ends -------------------- Proved ---------------------- Line 41.8: 1: a1 < a2 == Minus(a2) < Minus(a1) 2: Zero < Minus(a3) 3: a1 < a2 |- 1: a2 * a3 < a1 * a3 By 9 -------------------- Proved ---------------------- Line 41.9: 1: (a1 < a2 -> Minus(a2) < Minus(a1)) & ( Minus(a2) < Minus(a1) -> a1 < a2) 2: Zero < Minus(a3) 3: a1 < a2 |- 1: a2 * a3 < a1 * a3 By 10 -------------------- Proved ---------------------- Line 41.10: 1: a1 < a2 -> Minus(a2) < Minus(a1) 2: Minus(a2) < Minus(a1) -> a1 < a2 3: Zero < Minus(a3) 4: a1 < a2 |- 1: a2 * a3 < a1 * a3 By 11, 12 -------------------- Proved ---------------------- Line 41.11: 1: Minus(a2) < Minus(a1) -> a1 < a2 2: a1 < a2 |- 1: a1 < a2 By 13, 14 -------------------- Trivial --------------------- Line 41.13: 1: a1 < a2 |- 1: a1 < a2 -------------------- Trivial --------------------- Line 41.14: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 41.12: 1: Minus(a2) < Minus(a1) -> a1 < a2 2: Zero < Minus(a3) 3: Minus(a2) < Minus(a1) |- 1: a2 * a3 < a1 * a3 By 15, 16 -------------------- Trivial --------------------- Line 41.15: 1: Minus(a2) < Minus(a1) |- 1: Minus(a2) < Minus(a1) -------------------- Proved ---------------------- Line 41.16: 1: Zero < Minus(a3) 2: Minus(a2) < Minus(a1) |- 1: a2 * a3 < a1 * a3 By 17, 18 -------------------- Proved ---------------------- Line 41.17: |- 1: Zero < Minus(a3) & Minus(a2) < Minus(a1) -> Minus(a2) * Minus(a3) < Minus(a1) * Minus(a3) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 41.18: 1: Zero < Minus(a3) & Minus(a2) < Minus(a1) -> Minus(a2) * Minus(a3) < Minus(a1) * Minus(a3) 2: Zero < Minus(a3) 3: Minus(a2) < Minus(a1) |- 1: a2 * a3 < a1 * a3 By 19, 20 -------------------- Proved ---------------------- Line 41.19: 1: Zero < Minus(a3) 2: Minus(a2) < Minus(a1) |- 1: Zero < Minus(a3) & Minus(a2) < Minus(a1) By 21, 22 -------------------- Trivial --------------------- Line 41.21: 1: Zero < Minus(a3) |- 1: Zero < Minus(a3) -------------------- Trivial --------------------- Line 41.22: 1: Minus(a2) < Minus(a1) |- 1: Minus(a2) < Minus(a1) -------------------- Proved ---------------------- Line 41.20: 1: Minus(a2) * Minus(a3) < Minus(a1) * Minus(a3) |- 1: a2 * a3 < a1 * a3 By 23, 24 -------------------- Proved ---------------------- Line 41.23: |- 1: Minus(a2) * Minus(a3) = a2 * a3 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG begins -------------------- Proved ---------------------- Line 40.1: |- 1: Minus(a1) * Minus(a2) = a1 * a2 By 2, 3 -------------------- Proved ---------------------- Line 40.2: |- 1: Minus(a1) * Minus(a2) = Minus(Minus(a1) * a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM begins -------------------- Proved ---------------------- Line 38.3: |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 4, 5 -------------------- Proved ---------------------- Line 38.4: |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 6, 7 -------------------- Proved ---------------------- Line 38.6: |- 1: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.DISTAlready shown (11) -------------------- Proved ---------------------- Line 38.7: 1: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 9 -------------------- Proved ---------------------- Line 38.9: 1: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 10, 11 -------------------- Proved ---------------------- Line 38.10: |- 1: a2 + Minus(a2) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 38.11: 1: a2 + Minus(a2) = Zero 2: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 12 -------------------- Proved ---------------------- Line 38.12: 1: a2 + Minus(a2) = Zero 2: a1 * Zero = a1 * a2 + a1 * Minus(a2) |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 13, 14 -------------------- Proved ---------------------- Line 38.13: |- 1: a1 * Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.MZEROAlready shown (30) -------------------- Proved ---------------------- Line 38.14: 1: a1 * Zero = Zero 2: a1 * Zero = a1 * a2 + a1 * Minus(a2) 3: a2 + Minus(a2) = Zero |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 15 -------------------- Proved ---------------------- Line 38.15: 1: a1 * Zero = Zero 2: Zero = a1 * a2 + a1 * Minus(a2) 3: a2 + Minus(a2) = Zero |- 1: a1 * Minus(a2) = Minus(a1 * a2) By 16, 17, 18, 19 -------------------- Proved ---------------------- Line 38.16: 1: Real(a1 * Minus(a2)) 2: a1 * a2 + a1 * Minus(a2) = Zero |- 1: a1 * Minus(a2) = Minus(a1 * a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV begins -------------------- Proved ---------------------- Line 32.1: 1: Real(a2) 2: a1 + a2 = Zero |- 1: a2 = Minus(a1) By 2, 3 -------------------- Proved ---------------------- Line 32.2: |- 1: Real(a2) & a1 + a2 = Zero -> a2 = Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV begins -------------------- Proved ---------------------- Line 31.1: |- 1: Real(a2) & a1 + a2 = Zero -> a2 = Minus(a1) By 2, 3 -------------------- Proved ---------------------- Line 31.2: |- 1: a1 + a2 = a2 + a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 31.3: 1: a1 + a2 = a2 + a1 |- 1: Real(a2) & a1 + a2 = Zero -> a2 = Minus(a1) By 4 -------------------- Proved ---------------------- Line 31.4: |- 1: Real(a2) & a2 + a1 = Zero -> a2 = Minus(a1) By 5 -------------------- Proved ---------------------- Line 31.5: 1: Real(a2) & a2 + a1 = Zero |- 1: a2 = Minus(a1) By 6 -------------------- Proved ---------------------- Line 31.6: 1: Real(a2) 2: a2 + a1 = Zero |- 1: a2 = Minus(a1) By 7, 8 -------------------- Proved ---------------------- Line 31.7: |- 1: [a2 + a1] + Minus(a1) = [a2 + a1] + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.TRIV Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.TRIV begins -------------------- Trivial --------------------- Line 27.1: |- 1: a1 = a1 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.TRIV ends -------------------- Proved ---------------------- Line 31.8: 1: a2 + a1 = Zero 2: [a2 + a1] + Minus(a1) = [a2 + a1] + Minus(a1) 3: Real(a2) |- 1: a2 = Minus(a1) By 9 -------------------- Proved ---------------------- Line 31.9: 1: a2 + a1 = Zero 2: [a2 + a1] + Minus(a1) = Zero + Minus(a1) 3: Real(a2) |- 1: a2 = Minus(a1) By 10, 11 -------------------- Proved ---------------------- Line 31.10: |- 1: Zero + Minus(a1) = Minus(a1) + Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 31.11: 1: Zero + Minus(a1) = Minus(a1) + Zero 2: a2 + a1 = Zero 3: [a2 + a1] + Minus(a1) = Zero + Minus(a1) 4: Real(a2) |- 1: a2 = Minus(a1) By 12, 13 -------------------- Proved ---------------------- Line 31.12: |- 1: Real(Minus(a1)) -> Minus(a1) + Zero = Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 31.13: 1: Real(Minus(a1)) -> Minus(a1) + Zero = Minus(a1) 2: Zero + Minus(a1) = Minus(a1) + Zero 3: a2 + a1 = Zero 4: [a2 + a1] + Minus(a1) = Zero + Minus(a1) 5: Real(a2) |- 1: a2 = Minus(a1) By 14, 15 -------------------- Proved ---------------------- Line 31.14: |- 1: Real(Minus(a1)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.RMINUSAlready shown (4) -------------------- Proved ---------------------- Line 31.15: 1: Minus(a1) + Zero = Minus(a1) 2: Zero + Minus(a1) = Minus(a1) + Zero 3: a2 + a1 = Zero 4: [a2 + a1] + Minus(a1) = Zero + Minus(a1) 5: Real(a2) |- 1: a2 = Minus(a1) By 16 -------------------- Proved ---------------------- Line 31.16: 1: Zero + Minus(a1) = Minus(a1) 2: [a2 + a1] + Minus(a1) = Zero + Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero |- 1: a2 = Minus(a1) By 17 -------------------- Proved ---------------------- Line 31.17: 1: Zero + Minus(a1) = Minus(a1) 2: [a2 + a1] + Minus(a1) = Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero |- 1: a2 = Minus(a1) By 18, 19 -------------------- Proved ---------------------- Line 31.18: |- 1: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 31.19: 1: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) 2: [a2 + a1] + Minus(a1) = Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero 6: Zero + Minus(a1) = Minus(a1) |- 1: a2 = Minus(a1) By 20 -------------------- Proved ---------------------- Line 31.20: 1: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) 2: a2 + a1 + Minus(a1) = Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero 6: Zero + Minus(a1) = Minus(a1) |- 1: a2 = Minus(a1) By 21, 22 -------------------- Proved ---------------------- Line 31.21: |- 1: a1 + Minus(a1) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 31.22: 1: a1 + Minus(a1) = Zero 2: a2 + a1 + Minus(a1) = Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero 6: Zero + Minus(a1) = Minus(a1) 7: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) |- 1: a2 = Minus(a1) By 23 -------------------- Proved ---------------------- Line 31.23: 1: a1 + Minus(a1) = Zero 2: a2 + Zero = Minus(a1) 3: Real(a2) 4: Minus(a1) + Zero = Minus(a1) 5: a2 + a1 = Zero 6: Zero + Minus(a1) = Minus(a1) 7: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) |- 1: a2 = Minus(a1) By 24, 25 -------------------- Proved ---------------------- Line 31.24: |- 1: Real(a2) -> a2 + Zero = a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV.IPLUSAlready shown (12) -------------------- Proved ---------------------- Line 31.25: 1: Real(a2) -> a2 + Zero = a2 2: a1 + Minus(a1) = Zero 3: a2 + Zero = Minus(a1) 4: Real(a2) 5: Minus(a1) + Zero = Minus(a1) 6: a2 + a1 = Zero 7: Zero + Minus(a1) = Minus(a1) 8: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) |- 1: a2 = Minus(a1) By 26, 27 -------------------- Trivial --------------------- Line 31.26: 1: Real(a2) |- 1: Real(a2) -------------------- Proved ---------------------- Line 31.27: 1: a2 + Zero = a2 2: a2 + Zero = Minus(a1) 3: Minus(a1) + Zero = Minus(a1) 4: a2 + a1 = Zero 5: Zero + Minus(a1) = Minus(a1) 6: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) 7: a1 + Minus(a1) = Zero |- 1: a2 = Minus(a1) By 28 -------------------- Trivial --------------------- Line 31.28: 1: a2 = Minus(a1) 2: Minus(a1) + Zero = Minus(a1) 3: a2 + a1 = Zero 4: Zero + Minus(a1) = Minus(a1) 5: [a2 + a1] + Minus(a1) = a2 + a1 + Minus(a1) 6: a1 + Minus(a1) = Zero 7: a2 + Zero = a2 |- 1: a2 = Minus(a1) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV.UNIQUEINV ends -------------------- Proved ---------------------- Line 32.3: 1: Real(a2) & a1 + a2 = Zero -> a2 = Minus(a1) 2: Real(a2) 3: a1 + a2 = Zero |- 1: a2 = Minus(a1) By 4, 5 -------------------- Proved ---------------------- Line 32.4: 1: Real(a2) 2: a1 + a2 = Zero |- 1: Real(a2) & a1 + a2 = Zero By 6, 7 -------------------- Trivial --------------------- Line 32.6: 1: Real(a2) |- 1: Real(a2) -------------------- Trivial --------------------- Line 32.7: 1: a1 + a2 = Zero |- 1: a1 + a2 = Zero -------------------- Trivial --------------------- Line 32.5: 1: a2 = Minus(a1) |- 1: a2 = Minus(a1) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.UINV ends -------------------- Trivial --------------------- Line 38.17: 1: a1 * Minus(a2) = Minus(a1 * a2) |- 1: a1 * Minus(a2) = Minus(a1 * a2) -------------------- Proved ---------------------- Line 38.18: |- 1: Real(a1 * Minus(a2)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 38.19: 1: Zero = a1 * a2 + a1 * Minus(a2) 2: a2 + Minus(a2) = Zero 3: a1 * Zero = Zero |- 1: a1 * a2 + a1 * Minus(a2) = Zero By 20 -------------------- Trivial --------------------- Line 38.20: 1: a2 + Minus(a2) = Zero 2: a1 * Zero = Zero |- 1: a1 * a2 + a1 * Minus(a2) = a1 * a2 + a1 * Minus(a2) -------------------- Proved ---------------------- Line 38.5: |- 1: a1 * a2 + a1 * Minus(a2) = Zero By 21, 22 -------------------- Proved ---------------------- Line 38.21: |- 1: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.DISTAlready shown (11) -------------------- Proved ---------------------- Line 38.22: 1: a1 * [a2 + Minus(a2)] = a1 * a2 + a1 * Minus(a2) |- 1: a1 * a2 + a1 * Minus(a2) = Zero By 23 -------------------- Proved ---------------------- Line 38.23: |- 1: a1 * [a2 + Minus(a2)] = Zero By 24, 25 -------------------- Proved ---------------------- Line 38.24: |- 1: a2 + Minus(a2) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 38.25: 1: a2 + Minus(a2) = Zero |- 1: a1 * [a2 + Minus(a2)] = Zero By 26 -------------------- Proved ---------------------- Line 38.26: |- 1: a1 * Zero = Zero By 27, 28 -------------------- Proved ---------------------- Line 38.27: |- 1: a1 * Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM.MZEROAlready shown (30) -------------------- Trivial --------------------- Line 38.28: 1: a1 * Zero = Zero |- 1: a1 * Zero = Zero Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMM ends -------------------- Proved ---------------------- Line 40.3: 1: Minus(a1) * Minus(a2) = Minus(Minus(a1) * a2) |- 1: Minus(a1) * Minus(a2) = a1 * a2 By 4 -------------------- Proved ---------------------- Line 40.4: |- 1: Minus(Minus(a1) * a2) = a1 * a2 By 5, 6 -------------------- Proved ---------------------- Line 40.5: |- 1: Minus(a1) * a2 = a2 * Minus(a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 40.6: 1: Minus(a1) * a2 = a2 * Minus(a1) |- 1: Minus(Minus(a1) * a2) = a1 * a2 By 7 -------------------- Proved ---------------------- Line 40.7: |- 1: Minus(a2 * Minus(a1)) = a1 * a2 By 8, 9 -------------------- Proved ---------------------- Line 40.8: |- 1: a2 * Minus(a1) = Minus(a2 * a1) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.NEGCOMMAlready shown (38) -------------------- Proved ---------------------- Line 40.9: 1: a2 * Minus(a1) = Minus(a2 * a1) |- 1: Minus(a2 * Minus(a1)) = a1 * a2 By 10 -------------------- Proved ---------------------- Line 40.10: |- 1: Minus(Minus(a2 * a1)) = a1 * a2 By 11, 12 -------------------- Proved ---------------------- Line 40.11: |- 1: a2 * a1 = a1 * a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 40.12: 1: a2 * a1 = a1 * a2 |- 1: Minus(Minus(a2 * a1)) = a1 * a2 By 13 -------------------- Proved ---------------------- Line 40.13: |- 1: Minus(Minus(a1 * a2)) = a1 * a2 By 14, 15, 16 -------------------- Proved ---------------------- Line 40.14: 1: Real(a1 * a2) |- 1: Minus(Minus(a1 * a2)) = a1 * a2 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG begins -------------------- Proved ---------------------- Line 39.2: 1: Real(a1) |- 1: Minus(Minus(a1)) = a1 By 3, 4 -------------------- Proved ---------------------- Line 39.3: |- 1: a1 + Minus(a1) = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG.MINUSAlready shown (14) -------------------- Proved ---------------------- Line 39.4: 1: a1 + Minus(a1) = Zero 2: Real(a1) |- 1: Minus(Minus(a1)) = a1 By 5, 6 -------------------- Proved ---------------------- Line 39.5: |- 1: a1 + Minus(a1) = Minus(a1) + a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 39.6: 1: a1 + Minus(a1) = Minus(a1) + a1 2: a1 + Minus(a1) = Zero 3: Real(a1) |- 1: Minus(Minus(a1)) = a1 By 7 -------------------- Proved ---------------------- Line 39.7: 1: a1 + Minus(a1) = Minus(a1) + a1 2: Minus(a1) + a1 = Zero 3: Real(a1) |- 1: Minus(Minus(a1)) = a1 By 8, 9, 10, 11 -------------------- Proved ---------------------- Line 39.8: 1: Real(a1) 2: Minus(a1) + a1 = Zero |- 1: a1 = Minus(Minus(a1)) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG.UINVAlready shown (32) -------------------- Proved ---------------------- Line 39.9: 1: a1 = Minus(Minus(a1)) |- 1: Minus(Minus(a1)) = a1 By 12 -------------------- Trivial --------------------- Line 39.12: |- 1: a1 = a1 -------------------- Trivial --------------------- Line 39.10: 1: Real(a1) |- 1: Real(a1) -------------------- Trivial --------------------- Line 39.11: 1: Minus(a1) + a1 = Zero 2: a1 + Minus(a1) = Minus(a1) + a1 |- 1: Minus(a1) + a1 = Zero Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.DOUBLENEG ends -------------------- Trivial --------------------- Line 40.15: 1: Minus(Minus(a1 * a2)) = a1 * a2 |- 1: Minus(Minus(a1 * a2)) = a1 * a2 -------------------- Proved ---------------------- Line 40.16: |- 1: Real(a1 * a2) By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG.RTIMESAlready shown (3) Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEG ends -------------------- Proved ---------------------- Line 41.24: 1: Minus(a2) * Minus(a3) = a2 * a3 2: Minus(a2) * Minus(a3) < Minus(a1) * Minus(a3) |- 1: a2 * a3 < a1 * a3 By 25 -------------------- Proved ---------------------- Line 41.25: 1: Minus(a2) * Minus(a3) = a2 * a3 2: a2 * a3 < Minus(a1) * Minus(a3) |- 1: a2 * a3 < a1 * a3 By 26, 27 -------------------- Proved ---------------------- Line 41.26: |- 1: Minus(a1) * Minus(a3) = a1 * a3 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG.TIMESDOUBNEGAlready shown (40) -------------------- Proved ---------------------- Line 41.27: 1: Minus(a1) * Minus(a3) = a1 * a3 2: a2 * a3 < Minus(a1) * Minus(a3) 3: Minus(a2) * Minus(a3) = a2 * a3 |- 1: a2 * a3 < a1 * a3 By 28 -------------------- Trivial --------------------- Line 41.28: 1: a2 * a3 < a1 * a3 2: Minus(a2) * Minus(a3) = a2 * a3 3: Minus(a1) * Minus(a3) = a1 * a3 |- 1: a2 * a3 < a1 * a3 -------------------- Trivial --------------------- Line 41.6: 1: a3 < Zero |- 1: a3 < Zero Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESNEG ends -------------------- Proved ---------------------- Line 42.22: 1: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 25, 26 -------------------- Proved ---------------------- Line 42.25: |- 1: a1 * Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MZEROAlready shown (30) -------------------- Proved ---------------------- Line 42.26: 1: a1 * Zero = Zero 2: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 27, 28 -------------------- Proved ---------------------- Line 42.27: |- 1: a1 * Zero = Zero * a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 42.28: 1: a1 * Zero = Zero * a1 2: a1 * Zero = Zero 3: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 29 -------------------- Proved ---------------------- Line 42.29: 1: Zero * a1 = Zero 2: Zero * a1 < a1 * a1 3: a1 * Zero = Zero * a1 |- 1: Zero < a1 * a1 By 30 -------------------- Trivial --------------------- Line 42.30: 1: Zero < a1 * a1 2: a1 * Zero = Zero * a1 3: Zero * a1 = Zero |- 1: Zero < a1 * a1 -------------------- Trivial --------------------- Line 42.23: 1: a1 < Zero |- 1: a1 < Zero -------------------- Trivial --------------------- Line 42.24: 1: a1 < Zero |- 1: a1 < Zero -------------------- Proved ---------------------- Line 42.20: 1: Zero < a1 |- 1: Zero < a1 * a1 By 31, 32 -------------------- Proved ---------------------- Line 42.31: |- 1: Zero < a1 & Zero < a1 -> Zero * a1 < a1 * a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 42.32: 1: Zero < a1 & Zero < a1 -> Zero * a1 < a1 * a1 2: Zero < a1 |- 1: Zero < a1 * a1 By 33, 34 -------------------- Proved ---------------------- Line 42.33: 1: Zero < a1 |- 1: Zero < a1 & Zero < a1 By 35, 36 -------------------- Trivial --------------------- Line 42.35: 1: Zero < a1 |- 1: Zero < a1 -------------------- Trivial --------------------- Line 42.36: 1: Zero < a1 |- 1: Zero < a1 -------------------- Proved ---------------------- Line 42.34: 1: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 37, 38 -------------------- Proved ---------------------- Line 42.37: |- 1: a1 * Zero = Zero By THREEPOS.ZEROLESSONE.SQUARENONNEG.MZEROAlready shown (30) -------------------- Proved ---------------------- Line 42.38: 1: a1 * Zero = Zero 2: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 39, 40 -------------------- Proved ---------------------- Line 42.39: |- 1: a1 * Zero = Zero * a1 By THREEPOS.ZEROLESSONE.SQUARENONNEG.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 42.40: 1: a1 * Zero = Zero * a1 2: a1 * Zero = Zero 3: Zero * a1 < a1 * a1 |- 1: Zero < a1 * a1 By 41 -------------------- Proved ---------------------- Line 42.41: 1: Zero * a1 = Zero 2: Zero * a1 < a1 * a1 3: a1 * Zero = Zero * a1 |- 1: Zero < a1 * a1 By 42 -------------------- Trivial --------------------- Line 42.42: 1: Zero < a1 * a1 2: a1 * Zero = Zero * a1 3: Zero * a1 = Zero |- 1: Zero < a1 * a1 Proof of THREEPOS.ZEROLESSONE.SQUARENONNEG ends -------------------- Proved ---------------------- Line 49.3: 1: Zero <= One * One |- 1: Zero < One By 5, 6 -------------------- Proved ---------------------- Line 49.5: |- 1: Real(One) -> One * One = One By THREEPOS.ZEROLESSONE.ITIMESAlready shown (13) -------------------- Proved ---------------------- Line 49.6: 1: Real(One) -> One * One = One 2: Zero <= One * One |- 1: Zero < One By 7, 8 -------------------- Proved ---------------------- Line 49.7: |- 1: Real(One) By THREEPOS.ZEROLESSONE.REALONE Proof of THREEPOS.ZEROLESSONE.REALONE begins -------------------- Proved ---------------------- Line 45.1: |- 1: Real(One) By 2, 3 -------------------- Proved ---------------------- Line 45.2: |- 1: One Equal Zero v One * Inv(One) = One By THREEPOS.ZEROLESSONE.REALONE.INVAlready shown (15) -------------------- Proved ---------------------- Line 45.3: 1: One Equal Zero v One * Inv(One) = One |- 1: Real(One) By 4, 5 -------------------- Proved ---------------------- Line 45.4: |- 1: ~Zero Equal One By THREEPOS.ZEROLESSONE.REALONE.NONTRIV Proof of THREEPOS.ZEROLESSONE.REALONE.NONTRIV begins -------------------- Proved ---------------------- Line 16.16: |- 1: ~Zero Equal One By THREEPOS.ZEROLESSONE.REALONE.NONTRIV.NONTRIVTheorem reference error Proof of THREEPOS.ZEROLESSONE.REALONE.NONTRIV ends -------------------- Proved ---------------------- Line 45.5: 1: ~Zero Equal One 2: One Equal Zero v One * Inv(One) = One |- 1: Real(One) By 6 -------------------- Proved ---------------------- Line 45.6: 1: One Equal Zero v One * Inv(One) = One |- 1: Zero Equal One 2: Real(One) By 7, 8 -------------------- Proved ---------------------- Line 45.7: 1: One Equal Zero |- 1: Zero Equal One By THREEPOS.ZEROLESSONE.REALONE.EqualSymm Proof of THREEPOS.ZEROLESSONE.REALONE.EqualSymm begins -------------------- Proved ---------------------- Line 44.2: 1: a1 Equal a2 |- 1: a2 Equal a1 By 3 -------------------- Proved ---------------------- Line 44.3: 1: a1 Equal a2 |- 1: a2 + Zero = a1 + Zero By 4 -------------------- Proved ---------------------- Line 44.4: 1: a1 + Zero = a2 + Zero |- 1: a2 + Zero = a1 + Zero By 5 -------------------- Trivial --------------------- Line 44.5: |- 1: a2 + Zero = a2 + Zero Proof of THREEPOS.ZEROLESSONE.REALONE.EqualSymm ends -------------------- Proved ---------------------- Line 45.8: 1: One * Inv(One) = One |- 1: Real(One) By 9 -------------------- Proved ---------------------- Line 45.9: |- 1: Real(One * Inv(One)) By THREEPOS.ZEROLESSONE.REALONE.RTIMESAlready shown (3) Proof of THREEPOS.ZEROLESSONE.REALONE ends -------------------- Proved ---------------------- Line 49.8: 1: One * One = One 2: Zero <= One * One |- 1: Zero < One By 9 -------------------- Proved ---------------------- Line 49.9: 1: Zero <= One * One 2: One * One = One |- 1: Zero < One * One By 10 -------------------- Proved ---------------------- Line 49.10: 1: Zero Equal One * One v Zero < One * One 2: One * One = One |- 1: Zero < One * One By 11, 12 -------------------- Proved ---------------------- Line 49.11: 1: One * One = One 2: Zero Equal One * One |- By 13 -------------------- Proved ---------------------- Line 49.13: 1: One * One = One 2: Zero Equal One |- By 14, 15 -------------------- Proved ---------------------- Line 49.14: |- 1: ~Zero Equal One By THREEPOS.ZEROLESSONE.NONTRIVAlready shown (16) -------------------- Proved ---------------------- Line 49.15: 1: ~Zero Equal One 2: One * One = One 3: Zero Equal One |- By 16 -------------------- Trivial --------------------- Line 49.16: 1: Zero Equal One 2: One * One = One |- 1: Zero Equal One -------------------- Trivial --------------------- Line 49.12: 1: Zero < One * One |- 1: Zero < One * One -------------------- Proved ---------------------- Line 49.4: |- 1: Real(One) By THREEPOS.ZEROLESSONE.REALONEAlready shown (45) Proof of THREEPOS.ZEROLESSONE ends -------------------- Proved ---------------------- Line 92.3: 1: Zero < One |- 1: Zero < Three By 4, 5 -------------------- Proved ---------------------- Line 92.4: |- 1: Zero < One -> Zero + One < One + One By THREEPOS.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 92.5: 1: Zero < One -> Zero + One < One + One 2: Zero < One |- 1: Zero < Three By 6, 7 -------------------- Trivial --------------------- Line 92.6: 1: Zero < One |- 1: Zero < One -------------------- Proved ---------------------- Line 92.7: 1: Zero + One < One + One 2: Zero < One |- 1: Zero < Three By 8, 9, 10 -------------------- Proved ---------------------- Line 92.8: 1: Real(One) |- 1: Zero + One = One By THREEPOS.IPLUSbAlready shown (33) -------------------- Proved ---------------------- Line 92.9: 1: Zero + One = One 2: Zero + One < One + One 3: Zero < One |- 1: Zero < Three By 11 -------------------- Proved ---------------------- Line 92.11: 1: Zero + One = One 2: One < One + One 3: Zero < One |- 1: Zero < Three By 12, 13 -------------------- Proved ---------------------- Line 92.12: |- 1: Zero < One & One < One + One -> Zero < One + One By THREEPOS.TRANSAlready shown (19) -------------------- Proved ---------------------- Line 92.13: 1: Zero < One & One < One + One -> Zero < One + One 2: Zero + One = One 3: One < One + One 4: Zero < One |- 1: Zero < Three By 14, 15 -------------------- Proved ---------------------- Line 92.14: 1: Zero + One = One 2: One < One + One 3: Zero < One |- 1: Zero < One & One < One + One By 16, 17 -------------------- Trivial --------------------- Line 92.16: 1: Zero < One |- 1: Zero < One -------------------- Trivial --------------------- Line 92.17: 1: One < One + One 2: Zero + One = One |- 1: One < One + One -------------------- Proved ---------------------- Line 92.15: 1: Zero < One + One 2: Zero < One |- 1: Zero < Three By 18, 19 -------------------- Proved ---------------------- Line 92.18: |- 1: Zero < One + One -> Zero + One < [One + One] + One By THREEPOS.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 92.19: 1: Zero < One + One -> Zero + One < [One + One] + One 2: Zero < One + One 3: Zero < One |- 1: Zero < Three By 20, 21 -------------------- Trivial --------------------- Line 92.20: 1: Zero < One + One |- 1: Zero < One + One -------------------- Proved ---------------------- Line 92.21: 1: Zero + One < [One + One] + One 2: Zero < One |- 1: Zero < Three By 22, 23, 24 -------------------- Proved ---------------------- Line 92.22: 1: Real(One) |- 1: Zero + One = One By THREEPOS.IPLUSbAlready shown (33) -------------------- Proved ---------------------- Line 92.23: 1: Zero + One = One 2: Zero + One < [One + One] + One 3: Zero < One |- 1: Zero < Three By 25 -------------------- Proved ---------------------- Line 92.25: 1: Zero + One = One 2: One < [One + One] + One 3: Zero < One |- 1: Zero < Three By 26, 27 -------------------- Proved ---------------------- Line 92.26: |- 1: Zero < One & One < [One + One] + One -> Zero < [ One + One] + One By THREEPOS.TRANSAlready shown (19) -------------------- Proved ---------------------- Line 92.27: 1: Zero < One & One < [One + One] + One -> Zero < [ One + One] + One 2: Zero + One = One 3: One < [One + One] + One 4: Zero < One |- 1: Zero < Three By 28, 29 -------------------- Proved ---------------------- Line 92.28: 1: Zero + One = One 2: One < [One + One] + One 3: Zero < One |- 1: Zero < One & One < [One + One] + One By 30, 31 -------------------- Trivial --------------------- Line 92.30: 1: Zero < One |- 1: Zero < One -------------------- Trivial --------------------- Line 92.31: 1: One < [One + One] + One 2: Zero + One = One |- 1: One < [One + One] + One -------------------- Proved ---------------------- Line 92.29: 1: Zero < [One + One] + One |- 1: Zero < Three By 32, 33 -------------------- Proved ---------------------- Line 92.32: |- 1: [One + One] + One = One + One + One By THREEPOS.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 92.33: 1: [One + One] + One = One + One + One 2: Zero < [One + One] + One |- 1: Zero < Three By 34 -------------------- Proved ---------------------- Line 92.34: 1: [One + One] + One = One + One + One 2: Zero < One + One + One |- 1: Zero < Three By 35, 36 -------------------- Proved ---------------------- Line 92.35: 1: Three = One + One + One 2: [One + One] + One = One + One + One 3: Zero < One + One + One |- 1: Zero < Three By 38 -------------------- Trivial --------------------- Line 92.38: 1: Zero < One + One + One 2: [One + One] + One = One + One + One |- 1: Zero < One + One + One -------------------- Proved ---------------------- Line 92.36: |- 1: Three = One + One + One By 39 -------------------- Trivial --------------------- Line 92.39: |- 1: One + One + One = One + One + One -------------------- Proved ---------------------- Line 92.24: |- 1: Real(One) By THREEPOS.RONE Proof of THREEPOS.RONE begins -------------------- Proved ---------------------- Line 85.1: |- 1: Real(One) By 2, 3 -------------------- Proved ---------------------- Line 85.2: 1: One = Two * Half |- 1: Real(One) By 4 -------------------- Proved ---------------------- Line 85.4: |- 1: Real(Two * Half) By THREEPOS.RONE.RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 85.3: |- 1: One = Two * Half By 5, 6 -------------------- Proved ---------------------- Line 85.5: 1: Half = Inv(Two) |- 1: One = Two * Half By 7 -------------------- Proved ---------------------- Line 85.7: |- 1: One = Two * Inv(Two) By 8, 9 -------------------- Proved ---------------------- Line 85.8: |- 1: Two Equal Zero v Two * Inv(Two) = One By THREEPOS.RONE.INVAlready shown (15) -------------------- Proved ---------------------- Line 85.9: 1: Two Equal Zero v Two * Inv(Two) = One |- 1: One = Two * Inv(Two) By 10, 11 -------------------- Proved ---------------------- Line 85.10: 1: Two Equal Zero |- By 17, 18 -------------------- Proved ---------------------- Line 85.17: |- 1: Zero < Two By THREEPOS.RONE.TWOPOS Proof of THREEPOS.RONE.TWOPOS begins -------------------- Proved ---------------------- Line 78.1: |- 1: Zero < Two By 2, 3 -------------------- Proved ---------------------- Line 78.2: |- 1: Zero < One By THREEPOS.RONE.TWOPOS.ZEROLESSONEAlready shown (49) -------------------- Proved ---------------------- Line 78.3: 1: Zero < One |- 1: Zero < Two By 6, 7 -------------------- Proved ---------------------- Line 78.6: |- 1: One < One + One By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER begins -------------------- Proved ---------------------- Line 75.1: |- 1: a1 < a1 + One By 2, 3 -------------------- Proved ---------------------- Line 75.2: |- 1: Zero < One By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.ZEROLESSONEAlready shown (49) -------------------- Proved ---------------------- Line 75.3: 1: Zero < One |- 1: a1 < a1 + One By 4, 5 -------------------- Proved ---------------------- Line 75.4: |- 1: Zero < One -> Zero + a1 < One + a1 By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 75.5: 1: Zero < One -> Zero + a1 < One + a1 2: Zero < One |- 1: a1 < a1 + One By 6, 7 -------------------- Trivial --------------------- Line 75.6: 1: Zero < One |- 1: Zero < One -------------------- Proved ---------------------- Line 75.7: 1: Zero + a1 < One + a1 |- 1: a1 < a1 + One By 8, 9 -------------------- Proved ---------------------- Line 75.8: |- 1: One + a1 = a1 + One By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 75.9: 1: One + a1 = a1 + One 2: Zero + a1 < One + a1 |- 1: a1 < a1 + One By 10 -------------------- Proved ---------------------- Line 75.10: 1: One + a1 = a1 + One 2: Zero + a1 < a1 + One |- 1: a1 < a1 + One By 11, 12, 13 -------------------- Proved ---------------------- Line 75.11: 1: Real(a1 + One) |- 1: [a1 + One] + Zero = a1 + One By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.IPLUSaAlready shown (26) -------------------- Proved ---------------------- Line 75.12: 1: [a1 + One] + Zero = a1 + One 2: Zero + a1 < a1 + One 3: One + a1 = a1 + One |- 1: a1 < a1 + One By 14 -------------------- Proved ---------------------- Line 75.14: 1: Zero + a1 < [a1 + One] + Zero 2: One + a1 = a1 + One |- 1: a1 < a1 + One By 15, 16 -------------------- Proved ---------------------- Line 75.15: |- 1: Zero + a1 = a1 + Zero By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.CPLUSAlready shown (7) -------------------- Proved ---------------------- Line 75.16: 1: Zero + a1 = a1 + Zero 2: Zero + a1 < [a1 + One] + Zero 3: One + a1 = a1 + One |- 1: a1 < a1 + One By 17 -------------------- Proved ---------------------- Line 75.17: 1: a1 + Zero < [a1 + One] + Zero |- 1: a1 < a1 + One By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa begins -------------------- Proved ---------------------- Line 68.1: 1: a1 + Zero < a2 + Zero |- 1: a1 < a2 By 2, 3 -------------------- Proved ---------------------- Line 68.2: |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE begins -------------------- Proved ---------------------- Line 67.1: |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 2, 3 -------------------- Proved ---------------------- Line 67.2: |- 1: a1 Equal a2 v a1 < a2 v a2 < a1 By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE.TRIAlready shown (18) -------------------- Proved ---------------------- Line 67.3: 1: a1 Equal a2 v a1 < a2 v a2 < a1 |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 4, 5 -------------------- Proved ---------------------- Line 67.4: 1: a1 Equal a2 |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 6 -------------------- Proved ---------------------- Line 67.6: 1: a1 + Zero = a2 + Zero |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 7 -------------------- Proved ---------------------- Line 67.7: |- 1: a2 + Zero < a2 + Zero -> a1 < a2 By 8 -------------------- Proved ---------------------- Line 67.8: 1: a2 + Zero < a2 + Zero |- By 9, 10 -------------------- Proved ---------------------- Line 67.9: |- 1: ~a2 + Zero < a2 + Zero By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE.IRRAlready shown (17) -------------------- Proved ---------------------- Line 67.10: 1: ~a2 + Zero < a2 + Zero 2: a2 + Zero < a2 + Zero |- By 11 -------------------- Trivial --------------------- Line 67.11: 1: a2 + Zero < a2 + Zero |- 1: a2 + Zero < a2 + Zero -------------------- Proved ---------------------- Line 67.5: 1: a1 < a2 v a2 < a1 |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 12, 13 -------------------- Proved ---------------------- Line 67.12: 1: a1 < a2 |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 14 -------------------- Trivial --------------------- Line 67.14: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 67.13: 1: a2 < a1 |- 1: a1 + Zero < a2 + Zero -> a1 < a2 By 15 -------------------- Proved ---------------------- Line 67.15: 1: a1 + Zero < a2 + Zero 2: a2 < a1 |- By 16, 17 -------------------- Proved ---------------------- Line 67.16: |- 1: a2 < a1 -> a2 + Zero < a1 + Zero By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 67.17: 1: a2 < a1 -> a2 + Zero < a1 + Zero 2: a1 + Zero < a2 + Zero 3: a2 < a1 |- By 18, 19 -------------------- Trivial --------------------- Line 67.18: 1: a2 < a1 |- 1: a2 < a1 -------------------- Proved ---------------------- Line 67.19: 1: a2 + Zero < a1 + Zero 2: a1 + Zero < a2 + Zero |- By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE.NOTBOTHAlready shown (46) Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa.LESSTYPE ends -------------------- Proved ---------------------- Line 68.3: 1: a1 + Zero < a2 + Zero -> a1 < a2 2: a1 + Zero < a2 + Zero |- 1: a1 < a2 By 4, 5 -------------------- Trivial --------------------- Line 68.4: 1: a1 + Zero < a2 + Zero |- 1: a1 + Zero < a2 + Zero -------------------- Trivial --------------------- Line 68.5: 1: a1 < a2 |- 1: a1 < a2 Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.LESSTYPEa ends -------------------- Proved ---------------------- Line 75.13: |- 1: Real(a1 + One) By THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER.RPLUSAlready shown (2) Proof of THREEPOS.RONE.TWOPOS.SOMETHINGBIGGER ends -------------------- Proved ---------------------- Line 78.7: 1: One < One + One 2: Zero < One |- 1: Zero < Two By 8, 9 -------------------- Proved ---------------------- Line 78.8: |- 1: Zero < One & One < One + One -> Zero < One + One By THREEPOS.RONE.TWOPOS.TRANSAlready shown (19) -------------------- Proved ---------------------- Line 78.9: 1: Zero < One & One < One + One -> Zero < One + One 2: One < One + One 3: Zero < One |- 1: Zero < Two By 10, 11 -------------------- Proved ---------------------- Line 78.10: 1: One < One + One 2: Zero < One |- 1: Zero < One & One < One + One By 12, 13 -------------------- Trivial --------------------- Line 78.12: 1: Zero < One |- 1: Zero < One -------------------- Trivial --------------------- Line 78.13: 1: One < One + One |- 1: One < One + One -------------------- Proved ---------------------- Line 78.11: 1: Zero < One + One |- 1: Zero < Two By 14, 15 -------------------- Proved ---------------------- Line 78.14: 1: One + One = Two 2: Zero < One + One |- 1: Zero < Two By 16 -------------------- Trivial --------------------- Line 78.16: 1: Zero < One + One |- 1: Zero < One + One -------------------- Proved ---------------------- Line 78.15: |- 1: One + One = Two By 17 -------------------- Trivial --------------------- Line 78.17: |- 1: One + One = One + One Proof of THREEPOS.RONE.TWOPOS ends -------------------- Proved ---------------------- Line 85.18: 1: Zero < Two 2: Two Equal Zero |- By THREEPOS.RONE.NOTBOTH3 Proof of THREEPOS.RONE.NOTBOTH3 begins -------------------- Proved ---------------------- Line 48.3: 1: a2 < a1 2: a1 Equal a2 |- By 4, 5, 6 -------------------- Proved ---------------------- Line 48.4: 1: a1 Equal a2 |- 1: a2 Equal a1 By THREEPOS.RONE.NOTBOTH3.EqualSymmAlready shown (44) -------------------- Proved ---------------------- Line 48.5: 1: a2 Equal a1 2: a2 < a1 |- By THREEPOS.RONE.NOTBOTH3.NOTBOTH2 Proof of THREEPOS.RONE.NOTBOTH3.NOTBOTH2 begins -------------------- Proved ---------------------- Line 47.3: 1: a1 Equal a2 2: a1 < a2 |- By 4 -------------------- Proved ---------------------- Line 47.4: 1: a1 + Zero = a2 + Zero 2: a1 < a2 |- By 5, 6 -------------------- Proved ---------------------- Line 47.5: |- 1: a1 < a2 -> a1 + Zero < a2 + Zero By THREEPOS.RONE.NOTBOTH3.NOTBOTH2.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 47.6: 1: a1 < a2 -> a1 + Zero < a2 + Zero 2: a1 + Zero = a2 + Zero 3: a1 < a2 |- By 7, 8 -------------------- Trivial --------------------- Line 47.7: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 47.8: 1: a1 + Zero = a2 + Zero 2: a1 + Zero < a2 + Zero |- By 9 -------------------- Proved ---------------------- Line 47.9: 1: a1 + Zero = a2 + Zero 2: a2 + Zero < a2 + Zero |- By 10, 11 -------------------- Proved ---------------------- Line 47.10: |- 1: ~a2 + Zero < a2 + Zero By THREEPOS.RONE.NOTBOTH3.NOTBOTH2.IRRAlready shown (17) -------------------- Proved ---------------------- Line 47.11: 1: ~a2 + Zero < a2 + Zero 2: a1 + Zero = a2 + Zero 3: a2 + Zero < a2 + Zero |- By 12 -------------------- Trivial --------------------- Line 47.12: 1: a2 + Zero < a2 + Zero 2: a1 + Zero = a2 + Zero |- 1: a2 + Zero < a2 + Zero Proof of THREEPOS.RONE.NOTBOTH3.NOTBOTH2 ends -------------------- Trivial --------------------- Line 48.6: 1: a1 Equal a2 |- 1: a1 Equal a2 Proof of THREEPOS.RONE.NOTBOTH3 ends -------------------- Proved ---------------------- Line 85.11: 1: Two * Inv(Two) = One |- 1: One = Two * Inv(Two) By 12, 13 -------------------- Proved ---------------------- Line 85.12: |- 1: Two * Inv(Two) = Inv(Two) * Two By THREEPOS.RONE.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 85.13: 1: Two * Inv(Two) = Inv(Two) * Two 2: Two * Inv(Two) = One |- 1: One = Two * Inv(Two) By 14 -------------------- Proved ---------------------- Line 85.14: 1: Two * Inv(Two) = Inv(Two) * Two 2: Inv(Two) * Two = One |- 1: One = Two * Inv(Two) By 15 -------------------- Proved ---------------------- Line 85.15: 1: Inv(Two) * Two = One |- 1: One = Inv(Two) * Two By 16 -------------------- Trivial --------------------- Line 85.16: |- 1: One = One -------------------- Proved ---------------------- Line 85.6: |- 1: Half = Inv(Two) By 19 -------------------- Trivial --------------------- Line 85.19: |- 1: Inv(Two) = Inv(Two) Proof of THREEPOS.RONE ends -------------------- Proved ---------------------- Line 92.10: |- 1: Real(One) By THREEPOS.RONEAlready shown (85) Proof of THREEPOS ends -------------------- Proved ---------------------- Line 310: 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) By POSINV Proof of POSINV begins -------------------- Proved ---------------------- Line 73.2: 1: Zero < a1 |- 1: Zero < Inv(a1) By 3, 4 -------------------- Proved ---------------------- Line 73.3: |- 1: Zero Equal Inv(a1) v Zero < Inv(a1) v Inv(a1) < Zero By POSINV.TRIAlready shown (18) -------------------- Proved ---------------------- Line 73.4: 1: Zero Equal Inv(a1) v Zero < Inv(a1) v Inv(a1) < Zero 2: Zero < a1 |- 1: Zero < Inv(a1) By 5, 6 -------------------- Proved ---------------------- Line 73.5: 1: Zero Equal Inv(a1) 2: Zero < a1 |- By 28, 29 -------------------- Proved ---------------------- Line 73.28: |- 1: a1 Equal Zero v a1 * Inv(a1) = One By POSINV.INVAlready shown (15) -------------------- Proved ---------------------- Line 73.29: 1: a1 Equal Zero v a1 * Inv(a1) = One 2: Zero Equal Inv(a1) 3: Zero < a1 |- By 30, 31 -------------------- Proved ---------------------- Line 73.30: 1: Zero < a1 2: a1 Equal Zero |- By POSINV.NOTBOTH3Already shown (48) -------------------- Proved ---------------------- Line 73.31: 1: Zero Equal Inv(a1) 2: a1 * Inv(a1) = One |- By 32, 33, 34, 35, 36 -------------------- Proved ---------------------- Line 73.32: 1: Zero Equal Inv(a1) 2: Real(Zero) 3: Real(Inv(a1)) |- 1: Zero = Inv(a1) By POSINV.EQUALITYAlready shown (36) -------------------- Proved ---------------------- Line 73.33: 1: Zero = Inv(a1) 2: a1 * Inv(a1) = One |- By 37 -------------------- Proved ---------------------- Line 73.37: 1: a1 * Zero = One |- By 38, 39 -------------------- Proved ---------------------- Line 73.38: |- 1: a1 * Zero = Zero By POSINV.MZEROAlready shown (30) -------------------- Proved ---------------------- Line 73.39: 1: a1 * Zero = Zero 2: a1 * Zero = One |- By 40 -------------------- Proved ---------------------- Line 73.40: 1: Zero = One |- By 41, 42 -------------------- Proved ---------------------- Line 73.41: |- 1: Zero < One By POSINV.ZEROLESSONEAlready shown (49) -------------------- Proved ---------------------- Line 73.42: 1: Zero = One 2: Zero < One |- By 43 -------------------- Proved ---------------------- Line 73.43: 1: Zero = One 2: One < One |- By 44, 45 -------------------- Proved ---------------------- Line 73.44: |- 1: ~One < One By POSINV.IRRAlready shown (17) -------------------- Proved ---------------------- Line 73.45: 1: ~One < One 2: Zero = One 3: One < One |- By 46 -------------------- Trivial --------------------- Line 73.46: 1: One < One 2: Zero = One |- 1: One < One -------------------- Trivial --------------------- Line 73.34: 1: Zero Equal Inv(a1) |- 1: Zero Equal Inv(a1) -------------------- Proved ---------------------- Line 73.35: |- 1: Real(Zero) By POSINV.RZEROAlready shown (29) -------------------- Proved ---------------------- Line 73.36: |- 1: Real(Inv(a1)) By POSINV.RINV Proof of POSINV.RINV begins -------------------- Proved ---------------------- Line 5.5: |- 1: Real(Inv(a1)) By POSINV.RINV.RINVTheorem reference error Proof of POSINV.RINV ends -------------------- Proved ---------------------- Line 73.6: 1: Zero < Inv(a1) v Inv(a1) < Zero 2: Zero < a1 |- 1: Zero < Inv(a1) By 7, 8 -------------------- Trivial --------------------- Line 73.7: 1: Zero < Inv(a1) |- 1: Zero < Inv(a1) -------------------- Proved ---------------------- Line 73.8: 1: Inv(a1) < Zero 2: Zero < a1 |- By 9, 10 -------------------- Proved ---------------------- Line 73.9: |- 1: Zero < a1 & Inv(a1) < Zero -> Inv(a1) * a1 < Zero * a1 By POSINV.MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 73.10: 1: Zero < a1 & Inv(a1) < Zero -> Inv(a1) * a1 < Zero * a1 2: Inv(a1) < Zero 3: Zero < a1 |- By 11, 12 -------------------- Proved ---------------------- Line 73.11: 1: Inv(a1) < Zero 2: Zero < a1 |- 1: Zero < a1 & Inv(a1) < Zero By 13, 14 -------------------- Trivial --------------------- Line 73.13: 1: Zero < a1 |- 1: Zero < a1 -------------------- Trivial --------------------- Line 73.14: 1: Inv(a1) < Zero |- 1: Inv(a1) < Zero -------------------- Proved ---------------------- Line 73.12: 1: Inv(a1) * a1 < Zero * a1 2: Zero < a1 |- By 15, 16 -------------------- Proved ---------------------- Line 73.15: |- 1: a1 Equal Zero v a1 * Inv(a1) = One By POSINV.INVAlready shown (15) -------------------- Proved ---------------------- Line 73.16: 1: a1 Equal Zero v a1 * Inv(a1) = One 2: Inv(a1) * a1 < Zero * a1 3: Zero < a1 |- By 17, 18 -------------------- Proved ---------------------- Line 73.17: 1: Zero < a1 2: a1 Equal Zero |- By POSINV.NOTBOTH3Already shown (48) -------------------- Proved ---------------------- Line 73.18: 1: a1 * Inv(a1) = One 2: Inv(a1) * a1 < Zero * a1 |- By 19, 20 -------------------- Proved ---------------------- Line 73.19: |- 1: a1 * Inv(a1) = Inv(a1) * a1 By POSINV.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 73.20: 1: a1 * Inv(a1) = Inv(a1) * a1 2: a1 * Inv(a1) = One 3: Inv(a1) * a1 < Zero * a1 |- By 21 -------------------- Proved ---------------------- Line 73.21: 1: Inv(a1) * a1 = One 2: Inv(a1) * a1 < Zero * a1 |- By 22 -------------------- Proved ---------------------- Line 73.22: 1: One < Zero * a1 |- By 23, 24 -------------------- Proved ---------------------- Line 73.23: |- 1: Zero * a1 = Zero By POSINV.MZERO2Already shown (43) -------------------- Proved ---------------------- Line 73.24: 1: Zero * a1 = Zero 2: One < Zero * a1 |- By 25 -------------------- Proved ---------------------- Line 73.25: 1: One < Zero |- By 26, 27 -------------------- Proved ---------------------- Line 73.26: |- 1: Zero < One By POSINV.ZEROLESSONEAlready shown (49) -------------------- Proved ---------------------- Line 73.27: 1: Zero < One 2: One < Zero |- By POSINV.NOTBOTHAlready shown (46) Proof of POSINV ends -------------------- Proved ---------------------- Line 304: |- 1: Zero < Inv(Three * Sup({x34|Zero < x34 & x34 * x34 < a1})) & a1 + Minus(Sup({x30| Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1})) < Three * Sup({x35|Zero < x35 & x35 * x35 < a1}) * a23 -> [ a1 + Minus( Sup({x30|Zero < x30 & x30 * x30 < a1}) * Sup({x31| Zero < x31 & x31 * x31 < a1}))] * Inv(Three * Sup({x36| Zero < x36 & x36 * x36 < a1})) < [ Three * Sup({x37|Zero < x37 & x37 * x37 < a1}) * a23] * Inv( Three * Sup({x38|Zero < x38 & x38 * x38 < a1})) By MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 301: |- 1: a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) = Sup({x26| Zero < x26 & x26 * x26 < a1}) * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 299: |- 1: Real(a23 * Sup({x24|Zero < x24 & x24 * x24 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 297: 1: Real(a23 * Sup({x24|Zero < x24 & x24 * x24 < a1})) |- 1: a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) = Three * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) By LIKETERMS1 Proof of LIKETERMS1 begins -------------------- Proved ---------------------- Line 94.2: 1: Real(a1) |- 1: a1 + Two * a1 = Three * a1 By 3, 4 -------------------- Proved ---------------------- Line 94.3: |- 1: Real(a1) -> a1 * One = a1 By LIKETERMS1.ITIMESAlready shown (13) -------------------- Proved ---------------------- Line 94.4: 1: Real(a1) -> a1 * One = a1 2: Real(a1) |- 1: a1 + Two * a1 = Three * a1 By 5, 6 -------------------- Trivial --------------------- Line 94.5: 1: Real(a1) |- 1: Real(a1) -------------------- Proved ---------------------- Line 94.6: 1: a1 * One = a1 |- 1: a1 + Two * a1 = Three * a1 By 7 -------------------- Proved ---------------------- Line 94.7: |- 1: a1 * One + Two * a1 = Three * a1 By 8, 9 -------------------- Proved ---------------------- Line 94.8: |- 1: a1 * One = One * a1 By LIKETERMS1.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 94.9: 1: a1 * One = One * a1 |- 1: a1 * One + Two * a1 = Three * a1 By 10 -------------------- Proved ---------------------- Line 94.10: |- 1: One * a1 + Two * a1 = Three * a1 By 11, 12 -------------------- Proved ---------------------- Line 94.11: |- 1: [One + Two] * a1 = One * a1 + Two * a1 By LIKETERMS1.DIST2 Proof of LIKETERMS1.DIST2 begins -------------------- Proved ---------------------- Line 70.1: |- 1: [a1 + a2] * a3 = a1 * a3 + a2 * a3 By 2, 3 -------------------- Proved ---------------------- Line 70.2: |- 1: [a1 + a2] * a3 = a3 * [a1 + a2] By LIKETERMS1.DIST2.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 70.3: 1: [a1 + a2] * a3 = a3 * [a1 + a2] |- 1: [a1 + a2] * a3 = a1 * a3 + a2 * a3 By 4 -------------------- Proved ---------------------- Line 70.4: |- 1: a3 * [a1 + a2] = a1 * a3 + a2 * a3 By 5, 6 -------------------- Proved ---------------------- Line 70.5: |- 1: a3 * [a1 + a2] = a3 * a1 + a3 * a2 By LIKETERMS1.DIST2.DISTAlready shown (11) -------------------- Proved ---------------------- Line 70.6: 1: a3 * [a1 + a2] = a3 * a1 + a3 * a2 |- 1: a3 * [a1 + a2] = a1 * a3 + a2 * a3 By 7 -------------------- Proved ---------------------- Line 70.7: |- 1: a3 * a1 + a3 * a2 = a1 * a3 + a2 * a3 By 8, 9 -------------------- Proved ---------------------- Line 70.8: |- 1: a3 * a1 = a1 * a3 By LIKETERMS1.DIST2.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 70.9: 1: a3 * a1 = a1 * a3 |- 1: a3 * a1 + a3 * a2 = a1 * a3 + a2 * a3 By 10 -------------------- Proved ---------------------- Line 70.10: |- 1: a1 * a3 + a3 * a2 = a1 * a3 + a2 * a3 By 11, 12 -------------------- Proved ---------------------- Line 70.11: |- 1: a3 * a2 = a2 * a3 By LIKETERMS1.DIST2.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 70.12: 1: a3 * a2 = a2 * a3 |- 1: a1 * a3 + a3 * a2 = a1 * a3 + a2 * a3 By 13 -------------------- Trivial --------------------- Line 70.13: |- 1: a1 * a3 + a2 * a3 = a1 * a3 + a2 * a3 Proof of LIKETERMS1.DIST2 ends -------------------- Proved ---------------------- Line 94.12: 1: [One + Two] * a1 = One * a1 + Two * a1 |- 1: One * a1 + Two * a1 = Three * a1 By 13 -------------------- Proved ---------------------- Line 94.13: |- 1: [One + Two] * a1 = Three * a1 By 14, 15 -------------------- Proved ---------------------- Line 94.14: 1: Two = One + One |- 1: [One + Two] * a1 = Three * a1 By 16 -------------------- Proved ---------------------- Line 94.16: |- 1: [One + One + One] * a1 = Three * a1 By 17, 18 -------------------- Proved ---------------------- Line 94.17: 1: Three = One + One + One |- 1: [One + One + One] * a1 = Three * a1 By 19 -------------------- Trivial --------------------- Line 94.19: |- 1: Three * a1 = Three * a1 -------------------- Proved ---------------------- Line 94.18: |- 1: Three = One + One + One By 20 -------------------- Trivial --------------------- Line 94.20: |- 1: One + One + One = One + One + One -------------------- Proved ---------------------- Line 94.15: |- 1: Two = One + One By 21 -------------------- Trivial --------------------- Line 94.21: |- 1: One + One = One + One Proof of LIKETERMS1 ends -------------------- Proved ---------------------- Line 295: |- 1: Real(Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 293: 1: Real(Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1})) |- 1: Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Zero = Two * a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) By IPLUSaAlready shown (26) -------------------- Proved ---------------------- Line 290: |- 1: Sup({x25|Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1}) + Minus(Sup({x25| Zero < x25 & x25 * x25 < a1}) * Sup({x26| Zero < x26 & x26 * x26 < a1})) = Zero By MINUSAlready shown (14) -------------------- Proved ---------------------- Line 287: |- 1: [Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x18|Zero < x18 & x18 * x18 < a1}) * Sup({x19|Zero < x19 & x19 * x19 < a1})) = Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x20| Zero < x20 & x20 * x20 < a1}) * Sup({x21| Zero < x21 & x21 * x21 < a1}) + Minus(Sup({x22| Zero < x22 & x22 * x22 < a1}) * Sup({x23| Zero < x23 & x23 * x23 < a1})) By APLUSAlready shown (9) -------------------- Proved ---------------------- Line 284: |- 1: [a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1})] + Minus(Sup({x11| Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1})) = a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + [Two * a23 * Sup({x13|Zero < x13 & x13 * x13 < a1}) + Sup({x14| Zero < x14 & x14 * x14 < a1}) * Sup({x15| Zero < x15 & x15 * x15 < a1})] + Minus( Sup({x16|Zero < x16 & x16 * x16 < a1}) * Sup({x17|Zero < x17 & x17 * x17 < a1})) By APLUSAlready shown (9) -------------------- Trivial --------------------- Line 282: 1: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 11: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) |- 1: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 280: |- 1: a1 < a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1}) -> a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < [a23 * Sup({x10|Zero < x10 & x10 * x10 < a1}) + Two * a23 * Sup({x11| Zero < x11 & x11 * x11 < a1}) + Sup({x12|Zero < x12 & x12 * x12 < a1}) * Sup({x13|Zero < x13 & x13 * x13 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) By MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 277: |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) = Zero By MINUSAlready shown (14) -------------------- Trivial --------------------- Line 275: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 2: Real(a1) 3: Zero < a1 4: Zero < a23 5: a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 273: |- 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) < a1 -> Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) By MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 269: 1: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: a1 < a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) & a23 * a23 + Two * a23 * Sup({x9|Zero < x9 & x9 * x9 < a1}) + Sup({x10| Zero < x10 & x10 * x10 < a1}) * Sup({x11| Zero < x11 & x11 * x11 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 272, 271 -------------------- Trivial --------------------- Line 272: 1: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: a23 * a23 + Two * a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 271: 1: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 11: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) |- 1: a1 < a23 * a23 + Two * a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 267: |- 1: a1 < a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) & a23 * a23 + Two * a23 * Sup({x9|Zero < x9 & x9 * x9 < a1}) + Sup({x10| Zero < x10 & x10 * x10 < a1}) * Sup({x11| Zero < x11 & x11 * x11 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) -> a1 < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Trivial --------------------- Line 265: 1: Zero < a23 2: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 |- 1: Zero < a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 266: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: a1 < a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x5| Zero < x5 & x5 * x5 < a1}) * Sup({x6| Zero < x6 & x6 * x6 < a1}) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 263: 1: Zero < a23 2: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) By ORDERLEMMA1 Proof of ORDERLEMMA1 begins -------------------- Proved ---------------------- Line 91.3: 1: Zero < a1 2: a1 < a2 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 4, 5 -------------------- Proved ---------------------- Line 91.4: |- 1: Zero < a1 & a1 < a2 -> a1 * a1 < a2 * a1 By ORDERLEMMA1.MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 91.5: 1: Zero < a1 & a1 < a2 -> a1 * a1 < a2 * a1 2: Zero < a1 3: a1 < a2 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 6, 7 -------------------- Proved ---------------------- Line 91.6: 1: Zero < a1 2: a1 < a2 |- 1: Zero < a1 & a1 < a2 By 8, 9 -------------------- Trivial --------------------- Line 91.8: 1: Zero < a1 |- 1: Zero < a1 -------------------- Trivial --------------------- Line 91.9: 1: a1 < a2 |- 1: a1 < a2 -------------------- Proved ---------------------- Line 91.7: 1: a1 * a1 < a2 * a1 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 10, 11 -------------------- Proved ---------------------- Line 91.10: |- 1: a1 * a1 < a2 * a1 -> a1 * a1 + Two * a1 * a2 + a2 * a2 < a2 * a1 + Two * a1 * a2 + a2 * a2 By ORDERLEMMA1.MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 91.11: 1: a1 * a1 < a2 * a1 -> a1 * a1 + Two * a1 * a2 + a2 * a2 < a2 * a1 + Two * a1 * a2 + a2 * a2 2: a1 * a1 < a2 * a1 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 12, 13 -------------------- Trivial --------------------- Line 91.12: 1: a1 * a1 < a2 * a1 |- 1: a1 * a1 < a2 * a1 -------------------- Proved ---------------------- Line 91.13: 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a2 * a1 + Two * a1 * a2 + a2 * a2 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 14, 15 -------------------- Proved ---------------------- Line 91.14: |- 1: a2 * a1 = a1 * a2 By ORDERLEMMA1.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 91.15: 1: a2 * a1 = a1 * a2 2: a1 * a1 + Two * a1 * a2 + a2 * a2 < a2 * a1 + Two * a1 * a2 + a2 * a2 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 By 17 -------------------- Trivial --------------------- Line 91.17: 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 2: a2 * a1 = a1 * a2 |- 1: a1 * a1 + Two * a1 * a2 + a2 * a2 < a1 * a2 + Two * a1 * a2 + a2 * a2 Proof of ORDERLEMMA1 ends -------------------- Proved ---------------------- Line 260: |- 1: [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) By FOIL Proof of FOIL begins -------------------- Proved ---------------------- Line 88.1: |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + Two * a1 * a2 + a2 * a2 By 2, 3 -------------------- Proved ---------------------- Line 88.2: 1: Two * a1 * a2 = a1 * a2 + a1 * a2 |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + Two * a1 * a2 + a2 * a2 By 4 -------------------- Proved ---------------------- Line 88.4: |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + [ a1 * a2 + a1 * a2] + a2 * a2 By 5, 6 -------------------- Proved ---------------------- Line 88.5: |- 1: [a1 * a2 + a1 * a2] + a2 * a2 = a1 * a2 + a1 * a2 + a2 * a2 By FOIL.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 88.6: 1: [a1 * a2 + a1 * a2] + a2 * a2 = a1 * a2 + a1 * a2 + a2 * a2 |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + [ a1 * a2 + a1 * a2] + a2 * a2 By 7 -------------------- Proved ---------------------- Line 88.7: |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + a1 * a2 + a1 * a2 + a2 * a2 By 8, 9 -------------------- Proved ---------------------- Line 88.8: |- 1: [a1 * a1 + a1 * a2] + a1 * a2 + a2 * a2 = a1 * a1 + a1 * a2 + a1 * a2 + a2 * a2 By FOIL.APLUSAlready shown (9) -------------------- Proved ---------------------- Line 88.9: 1: [a1 * a1 + a1 * a2] + a1 * a2 + a2 * a2 = a1 * a1 + a1 * a2 + a1 * a2 + a2 * a2 |- 1: [a1 + a2] * [a1 + a2] = a1 * a1 + a1 * a2 + a1 * a2 + a2 * a2 By 10 -------------------- Proved ---------------------- Line 88.10: |- 1: [a1 + a2] * [a1 + a2] = [a1 * a1 + a1 * a2] + a1 * a2 + a2 * a2 By 11, 12 -------------------- Proved ---------------------- Line 88.11: |- 1: a1 * [a1 + a2] = a1 * a1 + a1 * a2 By FOIL.DISTAlready shown (11) -------------------- Proved ---------------------- Line 88.12: 1: a1 * [a1 + a2] = a1 * a1 + a1 * a2 |- 1: [a1 + a2] * [a1 + a2] = [a1 * a1 + a1 * a2] + a1 * a2 + a2 * a2 By 13 -------------------- Proved ---------------------- Line 88.13: |- 1: [a1 + a2] * [a1 + a2] = a1 * [a1 + a2] + a1 * a2 + a2 * a2 By 14, 15 -------------------- Proved ---------------------- Line 88.14: |- 1: [a1 + a2] * a2 = a1 * a2 + a2 * a2 By FOIL.DIST2Already shown (70) -------------------- Proved ---------------------- Line 88.15: 1: [a1 + a2] * a2 = a1 * a2 + a2 * a2 |- 1: [a1 + a2] * [a1 + a2] = a1 * [a1 + a2] + a1 * a2 + a2 * a2 By 16 -------------------- Proved ---------------------- Line 88.16: |- 1: [a1 + a2] * [a1 + a2] = a1 * [a1 + a2] + [a1 + a2] * a2 By 17, 18 -------------------- Proved ---------------------- Line 88.17: |- 1: a1 * [a1 + a2] = [a1 + a2] * a1 By FOIL.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 88.18: 1: a1 * [a1 + a2] = [a1 + a2] * a1 |- 1: [a1 + a2] * [a1 + a2] = a1 * [a1 + a2] + [a1 + a2] * a2 By 19 -------------------- Proved ---------------------- Line 88.19: |- 1: [a1 + a2] * [a1 + a2] = [a1 + a2] * a1 + [a1 + a2] * a2 By 20, 21 -------------------- Proved ---------------------- Line 88.20: |- 1: [a1 + a2] * [a1 + a2] = [a1 + a2] * a1 + [a1 + a2] * a2 By FOIL.DISTAlready shown (11) -------------------- Proved ---------------------- Line 88.21: 1: [a1 + a2] * [a1 + a2] = [a1 + a2] * a1 + [a1 + a2] * a2 |- 1: [a1 + a2] * [a1 + a2] = [a1 + a2] * a1 + [a1 + a2] * a2 By 22 -------------------- Trivial --------------------- Line 88.22: |- 1: [a1 + a2] * [a1 + a2] = [a1 + a2] * [a1 + a2] -------------------- Proved ---------------------- Line 88.3: |- 1: Two * a1 * a2 = a1 * a2 + a1 * a2 By 23, 24 -------------------- Proved ---------------------- Line 88.23: 1: Two = One + One |- 1: Two * a1 * a2 = a1 * a2 + a1 * a2 By 25 -------------------- Proved ---------------------- Line 88.25: |- 1: [One + One] * a1 * a2 = a1 * a2 + a1 * a2 By 26, 27 -------------------- Proved ---------------------- Line 88.26: |- 1: [One + One] * a1 * a2 = One * a1 * a2 + One * a1 * a2 By FOIL.DIST2Already shown (70) -------------------- Proved ---------------------- Line 88.27: 1: [One + One] * a1 * a2 = One * a1 * a2 + One * a1 * a2 |- 1: [One + One] * a1 * a2 = a1 * a2 + a1 * a2 By 28 -------------------- Proved ---------------------- Line 88.28: |- 1: One * a1 * a2 + One * a1 * a2 = a1 * a2 + a1 * a2 By 29, 30 -------------------- Proved ---------------------- Line 88.29: |- 1: One * a1 * a2 = [a1 * a2] * One By FOIL.CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 88.30: 1: One * a1 * a2 = [a1 * a2] * One |- 1: One * a1 * a2 + One * a1 * a2 = a1 * a2 + a1 * a2 By 31 -------------------- Proved ---------------------- Line 88.31: |- 1: [a1 * a2] * One + [a1 * a2] * One = a1 * a2 + a1 * a2 By 32, 33 -------------------- Proved ---------------------- Line 88.32: |- 1: Real(a1 * a2) -> [a1 * a2] * One = a1 * a2 By FOIL.ITIMESAlready shown (13) -------------------- Proved ---------------------- Line 88.33: 1: Real(a1 * a2) -> [a1 * a2] * One = a1 * a2 |- 1: [a1 * a2] * One + [a1 * a2] * One = a1 * a2 + a1 * a2 By 34, 35 -------------------- Proved ---------------------- Line 88.34: |- 1: Real(a1 * a2) By FOIL.RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 88.35: 1: [a1 * a2] * One = a1 * a2 |- 1: [a1 * a2] * One + [a1 * a2] * One = a1 * a2 + a1 * a2 By 37 -------------------- Trivial --------------------- Line 88.37: |- 1: a1 * a2 + a1 * a2 = a1 * a2 + a1 * a2 -------------------- Proved ---------------------- Line 88.24: |- 1: Two = One + One By 36 -------------------- Trivial --------------------- Line 88.36: |- 1: One + One = One + One Proof of FOIL ends -------------------- Proved ---------------------- Line 101: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 103, 102 -------------------- Proved ---------------------- Line 103: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) E {x3| Zero < x3 & x3 * x3 < a1} 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 115 -------------------- Proved ---------------------- Line 115: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) & [a23 + Sup({x8| Zero < x8 & x8 * x8 < a1})] * [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 117, 116 -------------------- Trivial --------------------- Line 117: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 116: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 119, 118 -------------------- Proved ---------------------- Line 119: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 121, 120 -------------------- Proved ---------------------- Line 121: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 125, 124 -------------------- Proved ---------------------- Line 125: 1: Zero < Sup({x6|Zero < x6 & x6 * x6 < a1}) & Sup({x7| Zero < x7 & x7 * x7 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) -> Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 127, 126 -------------------- Trivial --------------------- Line 127: 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 126: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: Zero < Sup({x6|Zero < x6 & x6 * x6 < a1}) & Sup({x7| Zero < x7 & x7 * x7 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 129, 128 -------------------- Trivial --------------------- Line 129: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 9: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] |- 1: Sup({x6|Zero < x6 & x6 * x6 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 128: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: Zero < Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 124: |- 1: Zero < Sup({x6|Zero < x6 & x6 * x6 < a1}) & Sup({x7| Zero < x7 & x7 * x7 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) -> Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 120: 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 123, 122 -------------------- Trivial --------------------- Line 123: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 122: 1: Zero < a23 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 |- 1: Zero < a23 2: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 118: |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 102: 1: a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) E {x3| Zero < x3 & x3 * x3 < a1} 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 104 -------------------- Proved ---------------------- Line 104: 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) & [a23 + Sup({x8| Zero < x8 & x8 * x8 < a1})] * [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 105 -------------------- Proved ---------------------- Line 105: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 5: Zero < a23 6: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 7: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 8: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) 12: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 106 -------------------- Proved ---------------------- Line 106: 1: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) E {x5| Zero < x5 & x5 * x5 < a1} -> a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) <= Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 8: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 9: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 12: Real(a23) 13: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 108, 107 -------------------- Proved ---------------------- Line 108: 1: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) <= Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 8: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) 12: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 112 -------------------- Proved ---------------------- Line 112: 1: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) Equal Sup({x6| Zero < x6 & x6 * x6 < a1}) v a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) < Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 8: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) 12: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 114, 113 -------------------- Proved ---------------------- Line 114: 1: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) < Sup({x6| Zero < x6 & x6 * x6 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) |- By NOTBOTHAlready shown (46) -------------------- Proved ---------------------- Line 113: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) Equal Sup({x6| Zero < x6 & x6 * x6 < a1}) |- By NOTBOTH3Already shown (48) -------------------- Proved ---------------------- Line 107: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 5: Zero < a23 6: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 7: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 8: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) 12: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) E {x5| Zero < x5 & x5 * x5 < a1} 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 109 -------------------- Proved ---------------------- Line 109: 1: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 3: Real(a1) 4: Zero < a1 5: Zero < a23 6: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 7: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 8: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) 12: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: Zero < a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) & [a23 + Sup({x3| Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 111, 110 -------------------- Trivial --------------------- Line 111: 1: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 2: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 12: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) |- 1: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 110: 1: Zero < a23 + Sup({x8|Zero < x8 & x8 * x8 < a1}) 2: [a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] * [ a23 + Sup({x8|Zero < x8 & x8 * x8 < a1})] < a1 3: [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: Zero < a23 + Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 98: 1: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 131, 132, 133, 134, 130 -------------------- Proved ---------------------- Line 131: 1: a1 = [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] 2: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 136, 135 -------------------- Proved ---------------------- Line 136: 1: [a23 + Sup({x6|Zero < x6 & x6 * x6 < a1})] * [ a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] = a23 * a23 + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1}) 2: a1 = [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] 3: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 137 -------------------- Proved ---------------------- Line 137: 1: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 139, 140, 141, 138 -------------------- Proved ---------------------- Line 139: 1: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) 3: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 142 -------------------- Proved ---------------------- Line 142: 1: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 3: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 12: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 144, 143 -------------------- Proved ---------------------- Line 144: 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) < a1 -> Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 3: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 146, 145 -------------------- Proved ---------------------- Line 146: 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 3: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 148, 147 -------------------- Proved ---------------------- Line 148: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) = Zero 2: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 149 -------------------- Proved ---------------------- Line 149: 1: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 2: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 3: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 151, 150 -------------------- Proved ---------------------- Line 151: 1: a1 < a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1}) -> a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x11|Zero < x11 & x11 * x11 < a1}) + Two * a23 * Sup({x12|Zero < x12 & x12 * x12 < a1}) + Sup({x13| Zero < x13 & x13 * x13 < a1}) * Sup({x14| Zero < x14 & x14 * x14 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 153, 152 -------------------- Proved ---------------------- Line 153: 1: a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 155, 154 -------------------- Proved ---------------------- Line 155: 1: [a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x9|Zero < x9 & x9 * x9 < a1}) + Sup({x10| Zero < x10 & x10 * x10 < a1}) * Sup({x11| Zero < x11 & x11 * x11 < a1})] + Minus(Sup({x12| Zero < x12 & x12 * x12 < a1}) * Sup({x13| Zero < x13 & x13 * x13 < a1})) = a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + [Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x17|Zero < x17 & x17 * x17 < a1}) * Sup({x18|Zero < x18 & x18 * x18 < a1})) 2: a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 156 -------------------- Proved ---------------------- Line 156: 1: a1 + Minus( Sup({x12|Zero < x12 & x12 * x12 < a1}) * Sup({x13| Zero < x13 & x13 * x13 < a1})) < a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + [ Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x17|Zero < x17 & x17 * x17 < a1}) * Sup({x18|Zero < x18 & x18 * x18 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 158, 157 -------------------- Proved ---------------------- Line 158: 1: [Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x16| Zero < x16 & x16 * x16 < a1}) * Sup({x17| Zero < x17 & x17 * x17 < a1})] + Minus( Sup({x19|Zero < x19 & x19 * x19 < a1}) * Sup({x20|Zero < x20 & x20 * x20 < a1})) = Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x21| Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1}) + Minus(Sup({x23| Zero < x23 & x23 * x23 < a1}) * Sup({x24| Zero < x24 & x24 * x24 < a1})) 2: a1 + Minus( Sup({x12|Zero < x12 & x12 * x12 < a1}) * Sup({x13| Zero < x13 & x13 * x13 < a1})) < a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + [ Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x17|Zero < x17 & x17 * x17 < a1}) * Sup({x18|Zero < x18 & x18 * x18 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 159 -------------------- Proved ---------------------- Line 159: 1: a1 + Minus( Sup({x21|Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1})) < a23 * Sup({x23|Zero < x23 & x23 * x23 < a1}) + Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x24| Zero < x24 & x24 * x24 < a1}) * Sup({x25| Zero < x25 & x25 * x25 < a1}) + Minus(Sup({x26| Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 161, 160 -------------------- Proved ---------------------- Line 161: 1: Sup({x26|Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1}) + Minus(Sup({x26| Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) = Zero 2: a1 + Minus( Sup({x21|Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1})) < a23 * Sup({x23|Zero < x23 & x23 * x23 < a1}) + Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x24| Zero < x24 & x24 * x24 < a1}) * Sup({x25| Zero < x25 & x25 * x25 < a1}) + Minus(Sup({x26| Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 162 -------------------- Proved ---------------------- Line 162: 1: a1 + Minus( Sup({x22|Zero < x22 & x22 * x22 < a1}) * Sup({x23| Zero < x23 & x23 * x23 < a1})) < a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Zero 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 164, 165, 163 -------------------- Proved ---------------------- Line 164: 1: Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Zero = Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 2: a1 + Minus( Sup({x22|Zero < x22 & x22 * x22 < a1}) * Sup({x23| Zero < x23 & x23 * x23 < a1})) < a23 * Sup({x24|Zero < x24 & x24 * x24 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Zero 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 166 -------------------- Proved ---------------------- Line 166: 1: a1 + Minus( Sup({x26|Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) < a23 * Sup({x28|Zero < x28 & x28 * x28 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 168, 169, 167 -------------------- Proved ---------------------- Line 168: 1: a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) = Three * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 2: a1 + Minus( Sup({x26|Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) < a23 * Sup({x28|Zero < x28 & x28 * x28 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 170 -------------------- Proved ---------------------- Line 170: 1: a1 + Minus( Sup({x28|Zero < x28 & x28 * x28 < a1}) * Sup({x29| Zero < x29 & x29 * x29 < a1})) < Three * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 172, 171 -------------------- Proved ---------------------- Line 172: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x28|Zero < x28 & x28 * x28 < a1}) * Sup({x29| Zero < x29 & x29 * x29 < a1})) < Three * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 173 -------------------- Proved ---------------------- Line 173: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 175, 174 -------------------- Proved ---------------------- Line 175: 1: Zero < Inv(Three * Sup({x35|Zero < x35 & x35 * x35 < a1})) & a1 + Minus(Sup({x31| Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 -> [ a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1}))] * Inv(Three * Sup({x37| Zero < x37 & x37 * x37 < a1})) < [ Three * Sup({x38|Zero < x38 & x38 * x38 < a1}) * a23] * Inv( Three * Sup({x39|Zero < x39 & x39 * x39 < a1})) 2: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 14: Real(a1) 15: Zero < a1 16: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 177, 176 -------------------- Proved ---------------------- Line 177: 1: [a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1}))] * Inv(Three * Sup({x35| Zero < x35 & x35 * x35 < a1})) < [ Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23] * Inv( Three * Sup({x37|Zero < x37 & x37 * x37 < a1})) 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 193, 192 -------------------- Proved ---------------------- Line 193: 1: [Three * Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x41|Zero < x41 & x41 * x41 < a1})) = Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) 2: [a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1}))] * Inv(Three * Sup({x35| Zero < x35 & x35 * x35 < a1})) < [ Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23] * Inv( Three * Sup({x37|Zero < x37 & x37 * x37 < a1})) 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 194 -------------------- Proved ---------------------- Line 194: 1: [Three * Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x41|Zero < x41 & x41 * x41 < a1})) = Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) 2: [a1 + Minus( Sup({x32|Zero < x32 & x32 * x32 < a1}) * Sup({x33| Zero < x33 & x33 * x33 < a1}))] * Inv(Three * Sup({x36| Zero < x36 & x36 * x36 < a1})) < [ Three * Sup({x37|Zero < x37 & x37 * x37 < a1}) * a23] * Inv( Three * Sup({x38|Zero < x38 & x38 * x38 < a1})) 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 195 -------------------- Proved ---------------------- Line 195: 1: [Three * Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x41|Zero < x41 & x41 * x41 < a1})) = Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) 2: [a1 + Minus( Sup({x33|Zero < x33 & x33 * x33 < a1}) * Sup({x34| Zero < x34 & x34 * x34 < a1}))] * Inv(Three * Sup({x37| Zero < x37 & x37 * x37 < a1})) < [ Three * Sup({x38|Zero < x38 & x38 * x38 < a1}) * a23] * Inv( Three * Sup({x39|Zero < x39 & x39 * x39 < a1})) 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 196 -------------------- Proved ---------------------- Line 196: 1: [a1 + Minus( Sup({x42|Zero < x42 & x42 * x42 < a1}) * Sup({x43| Zero < x43 & x43 * x43 < a1}))] * Inv(Three * Sup({x44| Zero < x44 & x44 * x44 < a1})) < Three * [ Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23] * Inv( Three * Sup({x46|Zero < x46 & x46 * x46 < a1})) 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 198, 197 -------------------- Proved ---------------------- Line 198: 1: [ Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23] * Inv( Three * Sup({x48|Zero < x48 & x48 * x48 < a1})) = Sup({x45| Zero < x45 & x45 * x45 < a1}) * a23 * Inv(Three * Sup({x49|Zero < x49 & x49 * x49 < a1})) 2: [a1 + Minus( Sup({x42|Zero < x42 & x42 * x42 < a1}) * Sup({x43| Zero < x43 & x43 * x43 < a1}))] * Inv(Three * Sup({x44| Zero < x44 & x44 * x44 < a1})) < Three * [ Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23] * Inv( Three * Sup({x46|Zero < x46 & x46 * x46 < a1})) 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 199 -------------------- Proved ---------------------- Line 199: 1: [a1 + Minus( Sup({x49|Zero < x49 & x49 * x49 < a1}) * Sup({x50| Zero < x50 & x50 * x50 < a1}))] * Inv(Three * Sup({x51| Zero < x51 & x51 * x51 < a1})) < Three * Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23 * Inv( Three * Sup({x52|Zero < x52 & x52 * x52 < a1})) 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 201, 200 -------------------- Proved ---------------------- Line 201: 1: a23 * Inv(Three * Sup({x53|Zero < x53 & x53 * x53 < a1})) = Inv( Three * Sup({x54|Zero < x54 & x54 * x54 < a1})) * a23 2: [a1 + Minus( Sup({x49|Zero < x49 & x49 * x49 < a1}) * Sup({x50| Zero < x50 & x50 * x50 < a1}))] * Inv(Three * Sup({x51| Zero < x51 & x51 * x51 < a1})) < Three * Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23 * Inv( Three * Sup({x52|Zero < x52 & x52 * x52 < a1})) 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 202 -------------------- Proved ---------------------- Line 202: 1: [a1 + Minus( Sup({x54|Zero < x54 & x54 * x54 < a1}) * Sup({x55| Zero < x55 & x55 * x55 < a1}))] * Inv(Three * Sup({x56| Zero < x56 & x56 * x56 < a1})) < Three * Sup({x57|Zero < x57 & x57 * x57 < a1}) * Inv(Three * Sup({x58| Zero < x58 & x58 * x58 < a1})) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 204, 203 -------------------- Proved ---------------------- Line 204: 1: [Three * Sup({x58|Zero < x58 & x58 * x58 < a1})] * Inv( Three * Sup({x60|Zero < x60 & x60 * x60 < a1})) * a23 = Three * Sup({x61|Zero < x61 & x61 * x61 < a1}) * Inv( Three * Sup({x62|Zero < x62 & x62 * x62 < a1})) * a23 2: [a1 + Minus( Sup({x54|Zero < x54 & x54 * x54 < a1}) * Sup({x55| Zero < x55 & x55 * x55 < a1}))] * Inv(Three * Sup({x56| Zero < x56 & x56 * x56 < a1})) < Three * Sup({x57|Zero < x57 & x57 * x57 < a1}) * Inv(Three * Sup({x58| Zero < x58 & x58 * x58 < a1})) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 205 -------------------- Proved ---------------------- Line 205: 1: [a1 + Minus( Sup({x61|Zero < x61 & x61 * x61 < a1}) * Sup({x62| Zero < x62 & x62 * x62 < a1}))] * Inv(Three * Sup({x63| Zero < x63 & x63 * x63 < a1})) < [ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x65|Zero < x65 & x65 * x65 < a1})) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 207, 206 -------------------- Proved ---------------------- Line 207: 1: [ [Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x66|Zero < x66 & x66 * x66 < a1}))] * a23 = [Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv(Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) * a23 2: [a1 + Minus( Sup({x61|Zero < x61 & x61 * x61 < a1}) * Sup({x62| Zero < x62 & x62 * x62 < a1}))] * Inv(Three * Sup({x63| Zero < x63 & x63 * x63 < a1})) < [ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x65|Zero < x65 & x65 * x65 < a1})) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 208 -------------------- Proved ---------------------- Line 208: 1: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 210, 209 -------------------- Proved ---------------------- Line 210: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero v [ Three * Sup({x70|Zero < x70 & x70 * x70 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1})) = One 2: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 211, 212 -------------------- Proved ---------------------- Line 211: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 2: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 248, 249, 250, 251, 247 -------------------- Proved ---------------------- Line 248: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 252 -------------------- Proved ---------------------- Line 252: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero 2: a1 + Minus( Sup({x32|Zero < x32 & x32 * x32 < a1}) * Sup({x33| Zero < x33 & x33 * x33 < a1})) < Three * Sup({x34|Zero < x34 & x34 * x34 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 254, 253 -------------------- Proved ---------------------- Line 254: 1: [Three * Sup({x35|Zero < x35 & x35 * x35 < a1})] * a23 = Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 2: a1 + Minus( Sup({x32|Zero < x32 & x32 * x32 < a1}) * Sup({x33| Zero < x33 & x33 * x33 < a1})) < Three * Sup({x34|Zero < x34 & x34 * x34 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 16: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 255 -------------------- Proved ---------------------- Line 255: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero 2: a1 + Minus( Sup({x36|Zero < x36 & x36 * x36 < a1}) * Sup({x37| Zero < x37 & x37 * x37 < a1})) < [ Three * Sup({x38|Zero < x38 & x38 * x38 < a1})] * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 16: [Three * Sup({x35|Zero < x35 & x35 * x35 < a1})] * a23 = Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 256 -------------------- Proved ---------------------- Line 256: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero 2: a1 + Minus( Sup({x37|Zero < x37 & x37 * x37 < a1}) * Sup({x38| Zero < x38 & x38 * x38 < a1})) < Zero * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 16: [Three * Sup({x35|Zero < x35 & x35 * x35 < a1})] * a23 = Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 258, 257 -------------------- Proved ---------------------- Line 258: 1: Zero * a23 = Zero 2: a1 + Minus( Sup({x37|Zero < x37 & x37 * x37 < a1}) * Sup({x38| Zero < x38 & x38 * x38 < a1})) < Zero * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 15: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 16: [Three * Sup({x35|Zero < x35 & x35 * x35 < a1})] * a23 = Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 17: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 259 -------------------- Proved ---------------------- Line 259: 1: a1 + Minus( Sup({x38|Zero < x38 & x38 * x38 < a1}) * Sup({x39| Zero < x39 & x39 * x39 < a1})) < Zero 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) |- By NOTBOTHAlready shown (46) -------------------- Proved ---------------------- Line 257: |- 1: Zero * a23 = Zero By MZERO2Already shown (43) -------------------- Proved ---------------------- Line 253: |- 1: [Three * Sup({x35|Zero < x35 & x35 * x35 < a1})] * a23 = Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 By ATIMESAlready shown (10) -------------------- Trivial --------------------- Line 249: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 2: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 250: |- 1: Real(Three * Sup({x70|Zero < x70 & x70 * x70 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 251: |- 1: Real(Zero) By RZEROAlready shown (29) -------------------- Proved ---------------------- Line 247: 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero 2: Real(Three * Sup({x70|Zero < x70 & x70 * x70 < a1})) 3: Real(Zero) |- 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) = Zero By EQUALITYAlready shown (36) -------------------- Proved ---------------------- Line 212: 1: [Three * Sup({x70|Zero < x70 & x70 * x70 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1})) = One 2: [a1 + Minus( Sup({x67|Zero < x67 & x67 * x67 < a1}) * Sup({x68| Zero < x68 & x68 * x68 < a1}))] * Inv(Three * Sup({x69| Zero < x69 & x69 * x69 < a1})) < [[ Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1}))] * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 213 -------------------- Proved ---------------------- Line 213: 1: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 215, 214 -------------------- Proved ---------------------- Line 215: 1: Real(a23) -> a23 * One = a23 2: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 216, 217 -------------------- Trivial --------------------- Line 216: 1: Real(a23) 2: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 5: Real(a1) 6: Zero < a1 7: Zero < a23 8: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 9: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 10: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 11: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) |- 1: Real(a23) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 217: 1: a23 * One = a23 2: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 219, 218 -------------------- Proved ---------------------- Line 219: 1: a23 * One = One * a23 2: a23 * One = a23 3: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 4: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 5: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 220 -------------------- Proved ---------------------- Line 220: 1: One * a23 = a23 2: [a1 + Minus( Sup({x68|Zero < x68 & x68 * x68 < a1}) * Sup({x69| Zero < x69 & x69 * x69 < a1}))] * Inv(Three * Sup({x70| Zero < x70 & x70 * x70 < a1})) < One * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 15: a23 * One = One * a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 221 -------------------- Proved ---------------------- Line 221: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 223, 222 -------------------- Proved ---------------------- Line 223: 1: Zero < Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) & Zero < a1 + Minus(Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1})) -> Zero * Inv(Three * Sup({x10| Zero < x10 & x10 * x10 < a1})) < [ a1 + Minus( Sup({x11|Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1}))] * Inv(Three * Sup({x13| Zero < x13 & x13 * x13 < a1})) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 225, 224 -------------------- Proved ---------------------- Line 225: 1: Zero * Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) < [a1 + Minus(Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1}))] * Inv( Three * Sup({x10|Zero < x10 & x10 * x10 < a1})) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 243, 242 -------------------- Proved ---------------------- Line 243: 1: Zero * Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) = Zero 2: Zero * Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) < [a1 + Minus(Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1}))] * Inv( Three * Sup({x10|Zero < x10 & x10 * x10 < a1})) 3: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 4: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 5: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 244 -------------------- Proved ---------------------- Line 244: 1: Zero < [a1 + Minus( Sup({x9|Zero < x9 & x9 * x9 < a1}) * Sup({x10|Zero < x10 & x10 * x10 < a1}))] * Inv(Three * Sup({x11|Zero < x11 & x11 * x11 < a1})) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 245 -------------------- Proved ---------------------- Line 245: 1: Zero < [a1 + Minus( Sup({x9|Zero < x9 & x9 * x9 < a1}) * Sup({x10|Zero < x10 & x10 * x10 < a1}))] * Inv(Three * Sup({x11|Zero < x11 & x11 * x11 < a1})) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) Equal a23 v [a1 + Minus(Sup({x71| Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv( Three * Sup({x73|Zero < x73 & x73 * x73 < a1})) < a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) By 246 -------------------- Trivial --------------------- Line 246: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 14: Zero < [a1 + Minus( Sup({x9|Zero < x9 & x9 * x9 < a1}) * Sup({x10|Zero < x10 & x10 * x10 < a1}))] * Inv(Three * Sup({x11|Zero < x11 & x11 * x11 < a1})) |- 1: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) < a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 3: (Ex3.x3 * x3 = a1) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) Equal a23 -------------------- Proved ---------------------- Line 242: |- 1: Zero * Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) = Zero By MZERO2Already shown (43) -------------------- Proved ---------------------- Line 224: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) & Zero < a1 + Minus(Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 227, 226 -------------------- Trivial --------------------- Line 227: 1: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 2: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 12: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 13: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 |- 1: Zero < a1 + Minus( Sup({x5|Zero < x5 & x5 * x5 < a1}) * Sup({x6|Zero < x6 & x6 * x6 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 226: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 229, 228 -------------------- Proved ---------------------- Line 229: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 231, 230 -------------------- Proved ---------------------- Line 231: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 233, 232 -------------------- Proved ---------------------- Line 233: 1: Zero < Three 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 235, 234 -------------------- Proved ---------------------- Line 235: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) By 237, 236 -------------------- Proved ---------------------- Line 237: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) By 239, 238 -------------------- Trivial --------------------- Line 239: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 238: 1: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 11: Real(a1) 12: Zero < a1 13: Zero < a23 |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 8: (Ex3.x3 * x3 = a1) By 241, 240 -------------------- Trivial --------------------- Line 241: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 8: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 9: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 10: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 13: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 8: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 240: 1: Zero < a23 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 |- 1: Zero < a23 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 8: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 236: |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 234: 1: Zero < Three 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) By TIMESPOSAlready shown (93) -------------------- Proved ---------------------- Line 232: |- 1: Zero < Three By THREEPOSAlready shown (92) -------------------- Proved ---------------------- Line 230: 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) By POSINVAlready shown (73) -------------------- Trivial --------------------- Line 228: 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: [a1 + Minus( Sup({x70|Zero < x70 & x70 * x70 < a1}) * Sup({x71| Zero < x71 & x71 * x71 < a1}))] * Inv(Three * Sup({x72| Zero < x72 & x72 * x72 < a1})) < a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 222: |- 1: Zero < Inv(Three * Sup({x7|Zero < x7 & x7 * x7 < a1})) & Zero < a1 + Minus(Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1})) -> Zero * Inv(Three * Sup({x10| Zero < x10 & x10 * x10 < a1})) < [ a1 + Minus( Sup({x11|Zero < x11 & x11 * x11 < a1}) * Sup({x12| Zero < x12 & x12 * x12 < a1}))] * Inv(Three * Sup({x13| Zero < x13 & x13 * x13 < a1})) By MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 218: |- 1: a23 * One = One * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 214: |- 1: Real(a23) -> a23 * One = a23 By ITIMESAlready shown (13) -------------------- Proved ---------------------- Line 209: |- 1: Three * Sup({x70|Zero < x70 & x70 * x70 < a1}) Equal Zero v [ Three * Sup({x70|Zero < x70 & x70 * x70 < a1})] * Inv( Three * Sup({x70|Zero < x70 & x70 * x70 < a1})) = One By INVAlready shown (15) -------------------- Proved ---------------------- Line 206: |- 1: [ [Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv( Three * Sup({x66|Zero < x66 & x66 * x66 < a1}))] * a23 = [Three * Sup({x64|Zero < x64 & x64 * x64 < a1})] * Inv(Three * Sup({x67|Zero < x67 & x67 * x67 < a1})) * a23 By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 203: |- 1: [Three * Sup({x58|Zero < x58 & x58 * x58 < a1})] * Inv( Three * Sup({x60|Zero < x60 & x60 * x60 < a1})) * a23 = Three * Sup({x61|Zero < x61 & x61 * x61 < a1}) * Inv( Three * Sup({x62|Zero < x62 & x62 * x62 < a1})) * a23 By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 200: |- 1: a23 * Inv(Three * Sup({x53|Zero < x53 & x53 * x53 < a1})) = Inv( Three * Sup({x54|Zero < x54 & x54 * x54 < a1})) * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 197: |- 1: [ Sup({x45|Zero < x45 & x45 * x45 < a1}) * a23] * Inv( Three * Sup({x48|Zero < x48 & x48 * x48 < a1})) = Sup({x45| Zero < x45 & x45 * x45 < a1}) * a23 * Inv(Three * Sup({x49|Zero < x49 & x49 * x49 < a1})) By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 192: |- 1: [Three * Sup({x39|Zero < x39 & x39 * x39 < a1}) * a23] * Inv( Three * Sup({x41|Zero < x41 & x41 * x41 < a1})) = Three * [ Sup({x42|Zero < x42 & x42 * x42 < a1}) * a23] * Inv( Three * Sup({x43|Zero < x43 & x43 * x43 < a1})) By ATIMESAlready shown (10) -------------------- Proved ---------------------- Line 176: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: Zero < Inv(Three * Sup({x35|Zero < x35 & x35 * x35 < a1})) & a1 + Minus(Sup({x31| Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 179, 178 -------------------- Trivial --------------------- Line 179: 1: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 2: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 3: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 4: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) 9: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 10: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 11: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 12: Real(a1) 13: Zero < a1 14: Zero < a23 15: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 |- 1: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x34|Zero < x34 & x34 * x34 < a1}) * a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 178: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 181, 180 -------------------- Proved ---------------------- Line 181: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 183, 182 -------------------- Proved ---------------------- Line 183: 1: Zero < Three 2: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 14: Real(a1) 15: Zero < a1 16: Zero < a23 |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 185, 184 -------------------- Proved ---------------------- Line 185: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 187, 186 -------------------- Proved ---------------------- Line 187: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 14: Real(a1) 15: Zero < a1 16: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) By 189, 188 -------------------- Trivial --------------------- Line 189: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 14: Real(a1) 15: Zero < a1 16: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 4: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 6: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 188: 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 2: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 3: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 4: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 5: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 9: Real(a23) 10: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 11: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 12: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 13: Real(a1) 14: Zero < a1 15: Zero < a23 |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) By 191, 190 -------------------- Trivial --------------------- Line 191: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 8: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 9: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 10: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 11: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 12: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 14: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 15: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 190: 1: Zero < a23 2: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 3: a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x33|Zero < x33 & x33 * x33 < a1}) * a23 4: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 13: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 14: Real(a1) 15: Zero < a1 |- 1: Zero < a23 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) 4: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 7: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 186: |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 184: 1: Zero < Three 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) By TIMESPOSAlready shown (93) -------------------- Proved ---------------------- Line 182: |- 1: Zero < Three By THREEPOSAlready shown (92) -------------------- Proved ---------------------- Line 180: 1: Zero < Three * Sup({x3|Zero < x3 & x3 * x3 < a1}) |- 1: Zero < Inv(Three * Sup({x3|Zero < x3 & x3 * x3 < a1})) By POSINVAlready shown (73) -------------------- Proved ---------------------- Line 174: |- 1: Zero < Inv(Three * Sup({x35|Zero < x35 & x35 * x35 < a1})) & a1 + Minus(Sup({x31| Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1})) < Three * Sup({x36|Zero < x36 & x36 * x36 < a1}) * a23 -> [ a1 + Minus( Sup({x31|Zero < x31 & x31 * x31 < a1}) * Sup({x32| Zero < x32 & x32 * x32 < a1}))] * Inv(Three * Sup({x37| Zero < x37 & x37 * x37 < a1})) < [ Three * Sup({x38|Zero < x38 & x38 * x38 < a1}) * a23] * Inv( Three * Sup({x39|Zero < x39 & x39 * x39 < a1})) By MTIMESAlready shown (22) -------------------- Proved ---------------------- Line 171: |- 1: a23 * Sup({x26|Zero < x26 & x26 * x26 < a1}) = Sup({x27| Zero < x27 & x27 * x27 < a1}) * a23 By CTIMESAlready shown (8) -------------------- Proved ---------------------- Line 169: |- 1: Real(a23 * Sup({x25|Zero < x25 & x25 * x25 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 167: 1: Real(a23 * Sup({x25|Zero < x25 & x25 * x25 < a1})) |- 1: a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) = Three * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) By LIKETERMS1Already shown (94) -------------------- Proved ---------------------- Line 165: |- 1: Real(Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1})) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 163: 1: Real(Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1})) |- 1: Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) + Zero = Two * a23 * Sup({x25|Zero < x25 & x25 * x25 < a1}) By IPLUSaAlready shown (26) -------------------- Proved ---------------------- Line 160: |- 1: Sup({x26|Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1}) + Minus(Sup({x26| Zero < x26 & x26 * x26 < a1}) * Sup({x27| Zero < x27 & x27 * x27 < a1})) = Zero By MINUSAlready shown (14) -------------------- Proved ---------------------- Line 157: |- 1: [Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x16| Zero < x16 & x16 * x16 < a1}) * Sup({x17| Zero < x17 & x17 * x17 < a1})] + Minus( Sup({x19|Zero < x19 & x19 * x19 < a1}) * Sup({x20|Zero < x20 & x20 * x20 < a1})) = Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x21| Zero < x21 & x21 * x21 < a1}) * Sup({x22| Zero < x22 & x22 * x22 < a1}) + Minus(Sup({x23| Zero < x23 & x23 * x23 < a1}) * Sup({x24| Zero < x24 & x24 * x24 < a1})) By APLUSAlready shown (9) -------------------- Proved ---------------------- Line 154: |- 1: [a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x9|Zero < x9 & x9 * x9 < a1}) + Sup({x10| Zero < x10 & x10 * x10 < a1}) * Sup({x11| Zero < x11 & x11 * x11 < a1})] + Minus(Sup({x12| Zero < x12 & x12 * x12 < a1}) * Sup({x13| Zero < x13 & x13 * x13 < a1})) = a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + [Two * a23 * Sup({x14|Zero < x14 & x14 * x14 < a1}) + Sup({x15| Zero < x15 & x15 * x15 < a1}) * Sup({x16| Zero < x16 & x16 * x16 < a1})] + Minus( Sup({x17|Zero < x17 & x17 * x17 < a1}) * Sup({x18|Zero < x18 & x18 * x18 < a1})) By APLUSAlready shown (9) -------------------- Trivial --------------------- Line 152: 1: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 2: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 5: Real(a23) 6: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 7: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 9: Real(a1) 10: Zero < a1 11: Zero < a23 12: Zero < a1 + Minus( Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1})) 13: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) |- 1: a1 < a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Two * a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Sup({x8| Zero < x8 & x8 * x8 < a1}) * Sup({x9| Zero < x9 & x9 * x9 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 150: |- 1: a1 < a23 * Sup({x7|Zero < x7 & x7 * x7 < a1}) + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1}) -> a1 + Minus( Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1})) < [ a23 * Sup({x11|Zero < x11 & x11 * x11 < a1}) + Two * a23 * Sup({x12|Zero < x12 & x12 * x12 < a1}) + Sup({x13| Zero < x13 & x13 * x13 < a1}) * Sup({x14| Zero < x14 & x14 * x14 < a1})] + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) By MPLUSAlready shown (21) -------------------- Proved ---------------------- Line 147: |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) + Minus(Sup({x3| Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) = Zero By MINUSAlready shown (14) -------------------- Trivial --------------------- Line 145: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 2: Real(a1) 3: Zero < a1 4: Zero < a23 5: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 6: a1 < a23 * Sup({x5|Zero < x5 & x5 * x5 < a1}) + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 7: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 8: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 10: Real(a23) 11: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 12: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 143: |- 1: Sup({x4|Zero < x4 & x4 * x4 < a1}) * Sup({x5|Zero < x5 & x5 * x5 < a1}) < a1 -> Sup({x6| Zero < x6 & x6 * x6 < a1}) * Sup({x7| Zero < x7 & x7 * x7 < a1}) + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) < a1 + Minus(Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3| Zero < x3 & x3 * x3 < a1})) By MPLUSAlready shown (21) -------------------- Trivial --------------------- Line 140: 1: Zero < a23 2: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 3: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 6: Real(a23) 7: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 8: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 10: Real(a1) 11: Zero < a1 |- 1: Zero < a23 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 141: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: a1 = a23 * a23 + Two * a23 * Sup({x6|Zero < x6 & x6 * x6 < a1}) + Sup({x7| Zero < x7 & x7 * x7 < a1}) * Sup({x8| Zero < x8 & x8 * x8 < a1}) 8: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 9: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 11: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 138: 1: Zero < a23 2: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) |- 1: a23 * a23 + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) < a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Two * a23 * Sup({x4|Zero < x4 & x4 * x4 < a1}) + Sup({x4| Zero < x4 & x4 * x4 < a1}) * Sup({x4| Zero < x4 & x4 * x4 < a1}) By ORDERLEMMA1Already shown (91) -------------------- Proved ---------------------- Line 135: |- 1: [a23 + Sup({x6|Zero < x6 & x6 * x6 < a1})] * [ a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] = a23 * a23 + Two * a23 * Sup({x8|Zero < x8 & x8 * x8 < a1}) + Sup({x9| Zero < x9 & x9 * x9 < a1}) * Sup({x10| Zero < x10 & x10 * x10 < a1}) By FOILAlready shown (88) -------------------- Trivial --------------------- Line 132: 1: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: a1 Equal [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 133: 1: Real(a1) 2: Zero < a1 3: Zero < a23 4: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < [a23 + Sup({x6| Zero < x6 & x6 * x6 < a1})] * [a23 + Sup({x7|Zero < x7 & x7 * x7 < a1})] 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 7: Real(a23) 8: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 9: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 10: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 |- 1: Real(a1) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 134: |- 1: Real([a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})]) By RTIMESAlready shown (3) -------------------- Proved ---------------------- Line 130: 1: a1 Equal [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] 2: Real(a1) 3: Real([a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})]) |- 1: a1 = [a23 + Sup({x4|Zero < x4 & x4 * x4 < a1})] * [ a23 + Sup({x5|Zero < x5 & x5 * x5 < a1})] By EQUALITYAlready shown (36) -------------------- Proved ---------------------- Line 96: |- 1: a1 Equal [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [ a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] v a1 < [a23 + Sup({x3| Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] v [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] * [a23 + Sup({x3|Zero < x3 & x3 * x3 < a1})] < a1 By TRIAlready shown (18) -------------------- Trivial --------------------- Line 75: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x6|Zero < x6 & x6 * x6 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 76: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 79, 78 -------------------- Proved ---------------------- Line 79: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 81, 80 -------------------- Trivial --------------------- Line 81: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 80: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) By 83, 82 -------------------- Trivial --------------------- Line 83: 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 4: Real(a1) 5: Zero < a1 6: Zero < a23 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 8: Real(a23) |- 1: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Trivial --------------------- Line 82: 1: Zero < a23 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 |- 1: Zero < a23 2: Zero < Sup({x3|Zero < x3 & x3 * x3 < a1}) 3: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 4: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 5: (Ex3.x3 * x3 = a1) -------------------- Proved ---------------------- Line 78: |- 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) By TRANSAlready shown (19) -------------------- Proved ---------------------- Line 77: 1: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Real(a23) 3: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 4: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 5: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 6: Real(a1) 7: Zero < a1 8: Zero < a23 |- 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 85, 84 -------------------- Proved ---------------------- Line 85: 1: Zero < a23 & a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) -> Zero < Sup({x4| Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 87, 86 -------------------- Proved ---------------------- Line 87: 1: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 2: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 3: Real(a23) 4: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 5: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 6: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 7: Real(a1) 8: Zero < a1 9: Zero < a23 |- 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 91, 90 -------------------- Proved ---------------------- Line 91: 1: Zero < Sup({x6|Zero < x6 & x6 * x6 < a1}) & Sup({x7| Zero < x7 & x7 * x7 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) -> Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4 E {x3|Zero < x3 & x3 * x3 < a1} -> x4 <= Sup({x3|Zero < x3 & x3 * x3 < a1})) 7: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) < a1 8: Real(a1) 9: Zero < a1 10: Zero < a23 |- 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: [a1 + Minus( Sup({x71|Zero < x71 & x71 * x71 < a1}) * Sup({x72| Zero < x72 & x72 * x72 < a1}))] * Inv(Three * Sup({x73| Zero < x73 & x73 * x73 < a1})) <= a23 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) * Sup({x3|Zero < x3 & x3 * x3 < a1}) = a1 4: (Ex3.x3 * x3 = a1) By 93, 92 -------------------- Trivial --------------------- Line 93: 1: Zero < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 2: Zero < Sup({x4|Zero < x4 & x4 * x4 < a1}) 3: Sup({x3|Zero < x3 & x3 * x3 < a1}) < a23 + Sup({x5|Zero < x5 & x5 * x5 < a1}) 4: Real(a23) 5: a23 < Sup({x4|Zero < x4 & x4 * x4 < a1}) 6: (Ax4. x4