%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : HEN011-3 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n010.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:19:01 AM UTC 2026
% Result : Unsatisfiable 6.03s 1.25s
% Output : Proof 7.64s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : HEN011-3 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.37 % Computer : n010.cluster.edu
% 0.10/0.37 % Model : x86_64 x86_64
% 0.10/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.37 % Memory : 8046.5625MB
% 0.10/0.37 % OS : Linux 6.8.0-71-generic
% 0.10/0.37 % CPULimit : 300
% 0.10/0.37 % WCLimit : 300
% 0.10/0.37 % DateTime : Sun Sep 27 10:52:01 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 6.03/1.25 Command-line arguments: --no-flatten-goal
% 6.03/1.25
% 6.03/1.25 % SZS status Unsatisfiable
% 6.03/1.25
% 6.82/1.37 % SZS output start Proof
% 6.82/1.37 Axiom 1 (identity_is_largest): less_equal(X, identity) = true.
% 6.82/1.37 Axiom 2 (zero_is_smallest): less_equal(zero, X) = true.
% 6.82/1.37 Axiom 3 (identity_divide_a): divide(identity, a) = c.
% 6.82/1.37 Axiom 4 (identity_divide_b): divide(identity, b) = d.
% 6.82/1.37 Axiom 5 (identity_divide_c): divide(identity, c) = e.
% 6.82/1.37 Axiom 6 (identity_divide_d): divide(identity, d) = g.
% 6.82/1.37 Axiom 7 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 6.82/1.37 Axiom 8 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 6.82/1.37 Axiom 9 (quotient_smaller_than_numerator): less_equal(divide(X, Y), X) = true.
% 6.82/1.37 Axiom 10 (quotient_less_equal2): ifeq2(divide(X, Y), zero, less_equal(X, Y), true) = true.
% 6.82/1.37 Axiom 11 (quotient_less_equal1): ifeq(less_equal(X, Y), true, divide(X, Y), zero) = zero.
% 6.82/1.37 Axiom 12 (less_equal_and_equal): ifeq(less_equal(X, Y), true, ifeq(less_equal(Y, X), true, Y, X), X) = X.
% 6.82/1.37 Axiom 13 (quotient_property): less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = true.
% 6.82/1.37
% 6.82/1.37 Lemma 14: divide(divide(X, Y), X) = zero.
% 6.82/1.37 Proof:
% 6.82/1.37 divide(divide(X, Y), X)
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(true, true, divide(divide(X, Y), X), zero)
% 6.82/1.37 = { by axiom 9 (quotient_smaller_than_numerator) R->L }
% 6.82/1.37 ifeq(less_equal(divide(X, Y), X), true, divide(divide(X, Y), X), zero)
% 6.82/1.37 = { by axiom 11 (quotient_less_equal1) }
% 6.82/1.37 zero
% 6.82/1.37
% 6.82/1.37 Lemma 15: divide(zero, X) = zero.
% 6.82/1.37 Proof:
% 6.82/1.37 divide(zero, X)
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(true, true, divide(zero, X), zero)
% 6.82/1.37 = { by axiom 2 (zero_is_smallest) R->L }
% 6.82/1.37 ifeq(less_equal(zero, X), true, divide(zero, X), zero)
% 6.82/1.37 = { by axiom 11 (quotient_less_equal1) }
% 6.82/1.37 zero
% 6.82/1.37
% 6.82/1.37 Lemma 16: divide(X, X) = zero.
% 6.82/1.37 Proof:
% 6.82/1.37 divide(X, X)
% 6.82/1.37 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.37 ifeq(less_equal(divide(X, X), zero), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 7 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(ifeq2(zero, zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(true, true, zero, divide(divide(X, X), divide(zero, X))), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 2 (zero_is_smallest) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(X, X), divide(zero, X))), true, zero, divide(divide(X, X), divide(zero, X))), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(X, X), divide(zero, X))), true, zero, divide(divide(X, X), divide(zero, X))), divide(divide(X, X), divide(zero, X))), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 13 (quotient_property) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(less_equal(divide(divide(X, X), divide(zero, X)), divide(divide(X, zero), X)), true, ifeq(less_equal(zero, divide(divide(X, X), divide(zero, X))), true, zero, divide(divide(X, X), divide(zero, X))), divide(divide(X, X), divide(zero, X))), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by lemma 14 }
% 6.82/1.37 ifeq(ifeq2(ifeq(less_equal(divide(divide(X, X), divide(zero, X)), zero), true, ifeq(less_equal(zero, divide(divide(X, X), divide(zero, X))), true, zero, divide(divide(X, X), divide(zero, X))), divide(divide(X, X), divide(zero, X))), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.37 ifeq(ifeq2(divide(divide(X, X), divide(zero, X)), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by lemma 15 }
% 6.82/1.37 ifeq(ifeq2(divide(divide(X, X), zero), zero, less_equal(divide(X, X), zero), true), true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 10 (quotient_less_equal2) }
% 6.82/1.37 ifeq(true, true, ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X)), divide(X, X))
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.37 ifeq(less_equal(zero, divide(X, X)), true, zero, divide(X, X))
% 6.82/1.37 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.37 ifeq(true, true, zero, divide(X, X))
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.37 zero
% 6.82/1.37
% 6.82/1.37 Lemma 17: divide(X, zero) = X.
% 6.82/1.37 Proof:
% 6.82/1.37 divide(X, zero)
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(true, true, divide(X, zero), X)
% 6.82/1.37 = { by axiom 9 (quotient_smaller_than_numerator) R->L }
% 6.82/1.37 ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X)
% 6.82/1.37 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(true, true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.37 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.37 ifeq(ifeq2(divide(X, divide(X, zero)), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.37 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(less_equal(divide(X, divide(X, zero)), zero), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.37 = { by axiom 7 (ifeq_axiom) R->L }
% 6.82/1.37 ifeq(ifeq2(ifeq(ifeq2(zero, zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(true, true, zero, divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 2 (zero_is_smallest) R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 13 (quotient_property) R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), divide(zero, divide(X, zero))), divide(divide(X, zero), divide(X, zero))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by lemma 15 }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), divide(divide(X, zero), divide(X, zero))), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by lemma 16 }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(ifeq(less_equal(divide(divide(X, divide(X, zero)), zero), zero), true, ifeq(less_equal(zero, divide(divide(X, divide(X, zero)), zero)), true, zero, divide(divide(X, divide(X, zero)), zero)), divide(divide(X, divide(X, zero)), zero)), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.38 ifeq(ifeq2(ifeq(ifeq2(divide(divide(X, divide(X, zero)), zero), zero, less_equal(divide(X, divide(X, zero)), zero), true), true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 10 (quotient_less_equal2) }
% 6.82/1.38 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.38 ifeq(ifeq2(ifeq(less_equal(zero, divide(X, divide(X, zero))), true, zero, divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.38 ifeq(ifeq2(ifeq(true, true, zero, divide(X, divide(X, zero))), zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.38 ifeq(ifeq2(zero, zero, less_equal(X, divide(X, zero)), true), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.38 ifeq(less_equal(X, divide(X, zero)), true, ifeq(less_equal(divide(X, zero), X), true, divide(X, zero), X), X)
% 6.82/1.38 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.38 X
% 6.82/1.38
% 6.82/1.38 Lemma 18: divide(X, identity) = zero.
% 6.82/1.38 Proof:
% 6.82/1.38 divide(X, identity)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.38 ifeq(true, true, divide(X, identity), zero)
% 6.82/1.38 = { by axiom 1 (identity_is_largest) R->L }
% 6.82/1.38 ifeq(less_equal(X, identity), true, divide(X, identity), zero)
% 6.82/1.38 = { by axiom 11 (quotient_less_equal1) }
% 6.82/1.38 zero
% 6.82/1.38
% 6.82/1.38 Lemma 19: divide(a, c) = e.
% 6.82/1.38 Proof:
% 6.82/1.38 divide(a, c)
% 6.82/1.38 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.38 ifeq(true, true, divide(a, c), e)
% 6.82/1.38 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.38 ifeq(ifeq2(divide(e, divide(a, c)), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.38 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(less_equal(divide(e, divide(a, c)), zero), true, ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.38 = { by lemma 16 R->L }
% 6.82/1.38 ifeq(ifeq2(ifeq(less_equal(divide(e, divide(a, c)), divide(c, c)), true, ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.38 = { by axiom 5 (identity_divide_c) R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(divide(identity, c), divide(a, c)), divide(c, c)), true, ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 3 (identity_divide_a) R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(divide(identity, c), divide(a, c)), divide(divide(identity, a), c)), true, ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 13 (quotient_property) }
% 6.82/1.39 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(zero, divide(e, divide(a, c))), true, zero, divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.39 ifeq(ifeq2(ifeq(true, true, zero, divide(e, divide(a, c))), zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(ifeq2(zero, zero, less_equal(e, divide(a, c)), true), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, divide(a, c), e)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(true, true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(divide(divide(a, c), e), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(a, c), e), zero), true, ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by lemma 15 R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(a, c), e), divide(zero, c)), true, ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by lemma 18 R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(a, c), e), divide(divide(a, identity), c)), true, ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 5 (identity_divide_c) R->L }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(a, c), divide(identity, c)), divide(divide(a, identity), c)), true, ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 13 (quotient_property) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(a, c), e)), true, zero, divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(ifeq(true, true, zero, divide(divide(a, c), e)), zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(ifeq2(zero, zero, less_equal(divide(a, c), e), true), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.39 ifeq(less_equal(e, divide(a, c)), true, ifeq(less_equal(divide(a, c), e), true, divide(a, c), e), e)
% 6.82/1.39 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.39 e
% 6.82/1.39
% 6.82/1.39 Lemma 20: divide(b, d) = g.
% 6.82/1.39 Proof:
% 6.82/1.39 divide(b, d)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.39 ifeq(true, true, divide(b, d), g)
% 6.82/1.39 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.39 ifeq(ifeq2(divide(g, divide(b, d)), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(g, divide(b, d)), zero), true, ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by lemma 16 R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(g, divide(b, d)), divide(d, d)), true, ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 6 (identity_divide_d) R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(divide(identity, d), divide(b, d)), divide(d, d)), true, ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 4 (identity_divide_b) R->L }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(divide(divide(identity, d), divide(b, d)), divide(divide(identity, b), d)), true, ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 13 (quotient_property) }
% 6.82/1.39 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(ifeq2(ifeq(less_equal(zero, divide(g, divide(b, d))), true, zero, divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.39 ifeq(ifeq2(ifeq(true, true, zero, divide(g, divide(b, d))), zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.39 ifeq(ifeq2(zero, zero, less_equal(g, divide(b, d)), true), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.39 ifeq(less_equal(g, divide(b, d)), true, divide(b, d), g)
% 6.82/1.39 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.39 ifeq(less_equal(g, divide(b, d)), true, ifeq(true, true, divide(b, d), g), g)
% 6.82/1.39 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.39 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(divide(divide(b, d), g), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.39 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.39 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(b, d), g), zero), true, ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.39 = { by lemma 15 R->L }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(b, d), g), divide(zero, d)), true, ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by lemma 18 R->L }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(b, d), g), divide(divide(b, identity), d)), true, ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 6 (identity_divide_d) R->L }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(less_equal(divide(divide(b, d), divide(identity, d)), divide(divide(b, identity), d)), true, ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 13 (quotient_property) }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(b, d), g)), true, zero, divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(ifeq(true, true, zero, divide(divide(b, d), g)), zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(ifeq2(zero, zero, less_equal(divide(b, d), g), true), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(g, divide(b, d)), true, ifeq(less_equal(divide(b, d), g), true, divide(b, d), g), g)
% 6.82/1.40 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.40 g
% 6.82/1.40
% 6.82/1.40 Lemma 21: divide(divide(divide(X, Y), Z), divide(X, Z)) = zero.
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(divide(X, Y), Z), divide(X, Z))
% 6.82/1.40 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.40 ifeq(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), zero), true, ifeq(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by lemma 15 R->L }
% 6.82/1.40 ifeq(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), divide(zero, Z)), true, ifeq(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by lemma 14 R->L }
% 6.82/1.40 ifeq(less_equal(divide(divide(divide(X, Y), Z), divide(X, Z)), divide(divide(divide(X, Y), X), Z)), true, ifeq(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by axiom 13 (quotient_property) }
% 6.82/1.40 ifeq(true, true, ifeq(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z))), divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(zero, divide(divide(divide(X, Y), Z), divide(X, Z))), true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by axiom 2 (zero_is_smallest) }
% 6.82/1.40 ifeq(true, true, zero, divide(divide(divide(X, Y), Z), divide(X, Z)))
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) }
% 6.82/1.40 zero
% 6.82/1.40
% 6.82/1.40 Lemma 22: divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)) = zero.
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y))
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.40 ifeq(true, true, divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)), zero)
% 6.82/1.40 = { by axiom 13 (quotient_property) R->L }
% 6.82/1.40 ifeq(less_equal(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)), true, divide(divide(divide(X, Y), divide(Z, Y)), divide(divide(X, Z), Y)), zero)
% 6.82/1.40 = { by axiom 11 (quotient_less_equal1) }
% 6.82/1.40 zero
% 6.82/1.40
% 6.82/1.40 Lemma 23: divide(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)) = zero.
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(X, Y), divide(divide(X, divide(Y, Z)), Y))
% 6.82/1.40 = { by lemma 17 R->L }
% 6.82/1.40 divide(divide(divide(X, Y), zero), divide(divide(X, divide(Y, Z)), Y))
% 6.82/1.40 = { by lemma 14 R->L }
% 6.82/1.40 divide(divide(divide(X, Y), divide(divide(Y, Z), Y)), divide(divide(X, divide(Y, Z)), Y))
% 6.82/1.40 = { by lemma 22 }
% 6.82/1.40 zero
% 6.82/1.40
% 6.82/1.40 Lemma 24: divide(divide(X, divide(Y, Z)), Y) = divide(X, Y).
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(X, divide(Y, Z)), Y)
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.40 ifeq(true, true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y))
% 6.82/1.40 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.40 ifeq(ifeq2(divide(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), zero, less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y))
% 6.82/1.40 = { by lemma 21 }
% 6.82/1.40 ifeq(ifeq2(zero, zero, less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y))
% 6.82/1.40 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y))
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.40 ifeq(true, true, ifeq(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), divide(X, Y))
% 6.82/1.40 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.40 ifeq(ifeq2(divide(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), zero, less_equal(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true), true, ifeq(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), divide(X, Y))
% 6.82/1.40 = { by lemma 23 }
% 6.82/1.40 ifeq(ifeq2(zero, zero, less_equal(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true), true, ifeq(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), divide(X, Y))
% 6.82/1.40 = { by axiom 7 (ifeq_axiom) }
% 6.82/1.40 ifeq(less_equal(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), true, ifeq(less_equal(divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), true, divide(divide(X, divide(Y, Z)), Y), divide(X, Y)), divide(X, Y))
% 6.82/1.40 = { by axiom 12 (less_equal_and_equal) }
% 6.82/1.40 divide(X, Y)
% 6.82/1.40
% 6.82/1.40 Lemma 25: divide(divide(X, g), b) = divide(X, b).
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(X, g), b)
% 6.82/1.40 = { by lemma 20 R->L }
% 6.82/1.40 divide(divide(X, divide(b, d)), b)
% 6.82/1.40 = { by lemma 24 }
% 6.82/1.40 divide(X, b)
% 6.82/1.40
% 6.82/1.40 Lemma 26: divide(divide(d, a), g) = divide(c, g).
% 6.82/1.40 Proof:
% 6.82/1.40 divide(divide(d, a), g)
% 6.82/1.40 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.40 ifeq(true, true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.40 = { by axiom 10 (quotient_less_equal2) R->L }
% 6.82/1.40 ifeq(ifeq2(divide(divide(divide(d, a), g), divide(c, g)), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.40 = { by axiom 12 (less_equal_and_equal) R->L }
% 6.82/1.40 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), zero), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.40 = { by lemma 15 R->L }
% 6.82/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(zero, g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.41 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(true, true, zero, divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.41 = { by axiom 2 (zero_is_smallest) R->L }
% 6.82/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.41 = { by axiom 8 (ifeq_axiom) R->L }
% 6.82/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(true, true, ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 6.82/1.41 = { by axiom 13 (quotient_property) R->L }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(less_equal(divide(divide(d, a), divide(identity, a)), divide(divide(d, identity), a)), true, ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 3 (identity_divide_a) }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(less_equal(divide(divide(d, a), c), divide(divide(d, identity), a)), true, ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by lemma 18 }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(less_equal(divide(divide(d, a), c), divide(zero, a)), true, ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by lemma 15 }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(ifeq(less_equal(divide(divide(d, a), c), zero), true, ifeq(less_equal(zero, divide(divide(d, a), c)), true, zero, divide(divide(d, a), c)), divide(divide(d, a), c)), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(d, a), g), divide(c, g)), divide(divide(divide(d, a), c), g)), true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 13 (quotient_property) }
% 7.64/1.41 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.41 ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(divide(d, a), g), divide(c, g))), true, zero, divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 2 (zero_is_smallest) }
% 7.64/1.41 ifeq(ifeq2(ifeq(true, true, zero, divide(divide(divide(d, a), g), divide(c, g))), zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.41 ifeq(ifeq2(zero, zero, less_equal(divide(divide(d, a), g), divide(c, g)), true), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.41 ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g))
% 7.64/1.41 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.41 ifeq(true, true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.41 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.41 ifeq(ifeq2(divide(divide(c, g), divide(divide(d, a), g)), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.41 = { by lemma 17 R->L }
% 7.64/1.41 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), zero), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.41 = { by lemma 23 R->L }
% 7.64/1.41 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(divide(divide(c, divide(d, a)), divide(g, a)), g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.41 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.41 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(ifeq(true, true, divide(divide(c, divide(d, a)), divide(g, a)), zero), g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 13 (quotient_property) R->L }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(ifeq(less_equal(divide(divide(identity, a), divide(d, a)), divide(divide(identity, d), a)), true, divide(divide(c, divide(d, a)), divide(g, a)), zero), g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 6 (identity_divide_d) }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(ifeq(less_equal(divide(divide(identity, a), divide(d, a)), divide(g, a)), true, divide(divide(c, divide(d, a)), divide(g, a)), zero), g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 3 (identity_divide_a) }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(ifeq(less_equal(divide(c, divide(d, a)), divide(g, a)), true, divide(divide(c, divide(d, a)), divide(g, a)), zero), g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 11 (quotient_less_equal1) }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), divide(zero, g))), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by lemma 15 }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(divide(c, divide(d, a)), g), zero)), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by lemma 17 }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, g), divide(divide(d, a), g)), divide(divide(c, divide(d, a)), g)), zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by lemma 22 }
% 7.64/1.42 ifeq(ifeq2(zero, zero, less_equal(divide(c, g), divide(divide(d, a), g)), true), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.42 ifeq(less_equal(divide(c, g), divide(divide(d, a), g)), true, ifeq(less_equal(divide(divide(d, a), g), divide(c, g)), true, divide(divide(d, a), g), divide(c, g)), divide(c, g))
% 7.64/1.42 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.42 divide(c, g)
% 7.64/1.42
% 7.64/1.42 Lemma 27: divide(divide(d, a), b) = divide(c, b).
% 7.64/1.42 Proof:
% 7.64/1.42 divide(divide(d, a), b)
% 7.64/1.42 = { by lemma 25 R->L }
% 7.64/1.42 divide(divide(divide(d, a), g), b)
% 7.64/1.42 = { by lemma 26 }
% 7.64/1.42 divide(divide(c, g), b)
% 7.64/1.42 = { by lemma 25 }
% 7.64/1.42 divide(c, b)
% 7.64/1.42
% 7.64/1.42 Lemma 28: divide(divide(X, e), a) = divide(X, a).
% 7.64/1.42 Proof:
% 7.64/1.42 divide(divide(X, e), a)
% 7.64/1.42 = { by lemma 19 R->L }
% 7.64/1.42 divide(divide(X, divide(a, c)), a)
% 7.64/1.42 = { by lemma 24 }
% 7.64/1.42 divide(X, a)
% 7.64/1.42
% 7.64/1.42 Lemma 29: divide(divide(c, b), e) = divide(d, e).
% 7.64/1.42 Proof:
% 7.64/1.42 divide(divide(c, b), e)
% 7.64/1.42 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.42 ifeq(true, true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.42 ifeq(ifeq2(divide(divide(divide(c, b), e), divide(d, e)), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by axiom 12 (less_equal_and_equal) R->L }
% 7.64/1.42 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), zero), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by lemma 15 R->L }
% 7.64/1.42 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(zero, e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.42 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(true, true, zero, divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by axiom 2 (zero_is_smallest) R->L }
% 7.64/1.42 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.42 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.42 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(true, true, ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 13 (quotient_property) R->L }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(less_equal(divide(divide(c, b), divide(identity, b)), divide(divide(c, identity), b)), true, ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 4 (identity_divide_b) }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(less_equal(divide(divide(c, b), d), divide(divide(c, identity), b)), true, ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by lemma 18 }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(less_equal(divide(divide(c, b), d), divide(zero, b)), true, ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by lemma 15 }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(ifeq(less_equal(divide(divide(c, b), d), zero), true, ifeq(less_equal(zero, divide(divide(c, b), d)), true, zero, divide(divide(c, b), d)), divide(divide(c, b), d)), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(divide(divide(divide(c, b), e), divide(d, e)), divide(divide(divide(c, b), d), e)), true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 13 (quotient_property) }
% 7.64/1.43 ifeq(ifeq2(ifeq(true, true, ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.43 ifeq(ifeq2(ifeq(less_equal(zero, divide(divide(divide(c, b), e), divide(d, e))), true, zero, divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 2 (zero_is_smallest) }
% 7.64/1.43 ifeq(ifeq2(ifeq(true, true, zero, divide(divide(divide(c, b), e), divide(d, e))), zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.43 ifeq(ifeq2(zero, zero, less_equal(divide(divide(c, b), e), divide(d, e)), true), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.43 ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e))
% 7.64/1.43 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.43 ifeq(true, true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.43 ifeq(ifeq2(divide(divide(d, e), divide(divide(c, b), e)), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by lemma 17 R->L }
% 7.64/1.43 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), zero), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by lemma 23 R->L }
% 7.64/1.43 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(divide(divide(d, divide(c, b)), divide(e, b)), e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.43 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(ifeq(true, true, divide(divide(d, divide(c, b)), divide(e, b)), zero), e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by axiom 13 (quotient_property) R->L }
% 7.64/1.43 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(ifeq(less_equal(divide(divide(identity, b), divide(c, b)), divide(divide(identity, c), b)), true, divide(divide(d, divide(c, b)), divide(e, b)), zero), e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.43 = { by axiom 5 (identity_divide_c) }
% 7.64/1.43 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(ifeq(less_equal(divide(divide(identity, b), divide(c, b)), divide(e, b)), true, divide(divide(d, divide(c, b)), divide(e, b)), zero), e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by axiom 4 (identity_divide_b) }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(ifeq(less_equal(divide(d, divide(c, b)), divide(e, b)), true, divide(divide(d, divide(c, b)), divide(e, b)), zero), e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by axiom 11 (quotient_less_equal1) }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), divide(zero, e))), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by lemma 15 }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(divide(d, divide(c, b)), e), zero)), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by lemma 17 }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, e), divide(divide(c, b), e)), divide(divide(d, divide(c, b)), e)), zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by lemma 22 }
% 7.64/1.44 ifeq(ifeq2(zero, zero, less_equal(divide(d, e), divide(divide(c, b), e)), true), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(d, e), divide(divide(c, b), e)), true, ifeq(less_equal(divide(divide(c, b), e), divide(d, e)), true, divide(divide(c, b), e), divide(d, e)), divide(d, e))
% 7.64/1.44 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.44 divide(d, e)
% 7.64/1.44
% 7.64/1.44 Lemma 30: divide(d, a) = divide(c, b).
% 7.64/1.44 Proof:
% 7.64/1.44 divide(d, a)
% 7.64/1.44 = { by axiom 12 (less_equal_and_equal) R->L }
% 7.64/1.44 ifeq(less_equal(divide(d, a), divide(divide(d, a), b)), true, ifeq(less_equal(divide(divide(d, a), b), divide(d, a)), true, divide(divide(d, a), b), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by axiom 9 (quotient_smaller_than_numerator) }
% 7.64/1.44 ifeq(less_equal(divide(d, a), divide(divide(d, a), b)), true, ifeq(true, true, divide(divide(d, a), b), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(d, a), divide(divide(d, a), b)), true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by lemma 27 }
% 7.64/1.44 ifeq(less_equal(divide(d, a), divide(c, b)), true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by lemma 28 R->L }
% 7.64/1.44 ifeq(less_equal(divide(divide(d, e), a), divide(c, b)), true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by lemma 29 R->L }
% 7.64/1.44 ifeq(less_equal(divide(divide(divide(c, b), e), a), divide(c, b)), true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by lemma 28 }
% 7.64/1.44 ifeq(less_equal(divide(divide(c, b), a), divide(c, b)), true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by axiom 9 (quotient_smaller_than_numerator) }
% 7.64/1.44 ifeq(true, true, divide(divide(d, a), b), divide(d, a))
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.44 divide(divide(d, a), b)
% 7.64/1.44 = { by lemma 27 }
% 7.64/1.44 divide(c, b)
% 7.64/1.44
% 7.64/1.44 Lemma 31: divide(divide(X, Y), divide(X, divide(Y, Z))) = zero.
% 7.64/1.44 Proof:
% 7.64/1.44 divide(divide(X, Y), divide(X, divide(Y, Z)))
% 7.64/1.44 = { by lemma 17 R->L }
% 7.64/1.44 divide(divide(divide(X, Y), divide(X, divide(Y, Z))), zero)
% 7.64/1.44 = { by lemma 15 R->L }
% 7.64/1.44 divide(divide(divide(X, Y), divide(X, divide(Y, Z))), divide(zero, divide(X, divide(Y, Z))))
% 7.64/1.44 = { by lemma 23 R->L }
% 7.64/1.44 divide(divide(divide(X, Y), divide(X, divide(Y, Z))), divide(divide(divide(X, Y), divide(divide(X, divide(Y, Z)), Y)), divide(X, divide(Y, Z))))
% 7.64/1.44 = { by lemma 23 }
% 7.64/1.44 zero
% 7.64/1.44
% 7.64/1.44 Lemma 32: divide(divide(X, Y), Y) = divide(X, Y).
% 7.64/1.44 Proof:
% 7.64/1.44 divide(divide(X, Y), Y)
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.44 ifeq(true, true, divide(divide(X, Y), Y), divide(X, Y))
% 7.64/1.44 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(X, Y), Y), divide(X, Y)), zero, less_equal(divide(divide(X, Y), Y), divide(X, Y)), true), true, divide(divide(X, Y), Y), divide(X, Y))
% 7.64/1.44 = { by lemma 21 }
% 7.64/1.44 ifeq(ifeq2(zero, zero, less_equal(divide(divide(X, Y), Y), divide(X, Y)), true), true, divide(divide(X, Y), Y), divide(X, Y))
% 7.64/1.44 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y))
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.44 ifeq(true, true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(X, Y), divide(divide(X, Y), Y)), zero, less_equal(divide(X, Y), divide(divide(X, Y), Y)), true), true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by lemma 17 R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(X, Y), zero), divide(divide(X, Y), Y)), zero, less_equal(divide(X, Y), divide(divide(X, Y), Y)), true), true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by lemma 16 R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(X, Y), divide(Y, Y)), divide(divide(X, Y), Y)), zero, less_equal(divide(X, Y), divide(divide(X, Y), Y)), true), true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by lemma 22 }
% 7.64/1.44 ifeq(ifeq2(zero, zero, less_equal(divide(X, Y), divide(divide(X, Y), Y)), true), true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(X, Y), divide(divide(X, Y), Y)), true, ifeq(less_equal(divide(divide(X, Y), Y), divide(X, Y)), true, divide(divide(X, Y), Y), divide(X, Y)), divide(X, Y))
% 7.64/1.44 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.44 divide(X, Y)
% 7.64/1.44
% 7.64/1.44 Lemma 33: divide(d, e) = divide(c, g).
% 7.64/1.44 Proof:
% 7.64/1.44 divide(d, e)
% 7.64/1.44 = { by lemma 29 R->L }
% 7.64/1.44 divide(divide(c, b), e)
% 7.64/1.44 = { by lemma 30 R->L }
% 7.64/1.44 divide(divide(d, a), e)
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.44 ifeq(true, true, divide(divide(d, a), e), divide(d, a))
% 7.64/1.44 = { by axiom 9 (quotient_smaller_than_numerator) R->L }
% 7.64/1.44 ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a))
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) R->L }
% 7.64/1.44 ifeq(true, true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by axiom 10 (quotient_less_equal2) R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, a), a), divide(divide(d, a), e)), zero, less_equal(divide(divide(d, a), a), divide(divide(d, a), e)), true), true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by lemma 19 R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(d, a), a), divide(divide(d, a), divide(a, c))), zero, less_equal(divide(divide(d, a), a), divide(divide(d, a), e)), true), true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by lemma 31 }
% 7.64/1.44 ifeq(ifeq2(zero, zero, less_equal(divide(divide(d, a), a), divide(divide(d, a), e)), true), true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by axiom 7 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(divide(d, a), a), divide(divide(d, a), e)), true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by lemma 32 }
% 7.64/1.44 ifeq(less_equal(divide(d, a), divide(divide(d, a), e)), true, ifeq(less_equal(divide(divide(d, a), e), divide(d, a)), true, divide(divide(d, a), e), divide(d, a)), divide(d, a))
% 7.64/1.44 = { by axiom 12 (less_equal_and_equal) }
% 7.64/1.44 divide(d, a)
% 7.64/1.44 = { by lemma 30 }
% 7.64/1.44 divide(c, b)
% 7.64/1.44 = { by axiom 12 (less_equal_and_equal) R->L }
% 7.64/1.44 ifeq(less_equal(divide(c, b), divide(divide(c, b), g)), true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by lemma 32 R->L }
% 7.64/1.44 ifeq(less_equal(divide(divide(c, b), b), divide(divide(c, b), g)), true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by axiom 7 (ifeq_axiom) R->L }
% 7.64/1.44 ifeq(ifeq2(zero, zero, less_equal(divide(divide(c, b), b), divide(divide(c, b), g)), true), true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by lemma 31 R->L }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(c, b), b), divide(divide(c, b), divide(b, d))), zero, less_equal(divide(divide(c, b), b), divide(divide(c, b), g)), true), true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by lemma 20 }
% 7.64/1.44 ifeq(ifeq2(divide(divide(divide(c, b), b), divide(divide(c, b), g)), zero, less_equal(divide(divide(c, b), b), divide(divide(c, b), g)), true), true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by axiom 10 (quotient_less_equal2) }
% 7.64/1.44 ifeq(true, true, ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b)), divide(c, b))
% 7.64/1.44 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.44 ifeq(less_equal(divide(divide(c, b), g), divide(c, b)), true, divide(divide(c, b), g), divide(c, b))
% 7.64/1.45 = { by axiom 9 (quotient_smaller_than_numerator) }
% 7.64/1.45 ifeq(true, true, divide(divide(c, b), g), divide(c, b))
% 7.64/1.45 = { by axiom 8 (ifeq_axiom) }
% 7.64/1.45 divide(divide(c, b), g)
% 7.64/1.45 = { by lemma 30 R->L }
% 7.64/1.45 divide(divide(d, a), g)
% 7.64/1.45 = { by lemma 26 }
% 7.64/1.45 divide(c, g)
% 7.64/1.45
% 7.64/1.45 Goal 1 (prove_commutativity): divide(c, g) = divide(d, e).
% 7.64/1.45 Proof:
% 7.64/1.45 divide(c, g)
% 7.64/1.45 = { by lemma 33 R->L }
% 7.64/1.45 divide(d, e)
% 7.64/1.45
% 7.64/1.45 Goal 2 (part_of_theorem): divide(divide(identity, a), divide(identity, divide(identity, b))) = divide(divide(identity, b), divide(identity, divide(identity, a))).
% 7.64/1.45 Proof:
% 7.64/1.45 divide(divide(identity, a), divide(identity, divide(identity, b)))
% 7.64/1.45 = { by axiom 3 (identity_divide_a) }
% 7.64/1.45 divide(c, divide(identity, divide(identity, b)))
% 7.64/1.45 = { by axiom 4 (identity_divide_b) }
% 7.64/1.45 divide(c, divide(identity, d))
% 7.64/1.45 = { by axiom 6 (identity_divide_d) }
% 7.64/1.45 divide(c, g)
% 7.64/1.45 = { by lemma 33 R->L }
% 7.64/1.45 divide(d, e)
% 7.64/1.45 = { by axiom 5 (identity_divide_c) R->L }
% 7.64/1.45 divide(d, divide(identity, c))
% 7.64/1.45 = { by axiom 4 (identity_divide_b) R->L }
% 7.64/1.45 divide(divide(identity, b), divide(identity, c))
% 7.64/1.45 = { by axiom 3 (identity_divide_a) R->L }
% 7.64/1.45 divide(divide(identity, b), divide(identity, divide(identity, a)))
% 7.64/1.45 % SZS output end Proof
% 7.64/1.45
% 7.64/1.45 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------