TSTP Solution File: RNG004-10 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG004-10 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:58:42 EDT 2023
% Result : Unsatisfiable 167.53s 21.76s
% Output : Proof 168.31s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : RNG004-10 : TPTP v8.1.2. Released v7.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n025.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 01:46:23 EDT 2023
% 0.14/0.34 % CPUTime :
% 167.53/21.76 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 167.53/21.76
% 167.53/21.76 % SZS status Unsatisfiable
% 167.53/21.76
% 168.31/21.89 % SZS output start Proof
% 168.31/21.89 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 168.31/21.89 Axiom 2 (additive_identity1): sum(additive_identity, X, X) = true.
% 168.31/21.89 Axiom 3 (a_times_b): product(a, b, c) = true.
% 168.31/21.89 Axiom 4 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y.
% 168.31/21.89 Axiom 5 (right_inverse): sum(X, additive_inverse(X), additive_identity) = true.
% 168.31/21.89 Axiom 6 (left_inverse): sum(additive_inverse(X), X, additive_identity) = true.
% 168.31/21.89 Axiom 7 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 168.31/21.89 Axiom 8 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 168.31/21.89 Axiom 9 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 168.31/21.89 Axiom 10 (a_inverse_times_b_inverse): product(additive_inverse(a), additive_inverse(b), d) = true.
% 168.31/21.89 Axiom 11 (commutativity_of_addition): ifeq(sum(X, Y, Z), true, sum(Y, X, Z), true) = true.
% 168.31/21.89 Axiom 12 (addition_is_well_defined): ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, W), true, W, Z), Z) = Z.
% 168.31/21.89 Axiom 13 (multiplication_is_well_defined): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 168.31/21.89 Axiom 14 (associativity_of_addition1): ifeq(sum(X, Y, Z), true, ifeq(sum(W, Y, V), true, ifeq(sum(U, W, X), true, sum(U, V, Z), true), true), true) = true.
% 168.31/21.89 Axiom 15 (associativity_of_addition2): ifeq(sum(X, Y, Z), true, ifeq(sum(W, Z, V), true, ifeq(sum(W, X, U), true, sum(U, Y, V), true), true), true) = true.
% 168.31/21.89 Axiom 16 (distributivity2): ifeq(product(X, Y, Z), true, ifeq(product(X, W, V), true, ifeq(sum(V, Z, U), true, ifeq(sum(W, Y, T), true, product(X, T, U), true), true), true), true) = true.
% 168.31/21.89 Axiom 17 (distributivity4): ifeq(product(X, Y, Z), true, ifeq(product(W, Y, V), true, ifeq(sum(V, Z, U), true, ifeq(sum(W, X, T), true, product(T, Y, U), true), true), true), true) = true.
% 168.31/21.89 Axiom 18 (distributivity3): ifeq(product(X, Y, Z), true, ifeq(product(W, Y, V), true, ifeq(product(U, Y, T), true, ifeq(sum(U, W, X), true, sum(T, V, Z), true), true), true), true) = true.
% 168.31/21.89
% 168.31/21.89 Lemma 19: ifeq2(sum(X, Y, Z), true, add(X, Y), Z) = Z.
% 168.31/21.89 Proof:
% 168.31/21.89 ifeq2(sum(X, Y, Z), true, add(X, Y), Z)
% 168.31/21.89 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.89 ifeq2(sum(X, Y, Z), true, ifeq2(true, true, add(X, Y), Z), Z)
% 168.31/21.89 = { by axiom 8 (closure_of_addition) R->L }
% 168.31/21.89 ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, add(X, Y)), true, add(X, Y), Z), Z)
% 168.31/21.89 = { by axiom 12 (addition_is_well_defined) }
% 168.31/21.89 Z
% 168.31/21.89
% 168.31/21.89 Lemma 20: add(X, Y) = add(Y, X).
% 168.31/21.89 Proof:
% 168.31/21.89 add(X, Y)
% 168.31/21.89 = { by lemma 19 R->L }
% 168.31/21.89 ifeq2(sum(Y, X, add(X, Y)), true, add(Y, X), add(X, Y))
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq2(ifeq(true, true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 168.31/21.89 = { by axiom 8 (closure_of_addition) R->L }
% 168.31/21.89 ifeq2(ifeq(sum(X, Y, add(X, Y)), true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 168.31/21.89 = { by axiom 11 (commutativity_of_addition) }
% 168.31/21.89 ifeq2(true, true, add(Y, X), add(X, Y))
% 168.31/21.89 = { by axiom 7 (ifeq_axiom) }
% 168.31/21.89 add(Y, X)
% 168.31/21.89
% 168.31/21.89 Lemma 21: ifeq(sum(X, Y, Z), true, ifeq(sum(W, X, additive_identity), true, sum(W, Z, Y), true), true) = true.
% 168.31/21.89 Proof:
% 168.31/21.89 ifeq(sum(X, Y, Z), true, ifeq(sum(W, X, additive_identity), true, sum(W, Z, Y), true), true)
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq(true, true, ifeq(sum(X, Y, Z), true, ifeq(sum(W, X, additive_identity), true, sum(W, Z, Y), true), true), true)
% 168.31/21.89 = { by axiom 2 (additive_identity1) R->L }
% 168.31/21.89 ifeq(sum(additive_identity, Y, Y), true, ifeq(sum(X, Y, Z), true, ifeq(sum(W, X, additive_identity), true, sum(W, Z, Y), true), true), true)
% 168.31/21.89 = { by axiom 14 (associativity_of_addition1) }
% 168.31/21.89 true
% 168.31/21.89
% 168.31/21.89 Lemma 22: sum(X, add(Y, additive_inverse(X)), Y) = true.
% 168.31/21.89 Proof:
% 168.31/21.89 sum(X, add(Y, additive_inverse(X)), Y)
% 168.31/21.89 = { by lemma 20 R->L }
% 168.31/21.89 sum(X, add(additive_inverse(X), Y), Y)
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq(true, true, sum(X, add(additive_inverse(X), Y), Y), true)
% 168.31/21.89 = { by axiom 8 (closure_of_addition) R->L }
% 168.31/21.89 ifeq(sum(additive_inverse(X), Y, add(additive_inverse(X), Y)), true, sum(X, add(additive_inverse(X), Y), Y), true)
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq(sum(additive_inverse(X), Y, add(additive_inverse(X), Y)), true, ifeq(true, true, sum(X, add(additive_inverse(X), Y), Y), true), true)
% 168.31/21.89 = { by axiom 5 (right_inverse) R->L }
% 168.31/21.89 ifeq(sum(additive_inverse(X), Y, add(additive_inverse(X), Y)), true, ifeq(sum(X, additive_inverse(X), additive_identity), true, sum(X, add(additive_inverse(X), Y), Y), true), true)
% 168.31/21.89 = { by lemma 21 }
% 168.31/21.89 true
% 168.31/21.89
% 168.31/21.89 Lemma 23: ifeq(sum(X, Y, Z), true, sum(additive_inverse(X), Z, Y), true) = true.
% 168.31/21.89 Proof:
% 168.31/21.89 ifeq(sum(X, Y, Z), true, sum(additive_inverse(X), Z, Y), true)
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq(sum(X, Y, Z), true, ifeq(true, true, sum(additive_inverse(X), Z, Y), true), true)
% 168.31/21.89 = { by axiom 6 (left_inverse) R->L }
% 168.31/21.89 ifeq(sum(X, Y, Z), true, ifeq(sum(additive_inverse(X), X, additive_identity), true, sum(additive_inverse(X), Z, Y), true), true)
% 168.31/21.89 = { by lemma 21 }
% 168.31/21.89 true
% 168.31/21.89
% 168.31/21.89 Lemma 24: sum(additive_inverse(X), add(X, Y), Y) = true.
% 168.31/21.89 Proof:
% 168.31/21.89 sum(additive_inverse(X), add(X, Y), Y)
% 168.31/21.89 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.89 ifeq(true, true, sum(additive_inverse(X), add(X, Y), Y), true)
% 168.31/21.89 = { by axiom 8 (closure_of_addition) R->L }
% 168.31/21.89 ifeq(sum(X, Y, add(X, Y)), true, sum(additive_inverse(X), add(X, Y), Y), true)
% 168.31/21.89 = { by lemma 23 }
% 168.31/21.89 true
% 168.31/21.89
% 168.31/21.89 Lemma 25: ifeq2(sum(X, additive_identity, Y), true, Y, X) = X.
% 168.31/21.89 Proof:
% 168.31/21.89 ifeq2(sum(X, additive_identity, Y), true, Y, X)
% 168.31/21.89 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.89 ifeq2(true, true, ifeq2(sum(X, additive_identity, Y), true, Y, X), X)
% 168.31/21.89 = { by axiom 1 (additive_identity2) R->L }
% 168.31/21.89 ifeq2(sum(X, additive_identity, X), true, ifeq2(sum(X, additive_identity, Y), true, Y, X), X)
% 168.31/21.89 = { by axiom 12 (addition_is_well_defined) }
% 168.31/21.89 X
% 168.31/21.89
% 168.31/21.89 Lemma 26: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 168.31/21.89 Proof:
% 168.31/21.89 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 168.31/21.89 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.89 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 168.31/21.89 = { by axiom 9 (closure_of_multiplication) R->L }
% 168.31/21.89 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 168.31/21.89 = { by axiom 13 (multiplication_is_well_defined) }
% 168.31/21.90 Z
% 168.31/21.90
% 168.31/21.90 Goal 1 (prove_c_equals_d): c = d.
% 168.31/21.90 Proof:
% 168.31/21.90 c
% 168.31/21.90 = { by lemma 25 R->L }
% 168.31/21.90 ifeq2(sum(c, additive_identity, d), true, d, c)
% 168.31/21.90 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.90 ifeq2(sum(c, ifeq2(true, true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.90 = { by axiom 14 (associativity_of_addition1) R->L }
% 168.31/21.90 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), additive_inverse(multiply(additive_inverse(a), additive_identity)), additive_identity), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), additive_inverse(multiply(additive_inverse(a), additive_identity)), additive_identity), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.90 = { by axiom 5 (right_inverse) }
% 168.31/21.90 ifeq2(sum(c, ifeq2(ifeq(true, true, ifeq(sum(multiply(additive_inverse(a), additive_identity), additive_inverse(multiply(additive_inverse(a), additive_identity)), additive_identity), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.90 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.90 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), additive_inverse(multiply(additive_inverse(a), additive_identity)), additive_identity), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.90 = { by axiom 5 (right_inverse) }
% 168.31/21.90 ifeq2(sum(c, ifeq2(ifeq(true, true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.90 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.90 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(true, true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 16 (distributivity2) R->L }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true, ifeq(sum(additive_identity, additive_identity, additive_identity), true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 1 (additive_identity2) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true, ifeq(true, true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 8 (closure_of_addition) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(true, true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 9 (closure_of_multiplication) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(true, true, ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(product(additive_inverse(a), additive_identity, multiply(additive_inverse(a), additive_identity)), true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 9 (closure_of_multiplication) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(ifeq(true, true, product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), ifeq2(product(additive_inverse(a), additive_identity, add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true, multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity)))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by lemma 26 }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity), add(multiply(additive_inverse(a), additive_identity), multiply(additive_inverse(a), additive_identity))), true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 8 (closure_of_addition) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(true, true, sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(sum(multiply(additive_inverse(a), additive_identity), additive_identity, additive_identity), true, additive_identity, multiply(additive_inverse(a), additive_identity)), d), true, d, c)
% 168.31/21.91 = { by lemma 25 }
% 168.31/21.91 ifeq2(sum(c, multiply(additive_inverse(a), additive_identity), d), true, d, c)
% 168.31/21.91 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.91 ifeq2(sum(c, ifeq2(true, true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 16 (distributivity2) R->L }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(product(additive_inverse(a), additive_inverse(b), d), true, ifeq(product(additive_inverse(a), b, multiply(additive_inverse(a), b)), true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, ifeq(sum(b, additive_inverse(b), additive_identity), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 5 (right_inverse) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(product(additive_inverse(a), additive_inverse(b), d), true, ifeq(product(additive_inverse(a), b, multiply(additive_inverse(a), b)), true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, ifeq(true, true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(product(additive_inverse(a), additive_inverse(b), d), true, ifeq(product(additive_inverse(a), b, multiply(additive_inverse(a), b)), true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 10 (a_inverse_times_b_inverse) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(true, true, ifeq(product(additive_inverse(a), b, multiply(additive_inverse(a), b)), true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.91 ifeq2(sum(c, ifeq2(ifeq(product(additive_inverse(a), b, multiply(additive_inverse(a), b)), true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.91 = { by axiom 9 (closure_of_multiplication) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(true, true, ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(multiply(additive_inverse(a), b), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(true, true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 14 (associativity_of_addition1) R->L }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(c, additive_inverse(multiply(add(a, a), b)), add(c, additive_inverse(multiply(add(a, a), b)))), true, ifeq(sum(multiply(add(a, a), b), additive_inverse(multiply(add(a, a), b)), additive_identity), true, ifeq(sum(additive_inverse(multiply(a, b)), multiply(add(a, a), b), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 5 (right_inverse) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(c, additive_inverse(multiply(add(a, a), b)), add(c, additive_inverse(multiply(add(a, a), b)))), true, ifeq(true, true, ifeq(sum(additive_inverse(multiply(a, b)), multiply(add(a, a), b), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(c, additive_inverse(multiply(add(a, a), b)), add(c, additive_inverse(multiply(add(a, a), b)))), true, ifeq(sum(additive_inverse(multiply(a, b)), multiply(add(a, a), b), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 8 (closure_of_addition) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(true, true, ifeq(sum(additive_inverse(multiply(a, b)), multiply(add(a, a), b), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), multiply(add(a, a), b), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(true, true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 17 (distributivity4) R->L }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, c), true, ifeq(product(a, b, multiply(a, b)), true, ifeq(sum(multiply(a, b), c, add(multiply(a, b), c)), true, ifeq(sum(a, a, add(a, a)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 3 (a_times_b) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(true, true, ifeq(product(a, b, multiply(a, b)), true, ifeq(sum(multiply(a, b), c, add(multiply(a, b), c)), true, ifeq(sum(a, a, add(a, a)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, multiply(a, b)), true, ifeq(sum(multiply(a, b), c, add(multiply(a, b), c)), true, ifeq(sum(a, a, add(a, a)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 8 (closure_of_addition) }
% 168.31/21.92 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, multiply(a, b)), true, ifeq(true, true, ifeq(sum(a, a, add(a, a)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.92 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, multiply(a, b)), true, ifeq(sum(a, a, add(a, a)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 8 (closure_of_addition) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, multiply(a, b)), true, ifeq(true, true, product(add(a, a), b, add(multiply(a, b), c)), true), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(product(a, b, multiply(a, b)), true, product(add(a, a), b, add(multiply(a, b), c)), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 9 (closure_of_multiplication) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(ifeq(true, true, product(add(a, a), b, add(multiply(a, b), c)), true), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(product(add(a, a), b, add(multiply(a, b), c)), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 20 }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), ifeq2(product(add(a, a), b, add(c, multiply(a, b))), true, multiply(add(a, a), b), add(c, multiply(a, b))), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 26 }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), add(c, multiply(a, b)), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 20 R->L }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), add(multiply(a, b), c), c), true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 24 }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(ifeq(true, true, sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, add(c, additive_inverse(multiply(add(a, a), b)))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(true, true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 23 R->L }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(true, true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.93 = { by lemma 22 R->L }
% 168.31/21.93 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(true, true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 9 (closure_of_multiplication) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(product(add(a, a), b, multiply(add(a, a), b)), true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(true, true, ifeq(product(add(a, a), b, multiply(add(a, a), b)), true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 9 (closure_of_multiplication) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(product(add(a, additive_inverse(add(a, a))), b, multiply(add(a, additive_inverse(add(a, a))), b)), true, ifeq(product(add(a, a), b, multiply(add(a, a), b)), true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 4 (ifeq_axiom_001) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(true, true, ifeq(product(add(a, additive_inverse(add(a, a))), b, multiply(add(a, additive_inverse(add(a, a))), b)), true, ifeq(product(add(a, a), b, multiply(add(a, a), b)), true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 3 (a_times_b) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(ifeq(product(a, b, c), true, ifeq(product(add(a, additive_inverse(add(a, a))), b, multiply(add(a, additive_inverse(add(a, a))), b)), true, ifeq(product(add(a, a), b, multiply(add(a, a), b)), true, ifeq(sum(add(a, a), add(a, additive_inverse(add(a, a))), a), true, sum(multiply(add(a, a), b), multiply(add(a, additive_inverse(add(a, a))), b), c), true), true), true), true), true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 18 (distributivity3) }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(ifeq(true, true, sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true, add(c, additive_inverse(multiply(add(a, a), b))), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by lemma 20 R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, ifeq2(sum(additive_inverse(multiply(add(a, a), b)), c, multiply(add(a, additive_inverse(add(a, a))), b)), true, add(additive_inverse(multiply(add(a, a), b)), c), multiply(add(a, additive_inverse(add(a, a))), b))), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by lemma 19 }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(add(a, additive_inverse(add(a, a))), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 7 (ifeq_axiom) R->L }
% 168.31/21.94 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(true, true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.94 = { by axiom 15 (associativity_of_addition2) R->L }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(sum(add(a, a), additive_inverse(add(a, a)), additive_identity), true, ifeq(sum(additive_inverse(a), additive_identity, additive_inverse(a)), true, ifeq(sum(additive_inverse(a), add(a, a), a), true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 1 (additive_identity2) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(sum(add(a, a), additive_inverse(add(a, a)), additive_identity), true, ifeq(true, true, ifeq(sum(additive_inverse(a), add(a, a), a), true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(sum(add(a, a), additive_inverse(add(a, a)), additive_identity), true, ifeq(sum(additive_inverse(a), add(a, a), a), true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 5 (right_inverse) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(true, true, ifeq(sum(additive_inverse(a), add(a, a), a), true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(sum(additive_inverse(a), add(a, a), a), true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by lemma 24 }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(ifeq(true, true, sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(ifeq2(sum(a, additive_inverse(add(a, a)), additive_inverse(a)), true, add(a, additive_inverse(add(a, a))), additive_inverse(a)), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by lemma 19 }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(ifeq2(sum(additive_inverse(multiply(a, b)), additive_identity, multiply(additive_inverse(a), b)), true, multiply(additive_inverse(a), b), additive_inverse(multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by lemma 25 }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(multiply(a, b)), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 13 (multiplication_is_well_defined) R->L }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(ifeq2(product(a, b, multiply(a, b)), true, ifeq2(product(a, b, c), true, c, multiply(a, b)), multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 3 (a_times_b) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(ifeq2(product(a, b, multiply(a, b)), true, ifeq2(true, true, c, multiply(a, b)), multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 7 (ifeq_axiom) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(ifeq2(product(a, b, multiply(a, b)), true, c, multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 9 (closure_of_multiplication) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(ifeq2(true, true, c, multiply(a, b))), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 7 (ifeq_axiom) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(sum(additive_inverse(c), d, add(additive_inverse(c), d)), true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 8 (closure_of_addition) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(ifeq(true, true, product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by axiom 4 (ifeq_axiom_001) }
% 168.31/21.95 ifeq2(sum(c, ifeq2(product(additive_inverse(a), additive_identity, add(additive_inverse(c), d)), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by lemma 20 }
% 168.31/21.95 ifeq2(sum(c, ifeq2(product(additive_inverse(a), additive_identity, add(d, additive_inverse(c))), true, multiply(additive_inverse(a), additive_identity), add(d, additive_inverse(c))), d), true, d, c)
% 168.31/21.95 = { by lemma 26 }
% 168.31/21.95 ifeq2(sum(c, add(d, additive_inverse(c)), d), true, d, c)
% 168.31/21.95 = { by lemma 22 }
% 168.31/21.95 ifeq2(true, true, d, c)
% 168.31/21.95 = { by axiom 7 (ifeq_axiom) }
% 168.31/21.95 d
% 168.31/21.95 % SZS output end Proof
% 168.31/21.95
% 168.31/21.95 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------