%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : BOO014-1 : TPTP v9.3.1. Bugfixed v1.2.1.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n019.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 09:35:24 AM UTC 2026
% Result : Unsatisfiable 256.07s 32.58s
% Output : Proof 260.84s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : BOO014-1 : TPTP v9.3.1. Bugfixed v1.2.1.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.18 % Computer : n019.cluster.edu
% 0.07/0.18 % Model : x86_64 x86_64
% 0.07/0.18 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.18 % Memory : 8046.5625MB
% 0.07/0.18 % OS : Linux 6.8.0-71-generic
% 0.07/0.18 % CPULimit : 300
% 0.07/0.18 % WCLimit : 300
% 0.07/0.18 % DateTime : Mon Sep 28 21:04:02 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 256.07/32.58 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 256.07/32.58
% 256.07/32.58 % SZS status Unsatisfiable
% 256.07/32.58
% 259.28/33.00 % SZS output start Proof
% 259.28/33.00 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 259.28/33.00 Axiom 2 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 259.28/33.00 Axiom 3 (additive_identity1): sum(additive_identity, X, X) = true.
% 259.28/33.00 Axiom 4 (x_plus_y): sum(x, y, x_plus_y) = true.
% 259.28/33.00 Axiom 5 (additive_inverse1): sum(inverse(X), X, multiplicative_identity) = true.
% 259.28/33.00 Axiom 6 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 259.28/33.00 Axiom 7 (multiplicative_inverse2): product(X, inverse(X), additive_identity) = true.
% 259.28/33.00 Axiom 8 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 259.28/33.00 Axiom 9 (x_inverse_times_y_inverse): product(inverse(x), inverse(y), x_inverse_times_y_inverse) = true.
% 259.28/33.00 Axiom 10 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 259.28/33.00 Axiom 11 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 259.28/33.00 Axiom 12 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 259.28/33.00 Axiom 13 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 259.28/33.00 Axiom 14 (commutativity_of_addition): ifeq(sum(X, Y, Z), true, sum(Y, X, Z), true) = true.
% 259.28/33.00 Axiom 15 (commutativity_of_multiplication): ifeq(product(X, Y, Z), true, product(Y, X, Z), true) = true.
% 259.28/33.00 Axiom 16 (addition_is_well_defined): ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, W), true, W, Z), Z) = Z.
% 259.28/33.00 Axiom 17 (multiplication_is_well_defined): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 259.28/33.00 Axiom 18 (distributivity7): ifeq(product(X, Y, Z), true, ifeq(sum(Z, W, V), true, ifeq(sum(Y, W, U), true, ifeq(sum(X, W, T), true, product(T, U, V), true), true), true), true) = true.
% 259.28/33.00 Axiom 19 (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.
% 259.28/33.00 Axiom 20 (distributivity1): ifeq(product(X, Y, Z), true, ifeq(product(X, W, V), true, ifeq(product(X, U, T), true, ifeq(sum(U, W, Y), true, sum(T, V, Z), true), true), true), true) = true.
% 259.28/33.00 Axiom 21 (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.
% 259.28/33.00 Axiom 22 (distributivity8): ifeq(product(X, Y, Z), true, ifeq(product(W, V, U), true, ifeq(sum(V, T, Y), true, ifeq(sum(W, T, X), true, sum(U, T, Z), true), true), true), true) = true.
% 259.28/33.00
% 259.28/33.00 Lemma 23: add(X, Y) = add(Y, X).
% 259.28/33.00 Proof:
% 259.28/33.00 add(X, Y)
% 259.28/33.00 = { by axiom 16 (addition_is_well_defined) R->L }
% 259.28/33.00 ifeq2(sum(Y, X, add(X, Y)), true, ifeq2(sum(Y, X, add(Y, X)), true, add(Y, X), add(X, Y)), add(X, Y))
% 259.28/33.00 = { by axiom 12 (closure_of_addition) }
% 259.28/33.00 ifeq2(sum(Y, X, add(X, Y)), true, ifeq2(true, true, add(Y, X), add(X, Y)), add(X, Y))
% 259.28/33.00 = { by axiom 11 (ifeq_axiom) }
% 259.28/33.00 ifeq2(sum(Y, X, add(X, Y)), true, add(Y, X), add(X, Y))
% 259.28/33.00 = { by axiom 10 (ifeq_axiom) R->L }
% 259.28/33.00 ifeq2(ifeq(true, true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 259.28/33.00 = { by axiom 12 (closure_of_addition) R->L }
% 259.28/33.00 ifeq2(ifeq(sum(X, Y, add(X, Y)), true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 259.28/33.00 = { by axiom 14 (commutativity_of_addition) }
% 259.28/33.00 ifeq2(true, true, add(Y, X), add(X, Y))
% 259.28/33.00 = { by axiom 11 (ifeq_axiom) }
% 259.28/33.00 add(Y, X)
% 259.28/33.00
% 259.28/33.00 Lemma 24: add(X, additive_identity) = X.
% 259.28/33.00 Proof:
% 259.28/33.00 add(X, additive_identity)
% 259.28/33.00 = { by axiom 11 (ifeq_axiom) R->L }
% 259.28/33.00 ifeq2(true, true, add(X, additive_identity), X)
% 259.28/33.00 = { by axiom 12 (closure_of_addition) R->L }
% 259.28/33.00 ifeq2(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 259.28/33.00 = { by axiom 11 (ifeq_axiom) R->L }
% 259.28/33.00 ifeq2(true, true, ifeq2(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X), X)
% 259.28/33.00 = { by axiom 1 (additive_identity2) R->L }
% 259.28/33.00 ifeq2(sum(X, additive_identity, X), true, ifeq2(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X), X)
% 259.28/33.00 = { by axiom 16 (addition_is_well_defined) }
% 259.28/33.00 X
% 259.28/33.00
% 259.28/33.00 Lemma 25: ifeq2(product(multiplicative_identity, X, Y), true, Y, X) = X.
% 259.28/33.00 Proof:
% 259.28/33.00 ifeq2(product(multiplicative_identity, X, Y), true, Y, X)
% 259.28/33.00 = { by axiom 11 (ifeq_axiom) R->L }
% 259.28/33.00 ifeq2(true, true, ifeq2(product(multiplicative_identity, X, Y), true, Y, X), X)
% 259.28/33.00 = { by axiom 8 (multiplicative_identity1) R->L }
% 259.28/33.00 ifeq2(product(multiplicative_identity, X, X), true, ifeq2(product(multiplicative_identity, X, Y), true, Y, X), X)
% 259.28/33.00 = { by axiom 17 (multiplication_is_well_defined) }
% 259.28/33.00 X
% 259.28/33.00
% 259.28/33.00 Lemma 26: add(X, multiplicative_identity) = multiplicative_identity.
% 259.28/33.00 Proof:
% 259.28/33.00 add(X, multiplicative_identity)
% 259.28/33.00 = { by lemma 25 R->L }
% 259.28/33.00 ifeq2(product(multiplicative_identity, add(X, multiplicative_identity), multiplicative_identity), true, multiplicative_identity, add(X, multiplicative_identity))
% 259.28/33.00 = { by lemma 23 R->L }
% 259.28/33.00 ifeq2(product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(ifeq(true, true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.01 ifeq2(ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(true, true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 5 (additive_inverse1) R->L }
% 260.10/33.01 ifeq2(ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(ifeq(true, true, ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 5 (additive_inverse1) R->L }
% 260.10/33.01 ifeq2(ifeq(sum(inverse(X), X, multiplicative_identity), true, ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(ifeq(true, true, ifeq(sum(inverse(X), X, multiplicative_identity), true, ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 6 (multiplicative_identity2) R->L }
% 260.10/33.01 ifeq2(ifeq(product(inverse(X), multiplicative_identity, inverse(X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, ifeq(sum(multiplicative_identity, X, add(multiplicative_identity, X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, X), multiplicative_identity), true), true), true), true), true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 18 (distributivity7) }
% 260.10/33.01 ifeq2(true, true, multiplicative_identity, add(X, multiplicative_identity))
% 260.10/33.01 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.01 multiplicative_identity
% 260.10/33.01
% 260.10/33.01 Lemma 27: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 260.10/33.01 Proof:
% 260.10/33.01 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 260.10/33.01 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 260.10/33.01 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.01 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 260.10/33.01 = { by axiom 17 (multiplication_is_well_defined) }
% 260.10/33.01 Z
% 260.10/33.01
% 260.10/33.01 Lemma 28: multiply(X, Y) = multiply(Y, X).
% 260.10/33.01 Proof:
% 260.10/33.01 multiply(X, Y)
% 260.10/33.01 = { by lemma 27 R->L }
% 260.10/33.01 ifeq2(product(Y, X, multiply(X, Y)), true, multiply(Y, X), multiply(X, Y))
% 260.10/33.01 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(ifeq(true, true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 260.10/33.01 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.01 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 260.10/33.01 = { by axiom 15 (commutativity_of_multiplication) }
% 260.10/33.01 ifeq2(true, true, multiply(Y, X), multiply(X, Y))
% 260.10/33.01 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.01 multiply(Y, X)
% 260.10/33.01
% 260.10/33.01 Lemma 29: ifeq2(product(X, inverse(X), Y), true, additive_identity, Y) = Y.
% 260.10/33.01 Proof:
% 260.10/33.01 ifeq2(product(X, inverse(X), Y), true, additive_identity, Y)
% 260.10/33.01 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.01 ifeq2(product(X, inverse(X), Y), true, ifeq2(true, true, additive_identity, Y), Y)
% 260.10/33.01 = { by axiom 7 (multiplicative_inverse2) R->L }
% 260.10/33.01 ifeq2(product(X, inverse(X), Y), true, ifeq2(product(X, inverse(X), additive_identity), true, additive_identity, Y), Y)
% 260.10/33.01 = { by axiom 17 (multiplication_is_well_defined) }
% 260.10/33.02 Y
% 260.10/33.02
% 260.10/33.02 Lemma 30: multiply(X, additive_identity) = additive_identity.
% 260.10/33.02 Proof:
% 260.10/33.02 multiply(X, additive_identity)
% 260.10/33.02 = { by lemma 29 R->L }
% 260.10/33.02 ifeq2(product(X, inverse(X), multiply(X, additive_identity)), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(ifeq(true, true, product(X, inverse(X), multiply(X, additive_identity)), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(ifeq(true, true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 7 (multiplicative_inverse2) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, ifeq(true, true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 3 (additive_identity1) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, ifeq(sum(additive_identity, inverse(X), inverse(X)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, ifeq(true, true, ifeq(sum(additive_identity, inverse(X), inverse(X)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 1 (additive_identity2) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, additive_identity, multiply(X, additive_identity)), true, ifeq(sum(multiply(X, additive_identity), additive_identity, multiply(X, additive_identity)), true, ifeq(sum(additive_identity, inverse(X), inverse(X)), true, product(X, inverse(X), multiply(X, additive_identity)), true), true), true), true), true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 19 (distributivity2) }
% 260.10/33.02 ifeq2(true, true, additive_identity, multiply(X, additive_identity))
% 260.10/33.02 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.02 additive_identity
% 260.10/33.02
% 260.10/33.02 Lemma 31: multiply(X, inverse(X)) = additive_identity.
% 260.10/33.02 Proof:
% 260.10/33.02 multiply(X, inverse(X))
% 260.10/33.02 = { by lemma 29 R->L }
% 260.10/33.02 ifeq2(product(X, inverse(X), multiply(X, inverse(X))), true, additive_identity, multiply(X, inverse(X)))
% 260.10/33.02 = { by axiom 13 (closure_of_multiplication) }
% 260.10/33.02 ifeq2(true, true, additive_identity, multiply(X, inverse(X)))
% 260.10/33.02 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.02 additive_identity
% 260.10/33.02
% 260.10/33.02 Lemma 32: ifeq2(product(X, multiplicative_identity, Y), true, Y, X) = X.
% 260.10/33.02 Proof:
% 260.10/33.02 ifeq2(product(X, multiplicative_identity, Y), true, Y, X)
% 260.10/33.02 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(true, true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 260.10/33.02 = { by axiom 6 (multiplicative_identity2) R->L }
% 260.10/33.02 ifeq2(product(X, multiplicative_identity, X), true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 260.10/33.02 = { by axiom 17 (multiplication_is_well_defined) }
% 260.10/33.02 X
% 260.10/33.02
% 260.10/33.02 Lemma 33: add(X, multiply(X, Y)) = X.
% 260.10/33.02 Proof:
% 260.10/33.02 add(X, multiply(X, Y))
% 260.10/33.02 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.02 ifeq2(true, true, add(X, multiply(X, Y)), X)
% 260.10/33.02 = { by axiom 19 (distributivity2) R->L }
% 260.10/33.02 ifeq2(ifeq(product(X, multiplicative_identity, X), true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, ifeq(sum(Y, multiplicative_identity, add(Y, multiplicative_identity)), true, product(X, add(Y, multiplicative_identity), add(multiply(X, Y), X)), true), true), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.02 = { by axiom 6 (multiplicative_identity2) }
% 260.10/33.02 ifeq2(ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, ifeq(sum(Y, multiplicative_identity, add(Y, multiplicative_identity)), true, product(X, add(Y, multiplicative_identity), add(multiply(X, Y), X)), true), true), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.02 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.02 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, ifeq(sum(Y, multiplicative_identity, add(Y, multiplicative_identity)), true, product(X, add(Y, multiplicative_identity), add(multiply(X, Y), X)), true), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 12 (closure_of_addition) }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, ifeq(true, true, product(X, add(Y, multiplicative_identity), add(multiply(X, Y), X)), true), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, add(Y, multiplicative_identity), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by lemma 26 }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, multiplicative_identity, add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 12 (closure_of_addition) }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(true, true, product(X, multiplicative_identity, add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiplicative_identity, add(multiply(X, Y), X)), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by lemma 23 }
% 260.10/33.03 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiplicative_identity, add(X, multiply(X, Y))), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 13 (closure_of_multiplication) }
% 260.10/33.03 ifeq2(ifeq(true, true, product(X, multiplicative_identity, add(X, multiply(X, Y))), true), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.03 ifeq2(product(X, multiplicative_identity, add(X, multiply(X, Y))), true, add(X, multiply(X, Y)), X)
% 260.10/33.03 = { by lemma 32 }
% 260.10/33.03 X
% 260.10/33.03
% 260.10/33.03 Lemma 34: ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)) = multiply(X, Y).
% 260.10/33.03 Proof:
% 260.10/33.03 ifeq2(product(X, Y, Z), true, Z, multiply(X, Y))
% 260.10/33.03 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.03 ifeq2(true, true, ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)), multiply(X, Y))
% 260.10/33.03 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.03 ifeq2(product(X, Y, multiply(X, Y)), true, ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)), multiply(X, Y))
% 260.10/33.03 = { by axiom 17 (multiplication_is_well_defined) }
% 260.10/33.03 multiply(X, Y)
% 260.10/33.03
% 260.10/33.03 Lemma 35: multiply(X, add(Y, X)) = X.
% 260.10/33.03 Proof:
% 260.10/33.03 multiply(X, add(Y, X))
% 260.10/33.03 = { by lemma 23 R->L }
% 260.10/33.03 multiply(X, add(X, Y))
% 260.10/33.03 = { by lemma 28 R->L }
% 260.10/33.03 multiply(add(X, Y), X)
% 260.10/33.03 = { by lemma 34 R->L }
% 260.10/33.03 ifeq2(product(add(X, Y), X, add(X, multiply(Y, additive_identity))), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by lemma 23 R->L }
% 260.10/33.03 ifeq2(product(add(X, Y), X, add(multiply(Y, additive_identity), X)), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by lemma 23 R->L }
% 260.10/33.03 ifeq2(product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.03 ifeq2(ifeq(true, true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.03 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.03 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(true, true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.03 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.03 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(true, true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.03 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(sum(multiply(Y, additive_identity), X, add(multiply(Y, additive_identity), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.03 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(sum(multiply(Y, additive_identity), X, add(multiply(Y, additive_identity), X)), true, ifeq(true, true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.04 = { by axiom 3 (additive_identity1) R->L }
% 260.10/33.04 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, ifeq(sum(multiply(Y, additive_identity), X, add(multiply(Y, additive_identity), X)), true, ifeq(sum(additive_identity, X, X), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true), true), true), true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.04 = { by axiom 18 (distributivity7) }
% 260.10/33.04 ifeq2(true, true, add(X, multiply(Y, additive_identity)), multiply(add(X, Y), X))
% 260.10/33.04 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.04 add(X, multiply(Y, additive_identity))
% 260.10/33.04 = { by lemma 30 }
% 260.10/33.04 add(X, additive_identity)
% 260.10/33.04 = { by lemma 24 }
% 260.10/33.04 X
% 260.10/33.04
% 260.10/33.04 Lemma 36: multiply(X, add(Y, inverse(X))) = multiply(X, Y).
% 260.10/33.04 Proof:
% 260.10/33.04 multiply(X, add(Y, inverse(X)))
% 260.10/33.04 = { by lemma 34 R->L }
% 260.10/33.04 ifeq2(product(X, add(Y, inverse(X)), multiply(X, Y)), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(ifeq(true, true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(true, true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(Y, inverse(X), add(Y, inverse(X))), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(ifeq(true, true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(Y, inverse(X), add(Y, inverse(X))), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 7 (multiplicative_inverse2) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(Y, inverse(X), add(Y, inverse(X))), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(true, true, ifeq(sum(Y, inverse(X), add(Y, inverse(X))), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 1 (additive_identity2) R->L }
% 260.10/33.04 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), additive_identity, multiply(X, Y)), true, ifeq(sum(Y, inverse(X), add(Y, inverse(X))), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true), true), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 19 (distributivity2) }
% 260.10/33.04 ifeq2(true, true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 260.10/33.04 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.04 multiply(X, Y)
% 260.10/33.04
% 260.10/33.04 Lemma 37: ifeq2(sum(X, Y, Z), true, Z, add(X, Y)) = add(X, Y).
% 260.10/33.04 Proof:
% 260.10/33.04 ifeq2(sum(X, Y, Z), true, Z, add(X, Y))
% 260.10/33.04 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(true, true, ifeq2(sum(X, Y, Z), true, Z, add(X, Y)), add(X, Y))
% 260.10/33.04 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.04 ifeq2(sum(X, Y, add(X, Y)), true, ifeq2(sum(X, Y, Z), true, Z, add(X, Y)), add(X, Y))
% 260.10/33.04 = { by axiom 16 (addition_is_well_defined) }
% 260.10/33.04 add(X, Y)
% 260.10/33.04
% 260.10/33.04 Lemma 38: add(inverse(X), add(X, Y)) = multiplicative_identity.
% 260.10/33.04 Proof:
% 260.10/33.04 add(inverse(X), add(X, Y))
% 260.10/33.04 = { by lemma 23 R->L }
% 260.10/33.04 add(inverse(X), add(Y, X))
% 260.10/33.04 = { by lemma 23 R->L }
% 260.10/33.04 add(add(Y, X), inverse(X))
% 260.10/33.04 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.04 ifeq2(true, true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.04 = { by axiom 20 (distributivity1) R->L }
% 260.10/33.04 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), multiplicative_identity, add(add(Y, X), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), X, multiply(add(add(Y, X), inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.04 = { by axiom 6 (multiplicative_identity2) }
% 260.10/33.04 ifeq2(ifeq(true, true, ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), X, multiply(add(add(Y, X), inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.05 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), X, multiply(add(add(Y, X), inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 2 (additive_inverse2) }
% 260.10/33.05 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), X, multiply(add(add(Y, X), inverse(X)), X)), true, ifeq(true, true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.05 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(product(add(add(Y, X), inverse(X)), X, multiply(add(add(Y, X), inverse(X)), X)), true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 13 (closure_of_multiplication) }
% 260.10/33.05 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, ifeq(true, true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.05 ifeq2(ifeq(product(add(add(Y, X), inverse(X)), inverse(X), multiply(add(add(Y, X), inverse(X)), inverse(X))), true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 13 (closure_of_multiplication) }
% 260.10/33.05 ifeq2(ifeq(true, true, sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.05 ifeq2(sum(multiply(add(add(Y, X), inverse(X)), X), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by lemma 28 }
% 260.10/33.05 ifeq2(sum(multiply(X, add(add(Y, X), inverse(X))), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by lemma 36 }
% 260.10/33.05 ifeq2(sum(multiply(X, add(Y, X)), multiply(add(add(Y, X), inverse(X)), inverse(X)), add(add(Y, X), inverse(X))), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by lemma 28 }
% 260.10/33.05 ifeq2(sum(multiply(X, add(Y, X)), multiply(inverse(X), add(add(Y, X), inverse(X))), add(add(Y, X), inverse(X))), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by lemma 35 }
% 260.10/33.05 ifeq2(sum(multiply(X, add(Y, X)), inverse(X), add(add(Y, X), inverse(X))), true, add(add(Y, X), inverse(X)), add(multiply(X, add(Y, X)), inverse(X)))
% 260.10/33.05 = { by lemma 37 }
% 260.10/33.05 add(multiply(X, add(Y, X)), inverse(X))
% 260.10/33.05 = { by lemma 23 }
% 260.10/33.05 add(inverse(X), multiply(X, add(Y, X)))
% 260.10/33.05 = { by lemma 35 }
% 260.10/33.05 add(inverse(X), X)
% 260.10/33.05 = { by axiom 16 (addition_is_well_defined) R->L }
% 260.10/33.05 ifeq2(sum(X, inverse(X), add(inverse(X), X)), true, ifeq2(sum(X, inverse(X), multiplicative_identity), true, multiplicative_identity, add(inverse(X), X)), add(inverse(X), X))
% 260.10/33.05 = { by axiom 2 (additive_inverse2) }
% 260.10/33.05 ifeq2(sum(X, inverse(X), add(inverse(X), X)), true, ifeq2(true, true, multiplicative_identity, add(inverse(X), X)), add(inverse(X), X))
% 260.10/33.05 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.05 ifeq2(sum(X, inverse(X), add(inverse(X), X)), true, multiplicative_identity, add(inverse(X), X))
% 260.10/33.05 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.05 ifeq2(ifeq(true, true, sum(X, inverse(X), add(inverse(X), X)), true), true, multiplicative_identity, add(inverse(X), X))
% 260.10/33.05 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.05 ifeq2(ifeq(sum(inverse(X), X, add(inverse(X), X)), true, sum(X, inverse(X), add(inverse(X), X)), true), true, multiplicative_identity, add(inverse(X), X))
% 260.10/33.05 = { by axiom 14 (commutativity_of_addition) }
% 260.10/33.05 ifeq2(true, true, multiplicative_identity, add(inverse(X), X))
% 260.10/33.05 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.05 multiplicative_identity
% 260.10/33.05
% 260.10/33.05 Lemma 39: multiply(add(X, Y), add(X, inverse(Y))) = X.
% 260.10/33.05 Proof:
% 260.10/33.05 multiply(add(X, Y), add(X, inverse(Y)))
% 260.10/33.05 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.05 ifeq2(true, true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.05 = { by axiom 18 (distributivity7) R->L }
% 260.10/33.05 ifeq2(ifeq(product(Y, inverse(Y), additive_identity), true, ifeq(sum(additive_identity, X, X), true, ifeq(sum(inverse(Y), X, add(inverse(Y), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.05 = { by axiom 3 (additive_identity1) }
% 260.10/33.06 ifeq2(ifeq(product(Y, inverse(Y), additive_identity), true, ifeq(true, true, ifeq(sum(inverse(Y), X, add(inverse(Y), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.06 ifeq2(ifeq(product(Y, inverse(Y), additive_identity), true, ifeq(sum(inverse(Y), X, add(inverse(Y), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 7 (multiplicative_inverse2) }
% 260.10/33.06 ifeq2(ifeq(true, true, ifeq(sum(inverse(Y), X, add(inverse(Y), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.06 ifeq2(ifeq(sum(inverse(Y), X, add(inverse(Y), X)), true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 12 (closure_of_addition) }
% 260.10/33.06 ifeq2(ifeq(true, true, ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.06 ifeq2(ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(inverse(Y), X), X), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by lemma 23 }
% 260.10/33.06 ifeq2(ifeq(sum(Y, X, add(Y, X)), true, product(add(Y, X), add(X, inverse(Y)), X), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 12 (closure_of_addition) }
% 260.10/33.06 ifeq2(ifeq(true, true, product(add(Y, X), add(X, inverse(Y)), X), true), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.06 ifeq2(product(add(Y, X), add(X, inverse(Y)), X), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by lemma 23 }
% 260.10/33.06 ifeq2(product(add(X, Y), add(X, inverse(Y)), X), true, multiply(add(X, Y), add(X, inverse(Y))), X)
% 260.10/33.06 = { by lemma 27 }
% 260.10/33.07 X
% 260.10/33.07
% 260.10/33.07 Goal 1 (prove_equation): inverse(x_plus_y) = x_inverse_times_y_inverse.
% 260.10/33.07 Proof:
% 260.10/33.07 inverse(x_plus_y)
% 260.10/33.07 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.07 ifeq2(true, true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 18 (distributivity7) R->L }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(sum(additive_identity, inverse(x_plus_y), inverse(x_plus_y)), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y))), true, ifeq(sum(x_plus_y, inverse(x_plus_y), multiplicative_identity), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 3 (additive_identity1) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(true, true, ifeq(sum(x_inverse_times_y_inverse, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y))), true, ifeq(sum(x_plus_y, inverse(x_plus_y), multiplicative_identity), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y))), true, ifeq(sum(x_plus_y, inverse(x_plus_y), multiplicative_identity), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 2 (additive_inverse2) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y))), true, ifeq(true, true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 12 (closure_of_addition) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, ifeq(true, true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, additive_identity), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by lemma 31 R->L }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, inverse(y))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.07 = { by lemma 32 R->L }
% 260.10/33.07 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(true, true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(true, true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 15 (commutativity_of_multiplication) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, product(inverse(y), inverse(x), x_inverse_times_y_inverse), true), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 9 (x_inverse_times_y_inverse) }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(ifeq(true, true, product(inverse(y), inverse(x), x_inverse_times_y_inverse), true), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(y))), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by lemma 26 R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, product(inverse(y), add(inverse(x), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(y))), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, ifeq(true, true, product(inverse(y), add(inverse(x), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(y))), true), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 12 (closure_of_addition) R->L }
% 260.10/33.08 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, ifeq(sum(inverse(x), multiplicative_identity, add(inverse(x), multiplicative_identity)), true, product(inverse(y), add(inverse(x), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(y))), true), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.08 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.09 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(true, true, ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, ifeq(sum(inverse(x), multiplicative_identity, add(inverse(x), multiplicative_identity)), true, product(inverse(y), add(inverse(x), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(y))), true), true), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.09 = { by axiom 6 (multiplicative_identity2) R->L }
% 260.10/33.09 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(ifeq(product(inverse(y), multiplicative_identity, inverse(y)), true, ifeq(product(inverse(y), inverse(x), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(y), add(x_inverse_times_y_inverse, inverse(y))), true, ifeq(sum(inverse(x), multiplicative_identity, add(inverse(x), multiplicative_identity)), true, product(inverse(y), add(inverse(x), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(y))), true), true), true), true), true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.09 = { by axiom 19 (distributivity2) }
% 260.10/33.09 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, ifeq2(true, true, add(x_inverse_times_y_inverse, inverse(y)), inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.09 = { by axiom 11 (ifeq_axiom) }
% 260.10/33.09 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, add(x_inverse_times_y_inverse, inverse(y)))), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.09 = { by lemma 36 }
% 260.10/33.09 ifeq2(ifeq(product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.09 = { by axiom 10 (ifeq_axiom) R->L }
% 260.10/33.10 ifeq2(ifeq(ifeq(true, true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, x_inverse_times_y_inverse)), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by lemma 36 R->L }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, add(x_inverse_times_y_inverse, inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 11 (ifeq_axiom) R->L }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(true, true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 19 (distributivity2) R->L }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(product(inverse(x), multiplicative_identity, inverse(x)), true, ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, ifeq(sum(inverse(y), multiplicative_identity, add(inverse(y), multiplicative_identity)), true, product(inverse(x), add(inverse(y), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(x))), true), true), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 6 (multiplicative_identity2) }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(true, true, ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, ifeq(sum(inverse(y), multiplicative_identity, add(inverse(y), multiplicative_identity)), true, product(inverse(x), add(inverse(y), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(x))), true), true), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, ifeq(sum(inverse(y), multiplicative_identity, add(inverse(y), multiplicative_identity)), true, product(inverse(x), add(inverse(y), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(x))), true), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 12 (closure_of_addition) }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, ifeq(true, true, product(inverse(x), add(inverse(y), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(x))), true), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by axiom 10 (ifeq_axiom) }
% 260.10/33.10 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, product(inverse(x), add(inverse(y), multiplicative_identity), add(x_inverse_times_y_inverse, inverse(x))), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.10/33.10 = { by lemma 26 }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, product(inverse(x), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x))), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 9 (x_inverse_times_y_inverse) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(true, true, ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, product(inverse(x), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x))), true), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(sum(x_inverse_times_y_inverse, inverse(x), add(x_inverse_times_y_inverse, inverse(x))), true, product(inverse(x), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x))), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 12 (closure_of_addition) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(ifeq(true, true, product(inverse(x), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x))), true), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, ifeq2(product(inverse(x), multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x))), true, add(x_inverse_times_y_inverse, inverse(x)), inverse(x)))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by lemma 32 }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, multiply(x, inverse(x))), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by lemma 31 }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) R->L }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(true, true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) R->L }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(true, true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 14 (commutativity_of_addition) R->L }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(ifeq(sum(x, y, x_plus_y), true, sum(y, x, x_plus_y), true), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 4 (x_plus_y) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(ifeq(true, true, sum(y, x, x_plus_y), true), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.11 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.11 = { by axiom 10 (ifeq_axiom) R->L }
% 260.84/33.12 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(true, true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 1 (additive_identity2) R->L }
% 260.84/33.12 ifeq2(ifeq(ifeq(product(x, x_inverse_times_y_inverse, additive_identity), true, ifeq(product(y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true, ifeq(sum(multiply(y, x_inverse_times_y_inverse), additive_identity, multiply(y, x_inverse_times_y_inverse)), true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, x_inverse_times_y_inverse, multiply(y, x_inverse_times_y_inverse)), true), true), true), true), true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 21 (distributivity4) }
% 260.84/33.12 ifeq2(ifeq(true, true, product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.12 ifeq2(product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), inverse(x_plus_y)), true, inverse(x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by lemma 25 }
% 260.84/33.12 add(x_inverse_times_y_inverse, inverse(x_plus_y))
% 260.84/33.12 = { by lemma 32 R->L }
% 260.84/33.12 ifeq2(product(add(x_inverse_times_y_inverse, inverse(x_plus_y)), multiplicative_identity, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)))), true, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y))), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 10 (ifeq_axiom) R->L }
% 260.84/33.12 ifeq2(ifeq(true, true, product(add(x_inverse_times_y_inverse, inverse(x_plus_y)), multiplicative_identity, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)))), true), true, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y))), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 13 (closure_of_multiplication) R->L }
% 260.84/33.12 ifeq2(ifeq(product(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)), multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)))), true, product(add(x_inverse_times_y_inverse, inverse(x_plus_y)), multiplicative_identity, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)))), true), true, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y))), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 15 (commutativity_of_multiplication) }
% 260.84/33.12 ifeq2(true, true, multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y))), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 11 (ifeq_axiom) }
% 260.84/33.12 multiply(multiplicative_identity, add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 11 (ifeq_axiom) R->L }
% 260.84/33.12 multiply(ifeq2(true, true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 22 (distributivity8) R->L }
% 260.84/33.12 multiply(ifeq2(ifeq(product(add(inverse(y), x_plus_y), multiplicative_identity, add(inverse(y), x_plus_y)), true, ifeq(product(inverse(y), inverse(x), multiply(inverse(y), inverse(x))), true, ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(y), inverse(x)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 6 (multiplicative_identity2) }
% 260.84/33.12 multiply(ifeq2(ifeq(true, true, ifeq(product(inverse(y), inverse(x), multiply(inverse(y), inverse(x))), true, ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(y), inverse(x)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.12 multiply(ifeq2(ifeq(product(inverse(y), inverse(x), multiply(inverse(y), inverse(x))), true, ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(y), inverse(x)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 13 (closure_of_multiplication) }
% 260.84/33.12 multiply(ifeq2(ifeq(true, true, ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(y), inverse(x)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(y), inverse(x)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by lemma 28 }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(sum(inverse(y), x_plus_y, add(inverse(y), x_plus_y)), true, sum(multiply(inverse(x), inverse(y)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 12 (closure_of_addition) }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, ifeq(true, true, sum(multiply(inverse(x), inverse(y)), x_plus_y, add(inverse(y), x_plus_y)), true), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(multiply(inverse(x), inverse(y)), x_plus_y, add(inverse(y), x_plus_y)), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by lemma 23 }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(multiply(inverse(x), inverse(y)), x_plus_y, add(x_plus_y, inverse(y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 11 (ifeq_axiom) R->L }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(ifeq2(true, true, multiply(inverse(x), inverse(y)), x_inverse_times_y_inverse), x_plus_y, add(x_plus_y, inverse(y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.12 = { by axiom 9 (x_inverse_times_y_inverse) R->L }
% 260.84/33.12 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(ifeq2(product(inverse(x), inverse(y), x_inverse_times_y_inverse), true, multiply(inverse(x), inverse(y)), x_inverse_times_y_inverse), x_plus_y, add(x_plus_y, inverse(y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 27 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(x_plus_y, inverse(y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 23 R->L }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), x_plus_y)), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 33 R->L }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(add(inverse(y), x_plus_y), multiply(add(inverse(y), x_plus_y), add(inverse(y), inverse(x_plus_y))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 39 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(add(inverse(y), x_plus_y), inverse(y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 23 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(inverse(y), x_plus_y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 23 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(x_plus_y, inverse(y)))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 33 R->L }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), multiply(add(x_plus_y, inverse(y)), y)))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 28 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), multiply(y, add(x_plus_y, inverse(y)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 36 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), multiply(y, x_plus_y)))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by lemma 28 }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), multiply(x_plus_y, y)))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by axiom 11 (ifeq_axiom) R->L }
% 260.84/33.13 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(true, true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.13 = { by axiom 18 (distributivity7) R->L }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(sum(multiply(x, additive_identity), y, add(multiply(x, additive_identity), y)), true, ifeq(sum(additive_identity, y, y), true, ifeq(sum(x, y, x_plus_y), true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 3 (additive_identity1) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(sum(multiply(x, additive_identity), y, add(multiply(x, additive_identity), y)), true, ifeq(true, true, ifeq(sum(x, y, x_plus_y), true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(sum(multiply(x, additive_identity), y, add(multiply(x, additive_identity), y)), true, ifeq(sum(x, y, x_plus_y), true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 4 (x_plus_y) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(sum(multiply(x, additive_identity), y, add(multiply(x, additive_identity), y)), true, ifeq(true, true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(sum(multiply(x, additive_identity), y, add(multiply(x, additive_identity), y)), true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 12 (closure_of_addition) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, ifeq(true, true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(product(x, additive_identity, multiply(x, additive_identity)), true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 13 (closure_of_multiplication) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(ifeq(true, true, product(x_plus_y, y, add(multiply(x, additive_identity), y)), true), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(product(x_plus_y, y, add(multiply(x, additive_identity), y)), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by lemma 23 }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(product(x_plus_y, y, add(y, multiply(x, additive_identity))), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by lemma 28 }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), ifeq2(product(x_plus_y, y, add(y, multiply(additive_identity, x))), true, multiply(x_plus_y, y), add(y, multiply(additive_identity, x)))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by lemma 27 }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), add(y, multiply(additive_identity, x))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by lemma 28 R->L }
% 260.84/33.14 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), add(y, multiply(x, additive_identity))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.14 = { by lemma 30 }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), add(y, additive_identity)))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 24 }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(add(x_plus_y, inverse(y)), y))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 23 }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, add(inverse(y), add(y, add(x_plus_y, inverse(y))))), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 38 }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), x_plus_y, multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 11 (ifeq_axiom) R->L }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), ifeq2(true, true, x_plus_y, add(x, y)), multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 12 (closure_of_addition) R->L }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), ifeq2(sum(x, y, add(x, y)), true, x_plus_y, add(x, y)), multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 11 (ifeq_axiom) R->L }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), ifeq2(sum(x, y, add(x, y)), true, ifeq2(true, true, x_plus_y, add(x, y)), add(x, y)), multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 4 (x_plus_y) R->L }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), ifeq2(sum(x, y, add(x, y)), true, ifeq2(sum(x, y, x_plus_y), true, x_plus_y, add(x, y)), add(x, y)), multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 16 (addition_is_well_defined) }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), add(x, y), multiplicative_identity), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 38 R->L }
% 260.84/33.15 multiply(ifeq2(ifeq(sum(inverse(x), add(x, y), add(inverse(x), add(x, y))), true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 12 (closure_of_addition) }
% 260.84/33.15 multiply(ifeq2(ifeq(true, true, sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by axiom 10 (ifeq_axiom) }
% 260.84/33.15 multiply(ifeq2(sum(x_inverse_times_y_inverse, x_plus_y, multiplicative_identity), true, multiplicative_identity, add(x_inverse_times_y_inverse, x_plus_y)), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 37 }
% 260.84/33.15 multiply(add(x_inverse_times_y_inverse, x_plus_y), add(x_inverse_times_y_inverse, inverse(x_plus_y)))
% 260.84/33.15 = { by lemma 39 }
% 260.84/33.15 x_inverse_times_y_inverse
% 260.84/33.15 % SZS output end Proof
% 260.84/33.15
% 260.84/33.15 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------