%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : BOO008-1 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n020.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 09:35:22 AM UTC 2026
% Result : Timeout 291.12s 49.14s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : BOO008-1 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.18 % Computer : n020.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 21:04:34 UTC 2026
% 0.07/0.18 % CPUTime :
% 0.07/0.18 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 291.12/49.14 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-regeneralise
% 291.12/49.14
% 291.12/49.14 % SZS status Unsatisfiable
% 291.12/49.14
% 293.12/49.46 % SZS output start Proof
% 293.12/49.46 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 293.12/49.46 Axiom 2 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 293.12/49.46 Axiom 3 (additive_identity1): sum(additive_identity, X, X) = true.
% 293.12/49.46 Axiom 4 (y_plus_z): sum(y, z, y_plus_z) = true.
% 293.12/49.46 Axiom 5 (x_plus__y_plus_z): sum(x, y_plus_z, x__plus_y_plus_z) = true.
% 293.12/49.46 Axiom 6 (x_plus_y): sum(x, y, x_plus_y) = true.
% 293.12/49.46 Axiom 7 (x_plus_y__plus_z): sum(x_plus_y, z, x_plus_y__plus_z) = true.
% 293.12/49.46 Axiom 8 (additive_inverse1): sum(inverse(X), X, multiplicative_identity) = true.
% 293.12/49.46 Axiom 9 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 293.12/49.46 Axiom 10 (multiplicative_inverse2): product(X, inverse(X), additive_identity) = true.
% 293.12/49.46 Axiom 11 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 293.12/49.46 Axiom 12 (multiplicative_inverse1): product(inverse(X), X, additive_identity) = true.
% 293.12/49.46 Axiom 13 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 293.12/49.46 Axiom 14 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 293.12/49.46 Axiom 15 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 293.12/49.46 Axiom 16 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 293.12/49.46 Axiom 17 (commutativity_of_addition): ifeq(sum(X, Y, Z), true, sum(Y, X, Z), true) = true.
% 293.12/49.46 Axiom 18 (commutativity_of_multiplication): ifeq(product(X, Y, Z), true, product(Y, X, Z), true) = true.
% 293.12/49.46 Axiom 19 (addition_is_well_defined): ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, W), true, W, Z), Z) = Z.
% 293.12/49.46 Axiom 20 (multiplication_is_well_defined): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 293.12/49.46 Axiom 21 (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.
% 293.12/49.46 Axiom 22 (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.
% 293.12/49.46 Axiom 23 (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.
% 293.12/49.46 Axiom 24 (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.
% 293.12/49.46
% 293.12/49.46 Lemma 25: add(X, Y) = add(Y, X).
% 293.12/49.46 Proof:
% 293.12/49.46 add(X, Y)
% 293.12/49.46 = { by axiom 19 (addition_is_well_defined) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 15 (closure_of_addition) }
% 293.12/49.46 ifeq2(sum(Y, X, add(X, Y)), true, ifeq2(true, true, add(Y, X), add(X, Y)), add(X, Y))
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.46 ifeq2(sum(Y, X, add(X, Y)), true, add(Y, X), add(X, Y))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(ifeq(true, true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 293.12/49.46 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.46 ifeq2(ifeq(sum(X, Y, add(X, Y)), true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 293.12/49.46 = { by axiom 17 (commutativity_of_addition) }
% 293.12/49.46 ifeq2(true, true, add(Y, X), add(X, Y))
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.46 add(Y, X)
% 293.12/49.46
% 293.12/49.46 Lemma 26: ifeq2(sum(X, additive_identity, Y), true, Y, X) = X.
% 293.12/49.46 Proof:
% 293.12/49.46 ifeq2(sum(X, additive_identity, Y), true, Y, X)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(true, true, ifeq2(sum(X, additive_identity, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 1 (additive_identity2) R->L }
% 293.12/49.46 ifeq2(sum(X, additive_identity, X), true, ifeq2(sum(X, additive_identity, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 19 (addition_is_well_defined) }
% 293.12/49.46 X
% 293.12/49.46
% 293.12/49.46 Lemma 27: add(X, additive_identity) = X.
% 293.12/49.46 Proof:
% 293.12/49.46 add(X, additive_identity)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(true, true, add(X, additive_identity), X)
% 293.12/49.46 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.46 ifeq2(sum(X, additive_identity, add(X, additive_identity)), true, add(X, additive_identity), X)
% 293.12/49.46 = { by lemma 26 }
% 293.12/49.46 X
% 293.12/49.46
% 293.12/49.46 Lemma 28: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 293.12/49.46 Proof:
% 293.12/49.46 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 293.12/49.46 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.46 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 293.12/49.46 = { by axiom 20 (multiplication_is_well_defined) }
% 293.12/49.46 Z
% 293.12/49.46
% 293.12/49.46 Lemma 29: multiply(X, Y) = multiply(Y, X).
% 293.12/49.46 Proof:
% 293.12/49.46 multiply(X, Y)
% 293.12/49.46 = { by lemma 28 R->L }
% 293.12/49.46 ifeq2(product(Y, X, multiply(X, Y)), true, multiply(Y, X), multiply(X, Y))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(ifeq(true, true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 293.12/49.46 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.46 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 293.12/49.46 = { by axiom 18 (commutativity_of_multiplication) }
% 293.12/49.46 ifeq2(true, true, multiply(Y, X), multiply(X, Y))
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.46 multiply(Y, X)
% 293.12/49.46
% 293.12/49.46 Lemma 30: ifeq2(product(X, inverse(X), Y), true, additive_identity, Y) = Y.
% 293.12/49.46 Proof:
% 293.12/49.46 ifeq2(product(X, inverse(X), Y), true, additive_identity, Y)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(product(X, inverse(X), Y), true, ifeq2(true, true, additive_identity, Y), Y)
% 293.12/49.46 = { by axiom 10 (multiplicative_inverse2) R->L }
% 293.12/49.46 ifeq2(product(X, inverse(X), Y), true, ifeq2(product(X, inverse(X), additive_identity), true, additive_identity, Y), Y)
% 293.12/49.46 = { by axiom 20 (multiplication_is_well_defined) }
% 293.12/49.46 Y
% 293.12/49.46
% 293.12/49.46 Lemma 31: multiply(X, additive_identity) = additive_identity.
% 293.12/49.46 Proof:
% 293.12/49.46 multiply(X, additive_identity)
% 293.12/49.46 = { by lemma 30 R->L }
% 293.12/49.46 ifeq2(product(X, inverse(X), multiply(X, additive_identity)), true, additive_identity, multiply(X, additive_identity))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(ifeq(true, true, product(X, inverse(X), multiply(X, additive_identity)), true), true, additive_identity, multiply(X, additive_identity))
% 293.12/49.46 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 10 (multiplicative_inverse2) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 3 (additive_identity1) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 1 (additive_identity2) R->L }
% 293.12/49.46 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))
% 293.12/49.46 = { by axiom 22 (distributivity2) }
% 293.12/49.46 ifeq2(true, true, additive_identity, multiply(X, additive_identity))
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.46 additive_identity
% 293.12/49.46
% 293.12/49.46 Lemma 32: ifeq2(product(multiplicative_identity, X, Y), true, Y, X) = X.
% 293.12/49.46 Proof:
% 293.12/49.46 ifeq2(product(multiplicative_identity, X, Y), true, Y, X)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(true, true, ifeq2(product(multiplicative_identity, X, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 11 (multiplicative_identity1) R->L }
% 293.12/49.46 ifeq2(product(multiplicative_identity, X, X), true, ifeq2(product(multiplicative_identity, X, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 20 (multiplication_is_well_defined) }
% 293.12/49.46 X
% 293.12/49.46
% 293.12/49.46 Lemma 33: ifeq2(product(X, multiplicative_identity, Y), true, Y, X) = X.
% 293.12/49.46 Proof:
% 293.12/49.46 ifeq2(product(X, multiplicative_identity, Y), true, Y, X)
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(true, true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 9 (multiplicative_identity2) R->L }
% 293.12/49.46 ifeq2(product(X, multiplicative_identity, X), true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 293.12/49.46 = { by axiom 20 (multiplication_is_well_defined) }
% 293.12/49.46 X
% 293.12/49.46
% 293.12/49.46 Lemma 34: inverse(inverse(X)) = X.
% 293.12/49.46 Proof:
% 293.12/49.46 inverse(inverse(X))
% 293.12/49.46 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(true, true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 21 (distributivity7) R->L }
% 293.12/49.46 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(sum(additive_identity, inverse(inverse(X)), inverse(inverse(X))), true, ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 3 (additive_identity1) }
% 293.12/49.46 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(true, true, ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.46 ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 10 (multiplicative_inverse2) }
% 293.12/49.46 ifeq2(ifeq(true, true, ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.46 ifeq2(ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 2 (additive_inverse2) }
% 293.12/49.46 ifeq2(ifeq(true, true, ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.46 ifeq2(ifeq(sum(X, inverse(inverse(X)), add(X, inverse(inverse(X)))), true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 15 (closure_of_addition) }
% 293.12/49.46 ifeq2(ifeq(true, true, product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.46 ifeq2(product(add(X, inverse(inverse(X))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by lemma 32 R->L }
% 293.12/49.46 ifeq2(product(ifeq2(product(multiplicative_identity, add(X, inverse(inverse(X))), X), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by lemma 25 R->L }
% 293.12/49.46 ifeq2(product(ifeq2(product(multiplicative_identity, add(inverse(inverse(X)), X), X), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(product(ifeq2(ifeq(true, true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.46 ifeq2(product(ifeq2(ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.46 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.46 ifeq2(product(ifeq2(ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(true, true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 8 (additive_inverse1) R->L }
% 293.12/49.47 ifeq2(product(ifeq2(ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(product(ifeq2(ifeq(true, true, ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 10 (multiplicative_inverse2) R->L }
% 293.12/49.47 ifeq2(product(ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), additive_identity), true, ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(product(ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), additive_identity), true, ifeq(true, true, ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 3 (additive_identity1) R->L }
% 293.12/49.47 ifeq2(product(ifeq2(ifeq(product(inverse(X), inverse(inverse(X)), additive_identity), true, ifeq(sum(additive_identity, X, X), true, ifeq(sum(inverse(inverse(X)), X, add(inverse(inverse(X)), X)), true, ifeq(sum(inverse(X), X, multiplicative_identity), true, product(multiplicative_identity, add(inverse(inverse(X)), X), X), true), true), true), true), true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 21 (distributivity7) }
% 293.12/49.47 ifeq2(product(ifeq2(true, true, X, add(X, inverse(inverse(X)))), multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.47 ifeq2(product(X, multiplicative_identity, inverse(inverse(X))), true, inverse(inverse(X)), X)
% 293.12/49.47 = { by lemma 33 }
% 293.12/49.47 X
% 293.12/49.47
% 293.12/49.47 Lemma 35: ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)) = multiply(X, Y).
% 293.12/49.47 Proof:
% 293.12/49.47 ifeq2(product(X, Y, Z), true, Z, multiply(X, Y))
% 293.12/49.47 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(true, true, ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)), multiply(X, Y))
% 293.12/49.47 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.47 ifeq2(product(X, Y, multiply(X, Y)), true, ifeq2(product(X, Y, Z), true, Z, multiply(X, Y)), multiply(X, Y))
% 293.12/49.47 = { by axiom 20 (multiplication_is_well_defined) }
% 293.12/49.47 multiply(X, Y)
% 293.12/49.47
% 293.12/49.47 Lemma 36: multiply(X, add(Y, inverse(X))) = multiply(X, Y).
% 293.12/49.47 Proof:
% 293.12/49.47 multiply(X, add(Y, inverse(X)))
% 293.12/49.47 = { by lemma 35 R->L }
% 293.12/49.47 ifeq2(product(X, add(Y, inverse(X)), multiply(X, Y)), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(ifeq(true, true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 293.12/49.47 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 10 (multiplicative_inverse2) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 1 (additive_identity2) R->L }
% 293.12/49.47 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))))
% 293.12/49.47 = { by axiom 22 (distributivity2) }
% 293.12/49.47 ifeq2(true, true, multiply(X, Y), multiply(X, add(Y, inverse(X))))
% 293.12/49.47 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.47 multiply(X, Y)
% 293.12/49.47
% 293.12/49.47 Lemma 37: multiply(inverse(X), add(Y, X)) = multiply(Y, inverse(X)).
% 293.12/49.47 Proof:
% 293.12/49.47 multiply(inverse(X), add(Y, X))
% 293.12/49.47 = { by lemma 34 R->L }
% 293.12/49.47 multiply(inverse(X), add(Y, inverse(inverse(X))))
% 293.12/49.47 = { by lemma 36 }
% 293.12/49.47 multiply(inverse(X), Y)
% 293.12/49.47 = { by lemma 29 }
% 293.12/49.47 multiply(Y, inverse(X))
% 293.12/49.47
% 293.12/49.47 Lemma 38: add(X, multiply(X, Y)) = X.
% 293.12/49.47 Proof:
% 293.12/49.47 add(X, multiply(X, Y))
% 293.12/49.47 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(true, true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 22 (distributivity2) R->L }
% 293.12/49.47 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)
% 293.12/49.47 = { by axiom 9 (multiplicative_identity2) }
% 293.12/49.47 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)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.47 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)
% 293.12/49.47 = { by axiom 15 (closure_of_addition) }
% 293.12/49.47 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)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.47 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)
% 293.12/49.47 = { by lemma 32 R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(product(multiplicative_identity, add(Y, multiplicative_identity), multiplicative_identity), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by lemma 25 R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(true, true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(true, true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 8 (additive_inverse1) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(true, true, ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.47 = { by axiom 8 (additive_inverse1) R->L }
% 293.12/49.47 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(sum(inverse(Y), Y, multiplicative_identity), true, ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(true, true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by axiom 9 (multiplicative_identity2) R->L }
% 293.12/49.48 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(ifeq(product(inverse(Y), multiplicative_identity, inverse(Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, ifeq(sum(multiplicative_identity, Y, add(multiplicative_identity, Y)), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, product(multiplicative_identity, add(multiplicative_identity, Y), multiplicative_identity), true), true), true), true), true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by axiom 21 (distributivity7) }
% 293.12/49.48 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, ifeq(sum(multiply(X, Y), X, add(multiply(X, Y), X)), true, product(X, ifeq2(true, true, multiplicative_identity, add(Y, multiplicative_identity)), add(multiply(X, Y), X)), true), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.48 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)
% 293.12/49.48 = { by axiom 15 (closure_of_addition) }
% 293.12/49.48 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)
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 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)
% 293.12/49.48 = { by lemma 25 }
% 293.12/49.48 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)
% 293.12/49.48 = { by axiom 16 (closure_of_multiplication) }
% 293.12/49.48 ifeq2(ifeq(true, true, product(X, multiplicative_identity, add(X, multiply(X, Y))), true), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 ifeq2(product(X, multiplicative_identity, add(X, multiply(X, Y))), true, add(X, multiply(X, Y)), X)
% 293.12/49.48 = { by lemma 33 }
% 293.12/49.48 X
% 293.12/49.48
% 293.12/49.48 Lemma 39: multiply(inverse(X), X) = additive_identity.
% 293.12/49.48 Proof:
% 293.12/49.48 multiply(inverse(X), X)
% 293.12/49.48 = { by lemma 30 R->L }
% 293.12/49.48 ifeq2(product(X, inverse(X), multiply(inverse(X), X)), true, additive_identity, multiply(inverse(X), X))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 ifeq2(ifeq(true, true, product(X, inverse(X), multiply(inverse(X), X)), true), true, additive_identity, multiply(inverse(X), X))
% 293.12/49.48 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.48 ifeq2(ifeq(product(inverse(X), X, multiply(inverse(X), X)), true, product(X, inverse(X), multiply(inverse(X), X)), true), true, additive_identity, multiply(inverse(X), X))
% 293.12/49.48 = { by axiom 18 (commutativity_of_multiplication) }
% 293.12/49.48 ifeq2(true, true, additive_identity, multiply(inverse(X), X))
% 293.12/49.48 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.48 additive_identity
% 293.12/49.48
% 293.12/49.48 Lemma 40: multiply(y_plus_z, inverse(x__plus_y_plus_z)) = additive_identity.
% 293.12/49.48 Proof:
% 293.12/49.48 multiply(y_plus_z, inverse(x__plus_y_plus_z))
% 293.12/49.48 = { by lemma 37 R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(y_plus_z, x__plus_y_plus_z))
% 293.12/49.48 = { by lemma 25 R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, y_plus_z))
% 293.12/49.48 = { by lemma 28 R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(product(add(y_plus_z, inverse(x)), x__plus_y_plus_z, y_plus_z), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by lemma 25 R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(true, true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(true, true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 12 (multiplicative_inverse1) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(true, true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 5 (x_plus__y_plus_z) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(x, y_plus_z, x__plus_y_plus_z), true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(true, true, ifeq(sum(x, y_plus_z, x__plus_y_plus_z), true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 3 (additive_identity1) R->L }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(additive_identity, y_plus_z, y_plus_z), true, ifeq(sum(x, y_plus_z, x__plus_y_plus_z), true, ifeq(sum(inverse(x), y_plus_z, add(inverse(x), y_plus_z)), true, product(add(inverse(x), y_plus_z), x__plus_y_plus_z, y_plus_z), true), true), true), true), true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 21 (distributivity7) }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(true, true, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z), y_plus_z)))
% 293.12/49.48 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, multiply(add(y_plus_z, inverse(x)), x__plus_y_plus_z)))
% 293.12/49.48 = { by lemma 29 }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, multiply(x__plus_y_plus_z, add(y_plus_z, inverse(x)))))
% 293.12/49.48 = { by lemma 38 }
% 293.12/49.48 multiply(inverse(x__plus_y_plus_z), x__plus_y_plus_z)
% 293.12/49.48 = { by lemma 39 }
% 293.12/49.48 additive_identity
% 293.12/49.48
% 293.12/49.48 Lemma 41: multiply(X, add(Y, X)) = X.
% 293.12/49.48 Proof:
% 293.12/49.48 multiply(X, add(Y, X))
% 293.12/49.48 = { by lemma 25 R->L }
% 293.12/49.48 multiply(X, add(X, Y))
% 293.12/49.48 = { by lemma 29 R->L }
% 293.12/49.48 multiply(add(X, Y), X)
% 293.12/49.48 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.48 ifeq2(true, true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 21 (distributivity7) R->L }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 3 (additive_identity1) }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 15 (closure_of_addition) }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 15 (closure_of_addition) }
% 293.12/49.48 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, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 ifeq2(ifeq(product(Y, additive_identity, multiply(Y, additive_identity)), true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 16 (closure_of_multiplication) }
% 293.12/49.48 ifeq2(ifeq(true, true, product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true), true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.48 ifeq2(product(add(Y, X), X, add(multiply(Y, additive_identity), X)), true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by lemma 25 }
% 293.12/49.48 ifeq2(product(add(X, Y), X, add(multiply(Y, additive_identity), X)), true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by lemma 25 }
% 293.12/49.48 ifeq2(product(add(X, Y), X, add(X, multiply(Y, additive_identity))), true, multiply(add(X, Y), X), add(X, multiply(Y, additive_identity)))
% 293.12/49.48 = { by lemma 28 }
% 293.12/49.48 add(X, multiply(Y, additive_identity))
% 293.12/49.48 = { by lemma 31 }
% 293.12/49.48 add(X, additive_identity)
% 293.12/49.48 = { by lemma 27 }
% 293.12/49.48 X
% 293.12/49.48
% 293.12/49.48 Lemma 42: ifeq2(sum(X, Y, Z), true, Z, add(X, Y)) = add(X, Y).
% 293.12/49.48 Proof:
% 293.12/49.48 ifeq2(sum(X, Y, Z), true, Z, add(X, Y))
% 293.12/49.48 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.48 ifeq2(true, true, ifeq2(sum(X, Y, Z), true, Z, add(X, Y)), add(X, Y))
% 293.12/49.48 = { by axiom 15 (closure_of_addition) R->L }
% 293.12/49.48 ifeq2(sum(X, Y, add(X, Y)), true, ifeq2(sum(X, Y, Z), true, Z, add(X, Y)), add(X, Y))
% 293.12/49.48 = { by axiom 19 (addition_is_well_defined) }
% 293.12/49.48 add(X, Y)
% 293.12/49.48
% 293.12/49.48 Lemma 43: add(X, multiply(Y, inverse(X))) = add(X, Y).
% 293.12/49.48 Proof:
% 293.12/49.48 add(X, multiply(Y, inverse(X)))
% 293.12/49.48 = { by lemma 37 R->L }
% 293.12/49.48 add(X, multiply(inverse(X), add(Y, X)))
% 293.12/49.48 = { by lemma 29 R->L }
% 293.12/49.48 add(X, multiply(add(Y, X), inverse(X)))
% 293.12/49.48 = { by lemma 41 R->L }
% 293.12/49.48 add(multiply(X, add(Y, X)), multiply(add(Y, X), inverse(X)))
% 293.12/49.48 = { by lemma 29 R->L }
% 293.12/49.48 add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)))
% 293.12/49.48 = { by lemma 42 R->L }
% 293.12/49.48 ifeq2(sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.48 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.48 ifeq2(ifeq(true, true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.48 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.48 ifeq2(ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(true, true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(product(add(Y, X), X, multiply(add(Y, X), X)), true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(product(add(Y, X), X, multiply(add(Y, X), X)), true, ifeq(true, true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 2 (additive_inverse2) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(product(add(Y, X), X, multiply(add(Y, X), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(true, true, ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(product(add(Y, X), X, multiply(add(Y, X), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 9 (multiplicative_identity2) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, X), multiplicative_identity, add(Y, X)), true, ifeq(product(add(Y, X), inverse(X), multiply(add(Y, X), inverse(X))), true, ifeq(product(add(Y, X), X, multiply(add(Y, X), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X)), add(Y, X)), true), true), true), true), true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 23 (distributivity1) }
% 293.12/49.49 ifeq2(true, true, add(Y, X), add(multiply(add(Y, X), X), multiply(add(Y, X), inverse(X))))
% 293.12/49.49 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.49 add(Y, X)
% 293.12/49.49 = { by lemma 25 }
% 293.12/49.49 add(X, Y)
% 293.12/49.49
% 293.12/49.49 Lemma 44: add(inverse(X), multiply(Y, X)) = add(Y, inverse(X)).
% 293.12/49.49 Proof:
% 293.12/49.49 add(inverse(X), multiply(Y, X))
% 293.12/49.49 = { by lemma 29 R->L }
% 293.12/49.49 add(inverse(X), multiply(X, Y))
% 293.12/49.49 = { by lemma 25 R->L }
% 293.12/49.49 add(multiply(X, Y), inverse(X))
% 293.12/49.49 = { by lemma 42 R->L }
% 293.12/49.49 ifeq2(sum(multiply(X, Y), inverse(X), add(Y, inverse(X))), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by lemma 41 R->L }
% 293.12/49.49 ifeq2(sum(multiply(X, Y), multiply(inverse(X), add(Y, inverse(X))), add(Y, inverse(X))), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by lemma 29 R->L }
% 293.12/49.49 ifeq2(sum(multiply(X, Y), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by lemma 36 R->L }
% 293.12/49.49 ifeq2(sum(multiply(X, add(Y, inverse(X))), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by lemma 29 R->L }
% 293.12/49.49 ifeq2(sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(true, true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(true, true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(product(add(Y, inverse(X)), X, multiply(add(Y, inverse(X)), X)), true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(product(add(Y, inverse(X)), X, multiply(add(Y, inverse(X)), X)), true, ifeq(true, true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 2 (additive_inverse2) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(product(add(Y, inverse(X)), X, multiply(add(Y, inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(ifeq(true, true, ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(product(add(Y, inverse(X)), X, multiply(add(Y, inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 9 (multiplicative_identity2) R->L }
% 293.12/49.49 ifeq2(ifeq(product(add(Y, inverse(X)), multiplicative_identity, add(Y, inverse(X))), true, ifeq(product(add(Y, inverse(X)), inverse(X), multiply(add(Y, inverse(X)), inverse(X))), true, ifeq(product(add(Y, inverse(X)), X, multiply(add(Y, inverse(X)), X)), true, ifeq(sum(X, inverse(X), multiplicative_identity), true, sum(multiply(add(Y, inverse(X)), X), multiply(add(Y, inverse(X)), inverse(X)), add(Y, inverse(X))), true), true), true), true), true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 23 (distributivity1) }
% 293.12/49.49 ifeq2(true, true, add(Y, inverse(X)), add(multiply(X, Y), inverse(X)))
% 293.12/49.49 = { by axiom 14 (ifeq_axiom) }
% 293.12/49.49 add(Y, inverse(X))
% 293.12/49.49
% 293.12/49.49 Lemma 45: add(multiply(X, y), multiply(X, z)) = multiply(X, y_plus_z).
% 293.12/49.49 Proof:
% 293.12/49.49 add(multiply(X, y), multiply(X, z))
% 293.12/49.49 = { by axiom 14 (ifeq_axiom) R->L }
% 293.12/49.49 ifeq2(true, true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 22 (distributivity2) R->L }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, ifeq(sum(multiply(X, y), multiply(X, z), add(multiply(X, y), multiply(X, z))), true, ifeq(sum(y, z, y_plus_z), true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 4 (y_plus_z) }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, ifeq(sum(multiply(X, y), multiply(X, z), add(multiply(X, y), multiply(X, z))), true, ifeq(true, true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, ifeq(sum(multiply(X, y), multiply(X, z), add(multiply(X, y), multiply(X, z))), true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 15 (closure_of_addition) }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, ifeq(true, true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by lemma 25 }
% 293.12/49.49 ifeq2(ifeq(product(X, z, multiply(X, z)), true, ifeq(product(X, y, multiply(X, y)), true, product(X, y_plus_z, add(multiply(X, z), multiply(X, y))), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 16 (closure_of_multiplication) }
% 293.12/49.49 ifeq2(ifeq(true, true, ifeq(product(X, y, multiply(X, y)), true, product(X, y_plus_z, add(multiply(X, z), multiply(X, y))), true), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.49 ifeq2(ifeq(product(X, y, multiply(X, y)), true, product(X, y_plus_z, add(multiply(X, z), multiply(X, y))), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by lemma 25 }
% 293.12/49.49 ifeq2(ifeq(product(X, y, multiply(X, y)), true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 16 (closure_of_multiplication) }
% 293.12/49.49 ifeq2(ifeq(true, true, product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by axiom 13 (ifeq_axiom) }
% 293.12/49.49 ifeq2(product(X, y_plus_z, add(multiply(X, y), multiply(X, z))), true, add(multiply(X, y), multiply(X, z)), multiply(X, y_plus_z))
% 293.12/49.49 = { by lemma 35 }
% 293.52/49.50 multiply(X, y_plus_z)
% 293.52/49.50
% 293.52/49.50 Goal 1 (prove_equality): x__plus_y_plus_z = x_plus_y__plus_z.
% 293.52/49.50 Proof:
% 293.52/49.50 x__plus_y_plus_z
% 293.52/49.50 = { by axiom 14 (ifeq_axiom) R->L }
% 293.52/49.50 ifeq2(true, true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 21 (distributivity7) R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(sum(additive_identity, x__plus_y_plus_z, x__plus_y_plus_z), true, ifeq(sum(inverse(x__plus_y_plus_z), x__plus_y_plus_z, multiplicative_identity), true, ifeq(sum(x_plus_y, x__plus_y_plus_z, add(x_plus_y, x__plus_y_plus_z)), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 3 (additive_identity1) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(true, true, ifeq(sum(inverse(x__plus_y_plus_z), x__plus_y_plus_z, multiplicative_identity), true, ifeq(sum(x_plus_y, x__plus_y_plus_z, add(x_plus_y, x__plus_y_plus_z)), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(sum(inverse(x__plus_y_plus_z), x__plus_y_plus_z, multiplicative_identity), true, ifeq(sum(x_plus_y, x__plus_y_plus_z, add(x_plus_y, x__plus_y_plus_z)), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 8 (additive_inverse1) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(true, true, ifeq(sum(x_plus_y, x__plus_y_plus_z, add(x_plus_y, x__plus_y_plus_z)), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(sum(x_plus_y, x__plus_y_plus_z, add(x_plus_y, x__plus_y_plus_z)), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 15 (closure_of_addition) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(true, true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, x__plus_y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 25 }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x__plus_y_plus_z, x_plus_y), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 34 R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x__plus_y_plus_z, inverse(inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 44 R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), multiply(x__plus_y_plus_z, inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 29 R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), multiply(inverse(x_plus_y), x__plus_y_plus_z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 35 R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x))), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by lemma 25 R->L }
% 293.52/49.50 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.50 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(true, true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 25 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x))), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(true, true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x))), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x))), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 25 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, ifeq(true, true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 15 (closure_of_addition) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, ifeq(sum(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z), add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, ifeq(sum(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z), add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true, ifeq(true, true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 5 (x_plus__y_plus_z) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(ifeq(product(inverse(x_plus_y), y_plus_z, multiply(inverse(x_plus_y), y_plus_z)), true, ifeq(product(inverse(x_plus_y), x, multiply(inverse(x_plus_y), x)), true, ifeq(sum(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z), add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true, ifeq(sum(x, y_plus_z, x__plus_y_plus_z), true, product(inverse(x_plus_y), x__plus_y_plus_z, add(multiply(inverse(x_plus_y), x), multiply(inverse(x_plus_y), y_plus_z))), true), true), true), true), true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 22 (distributivity2) }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), ifeq2(true, true, add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x)), multiply(inverse(x_plus_y), x__plus_y_plus_z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 29 }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(x, inverse(x_plus_y)))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 37 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x, x_plus_y)))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 25 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, x)))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 28 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(product(add(x, inverse(y)), x_plus_y, x), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by lemma 25 R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(product(add(inverse(y), x), x_plus_y, x), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(true, true, product(add(inverse(y), x), x_plus_y, x), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.51 = { by axiom 15 (closure_of_addition) R->L }
% 293.52/49.51 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(true, true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 12 (multiplicative_inverse1) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(true, true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 17 (commutativity_of_addition) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(ifeq(sum(x, y, x_plus_y), true, sum(y, x, x_plus_y), true), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 6 (x_plus_y) }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(ifeq(true, true, sum(y, x, x_plus_y), true), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(sum(y, x, x_plus_y), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(true, true, ifeq(sum(y, x, x_plus_y), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 3 (additive_identity1) R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(sum(additive_identity, x, x), true, ifeq(sum(y, x, x_plus_y), true, ifeq(sum(inverse(y), x, add(inverse(y), x)), true, product(add(inverse(y), x), x_plus_y, x), true), true), true), true), true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 21 (distributivity7) }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(true, true, multiply(add(x, inverse(y)), x_plus_y), x))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, multiply(add(x, inverse(y)), x_plus_y))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 29 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), add(x_plus_y, multiply(x_plus_y, add(x, inverse(y))))))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 38 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), multiply(inverse(x_plus_y), x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 39 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), add(multiply(inverse(x_plus_y), y_plus_z), additive_identity)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 27 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), multiply(inverse(x_plus_y), y_plus_z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 29 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(inverse(inverse(x_plus_y)), multiply(y_plus_z, inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 44 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(y_plus_z, inverse(inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 34 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(y_plus_z, x_plus_y), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 25 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, y_plus_z), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 43 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, multiply(y_plus_z, inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 29 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, multiply(inverse(x_plus_y), y_plus_z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 45 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), y), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 29 }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(y, inverse(x_plus_y)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 37 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(y, x_plus_y)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 25 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, y)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 28 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(product(add(y, inverse(x)), x_plus_y, y), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.52 = { by lemma 25 R->L }
% 293.52/49.52 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(product(add(inverse(x), y), x_plus_y, y), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(true, true, product(add(inverse(x), y), x_plus_y, y), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 15 (closure_of_addition) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(true, true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 12 (multiplicative_inverse1) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(true, true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 6 (x_plus_y) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(x, y, x_plus_y), true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(true, true, ifeq(sum(x, y, x_plus_y), true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 3 (additive_identity1) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(ifeq(product(inverse(x), x, additive_identity), true, ifeq(sum(additive_identity, y, y), true, ifeq(sum(x, y, x_plus_y), true, ifeq(sum(inverse(x), y, add(inverse(x), y)), true, product(add(inverse(x), y), x_plus_y, y), true), true), true), true), true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 21 (distributivity7) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, ifeq2(true, true, multiply(add(y, inverse(x)), x_plus_y), y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, multiply(add(y, inverse(x)), x_plus_y))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 29 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), add(x_plus_y, multiply(x_plus_y, add(y, inverse(x))))), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 38 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(multiply(inverse(x_plus_y), x_plus_y), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 39 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, add(additive_identity, multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 14 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, ifeq2(true, true, add(additive_identity, multiply(inverse(x_plus_y), z)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 17 (commutativity_of_addition) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, ifeq2(ifeq(sum(additive_identity, multiply(inverse(x_plus_y), z), add(additive_identity, multiply(inverse(x_plus_y), z))), true, sum(multiply(inverse(x_plus_y), z), additive_identity, add(additive_identity, multiply(inverse(x_plus_y), z))), true), true, add(additive_identity, multiply(inverse(x_plus_y), z)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 15 (closure_of_addition) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, ifeq2(ifeq(true, true, sum(multiply(inverse(x_plus_y), z), additive_identity, add(additive_identity, multiply(inverse(x_plus_y), z))), true), true, add(additive_identity, multiply(inverse(x_plus_y), z)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, ifeq2(sum(multiply(inverse(x_plus_y), z), additive_identity, add(additive_identity, multiply(inverse(x_plus_y), z))), true, add(additive_identity, multiply(inverse(x_plus_y), z)), multiply(inverse(x_plus_y), z))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 26 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, multiply(inverse(x_plus_y), z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 29 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, multiply(z, inverse(x_plus_y))), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 43 }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(add(x_plus_y, z), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 19 (addition_is_well_defined) R->L }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(ifeq2(sum(x_plus_y, z, add(x_plus_y, z)), true, ifeq2(sum(x_plus_y, z, x_plus_y__plus_z), true, x_plus_y__plus_z, add(x_plus_y, z)), add(x_plus_y, z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 7 (x_plus_y__plus_z) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(ifeq2(sum(x_plus_y, z, add(x_plus_y, z)), true, ifeq2(true, true, x_plus_y__plus_z, add(x_plus_y, z)), add(x_plus_y, z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(ifeq2(sum(x_plus_y, z, add(x_plus_y, z)), true, x_plus_y__plus_z, add(x_plus_y, z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 15 (closure_of_addition) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(ifeq2(true, true, x_plus_y__plus_z, add(x_plus_y, z)), multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.53 ifeq2(ifeq(product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.53 ifeq2(ifeq(ifeq(true, true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.52/49.53 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(x, inverse(x__plus_y_plus_z))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 37 R->L }
% 293.52/49.53 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x, x__plus_y_plus_z))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.53 = { by lemma 25 R->L }
% 293.52/49.53 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, x))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 28 R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(product(add(x, inverse(y_plus_z)), x__plus_y_plus_z, x), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 25 R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(true, true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 15 (closure_of_addition) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(true, true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 12 (multiplicative_inverse1) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(true, true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 17 (commutativity_of_addition) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(ifeq(sum(x, y_plus_z, x__plus_y_plus_z), true, sum(y_plus_z, x, x__plus_y_plus_z), true), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 5 (x_plus__y_plus_z) }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(ifeq(true, true, sum(y_plus_z, x, x__plus_y_plus_z), true), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(sum(y_plus_z, x, x__plus_y_plus_z), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(true, true, ifeq(sum(y_plus_z, x, x__plus_y_plus_z), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 3 (additive_identity1) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(ifeq(product(inverse(y_plus_z), y_plus_z, additive_identity), true, ifeq(sum(additive_identity, x, x), true, ifeq(sum(y_plus_z, x, x__plus_y_plus_z), true, ifeq(sum(inverse(y_plus_z), x, add(inverse(y_plus_z), x)), true, product(add(inverse(y_plus_z), x), x__plus_y_plus_z, x), true), true), true), true), true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 21 (distributivity7) }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, ifeq2(true, true, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z), x)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 14 (ifeq_axiom) }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, multiply(add(x, inverse(y_plus_z)), x__plus_y_plus_z)))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 29 }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), add(x__plus_y_plus_z, multiply(x__plus_y_plus_z, add(x, inverse(y_plus_z)))))), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 38 }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), multiply(inverse(x__plus_y_plus_z), x__plus_y_plus_z)), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 39 }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), additive_identity), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 14 (ifeq_axiom) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(true, true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by axiom 15 (closure_of_addition) R->L }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z), add(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z))), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.54 = { by lemma 45 }
% 293.52/49.54 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z), multiply(inverse(x__plus_y_plus_z), y_plus_z)), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 29 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z), multiply(y_plus_z, inverse(x__plus_y_plus_z))), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 40 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 29 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(inverse(x__plus_y_plus_z), z), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 41 R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(multiply(inverse(x__plus_y_plus_z), z), add(multiply(inverse(x__plus_y_plus_z), y), multiply(inverse(x__plus_y_plus_z), z))), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 45 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(multiply(inverse(x__plus_y_plus_z), z), multiply(inverse(x__plus_y_plus_z), y_plus_z)), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 29 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(multiply(inverse(x__plus_y_plus_z), y_plus_z), multiply(inverse(x__plus_y_plus_z), z)), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 29 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(multiply(y_plus_z, inverse(x__plus_y_plus_z)), multiply(inverse(x__plus_y_plus_z), z)), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 40 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(additive_identity, multiply(inverse(x__plus_y_plus_z), z)), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 29 R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), multiply(multiply(inverse(x__plus_y_plus_z), z), additive_identity), additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 31 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), ifeq2(sum(multiply(y, inverse(x__plus_y_plus_z)), additive_identity, additive_identity), true, additive_identity, multiply(y, inverse(x__plus_y_plus_z)))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 26 }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(true, true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 16 (closure_of_multiplication) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(true, true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 17 (commutativity_of_addition) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(ifeq(sum(x, y, x_plus_y), true, sum(y, x, x_plus_y), true), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 6 (x_plus_y) }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(ifeq(true, true, sum(y, x, x_plus_y), true), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 13 (ifeq_axiom) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(true, true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 1 (additive_identity2) R->L }
% 293.52/49.55 ifeq2(ifeq(ifeq(product(x, inverse(x__plus_y_plus_z), additive_identity), true, ifeq(product(y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(sum(multiply(y, inverse(x__plus_y_plus_z)), additive_identity, multiply(y, inverse(x__plus_y_plus_z))), true, ifeq(sum(y, x, x_plus_y), true, product(x_plus_y, inverse(x__plus_y_plus_z), multiply(y, inverse(x__plus_y_plus_z))), true), true), true), true), true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 24 (distributivity4) }
% 293.52/49.55 ifeq2(ifeq(true, true, product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by axiom 13 (ifeq_axiom) }
% 293.52/49.55 ifeq2(product(x_plus_y__plus_z, multiplicative_identity, x__plus_y_plus_z), true, x__plus_y_plus_z, x_plus_y__plus_z)
% 293.52/49.55 = { by lemma 33 }
% 293.52/49.55 x_plus_y__plus_z
% 293.52/49.55 % SZS output end Proof
% 293.52/49.55
% 293.52/49.55 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------