TSTP Solution File: BOO015-10 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO015-10 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n027.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 18:11:25 EDT 2023
% Result : Unsatisfiable 132.67s 17.70s
% Output : Proof 138.76s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : BOO015-10 : TPTP v8.1.2. Released v7.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.32 % Computer : n027.cluster.edu
% 0.14/0.32 % Model : x86_64 x86_64
% 0.14/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.32 % Memory : 8042.1875MB
% 0.14/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.32 % CPULimit : 300
% 0.14/0.32 % WCLimit : 300
% 0.14/0.32 % DateTime : Sun Aug 27 08:13:35 EDT 2023
% 0.14/0.33 % CPUTime :
% 132.67/17.70 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 132.67/17.70
% 132.67/17.70 % SZS status Unsatisfiable
% 132.67/17.70
% 137.44/18.04 % SZS output start Proof
% 137.44/18.04 Axiom 1 (additive_identity2): sum(X, additive_identity, X) = true.
% 137.44/18.04 Axiom 2 (additive_identity1): sum(additive_identity, X, X) = true.
% 137.44/18.04 Axiom 3 (multiplicative_identity2): product(X, multiplicative_identity, X) = true.
% 137.44/18.04 Axiom 4 (x_times_y): product(x, y, x_times_y) = true.
% 137.44/18.04 Axiom 5 (multiplicative_identity1): product(multiplicative_identity, X, X) = true.
% 137.44/18.04 Axiom 6 (ifeq_axiom): ifeq2(X, X, Y, Z) = Y.
% 137.44/18.04 Axiom 7 (additive_inverse2): sum(X, inverse(X), multiplicative_identity) = true.
% 137.96/18.04 Axiom 8 (additive_inverse1): sum(inverse(X), X, multiplicative_identity) = true.
% 137.96/18.04 Axiom 9 (multiplicative_inverse2): product(X, inverse(X), additive_identity) = true.
% 137.96/18.04 Axiom 10 (multiplicative_inverse1): product(inverse(X), X, additive_identity) = true.
% 137.96/18.04 Axiom 11 (ifeq_axiom_001): ifeq(X, X, Y, Z) = Y.
% 137.96/18.04 Axiom 12 (closure_of_addition): sum(X, Y, add(X, Y)) = true.
% 137.96/18.04 Axiom 13 (x_inverse_plus_y_inverse): sum(inverse(x), inverse(y), x_inverse_plus_y_inverse) = true.
% 137.96/18.04 Axiom 14 (closure_of_multiplication): product(X, Y, multiply(X, Y)) = true.
% 137.96/18.05 Axiom 15 (commutativity_of_addition): ifeq(sum(X, Y, Z), true, sum(Y, X, Z), true) = true.
% 137.96/18.05 Axiom 16 (commutativity_of_multiplication): ifeq(product(X, Y, Z), true, product(Y, X, Z), true) = true.
% 137.96/18.05 Axiom 17 (addition_is_well_defined): ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, W), true, W, Z), Z) = Z.
% 137.96/18.05 Axiom 18 (multiplication_is_well_defined): ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, W), true, W, Z), Z) = Z.
% 137.96/18.05 Axiom 19 (distributivity2): ifeq(product(X, Y, Z), true, ifeq(product(X, W, V), true, ifeq(sum(V, Z, U), true, ifeq(sum(W, Y, T), true, product(X, T, U), true), true), true), true) = true.
% 137.96/18.05 Axiom 20 (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.
% 137.96/18.05 Axiom 21 (distributivity8): ifeq(product(X, Y, Z), true, ifeq(product(W, V, U), true, ifeq(sum(V, T, Y), true, ifeq(sum(W, T, X), true, sum(U, T, Z), true), true), true), true) = true.
% 137.96/18.05
% 137.96/18.05 Lemma 22: ifeq(product(X, Y, Z), true, ifeq(sum(Y, W, multiplicative_identity), true, ifeq(sum(X, W, V), true, sum(Z, W, V), true), true), true) = true.
% 137.96/18.05 Proof:
% 137.96/18.05 ifeq(product(X, Y, Z), true, ifeq(sum(Y, W, multiplicative_identity), true, ifeq(sum(X, W, V), true, sum(Z, W, V), true), true), true)
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq(true, true, ifeq(product(X, Y, Z), true, ifeq(sum(Y, W, multiplicative_identity), true, ifeq(sum(X, W, V), true, sum(Z, W, V), true), true), true), true)
% 137.96/18.05 = { by axiom 3 (multiplicative_identity2) R->L }
% 137.96/18.05 ifeq(product(V, multiplicative_identity, V), true, ifeq(product(X, Y, Z), true, ifeq(sum(Y, W, multiplicative_identity), true, ifeq(sum(X, W, V), true, sum(Z, W, V), true), true), true), true)
% 137.96/18.05 = { by axiom 21 (distributivity8) }
% 137.96/18.05 true
% 137.96/18.05
% 137.96/18.05 Lemma 23: ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(sum(X, Y, Z), true, sum(additive_identity, Y, Z), true), true) = true.
% 137.96/18.05 Proof:
% 137.96/18.05 ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(sum(X, Y, Z), true, sum(additive_identity, Y, Z), true), true)
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq(true, true, ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(sum(X, Y, Z), true, sum(additive_identity, Y, Z), true), true), true)
% 137.96/18.05 = { by axiom 9 (multiplicative_inverse2) R->L }
% 137.96/18.05 ifeq(product(X, inverse(X), additive_identity), true, ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(sum(X, Y, Z), true, sum(additive_identity, Y, Z), true), true), true)
% 137.96/18.05 = { by lemma 22 }
% 137.96/18.05 true
% 137.96/18.05
% 137.96/18.05 Lemma 24: ifeq2(product(X, multiplicative_identity, Y), true, Y, X) = X.
% 137.96/18.05 Proof:
% 137.96/18.05 ifeq2(product(X, multiplicative_identity, Y), true, Y, X)
% 137.96/18.05 = { by axiom 6 (ifeq_axiom) R->L }
% 137.96/18.05 ifeq2(true, true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 137.96/18.05 = { by axiom 3 (multiplicative_identity2) R->L }
% 137.96/18.05 ifeq2(product(X, multiplicative_identity, X), true, ifeq2(product(X, multiplicative_identity, Y), true, Y, X), X)
% 137.96/18.05 = { by axiom 18 (multiplication_is_well_defined) }
% 137.96/18.05 X
% 137.96/18.05
% 137.96/18.05 Lemma 25: ifeq(product(X, Y, additive_identity), true, ifeq(product(X, Z, W), true, ifeq(sum(Z, Y, V), true, product(X, V, W), true), true), true) = true.
% 137.96/18.05 Proof:
% 137.96/18.05 ifeq(product(X, Y, additive_identity), true, ifeq(product(X, Z, W), true, ifeq(sum(Z, Y, V), true, product(X, V, W), true), true), true)
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq(product(X, Y, additive_identity), true, ifeq(product(X, Z, W), true, ifeq(true, true, ifeq(sum(Z, Y, V), true, product(X, V, W), true), true), true), true)
% 137.96/18.05 = { by axiom 1 (additive_identity2) R->L }
% 137.96/18.05 ifeq(product(X, Y, additive_identity), true, ifeq(product(X, Z, W), true, ifeq(sum(W, additive_identity, W), true, ifeq(sum(Z, Y, V), true, product(X, V, W), true), true), true), true)
% 137.96/18.05 = { by axiom 19 (distributivity2) }
% 137.96/18.05 true
% 137.96/18.05
% 137.96/18.05 Lemma 26: ifeq(product(X, inverse(Y), additive_identity), true, ifeq(product(X, Y, Z), true, product(X, multiplicative_identity, Z), true), true) = true.
% 137.96/18.05 Proof:
% 137.96/18.05 ifeq(product(X, inverse(Y), additive_identity), true, ifeq(product(X, Y, Z), true, product(X, multiplicative_identity, Z), true), true)
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq(product(X, inverse(Y), additive_identity), true, ifeq(product(X, Y, Z), true, ifeq(true, true, product(X, multiplicative_identity, Z), true), true), true)
% 137.96/18.05 = { by axiom 7 (additive_inverse2) R->L }
% 137.96/18.05 ifeq(product(X, inverse(Y), additive_identity), true, ifeq(product(X, Y, Z), true, ifeq(sum(Y, inverse(Y), multiplicative_identity), true, product(X, multiplicative_identity, Z), true), true), true)
% 137.96/18.05 = { by lemma 25 }
% 137.96/18.05 true
% 137.96/18.05
% 137.96/18.05 Lemma 27: inverse(inverse(X)) = X.
% 137.96/18.05 Proof:
% 137.96/18.05 inverse(inverse(X))
% 137.96/18.05 = { by lemma 24 R->L }
% 137.96/18.05 ifeq2(product(inverse(inverse(X)), multiplicative_identity, X), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq2(ifeq(true, true, product(inverse(inverse(X)), multiplicative_identity, X), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq2(ifeq(true, true, ifeq(true, true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 10 (multiplicative_inverse1) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(true, true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 16 (commutativity_of_multiplication) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), X), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by lemma 24 R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(true, true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 14 (closure_of_multiplication) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(true, true, ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 9 (multiplicative_inverse2) R->L }
% 137.96/18.05 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.05 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, ifeq(true, true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by axiom 8 (additive_inverse1) R->L }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, ifeq(sum(inverse(inverse(X)), inverse(X), multiplicative_identity), true, product(X, multiplicative_identity, multiply(X, inverse(inverse(X)))), true), true), true), true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by lemma 25 }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), ifeq2(true, true, multiply(X, inverse(inverse(X))), X)), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(product(X, inverse(inverse(X)), multiply(X, inverse(inverse(X)))), true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by axiom 14 (closure_of_multiplication) }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(ifeq(true, true, product(inverse(inverse(X)), X, X), true), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by axiom 11 (ifeq_axiom_001) }
% 137.96/18.06 ifeq2(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(product(inverse(inverse(X)), X, X), true, product(inverse(inverse(X)), multiplicative_identity, X), true), true), true, X, inverse(inverse(X)))
% 137.96/18.06 = { by lemma 26 }
% 137.96/18.06 ifeq2(true, true, X, inverse(inverse(X)))
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) }
% 137.96/18.06 X
% 137.96/18.06
% 137.96/18.06 Lemma 28: ifeq2(sum(additive_identity, X, Y), true, Y, X) = X.
% 137.96/18.06 Proof:
% 137.96/18.06 ifeq2(sum(additive_identity, X, Y), true, Y, X)
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 137.96/18.06 ifeq2(true, true, ifeq2(sum(additive_identity, X, Y), true, Y, X), X)
% 137.96/18.06 = { by axiom 2 (additive_identity1) R->L }
% 137.96/18.06 ifeq2(sum(additive_identity, X, X), true, ifeq2(sum(additive_identity, X, Y), true, Y, X), X)
% 137.96/18.06 = { by axiom 17 (addition_is_well_defined) }
% 137.96/18.06 X
% 137.96/18.06
% 137.96/18.06 Lemma 29: add(X, X) = X.
% 137.96/18.06 Proof:
% 137.96/18.06 add(X, X)
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 137.96/18.06 ifeq2(true, true, add(X, X), X)
% 137.96/18.06 = { by lemma 23 R->L }
% 137.96/18.06 ifeq2(ifeq(sum(inverse(X), inverse(inverse(X)), multiplicative_identity), true, ifeq(sum(X, inverse(inverse(X)), add(X, X)), true, sum(additive_identity, inverse(inverse(X)), add(X, X)), true), true), true, add(X, X), X)
% 137.96/18.06 = { by axiom 7 (additive_inverse2) }
% 137.96/18.06 ifeq2(ifeq(true, true, ifeq(sum(X, inverse(inverse(X)), add(X, X)), true, sum(additive_identity, inverse(inverse(X)), add(X, X)), true), true), true, add(X, X), X)
% 137.96/18.06 = { by axiom 11 (ifeq_axiom_001) }
% 137.96/18.06 ifeq2(ifeq(sum(X, inverse(inverse(X)), add(X, X)), true, sum(additive_identity, inverse(inverse(X)), add(X, X)), true), true, add(X, X), X)
% 137.96/18.06 = { by lemma 27 }
% 137.96/18.06 ifeq2(ifeq(sum(X, inverse(inverse(X)), add(X, X)), true, sum(additive_identity, X, add(X, X)), true), true, add(X, X), X)
% 137.96/18.06 = { by lemma 27 }
% 137.96/18.06 ifeq2(ifeq(sum(X, X, add(X, X)), true, sum(additive_identity, X, add(X, X)), true), true, add(X, X), X)
% 137.96/18.06 = { by axiom 12 (closure_of_addition) }
% 137.96/18.06 ifeq2(ifeq(true, true, sum(additive_identity, X, add(X, X)), true), true, add(X, X), X)
% 137.96/18.06 = { by axiom 11 (ifeq_axiom_001) }
% 137.96/18.06 ifeq2(sum(additive_identity, X, add(X, X)), true, add(X, X), X)
% 137.96/18.06 = { by lemma 28 }
% 137.96/18.06 X
% 137.96/18.06
% 137.96/18.06 Lemma 30: ifeq2(sum(X, Y, Z), true, add(X, Y), Z) = Z.
% 137.96/18.06 Proof:
% 137.96/18.06 ifeq2(sum(X, Y, Z), true, add(X, Y), Z)
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 137.96/18.06 ifeq2(sum(X, Y, Z), true, ifeq2(true, true, add(X, Y), Z), Z)
% 137.96/18.06 = { by axiom 12 (closure_of_addition) R->L }
% 137.96/18.06 ifeq2(sum(X, Y, Z), true, ifeq2(sum(X, Y, add(X, Y)), true, add(X, Y), Z), Z)
% 137.96/18.06 = { by axiom 17 (addition_is_well_defined) }
% 137.96/18.06 Z
% 137.96/18.06
% 137.96/18.06 Lemma 31: add(X, Y) = add(Y, X).
% 137.96/18.06 Proof:
% 137.96/18.06 add(X, Y)
% 137.96/18.06 = { by lemma 30 R->L }
% 137.96/18.06 ifeq2(sum(Y, X, add(X, Y)), true, add(Y, X), add(X, Y))
% 137.96/18.06 = { by axiom 11 (ifeq_axiom_001) R->L }
% 137.96/18.06 ifeq2(ifeq(true, true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 137.96/18.06 = { by axiom 12 (closure_of_addition) R->L }
% 137.96/18.06 ifeq2(ifeq(sum(X, Y, add(X, Y)), true, sum(Y, X, add(X, Y)), true), true, add(Y, X), add(X, Y))
% 137.96/18.06 = { by axiom 15 (commutativity_of_addition) }
% 137.96/18.06 ifeq2(true, true, add(Y, X), add(X, Y))
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) }
% 137.96/18.06 add(Y, X)
% 137.96/18.06
% 137.96/18.06 Lemma 32: ifeq2(product(X, Y, Z), true, multiply(X, Y), Z) = Z.
% 137.96/18.06 Proof:
% 137.96/18.06 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 137.96/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 137.96/18.06 ifeq2(product(X, Y, Z), true, ifeq2(true, true, multiply(X, Y), Z), Z)
% 138.07/18.06 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.06 ifeq2(product(X, Y, Z), true, ifeq2(product(X, Y, multiply(X, Y)), true, multiply(X, Y), Z), Z)
% 138.07/18.06 = { by axiom 18 (multiplication_is_well_defined) }
% 138.07/18.06 Z
% 138.07/18.06
% 138.07/18.06 Lemma 33: multiply(X, Y) = multiply(Y, X).
% 138.07/18.06 Proof:
% 138.07/18.06 multiply(X, Y)
% 138.07/18.06 = { by lemma 32 R->L }
% 138.07/18.06 ifeq2(product(Y, X, multiply(X, Y)), true, multiply(Y, X), multiply(X, Y))
% 138.07/18.06 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.06 ifeq2(ifeq(true, true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 138.07/18.06 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.06 ifeq2(ifeq(product(X, Y, multiply(X, Y)), true, product(Y, X, multiply(X, Y)), true), true, multiply(Y, X), multiply(X, Y))
% 138.07/18.06 = { by axiom 16 (commutativity_of_multiplication) }
% 138.07/18.06 ifeq2(true, true, multiply(Y, X), multiply(X, Y))
% 138.07/18.06 = { by axiom 6 (ifeq_axiom) }
% 138.07/18.06 multiply(Y, X)
% 138.07/18.06
% 138.07/18.06 Lemma 34: ifeq2(sum(X, inverse(X), Y), true, multiplicative_identity, Y) = Y.
% 138.07/18.06 Proof:
% 138.07/18.06 ifeq2(sum(X, inverse(X), Y), true, multiplicative_identity, Y)
% 138.07/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.06 ifeq2(sum(X, inverse(X), Y), true, ifeq2(true, true, multiplicative_identity, Y), Y)
% 138.07/18.06 = { by axiom 7 (additive_inverse2) R->L }
% 138.07/18.06 ifeq2(sum(X, inverse(X), Y), true, ifeq2(sum(X, inverse(X), multiplicative_identity), true, multiplicative_identity, Y), Y)
% 138.07/18.06 = { by axiom 17 (addition_is_well_defined) }
% 138.07/18.06 Y
% 138.07/18.06
% 138.07/18.06 Lemma 35: add(X, inverse(X)) = multiplicative_identity.
% 138.07/18.06 Proof:
% 138.07/18.06 add(X, inverse(X))
% 138.07/18.06 = { by lemma 34 R->L }
% 138.07/18.06 ifeq2(sum(X, inverse(X), add(X, inverse(X))), true, multiplicative_identity, add(X, inverse(X)))
% 138.07/18.06 = { by axiom 12 (closure_of_addition) }
% 138.07/18.06 ifeq2(true, true, multiplicative_identity, add(X, inverse(X)))
% 138.07/18.06 = { by axiom 6 (ifeq_axiom) }
% 138.07/18.06 multiplicative_identity
% 138.07/18.06
% 138.07/18.06 Lemma 36: ifeq2(product(X, inverse(X), Y), true, additive_identity, Y) = Y.
% 138.07/18.06 Proof:
% 138.07/18.06 ifeq2(product(X, inverse(X), Y), true, additive_identity, Y)
% 138.07/18.06 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.06 ifeq2(product(X, inverse(X), Y), true, ifeq2(true, true, additive_identity, Y), Y)
% 138.07/18.06 = { by axiom 9 (multiplicative_inverse2) R->L }
% 138.07/18.06 ifeq2(product(X, inverse(X), Y), true, ifeq2(product(X, inverse(X), additive_identity), true, additive_identity, Y), Y)
% 138.07/18.06 = { by axiom 18 (multiplication_is_well_defined) }
% 138.07/18.06 Y
% 138.07/18.06
% 138.07/18.06 Lemma 37: multiply(X, inverse(X)) = additive_identity.
% 138.07/18.06 Proof:
% 138.07/18.06 multiply(X, inverse(X))
% 138.07/18.06 = { by lemma 36 R->L }
% 138.07/18.06 ifeq2(product(X, inverse(X), multiply(X, inverse(X))), true, additive_identity, multiply(X, inverse(X)))
% 138.07/18.06 = { by axiom 14 (closure_of_multiplication) }
% 138.07/18.06 ifeq2(true, true, additive_identity, multiply(X, inverse(X)))
% 138.07/18.06 = { by axiom 6 (ifeq_axiom) }
% 138.07/18.06 additive_identity
% 138.07/18.06
% 138.07/18.06 Lemma 38: ifeq(product(X, Y, Z), true, ifeq(sum(Y, inverse(X), W), true, product(X, W, Z), true), true) = true.
% 138.07/18.06 Proof:
% 138.07/18.06 ifeq(product(X, Y, Z), true, ifeq(sum(Y, inverse(X), W), true, product(X, W, Z), true), true)
% 138.07/18.06 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.06 ifeq(true, true, ifeq(product(X, Y, Z), true, ifeq(sum(Y, inverse(X), W), true, product(X, W, Z), true), true), true)
% 138.07/18.06 = { by axiom 9 (multiplicative_inverse2) R->L }
% 138.07/18.06 ifeq(product(X, inverse(X), additive_identity), true, ifeq(product(X, Y, Z), true, ifeq(sum(Y, inverse(X), W), true, product(X, W, Z), true), true), true)
% 138.07/18.06 = { by lemma 25 }
% 138.07/18.06 true
% 138.07/18.06
% 138.07/18.06 Lemma 39: product(X, add(Y, inverse(X)), multiply(X, Y)) = true.
% 138.07/18.06 Proof:
% 138.07/18.06 product(X, add(Y, inverse(X)), multiply(X, Y))
% 138.07/18.06 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.06 ifeq(true, true, product(X, add(Y, inverse(X)), multiply(X, Y)), true)
% 138.07/18.06 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.06 ifeq(product(X, Y, multiply(X, Y)), true, product(X, add(Y, inverse(X)), multiply(X, Y)), true)
% 138.07/18.06 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.06 ifeq(product(X, Y, multiply(X, Y)), true, ifeq(true, true, product(X, add(Y, inverse(X)), multiply(X, Y)), true), true)
% 138.07/18.06 = { by axiom 12 (closure_of_addition) R->L }
% 138.07/18.06 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)
% 138.07/18.06 = { by lemma 38 }
% 138.07/18.06 true
% 138.07/18.06
% 138.07/18.06 Lemma 40: ifeq2(product(X, Y, Z), true, multiply(Y, X), Z) = Z.
% 138.07/18.06 Proof:
% 138.07/18.07 ifeq2(product(X, Y, Z), true, multiply(Y, X), Z)
% 138.07/18.07 = { by lemma 33 R->L }
% 138.07/18.07 ifeq2(product(X, Y, Z), true, multiply(X, Y), Z)
% 138.07/18.07 = { by lemma 32 }
% 138.07/18.07 Z
% 138.07/18.07
% 138.07/18.07 Lemma 41: multiply(X, add(Y, inverse(X))) = multiply(X, Y).
% 138.07/18.07 Proof:
% 138.07/18.07 multiply(X, add(Y, inverse(X)))
% 138.07/18.07 = { by lemma 33 R->L }
% 138.07/18.07 multiply(add(Y, inverse(X)), X)
% 138.07/18.07 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.07 ifeq2(true, true, multiply(add(Y, inverse(X)), X), multiply(X, Y))
% 138.07/18.07 = { by lemma 39 R->L }
% 138.07/18.07 ifeq2(product(X, add(Y, inverse(X)), multiply(X, Y)), true, multiply(add(Y, inverse(X)), X), multiply(X, Y))
% 138.07/18.07 = { by lemma 40 }
% 138.07/18.07 multiply(X, Y)
% 138.07/18.07
% 138.07/18.07 Lemma 42: ifeq(sum(y, X, multiplicative_identity), true, ifeq(sum(x, X, Y), true, sum(x_times_y, X, Y), true), true) = true.
% 138.07/18.07 Proof:
% 138.07/18.07 ifeq(sum(y, X, multiplicative_identity), true, ifeq(sum(x, X, Y), true, sum(x_times_y, X, Y), true), true)
% 138.07/18.07 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.07 ifeq(true, true, ifeq(sum(y, X, multiplicative_identity), true, ifeq(sum(x, X, Y), true, sum(x_times_y, X, Y), true), true), true)
% 138.07/18.07 = { by axiom 4 (x_times_y) R->L }
% 138.07/18.07 ifeq(product(x, y, x_times_y), true, ifeq(sum(y, X, multiplicative_identity), true, ifeq(sum(x, X, Y), true, sum(x_times_y, X, Y), true), true), true)
% 138.07/18.07 = { by lemma 22 }
% 138.07/18.07 true
% 138.07/18.07
% 138.07/18.07 Lemma 43: ifeq2(sum(X, Y, Z), true, add(Y, X), Z) = Z.
% 138.07/18.07 Proof:
% 138.07/18.07 ifeq2(sum(X, Y, Z), true, add(Y, X), Z)
% 138.07/18.07 = { by lemma 31 R->L }
% 138.07/18.07 ifeq2(sum(X, Y, Z), true, add(X, Y), Z)
% 138.07/18.07 = { by lemma 30 }
% 138.07/18.07 Z
% 138.07/18.07
% 138.07/18.07 Lemma 44: sum(y, inverse(x_times_y), multiplicative_identity) = true.
% 138.07/18.07 Proof:
% 138.07/18.07 sum(y, inverse(x_times_y), multiplicative_identity)
% 138.07/18.07 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(true, true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by lemma 22 R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(product(y, x_times_y, x_times_y), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(true, true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(x_times_y, y)), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by lemma 33 R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, x_times_y)), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by lemma 41 R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, add(x_times_y, inverse(y)))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by lemma 31 R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, add(inverse(y), x_times_y))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(true, true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by lemma 42 R->L }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(ifeq(sum(y, inverse(y), multiplicative_identity), true, ifeq(sum(x, inverse(y), add(x, inverse(y))), true, sum(x_times_y, inverse(y), add(x, inverse(y))), true), true), true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 7 (additive_inverse2) }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(ifeq(true, true, ifeq(sum(x, inverse(y), add(x, inverse(y))), true, sum(x_times_y, inverse(y), add(x, inverse(y))), true), true), true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(ifeq(sum(x, inverse(y), add(x, inverse(y))), true, sum(x_times_y, inverse(y), add(x, inverse(y))), true), true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 12 (closure_of_addition) }
% 138.07/18.07 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(ifeq(true, true, sum(x_times_y, inverse(y), add(x, inverse(y))), true), true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.07 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, ifeq2(sum(x_times_y, inverse(y), add(x, inverse(y))), true, add(inverse(y), x_times_y), add(x, inverse(y))))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by lemma 43 }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, add(x, inverse(y)))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by lemma 41 }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(y, x)), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by lemma 33 }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, multiply(x, y)), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 18 (multiplication_is_well_defined) R->L }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, ifeq2(product(x, y, multiply(x, y)), true, ifeq2(product(x, y, x_times_y), true, x_times_y, multiply(x, y)), multiply(x, y))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 4 (x_times_y) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, ifeq2(product(x, y, multiply(x, y)), true, ifeq2(true, true, x_times_y, multiply(x, y)), multiply(x, y))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 6 (ifeq_axiom) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, ifeq2(product(x, y, multiply(x, y)), true, x_times_y, multiply(x, y))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 14 (closure_of_multiplication) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, ifeq2(true, true, x_times_y, multiply(x, y))), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 6 (ifeq_axiom) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(ifeq(product(x_times_y, y, x_times_y), true, product(y, x_times_y, x_times_y), true), true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 16 (commutativity_of_multiplication) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(true, true, ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(sum(x_times_y, inverse(x_times_y), multiplicative_identity), true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 7 (additive_inverse2) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(true, true, ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(sum(y, inverse(x_times_y), add(y, inverse(x_times_y))), true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 12 (closure_of_addition) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(ifeq(true, true, sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.08 sum(y, inverse(x_times_y), ifeq2(sum(x_times_y, inverse(x_times_y), add(y, inverse(x_times_y))), true, multiplicative_identity, add(y, inverse(x_times_y))))
% 138.07/18.08 = { by lemma 34 }
% 138.07/18.08 sum(y, inverse(x_times_y), add(y, inverse(x_times_y)))
% 138.07/18.08 = { by axiom 12 (closure_of_addition) }
% 138.07/18.08 true
% 138.07/18.08
% 138.07/18.08 Lemma 45: ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true) = true.
% 138.07/18.08 Proof:
% 138.07/18.08 ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true)
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.08 ifeq(true, true, ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true), true)
% 138.07/18.08 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.08 ifeq(product(X, inverse(Y), multiply(X, inverse(Y))), true, ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true), true)
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.08 ifeq(product(X, inverse(Y), multiply(X, inverse(Y))), true, ifeq(true, true, ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true), true), true)
% 138.07/18.08 = { by axiom 8 (additive_inverse1) R->L }
% 138.07/18.08 ifeq(product(X, inverse(Y), multiply(X, inverse(Y))), true, ifeq(sum(inverse(Y), Y, multiplicative_identity), true, ifeq(sum(X, Y, Z), true, sum(multiply(X, inverse(Y)), Y, Z), true), true), true)
% 138.07/18.08 = { by lemma 22 }
% 138.07/18.08 true
% 138.07/18.08
% 138.07/18.08 Lemma 46: add(X, multiply(Y, inverse(X))) = add(X, Y).
% 138.07/18.08 Proof:
% 138.07/18.08 add(X, multiply(Y, inverse(X)))
% 138.07/18.08 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.08 ifeq2(true, true, add(X, multiply(Y, inverse(X))), add(Y, X))
% 138.07/18.08 = { by lemma 45 R->L }
% 138.07/18.08 ifeq2(ifeq(sum(Y, X, add(Y, X)), true, sum(multiply(Y, inverse(X)), X, add(Y, X)), true), true, add(X, multiply(Y, inverse(X))), add(Y, X))
% 138.07/18.08 = { by axiom 12 (closure_of_addition) }
% 138.07/18.08 ifeq2(ifeq(true, true, sum(multiply(Y, inverse(X)), X, add(Y, X)), true), true, add(X, multiply(Y, inverse(X))), add(Y, X))
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.08 ifeq2(sum(multiply(Y, inverse(X)), X, add(Y, X)), true, add(X, multiply(Y, inverse(X))), add(Y, X))
% 138.07/18.08 = { by lemma 43 }
% 138.07/18.08 add(Y, X)
% 138.07/18.08 = { by lemma 31 }
% 138.07/18.08 add(X, Y)
% 138.07/18.08
% 138.07/18.08 Lemma 47: add(inverse(X), multiply(X, Y)) = add(Y, inverse(X)).
% 138.07/18.08 Proof:
% 138.07/18.08 add(inverse(X), multiply(X, Y))
% 138.07/18.08 = { by lemma 33 R->L }
% 138.07/18.08 add(inverse(X), multiply(Y, X))
% 138.07/18.08 = { by lemma 27 R->L }
% 138.07/18.08 add(inverse(X), multiply(Y, inverse(inverse(X))))
% 138.07/18.08 = { by lemma 46 }
% 138.07/18.08 add(inverse(X), Y)
% 138.07/18.08 = { by lemma 31 }
% 138.07/18.08 add(Y, inverse(X))
% 138.07/18.08
% 138.07/18.08 Lemma 48: sum(inverse(y), inverse(x), x_inverse_plus_y_inverse) = true.
% 138.07/18.08 Proof:
% 138.07/18.08 sum(inverse(y), inverse(x), x_inverse_plus_y_inverse)
% 138.07/18.08 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.08 ifeq(true, true, sum(inverse(y), inverse(x), x_inverse_plus_y_inverse), true)
% 138.07/18.08 = { by axiom 13 (x_inverse_plus_y_inverse) R->L }
% 138.07/18.08 ifeq(sum(inverse(x), inverse(y), x_inverse_plus_y_inverse), true, sum(inverse(y), inverse(x), x_inverse_plus_y_inverse), true)
% 138.07/18.08 = { by axiom 15 (commutativity_of_addition) }
% 138.07/18.08 true
% 138.07/18.08
% 138.07/18.08 Lemma 49: multiply(add(X, Y), add(inverse(X), inverse(X))) = multiply(Y, add(inverse(X), inverse(X))).
% 138.07/18.08 Proof:
% 138.07/18.08 multiply(add(X, Y), add(inverse(X), inverse(X)))
% 138.07/18.08 = { by lemma 31 R->L }
% 138.07/18.08 multiply(add(Y, X), add(inverse(X), inverse(X)))
% 138.07/18.08 = { by lemma 33 R->L }
% 138.07/18.08 multiply(add(inverse(X), inverse(X)), add(Y, X))
% 138.07/18.08 = { by lemma 27 R->L }
% 138.07/18.08 multiply(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))))
% 138.07/18.08 = { by lemma 33 R->L }
% 138.07/18.09 multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X)))
% 138.07/18.09 = { by axiom 6 (ifeq_axiom) R->L }
% 138.07/18.09 ifeq2(true, true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 25 R->L }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, ifeq(sum(Y, inverse(inverse(X)), add(Y, inverse(inverse(X)))), true, product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 12 (closure_of_addition) }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, ifeq(true, true, product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 31 }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.09 ifeq2(ifeq(ifeq(true, true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 38 R->L }
% 138.07/18.09 ifeq2(ifeq(ifeq(ifeq(product(inverse(inverse(X)), inverse(X), additive_identity), true, ifeq(sum(inverse(X), inverse(inverse(inverse(X))), add(inverse(X), inverse(X))), true, product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true), true), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 10 (multiplicative_inverse1) }
% 138.07/18.09 ifeq2(ifeq(ifeq(ifeq(true, true, ifeq(sum(inverse(X), inverse(inverse(inverse(X))), add(inverse(X), inverse(X))), true, product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true), true), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.09 ifeq2(ifeq(ifeq(ifeq(sum(inverse(X), inverse(inverse(inverse(X))), add(inverse(X), inverse(X))), true, product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 27 }
% 138.07/18.09 ifeq2(ifeq(ifeq(ifeq(sum(inverse(X), inverse(X), add(inverse(X), inverse(X))), true, product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 12 (closure_of_addition) }
% 138.07/18.09 ifeq2(ifeq(ifeq(ifeq(true, true, product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.09 ifeq2(ifeq(ifeq(product(inverse(inverse(X)), add(inverse(X), inverse(X)), additive_identity), true, product(add(inverse(X), inverse(X)), inverse(inverse(X)), additive_identity), true), true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 16 (commutativity_of_multiplication) }
% 138.07/18.09 ifeq2(ifeq(true, true, ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(inverse(inverse(X)), Y), multiply(add(inverse(X), inverse(X)), Y)), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 31 }
% 138.07/18.09 ifeq2(ifeq(product(add(inverse(X), inverse(X)), Y, multiply(add(inverse(X), inverse(X)), Y)), true, product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 14 (closure_of_multiplication) }
% 138.07/18.09 ifeq2(ifeq(true, true, product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) }
% 138.07/18.09 ifeq2(product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(add(inverse(X), inverse(X)), Y)), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 33 }
% 138.07/18.09 ifeq2(product(add(inverse(X), inverse(X)), add(Y, inverse(inverse(X))), multiply(Y, add(inverse(X), inverse(X)))), true, multiply(add(Y, inverse(inverse(X))), add(inverse(X), inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 138.07/18.09 = { by lemma 40 }
% 138.07/18.09 multiply(Y, add(inverse(X), inverse(X)))
% 138.07/18.09
% 138.07/18.09 Lemma 50: ifeq(sum(y, X, multiplicative_identity), true, sum(x_times_y, X, add(X, x)), true) = true.
% 138.07/18.09 Proof:
% 138.07/18.09 ifeq(sum(y, X, multiplicative_identity), true, sum(x_times_y, X, add(X, x)), true)
% 138.07/18.09 = { by lemma 31 R->L }
% 138.07/18.09 ifeq(sum(y, X, multiplicative_identity), true, sum(x_times_y, X, add(x, X)), true)
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.09 ifeq(sum(y, X, multiplicative_identity), true, ifeq(true, true, sum(x_times_y, X, add(x, X)), true), true)
% 138.07/18.09 = { by axiom 12 (closure_of_addition) R->L }
% 138.07/18.09 ifeq(sum(y, X, multiplicative_identity), true, ifeq(sum(x, X, add(x, X)), true, sum(x_times_y, X, add(x, X)), true), true)
% 138.07/18.09 = { by lemma 42 }
% 138.07/18.09 true
% 138.07/18.09
% 138.07/18.09 Lemma 51: ifeq(sum(inverse(X), Y, multiplicative_identity), true, sum(additive_identity, Y, add(X, Y)), true) = true.
% 138.07/18.09 Proof:
% 138.07/18.09 ifeq(sum(inverse(X), Y, multiplicative_identity), true, sum(additive_identity, Y, add(X, Y)), true)
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.09 ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(true, true, sum(additive_identity, Y, add(X, Y)), true), true)
% 138.07/18.09 = { by axiom 12 (closure_of_addition) R->L }
% 138.07/18.09 ifeq(sum(inverse(X), Y, multiplicative_identity), true, ifeq(sum(X, Y, add(X, Y)), true, sum(additive_identity, Y, add(X, Y)), true), true)
% 138.07/18.09 = { by lemma 23 }
% 138.07/18.09 true
% 138.07/18.09
% 138.07/18.09 Lemma 52: ifeq(product(X, inverse(Y), additive_identity), true, product(X, multiplicative_identity, multiply(X, Y)), true) = true.
% 138.07/18.09 Proof:
% 138.07/18.09 ifeq(product(X, inverse(Y), additive_identity), true, product(X, multiplicative_identity, multiply(X, Y)), true)
% 138.07/18.09 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.09 ifeq(product(X, inverse(Y), additive_identity), true, ifeq(true, true, product(X, multiplicative_identity, multiply(X, Y)), true), true)
% 138.07/18.09 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.07/18.09 ifeq(product(X, inverse(Y), additive_identity), true, ifeq(product(X, Y, multiply(X, Y)), true, product(X, multiplicative_identity, multiply(X, Y)), true), true)
% 138.07/18.09 = { by lemma 26 }
% 138.07/18.10 true
% 138.07/18.10
% 138.07/18.10 Lemma 53: ifeq(product(X, inverse(y), additive_identity), true, ifeq(product(X, inverse(x), Y), true, product(X, x_inverse_plus_y_inverse, Y), true), true) = true.
% 138.07/18.10 Proof:
% 138.07/18.10 ifeq(product(X, inverse(y), additive_identity), true, ifeq(product(X, inverse(x), Y), true, product(X, x_inverse_plus_y_inverse, Y), true), true)
% 138.07/18.10 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.07/18.10 ifeq(product(X, inverse(y), additive_identity), true, ifeq(product(X, inverse(x), Y), true, ifeq(true, true, product(X, x_inverse_plus_y_inverse, Y), true), true), true)
% 138.07/18.10 = { by axiom 13 (x_inverse_plus_y_inverse) R->L }
% 138.07/18.10 ifeq(product(X, inverse(y), additive_identity), true, ifeq(product(X, inverse(x), Y), true, ifeq(sum(inverse(x), inverse(y), x_inverse_plus_y_inverse), true, product(X, x_inverse_plus_y_inverse, Y), true), true), true)
% 138.07/18.10 = { by lemma 25 }
% 138.51/18.11 true
% 138.51/18.11
% 138.51/18.11 Goal 1 (prove_equation): inverse(x_times_y) = x_inverse_plus_y_inverse.
% 138.51/18.11 Proof:
% 138.51/18.11 inverse(x_times_y)
% 138.51/18.11 = { by lemma 24 R->L }
% 138.51/18.11 ifeq2(product(inverse(x_times_y), multiplicative_identity, multiply(inverse(x_times_y), multiplicative_identity)), true, multiply(inverse(x_times_y), multiplicative_identity), inverse(x_times_y))
% 138.51/18.11 = { by axiom 14 (closure_of_multiplication) }
% 138.51/18.11 ifeq2(true, true, multiply(inverse(x_times_y), multiplicative_identity), inverse(x_times_y))
% 138.51/18.11 = { by axiom 6 (ifeq_axiom) }
% 138.51/18.11 multiply(inverse(x_times_y), multiplicative_identity)
% 138.51/18.11 = { by lemma 35 R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, inverse(x)))
% 138.51/18.11 = { by lemma 24 R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, ifeq2(product(inverse(x), multiplicative_identity, multiply(x_inverse_plus_y_inverse, inverse(x))), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.11 = { by lemma 33 R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, ifeq2(product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.11 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(true, true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.11 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(inverse(x), inverse(x_inverse_plus_y_inverse))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.11 = { by lemma 29 R->L }
% 138.51/18.11 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(inverse(x), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.11 = { by lemma 49 R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(add(x_inverse_plus_y_inverse, inverse(x)), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 47 R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 6 (ifeq_axiom) R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(true, true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 45 R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(ifeq(sum(inverse(y), inverse(x), x_inverse_plus_y_inverse), true, sum(multiply(inverse(y), inverse(inverse(x))), inverse(x), x_inverse_plus_y_inverse), true), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 48 }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(ifeq(true, true, sum(multiply(inverse(y), inverse(inverse(x))), inverse(x), x_inverse_plus_y_inverse), true), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 11 (ifeq_axiom_001) }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(inverse(y), inverse(inverse(x))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 27 }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(inverse(y), x), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 33 }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(x, inverse(y)), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by lemma 40 R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(true, true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(x, inverse(y), multiply(x, inverse(y))), true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(true, true, ifeq(product(x, inverse(y), multiply(x, inverse(y))), true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.12 = { by axiom 9 (multiplicative_inverse2) R->L }
% 138.51/18.12 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(x, inverse(x), additive_identity), true, ifeq(product(x, inverse(y), multiply(x, inverse(y))), true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(x, inverse(x), additive_identity), true, ifeq(product(x, inverse(y), multiply(x, inverse(y))), true, ifeq(true, true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 48 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(x, inverse(x), additive_identity), true, ifeq(product(x, inverse(y), multiply(x, inverse(y))), true, ifeq(sum(inverse(y), inverse(x), x_inverse_plus_y_inverse), true, product(x, x_inverse_plus_y_inverse, multiply(x, inverse(y))), true), true), true), true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 25 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(true, true, multiply(x_inverse_plus_y_inverse, x), multiply(x, inverse(y))), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by axiom 6 (ifeq_axiom) }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(x_inverse_plus_y_inverse, x), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 33 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(x, x_inverse_plus_y_inverse), inverse(x), x_inverse_plus_y_inverse), true, add(inverse(x), multiply(x, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 43 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(x_inverse_plus_y_inverse, add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 41 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), multiply(x_inverse_plus_y_inverse, inverse(x_inverse_plus_y_inverse))), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 37 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(ifeq(product(inverse(x), inverse(x_inverse_plus_y_inverse), additive_identity), true, product(inverse(x), multiplicative_identity, multiply(inverse(x), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by lemma 52 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, ifeq2(true, true, multiply(x_inverse_plus_y_inverse, inverse(x)), inverse(x))))
% 138.51/18.13 = { by axiom 6 (ifeq_axiom) }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, multiply(x_inverse_plus_y_inverse, inverse(x))))
% 138.51/18.13 = { by lemma 46 }
% 138.51/18.13 multiply(inverse(x_times_y), add(x, x_inverse_plus_y_inverse))
% 138.51/18.13 = { by lemma 43 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(sum(x_times_y, x_inverse_plus_y_inverse, add(x, x_inverse_plus_y_inverse)), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 31 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(true, true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by axiom 12 (closure_of_addition) R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, x_inverse_plus_y_inverse)), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 46 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, multiply(x_inverse_plus_y_inverse, inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by axiom 6 (ifeq_axiom) R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(true, true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 52 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), additive_identity), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 37 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(x_inverse_plus_y_inverse, inverse(x_inverse_plus_y_inverse))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 41 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(x_inverse_plus_y_inverse, add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.13 = { by lemma 43 R->L }
% 138.51/18.13 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(y, x_inverse_plus_y_inverse), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by lemma 33 R->L }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(x_inverse_plus_y_inverse, y), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by axiom 6 (ifeq_axiom) R->L }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(true, true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by lemma 53 R->L }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(y, inverse(y), additive_identity), true, ifeq(product(y, inverse(x), multiply(y, inverse(x))), true, product(y, x_inverse_plus_y_inverse, multiply(y, inverse(x))), true), true), true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by axiom 9 (multiplicative_inverse2) }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(true, true, ifeq(product(y, inverse(x), multiply(y, inverse(x))), true, product(y, x_inverse_plus_y_inverse, multiply(y, inverse(x))), true), true), true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by axiom 11 (ifeq_axiom_001) }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(product(y, inverse(x), multiply(y, inverse(x))), true, product(y, x_inverse_plus_y_inverse, multiply(y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by axiom 14 (closure_of_multiplication) }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(ifeq(true, true, product(y, x_inverse_plus_y_inverse, multiply(y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by axiom 11 (ifeq_axiom_001) }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(ifeq2(product(y, x_inverse_plus_y_inverse, multiply(y, inverse(x))), true, multiply(x_inverse_plus_y_inverse, y), multiply(y, inverse(x))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by lemma 40 }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(y, inverse(x)), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.14 = { by lemma 33 R->L }
% 138.51/18.14 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(inverse(x), y), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.15 = { by lemma 27 R->L }
% 138.51/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(sum(multiply(inverse(x), inverse(inverse(y))), inverse(y), x_inverse_plus_y_inverse), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.15 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.51/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(ifeq(true, true, sum(multiply(inverse(x), inverse(inverse(y))), inverse(y), x_inverse_plus_y_inverse), true), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.51/18.15 = { by axiom 13 (x_inverse_plus_y_inverse) R->L }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(ifeq(sum(inverse(x), inverse(y), x_inverse_plus_y_inverse), true, sum(multiply(inverse(x), inverse(inverse(y))), inverse(y), x_inverse_plus_y_inverse), true), true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 45 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(ifeq2(true, true, add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), x_inverse_plus_y_inverse), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(add(inverse(y), multiply(y, x_inverse_plus_y_inverse)), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 47 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(add(x_inverse_plus_y_inverse, inverse(y)), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 49 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(inverse(y), add(inverse(x_inverse_plus_y_inverse), inverse(x_inverse_plus_y_inverse)))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 29 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(product(inverse(y), inverse(x_inverse_plus_y_inverse), multiply(inverse(y), inverse(x_inverse_plus_y_inverse))), true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by axiom 14 (closure_of_multiplication) }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(ifeq(true, true, product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(product(inverse(y), multiplicative_identity, multiply(inverse(y), x_inverse_plus_y_inverse)), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 33 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, ifeq2(product(inverse(y), multiplicative_identity, multiply(x_inverse_plus_y_inverse, inverse(y))), true, multiply(x_inverse_plus_y_inverse, inverse(y)), inverse(y)))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 24 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, add(y, inverse(y))), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 35 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(ifeq(sum(y, x_inverse_plus_y_inverse, multiplicative_identity), true, sum(x_times_y, x_inverse_plus_y_inverse, add(x_inverse_plus_y_inverse, x)), true), true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.15 = { by lemma 50 }
% 138.76/18.15 multiply(inverse(x_times_y), ifeq2(true, true, add(x_inverse_plus_y_inverse, x_times_y), add(x, x_inverse_plus_y_inverse)))
% 138.76/18.16 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.16 multiply(inverse(x_times_y), add(x_inverse_plus_y_inverse, x_times_y))
% 138.76/18.16 = { by lemma 27 R->L }
% 138.76/18.16 multiply(inverse(x_times_y), add(x_inverse_plus_y_inverse, inverse(inverse(x_times_y))))
% 138.76/18.16 = { by lemma 41 }
% 138.76/18.16 multiply(inverse(x_times_y), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by lemma 33 }
% 138.76/18.16 multiply(x_inverse_plus_y_inverse, inverse(x_times_y))
% 138.76/18.16 = { by axiom 6 (ifeq_axiom) R->L }
% 138.76/18.16 ifeq2(true, true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 20 (distributivity4) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(product(inverse(x_times_y), x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, ifeq(sum(multiply(inverse(x_times_y), x_inverse_plus_y_inverse), additive_identity, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, ifeq(sum(inverse(x_times_y), x_times_y, multiplicative_identity), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 1 (additive_identity2) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(product(inverse(x_times_y), x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, ifeq(true, true, ifeq(sum(inverse(x_times_y), x_times_y, multiplicative_identity), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(product(inverse(x_times_y), x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, ifeq(sum(inverse(x_times_y), x_times_y, multiplicative_identity), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 8 (additive_inverse1) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(product(inverse(x_times_y), x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, ifeq(true, true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(product(inverse(x_times_y), x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 14 (closure_of_multiplication) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, ifeq(true, true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(inverse(x_times_y), x_inverse_plus_y_inverse)), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by lemma 33 }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, additive_identity), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 6 (ifeq_axiom) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(true, true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 16 (commutativity_of_multiplication) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(ifeq(product(inverse(x), add(x_times_y, inverse(inverse(x))), multiply(inverse(x), x_times_y)), true, product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(inverse(x), x_times_y)), true), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by lemma 39 }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(ifeq(true, true, product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(inverse(x), x_times_y)), true), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(inverse(x), x_times_y)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by lemma 33 }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(x_times_y, inverse(x))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by lemma 40 R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(true, true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), multiply(x_times_y, inverse(y))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 18 (multiplication_is_well_defined) R->L }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(product(inverse(y), y, multiply(x_times_y, inverse(y))), true, ifeq2(product(inverse(y), y, additive_identity), true, additive_identity, multiply(x_times_y, inverse(y))), multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 10 (multiplicative_inverse1) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(product(inverse(y), y, multiply(x_times_y, inverse(y))), true, ifeq2(true, true, additive_identity, multiply(x_times_y, inverse(y))), multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.16 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(product(inverse(y), y, multiply(x_times_y, inverse(y))), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.16 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(true, true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 12 (closure_of_addition) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, add(x_times_y, y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 6 (ifeq_axiom) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(true, true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by lemma 51 R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(ifeq(sum(inverse(x_times_y), y, multiplicative_identity), true, sum(additive_identity, y, add(x_times_y, y)), true), true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(ifeq(ifeq(true, true, sum(inverse(x_times_y), y, multiplicative_identity), true), true, sum(additive_identity, y, add(x_times_y, y)), true), true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by lemma 44 R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(ifeq(ifeq(sum(y, inverse(x_times_y), multiplicative_identity), true, sum(inverse(x_times_y), y, multiplicative_identity), true), true, sum(additive_identity, y, add(x_times_y, y)), true), true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 15 (commutativity_of_addition) }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(ifeq(true, true, sum(additive_identity, y, add(x_times_y, y)), true), true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, ifeq2(sum(additive_identity, y, add(x_times_y, y)), true, add(x_times_y, y), y)), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by lemma 28 }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(x_times_y, inverse(y))), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by lemma 33 R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(inverse(y), x_times_y)), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(true, true, ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(inverse(y), x_times_y)), true), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(product(inverse(y), x_times_y, multiply(inverse(y), x_times_y)), true, ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(inverse(y), x_times_y)), true), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.17 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(true, true, ifeq(product(inverse(y), x_times_y, multiply(inverse(y), x_times_y)), true, ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(inverse(y), x_times_y)), true), true), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.17 = { by axiom 10 (multiplicative_inverse1) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(ifeq(product(inverse(y), y, additive_identity), true, ifeq(product(inverse(y), x_times_y, multiply(inverse(y), x_times_y)), true, ifeq(sum(x_times_y, y, y), true, product(inverse(y), y, multiply(inverse(y), x_times_y)), true), true), true), true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 25 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), ifeq2(true, true, additive_identity, multiply(x_times_y, inverse(y)))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), additive_identity), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), additive_identity), true, ifeq(true, true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 14 (closure_of_multiplication) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(ifeq(product(x_times_y, inverse(y), additive_identity), true, ifeq(product(x_times_y, inverse(x), multiply(x_times_y, inverse(x))), true, product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, inverse(x))), true), true), true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 53 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), ifeq2(true, true, multiply(x_inverse_plus_y_inverse, x_times_y), multiply(x_times_y, inverse(x)))), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(x_inverse_plus_y_inverse, x_times_y)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 33 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, inverse(inverse(x))), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 27 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(add(x_times_y, x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 6 (ifeq_axiom) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(true, true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 51 R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(sum(inverse(x_times_y), x, multiplicative_identity), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(true, true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 12 (closure_of_addition) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), add(x, inverse(x_times_y))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 34 R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), ifeq2(sum(x_times_y, inverse(x_times_y), add(x, inverse(x_times_y))), true, multiplicative_identity, add(x, inverse(x_times_y)))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 31 R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), ifeq2(sum(x_times_y, inverse(x_times_y), add(inverse(x_times_y), x)), true, multiplicative_identity, add(x, inverse(x_times_y)))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 11 (ifeq_axiom_001) R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), ifeq2(ifeq(true, true, sum(x_times_y, inverse(x_times_y), add(inverse(x_times_y), x)), true), true, multiplicative_identity, add(x, inverse(x_times_y)))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 44 R->L }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), ifeq2(ifeq(sum(y, inverse(x_times_y), multiplicative_identity), true, sum(x_times_y, inverse(x_times_y), add(inverse(x_times_y), x)), true), true, multiplicative_identity, add(x, inverse(x_times_y)))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 50 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), ifeq2(true, true, multiplicative_identity, add(x, inverse(x_times_y)))), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 6 (ifeq_axiom) }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(ifeq(sum(x, inverse(x_times_y), multiplicative_identity), true, sum(inverse(x_times_y), x, multiplicative_identity), true), true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 15 (commutativity_of_addition) }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(ifeq(true, true, sum(additive_identity, x, add(x_times_y, x)), true), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(ifeq2(sum(additive_identity, x, add(x_times_y, x)), true, add(x_times_y, x), x), inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.18 = { by lemma 28 }
% 138.76/18.18 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, ifeq2(product(x, inverse(x), multiply(x_times_y, x_inverse_plus_y_inverse)), true, additive_identity, multiply(x_times_y, x_inverse_plus_y_inverse))), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by lemma 36 }
% 138.76/18.19 ifeq2(ifeq(product(x_times_y, x_inverse_plus_y_inverse, multiply(x_times_y, x_inverse_plus_y_inverse)), true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by axiom 14 (closure_of_multiplication) }
% 138.76/18.19 ifeq2(ifeq(true, true, product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by axiom 11 (ifeq_axiom_001) }
% 138.76/18.19 ifeq2(product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by axiom 6 (ifeq_axiom) R->L }
% 138.76/18.19 ifeq2(true, true, ifeq2(product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by axiom 5 (multiplicative_identity1) R->L }
% 138.76/18.19 ifeq2(product(multiplicative_identity, x_inverse_plus_y_inverse, x_inverse_plus_y_inverse), true, ifeq2(product(multiplicative_identity, x_inverse_plus_y_inverse, multiply(x_inverse_plus_y_inverse, inverse(x_times_y))), true, multiply(x_inverse_plus_y_inverse, inverse(x_times_y)), x_inverse_plus_y_inverse), x_inverse_plus_y_inverse)
% 138.76/18.19 = { by axiom 18 (multiplication_is_well_defined) }
% 138.76/18.19 x_inverse_plus_y_inverse
% 138.76/18.19 % SZS output end Proof
% 138.76/18.19
% 138.76/18.19 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------