TSTP Solution File: BOO023-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO023-1 : TPTP v8.1.2. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n025.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 18:11:28 EDT 2023
% Result : Unsatisfiable 9.11s 1.73s
% Output : Proof 11.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : BOO023-1 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n025.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Sun Aug 27 08:36:53 EDT 2023
% 0.14/0.34 % CPUTime :
% 9.11/1.73 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 9.11/1.73
% 9.11/1.73 % SZS status Unsatisfiable
% 9.11/1.73
% 10.92/1.78 % SZS output start Proof
% 10.92/1.78 Axiom 1 (additive_inverse): add(X, inverse(X)) = n1.
% 10.92/1.78 Axiom 2 (pixley1): pixley(X, X, Y) = Y.
% 10.92/1.78 Axiom 3 (pixley3): pixley(X, Y, X) = X.
% 10.92/1.78 Axiom 4 (pixley2): pixley(X, Y, Y) = X.
% 10.92/1.78 Axiom 5 (multiply_add): multiply(add(X, Y), Y) = Y.
% 10.92/1.78 Axiom 6 (multiply_add_property): multiply(X, add(Y, Z)) = add(multiply(Y, X), multiply(Z, X)).
% 10.92/1.78 Axiom 7 (pixley_defn): pixley(X, Y, Z) = add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z))).
% 10.92/1.78
% 10.92/1.78 Lemma 8: multiply(n1, inverse(X)) = inverse(X).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(n1, inverse(X))
% 10.92/1.78 = { by axiom 1 (additive_inverse) R->L }
% 10.92/1.78 multiply(add(X, inverse(X)), inverse(X))
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 inverse(X)
% 10.92/1.78
% 10.92/1.78 Lemma 9: add(multiply(X, inverse(Y)), multiply(Z, add(X, inverse(Y)))) = pixley(X, Y, Z).
% 10.92/1.78 Proof:
% 10.92/1.78 add(multiply(X, inverse(Y)), multiply(Z, add(X, inverse(Y))))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) }
% 10.92/1.78 add(multiply(X, inverse(Y)), add(multiply(X, Z), multiply(inverse(Y), Z)))
% 10.92/1.78 = { by axiom 7 (pixley_defn) R->L }
% 10.92/1.78 pixley(X, Y, Z)
% 10.92/1.78
% 10.92/1.78 Lemma 10: add(multiply(X, inverse(X)), multiply(Y, n1)) = Y.
% 10.92/1.78 Proof:
% 10.92/1.78 add(multiply(X, inverse(X)), multiply(Y, n1))
% 10.92/1.78 = { by axiom 1 (additive_inverse) R->L }
% 10.92/1.78 add(multiply(X, inverse(X)), multiply(Y, add(X, inverse(X))))
% 10.92/1.78 = { by lemma 9 }
% 10.92/1.78 pixley(X, X, Y)
% 10.92/1.78 = { by axiom 2 (pixley1) }
% 10.92/1.78 Y
% 10.92/1.78
% 10.92/1.78 Lemma 11: add(inverse(n1), multiply(X, n1)) = X.
% 10.92/1.78 Proof:
% 10.92/1.78 add(inverse(n1), multiply(X, n1))
% 10.92/1.78 = { by lemma 8 R->L }
% 10.92/1.78 add(multiply(n1, inverse(n1)), multiply(X, n1))
% 10.92/1.78 = { by lemma 10 }
% 10.92/1.78 X
% 10.92/1.78
% 10.92/1.78 Lemma 12: add(Y, n1) = add(X, n1).
% 10.92/1.78 Proof:
% 10.92/1.78 add(Y, n1)
% 10.92/1.78 = { by lemma 11 R->L }
% 10.92/1.78 add(inverse(n1), multiply(add(Y, n1), n1))
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 add(inverse(n1), n1)
% 10.92/1.78 = { by axiom 5 (multiply_add) R->L }
% 10.92/1.78 add(inverse(n1), multiply(add(X, n1), n1))
% 10.92/1.78 = { by lemma 11 }
% 10.92/1.78 add(X, n1)
% 10.92/1.78
% 10.92/1.78 Lemma 13: add(inverse(X), multiply(Y, inverse(X))) = multiply(inverse(X), add(n1, Y)).
% 10.92/1.78 Proof:
% 10.92/1.78 add(inverse(X), multiply(Y, inverse(X)))
% 10.92/1.78 = { by lemma 8 R->L }
% 10.92/1.78 add(multiply(n1, inverse(X)), multiply(Y, inverse(X)))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) R->L }
% 10.92/1.78 multiply(inverse(X), add(n1, Y))
% 10.92/1.78
% 10.92/1.78 Lemma 14: multiply(inverse(X), add(Y, n1)) = add(inverse(X), inverse(X)).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(inverse(X), add(Y, n1))
% 10.92/1.78 = { by lemma 12 }
% 10.92/1.78 multiply(inverse(X), add(n1, n1))
% 10.92/1.78 = { by lemma 13 R->L }
% 10.92/1.78 add(inverse(X), multiply(n1, inverse(X)))
% 10.92/1.78 = { by lemma 8 }
% 10.92/1.78 add(inverse(X), inverse(X))
% 10.92/1.78
% 10.92/1.78 Lemma 15: add(multiply(X, inverse(Y)), inverse(Y)) = multiply(inverse(Y), add(X, n1)).
% 10.92/1.78 Proof:
% 10.92/1.78 add(multiply(X, inverse(Y)), inverse(Y))
% 10.92/1.78 = { by lemma 8 R->L }
% 10.92/1.78 add(multiply(X, inverse(Y)), multiply(n1, inverse(Y)))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) R->L }
% 10.92/1.78 multiply(inverse(Y), add(X, n1))
% 10.92/1.78
% 10.92/1.78 Lemma 16: multiply(X, add(Y, add(Z, X))) = add(multiply(Y, X), X).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(X, add(Y, add(Z, X)))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) }
% 10.92/1.78 add(multiply(Y, X), multiply(add(Z, X), X))
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 add(multiply(Y, X), X)
% 10.92/1.78
% 10.92/1.78 Lemma 17: multiply(X, add(add(Y, X), Z)) = add(X, multiply(Z, X)).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(X, add(add(Y, X), Z))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) }
% 10.92/1.78 add(multiply(add(Y, X), X), multiply(Z, X))
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 add(X, multiply(Z, X))
% 10.92/1.78
% 10.92/1.78 Lemma 18: multiply(multiply(X, add(Y, Z)), multiply(Z, X)) = multiply(Z, X).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(multiply(X, add(Y, Z)), multiply(Z, X))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) }
% 10.92/1.78 multiply(add(multiply(Y, X), multiply(Z, X)), multiply(Z, X))
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 multiply(Z, X)
% 10.92/1.78
% 10.92/1.78 Lemma 19: multiply(add(X, Y), add(Y, Y)) = add(Y, Y).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(add(X, Y), add(Y, Y))
% 10.92/1.78 = { by axiom 5 (multiply_add) R->L }
% 10.92/1.78 multiply(add(X, Y), add(multiply(add(Z, Y), Y), Y))
% 10.92/1.78 = { by lemma 16 R->L }
% 10.92/1.78 multiply(add(X, Y), multiply(Y, add(add(Z, Y), add(X, Y))))
% 10.92/1.78 = { by axiom 5 (multiply_add) R->L }
% 10.92/1.78 multiply(multiply(add(add(Z, Y), add(X, Y)), add(X, Y)), multiply(Y, add(add(Z, Y), add(X, Y))))
% 10.92/1.78 = { by lemma 18 }
% 10.92/1.78 multiply(Y, add(add(Z, Y), add(X, Y)))
% 10.92/1.78 = { by lemma 16 }
% 10.92/1.78 add(multiply(add(Z, Y), Y), Y)
% 10.92/1.78 = { by axiom 5 (multiply_add) }
% 10.92/1.78 add(Y, Y)
% 10.92/1.78
% 10.92/1.78 Lemma 20: multiply(n1, add(inverse(X), inverse(X))) = add(inverse(X), inverse(X)).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(n1, add(inverse(X), inverse(X)))
% 10.92/1.78 = { by axiom 1 (additive_inverse) R->L }
% 10.92/1.78 multiply(add(X, inverse(X)), add(inverse(X), inverse(X)))
% 10.92/1.78 = { by lemma 19 }
% 10.92/1.78 add(inverse(X), inverse(X))
% 10.92/1.78
% 10.92/1.78 Lemma 21: add(multiply(inverse(X), inverse(X)), add(inverse(X), inverse(X))) = inverse(X).
% 10.92/1.78 Proof:
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), add(inverse(X), inverse(X)))
% 10.92/1.78 = { by lemma 20 R->L }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), multiply(n1, add(inverse(X), inverse(X))))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), add(multiply(inverse(X), n1), multiply(inverse(X), n1)))
% 10.92/1.78 = { by lemma 19 R->L }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), multiply(add(multiply(Y, inverse(Y)), multiply(inverse(X), n1)), add(multiply(inverse(X), n1), multiply(inverse(X), n1))))
% 10.92/1.78 = { by lemma 10 }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), multiply(inverse(X), add(multiply(inverse(X), n1), multiply(inverse(X), n1))))
% 10.92/1.78 = { by axiom 6 (multiply_add_property) R->L }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), multiply(inverse(X), multiply(n1, add(inverse(X), inverse(X)))))
% 10.92/1.78 = { by lemma 20 }
% 10.92/1.78 add(multiply(inverse(X), inverse(X)), multiply(inverse(X), add(inverse(X), inverse(X))))
% 10.92/1.78 = { by lemma 9 }
% 10.92/1.78 pixley(inverse(X), X, inverse(X))
% 10.92/1.78 = { by axiom 3 (pixley3) }
% 10.92/1.78 inverse(X)
% 10.92/1.78
% 10.92/1.78 Lemma 22: multiply(inverse(X), add(Y, n1)) = inverse(X).
% 10.92/1.78 Proof:
% 10.92/1.78 multiply(inverse(X), add(Y, n1))
% 10.92/1.79 = { by lemma 12 }
% 10.92/1.79 multiply(inverse(X), add(n1, n1))
% 10.92/1.79 = { by lemma 15 R->L }
% 10.92/1.79 add(multiply(n1, inverse(X)), inverse(X))
% 10.92/1.79 = { by lemma 16 R->L }
% 10.92/1.79 multiply(inverse(X), add(n1, add(inverse(X), inverse(X))))
% 10.92/1.79 = { by lemma 13 R->L }
% 10.92/1.79 add(inverse(X), multiply(add(inverse(X), inverse(X)), inverse(X)))
% 10.92/1.79 = { by lemma 17 R->L }
% 10.92/1.79 multiply(inverse(X), add(add(inverse(X), inverse(X)), add(inverse(X), inverse(X))))
% 10.92/1.79 = { by lemma 21 R->L }
% 10.92/1.79 multiply(add(multiply(inverse(X), inverse(X)), add(inverse(X), inverse(X))), add(add(inverse(X), inverse(X)), add(inverse(X), inverse(X))))
% 10.92/1.79 = { by lemma 19 }
% 10.92/1.79 add(add(inverse(X), inverse(X)), add(inverse(X), inverse(X)))
% 10.92/1.79 = { by lemma 14 R->L }
% 10.92/1.79 add(multiply(inverse(X), add(multiply(inverse(X), inverse(X)), n1)), add(inverse(X), inverse(X)))
% 10.92/1.79 = { by lemma 15 R->L }
% 10.92/1.79 add(add(multiply(multiply(inverse(X), inverse(X)), inverse(X)), inverse(X)), add(inverse(X), inverse(X)))
% 10.92/1.79 = { by lemma 16 R->L }
% 10.92/1.79 add(multiply(inverse(X), add(multiply(inverse(X), inverse(X)), add(inverse(X), inverse(X)))), add(inverse(X), inverse(X)))
% 10.92/1.79 = { by lemma 21 }
% 10.92/1.79 add(multiply(inverse(X), inverse(X)), add(inverse(X), inverse(X)))
% 10.92/1.79 = { by lemma 21 }
% 10.92/1.79 inverse(X)
% 10.92/1.79
% 10.92/1.79 Lemma 23: add(inverse(X), inverse(X)) = inverse(X).
% 10.92/1.79 Proof:
% 10.92/1.79 add(inverse(X), inverse(X))
% 10.92/1.79 = { by lemma 14 R->L }
% 10.92/1.79 multiply(inverse(X), add(Y, n1))
% 10.92/1.79 = { by lemma 22 }
% 10.92/1.79 inverse(X)
% 10.92/1.79
% 10.92/1.79 Lemma 24: multiply(inverse(X), add(Y, inverse(X))) = inverse(X).
% 10.92/1.79 Proof:
% 10.92/1.79 multiply(inverse(X), add(Y, inverse(X)))
% 10.92/1.79 = { by lemma 23 R->L }
% 10.92/1.79 multiply(inverse(X), add(Y, add(inverse(X), inverse(X))))
% 10.92/1.79 = { by lemma 16 }
% 10.92/1.79 add(multiply(Y, inverse(X)), inverse(X))
% 10.92/1.79 = { by lemma 15 }
% 10.92/1.79 multiply(inverse(X), add(Y, n1))
% 10.92/1.79 = { by lemma 12 R->L }
% 10.92/1.79 multiply(inverse(X), add(Z, n1))
% 10.92/1.79 = { by lemma 22 }
% 10.92/1.79 inverse(X)
% 10.92/1.79
% 10.92/1.79 Lemma 25: multiply(inverse(X), n1) = inverse(X).
% 10.92/1.79 Proof:
% 10.92/1.79 multiply(inverse(X), n1)
% 10.92/1.79 = { by axiom 1 (additive_inverse) R->L }
% 10.92/1.79 multiply(inverse(X), add(X, inverse(X)))
% 10.92/1.79 = { by lemma 24 }
% 10.92/1.79 inverse(X)
% 10.92/1.79
% 10.92/1.79 Lemma 26: multiply(n1, X) = multiply(X, n1).
% 10.92/1.79 Proof:
% 11.21/1.79 multiply(n1, X)
% 11.21/1.79 = { by lemma 11 R->L }
% 11.21/1.79 multiply(n1, add(inverse(n1), multiply(X, n1)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 add(multiply(inverse(n1), n1), multiply(multiply(X, n1), n1))
% 11.21/1.79 = { by lemma 25 }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), n1))
% 11.21/1.79 = { by lemma 11 }
% 11.21/1.79 multiply(X, n1)
% 11.21/1.79
% 11.21/1.79 Lemma 27: add(inverse(n1), inverse(X)) = inverse(X).
% 11.21/1.79 Proof:
% 11.21/1.79 add(inverse(n1), inverse(X))
% 11.21/1.79 = { by lemma 25 R->L }
% 11.21/1.79 add(inverse(n1), multiply(inverse(X), n1))
% 11.21/1.79 = { by lemma 11 }
% 11.21/1.79 inverse(X)
% 11.21/1.79
% 11.21/1.79 Lemma 28: add(X, n1) = n1.
% 11.21/1.79 Proof:
% 11.21/1.79 add(X, n1)
% 11.21/1.79 = { by lemma 12 }
% 11.21/1.79 add(inverse(n1), n1)
% 11.21/1.79 = { by axiom 1 (additive_inverse) R->L }
% 11.21/1.79 add(inverse(n1), add(inverse(n1), inverse(inverse(n1))))
% 11.21/1.79 = { by lemma 27 }
% 11.21/1.79 add(inverse(n1), inverse(inverse(n1)))
% 11.21/1.79 = { by axiom 1 (additive_inverse) }
% 11.21/1.79 n1
% 11.21/1.79
% 11.21/1.79 Lemma 29: multiply(X, add(Y, n1)) = add(X, multiply(n1, X)).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(X, add(Y, n1))
% 11.21/1.79 = { by lemma 12 }
% 11.21/1.79 multiply(X, add(add(Z, X), n1))
% 11.21/1.79 = { by lemma 17 }
% 11.21/1.79 add(X, multiply(n1, X))
% 11.21/1.79
% 11.21/1.79 Lemma 30: multiply(multiply(X, n1), multiply(X, n1)) = multiply(X, n1).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(multiply(X, n1), multiply(X, n1))
% 11.21/1.79 = { by lemma 28 R->L }
% 11.21/1.79 multiply(multiply(X, n1), multiply(X, add(n1, n1)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 multiply(multiply(X, n1), add(multiply(n1, X), multiply(n1, X)))
% 11.21/1.79 = { by lemma 28 R->L }
% 11.21/1.79 multiply(multiply(X, add(Y, n1)), add(multiply(n1, X), multiply(n1, X)))
% 11.21/1.79 = { by lemma 29 }
% 11.21/1.79 multiply(add(X, multiply(n1, X)), add(multiply(n1, X), multiply(n1, X)))
% 11.21/1.79 = { by lemma 19 }
% 11.21/1.79 add(multiply(n1, X), multiply(n1, X))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.79 multiply(X, add(n1, n1))
% 11.21/1.79 = { by lemma 28 }
% 11.21/1.79 multiply(X, n1)
% 11.21/1.79
% 11.21/1.79 Lemma 31: add(multiply(X, n1), multiply(Y, multiply(X, n1))) = multiply(multiply(X, n1), add(X, Y)).
% 11.21/1.79 Proof:
% 11.21/1.79 add(multiply(X, n1), multiply(Y, multiply(X, n1)))
% 11.21/1.79 = { by lemma 17 R->L }
% 11.21/1.79 multiply(multiply(X, n1), add(add(multiply(Z, inverse(Z)), multiply(X, n1)), Y))
% 11.21/1.79 = { by lemma 10 }
% 11.21/1.79 multiply(multiply(X, n1), add(X, Y))
% 11.21/1.79
% 11.21/1.79 Lemma 32: add(X, X) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 add(X, X)
% 11.21/1.79 = { by lemma 11 R->L }
% 11.21/1.79 add(inverse(n1), multiply(add(X, X), n1))
% 11.21/1.79 = { by lemma 26 R->L }
% 11.21/1.79 add(inverse(n1), multiply(n1, add(X, X)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 add(inverse(n1), add(multiply(X, n1), multiply(X, n1)))
% 11.21/1.79 = { by lemma 19 R->L }
% 11.21/1.79 add(inverse(n1), multiply(add(multiply(X, n1), multiply(X, n1)), add(multiply(X, n1), multiply(X, n1))))
% 11.21/1.79 = { by lemma 30 R->L }
% 11.21/1.79 add(inverse(n1), multiply(add(multiply(X, n1), multiply(X, n1)), add(multiply(X, n1), multiply(multiply(X, n1), multiply(X, n1)))))
% 11.21/1.79 = { by lemma 17 R->L }
% 11.21/1.79 add(inverse(n1), multiply(add(multiply(X, n1), multiply(X, n1)), multiply(multiply(X, n1), add(add(Y, multiply(X, n1)), multiply(X, n1)))))
% 11.21/1.79 = { by lemma 19 R->L }
% 11.21/1.79 add(inverse(n1), multiply(multiply(add(add(Y, multiply(X, n1)), multiply(X, n1)), add(multiply(X, n1), multiply(X, n1))), multiply(multiply(X, n1), add(add(Y, multiply(X, n1)), multiply(X, n1)))))
% 11.21/1.79 = { by lemma 18 }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), add(add(Y, multiply(X, n1)), multiply(X, n1))))
% 11.21/1.79 = { by lemma 17 }
% 11.21/1.79 add(inverse(n1), add(multiply(X, n1), multiply(multiply(X, n1), multiply(X, n1))))
% 11.21/1.79 = { by lemma 31 }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), add(X, multiply(X, n1))))
% 11.21/1.79 = { by lemma 26 R->L }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), add(X, multiply(n1, X))))
% 11.21/1.79 = { by lemma 29 R->L }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), multiply(X, add(Z, n1))))
% 11.21/1.79 = { by lemma 28 }
% 11.21/1.79 add(inverse(n1), multiply(multiply(X, n1), multiply(X, n1)))
% 11.21/1.79 = { by lemma 30 }
% 11.21/1.79 add(inverse(n1), multiply(X, n1))
% 11.21/1.79 = { by lemma 11 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 33: multiply(Y, X) = multiply(X, Y).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(Y, X)
% 11.21/1.79 = { by lemma 32 R->L }
% 11.21/1.79 add(multiply(Y, X), multiply(Y, X))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.79 multiply(X, add(Y, Y))
% 11.21/1.79 = { by lemma 32 }
% 11.21/1.79 multiply(X, Y)
% 11.21/1.79
% 11.21/1.79 Lemma 34: add(multiply(X, Y), X) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 add(multiply(X, Y), X)
% 11.21/1.79 = { by lemma 33 }
% 11.21/1.79 add(multiply(Y, X), X)
% 11.21/1.79 = { by lemma 16 R->L }
% 11.21/1.79 multiply(X, add(Y, add(X, X)))
% 11.21/1.79 = { by lemma 32 R->L }
% 11.21/1.79 multiply(add(X, X), add(Y, add(X, X)))
% 11.21/1.79 = { by lemma 32 R->L }
% 11.21/1.79 multiply(add(X, X), add(add(Y, add(X, X)), add(Y, add(X, X))))
% 11.21/1.79 = { by lemma 17 }
% 11.21/1.79 add(add(X, X), multiply(add(Y, add(X, X)), add(X, X)))
% 11.21/1.79 = { by axiom 5 (multiply_add) }
% 11.21/1.79 add(add(X, X), add(X, X))
% 11.21/1.79 = { by lemma 32 }
% 11.21/1.79 add(X, X)
% 11.21/1.79 = { by lemma 32 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 35: multiply(X, n1) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(X, n1)
% 11.21/1.79 = { by lemma 34 R->L }
% 11.21/1.79 add(multiply(multiply(X, n1), inverse(multiply(X, n1))), multiply(X, n1))
% 11.21/1.79 = { by lemma 10 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 36: multiply(n1, X) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(n1, X)
% 11.21/1.79 = { by lemma 26 }
% 11.21/1.79 multiply(X, n1)
% 11.21/1.79 = { by lemma 35 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 37: multiply(n1, add(X, inverse(Y))) = add(multiply(X, n1), inverse(Y)).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(n1, add(X, inverse(Y)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 add(multiply(X, n1), multiply(inverse(Y), n1))
% 11.21/1.79 = { by lemma 25 }
% 11.21/1.79 add(multiply(X, n1), inverse(Y))
% 11.21/1.79
% 11.21/1.79 Lemma 38: add(n1, inverse(X)) = n1.
% 11.21/1.79 Proof:
% 11.21/1.79 add(n1, inverse(X))
% 11.21/1.79 = { by lemma 34 R->L }
% 11.21/1.79 add(multiply(add(n1, inverse(X)), inverse(X)), add(n1, inverse(X)))
% 11.21/1.79 = { by axiom 5 (multiply_add) }
% 11.21/1.79 add(inverse(X), add(n1, inverse(X)))
% 11.21/1.79 = { by lemma 35 R->L }
% 11.21/1.79 add(inverse(X), add(multiply(n1, n1), inverse(X)))
% 11.21/1.79 = { by lemma 37 R->L }
% 11.21/1.79 add(inverse(X), multiply(n1, add(n1, inverse(X))))
% 11.21/1.79 = { by lemma 8 R->L }
% 11.21/1.79 add(multiply(n1, inverse(X)), multiply(n1, add(n1, inverse(X))))
% 11.21/1.79 = { by lemma 9 }
% 11.21/1.79 pixley(n1, X, n1)
% 11.21/1.79 = { by axiom 3 (pixley3) }
% 11.21/1.79 n1
% 11.21/1.79
% 11.21/1.79 Lemma 39: add(inverse(n1), X) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 add(inverse(n1), X)
% 11.21/1.79 = { by lemma 35 R->L }
% 11.21/1.79 add(inverse(n1), multiply(X, n1))
% 11.21/1.79 = { by lemma 11 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 40: multiply(inverse(X), add(n1, Y)) = pixley(inverse(X), X, Y).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(inverse(X), add(n1, Y))
% 11.21/1.79 = { by lemma 13 R->L }
% 11.21/1.79 add(inverse(X), multiply(Y, inverse(X)))
% 11.21/1.79 = { by lemma 23 R->L }
% 11.21/1.79 add(inverse(X), multiply(Y, add(inverse(X), inverse(X))))
% 11.21/1.79 = { by lemma 8 R->L }
% 11.21/1.79 add(multiply(n1, inverse(X)), multiply(Y, add(inverse(X), inverse(X))))
% 11.21/1.79 = { by lemma 18 R->L }
% 11.21/1.79 add(multiply(multiply(inverse(X), add(Z, n1)), multiply(n1, inverse(X))), multiply(Y, add(inverse(X), inverse(X))))
% 11.21/1.79 = { by lemma 8 }
% 11.21/1.79 add(multiply(multiply(inverse(X), add(Z, n1)), inverse(X)), multiply(Y, add(inverse(X), inverse(X))))
% 11.21/1.79 = { by lemma 22 }
% 11.21/1.79 add(multiply(inverse(X), inverse(X)), multiply(Y, add(inverse(X), inverse(X))))
% 11.21/1.79 = { by lemma 9 }
% 11.21/1.79 pixley(inverse(X), X, Y)
% 11.21/1.79
% 11.21/1.79 Lemma 41: multiply(X, add(X, Y)) = add(X, multiply(X, Y)).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(X, add(X, Y))
% 11.21/1.79 = { by lemma 32 R->L }
% 11.21/1.79 multiply(X, add(add(X, X), Y))
% 11.21/1.79 = { by lemma 17 }
% 11.21/1.79 add(X, multiply(Y, X))
% 11.21/1.79 = { by lemma 33 R->L }
% 11.21/1.79 add(X, multiply(X, Y))
% 11.21/1.79
% 11.21/1.79 Lemma 42: multiply(X, add(n1, Y)) = add(X, multiply(X, Y)).
% 11.21/1.79 Proof:
% 11.21/1.79 multiply(X, add(n1, Y))
% 11.21/1.79 = { by lemma 32 R->L }
% 11.21/1.79 multiply(add(X, X), add(n1, Y))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 add(multiply(n1, add(X, X)), multiply(Y, add(X, X)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) }
% 11.21/1.79 add(add(multiply(X, n1), multiply(X, n1)), multiply(Y, add(X, X)))
% 11.21/1.79 = { by axiom 5 (multiply_add) R->L }
% 11.21/1.79 add(add(multiply(X, n1), multiply(add(multiply(Z, inverse(Z)), multiply(X, n1)), multiply(X, n1))), multiply(Y, add(X, X)))
% 11.21/1.79 = { by lemma 10 }
% 11.21/1.79 add(add(multiply(X, n1), multiply(X, multiply(X, n1))), multiply(Y, add(X, X)))
% 11.21/1.79 = { by lemma 31 }
% 11.21/1.79 add(multiply(multiply(X, n1), add(X, X)), multiply(Y, add(X, X)))
% 11.21/1.79 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.79 multiply(add(X, X), add(multiply(X, n1), Y))
% 11.21/1.79 = { by lemma 32 }
% 11.21/1.79 multiply(X, add(multiply(X, n1), Y))
% 11.21/1.79 = { by lemma 35 }
% 11.21/1.79 multiply(X, add(X, Y))
% 11.21/1.79 = { by lemma 41 }
% 11.21/1.79 add(X, multiply(X, Y))
% 11.21/1.79
% 11.21/1.79 Lemma 43: add(X, multiply(X, multiply(X, inverse(Y)))) = add(X, multiply(X, inverse(Y))).
% 11.21/1.79 Proof:
% 11.21/1.79 add(X, multiply(X, multiply(X, inverse(Y))))
% 11.21/1.79 = { by lemma 33 }
% 11.21/1.79 add(X, multiply(X, multiply(inverse(Y), X)))
% 11.21/1.79 = { by lemma 42 R->L }
% 11.21/1.79 multiply(X, add(n1, multiply(inverse(Y), X)))
% 11.21/1.79 = { by lemma 35 R->L }
% 11.21/1.79 multiply(multiply(X, n1), add(n1, multiply(inverse(Y), X)))
% 11.21/1.79 = { by lemma 42 }
% 11.21/1.79 add(multiply(X, n1), multiply(multiply(X, n1), multiply(inverse(Y), X)))
% 11.21/1.79 = { by axiom 1 (additive_inverse) R->L }
% 11.21/1.79 add(multiply(X, n1), multiply(multiply(X, add(Y, inverse(Y))), multiply(inverse(Y), X)))
% 11.21/1.79 = { by lemma 18 }
% 11.21/1.79 add(multiply(X, n1), multiply(inverse(Y), X))
% 11.21/1.79 = { by lemma 35 }
% 11.21/1.79 add(X, multiply(inverse(Y), X))
% 11.21/1.79 = { by lemma 33 R->L }
% 11.21/1.79 add(X, multiply(X, inverse(Y)))
% 11.21/1.79
% 11.21/1.79 Lemma 44: add(X, multiply(X, inverse(Y))) = X.
% 11.21/1.79 Proof:
% 11.21/1.79 add(X, multiply(X, inverse(Y)))
% 11.21/1.79 = { by lemma 8 R->L }
% 11.21/1.79 add(X, multiply(X, multiply(n1, inverse(Y))))
% 11.21/1.79 = { by lemma 36 R->L }
% 11.21/1.79 add(X, multiply(X, multiply(n1, multiply(n1, inverse(Y)))))
% 11.21/1.79 = { by lemma 42 R->L }
% 11.21/1.79 multiply(X, add(n1, multiply(n1, multiply(n1, inverse(Y)))))
% 11.21/1.79 = { by lemma 43 }
% 11.21/1.79 multiply(X, add(n1, multiply(n1, inverse(Y))))
% 11.21/1.79 = { by lemma 8 }
% 11.21/1.79 multiply(X, add(n1, inverse(Y)))
% 11.21/1.79 = { by lemma 38 }
% 11.21/1.79 multiply(X, n1)
% 11.21/1.79 = { by lemma 35 }
% 11.21/1.79 X
% 11.21/1.79
% 11.21/1.79 Lemma 45: pixley(inverse(X), X, inverse(Y)) = inverse(X).
% 11.21/1.79 Proof:
% 11.21/1.79 pixley(inverse(X), X, inverse(Y))
% 11.21/1.79 = { by lemma 8 R->L }
% 11.21/1.79 pixley(inverse(X), X, multiply(n1, inverse(Y)))
% 11.21/1.79 = { by lemma 36 R->L }
% 11.21/1.79 pixley(inverse(X), X, multiply(n1, multiply(n1, inverse(Y))))
% 11.21/1.79 = { by lemma 40 R->L }
% 11.21/1.79 multiply(inverse(X), add(n1, multiply(n1, multiply(n1, inverse(Y)))))
% 11.21/1.79 = { by lemma 43 }
% 11.21/1.79 multiply(inverse(X), add(n1, multiply(n1, inverse(Y))))
% 11.21/1.79 = { by lemma 44 }
% 11.21/1.79 multiply(inverse(X), n1)
% 11.21/1.79 = { by lemma 25 }
% 11.21/1.79 inverse(X)
% 11.21/1.79
% 11.21/1.79 Lemma 46: add(add(X, inverse(n1)), inverse(n1)) = add(X, inverse(n1)).
% 11.21/1.79 Proof:
% 11.21/1.79 add(add(X, inverse(n1)), inverse(n1))
% 11.21/1.79 = { by axiom 5 (multiply_add) R->L }
% 11.21/1.79 add(add(X, inverse(n1)), multiply(add(X, inverse(n1)), inverse(n1)))
% 11.21/1.79 = { by lemma 42 R->L }
% 11.21/1.79 multiply(add(X, inverse(n1)), add(n1, inverse(n1)))
% 11.21/1.79 = { by axiom 1 (additive_inverse) }
% 11.21/1.79 multiply(add(X, inverse(n1)), n1)
% 11.21/1.79 = { by lemma 35 }
% 11.21/1.80 add(X, inverse(n1))
% 11.21/1.80
% 11.21/1.80 Lemma 47: add(inverse(X), inverse(n1)) = inverse(X).
% 11.21/1.80 Proof:
% 11.21/1.80 add(inverse(X), inverse(n1))
% 11.21/1.80 = { by lemma 45 R->L }
% 11.21/1.80 add(inverse(X), pixley(inverse(n1), n1, inverse(X)))
% 11.21/1.80 = { by lemma 22 R->L }
% 11.21/1.80 add(inverse(X), pixley(multiply(inverse(n1), add(Y, n1)), n1, inverse(X)))
% 11.21/1.80 = { by lemma 15 R->L }
% 11.21/1.80 add(inverse(X), pixley(add(multiply(Y, inverse(n1)), inverse(n1)), n1, inverse(X)))
% 11.21/1.80 = { by lemma 9 R->L }
% 11.21/1.80 add(inverse(X), add(multiply(add(multiply(Y, inverse(n1)), inverse(n1)), inverse(n1)), multiply(inverse(X), add(add(multiply(Y, inverse(n1)), inverse(n1)), inverse(n1)))))
% 11.21/1.80 = { by lemma 46 }
% 11.21/1.80 add(inverse(X), add(multiply(add(multiply(Y, inverse(n1)), inverse(n1)), inverse(n1)), multiply(inverse(X), add(multiply(Y, inverse(n1)), inverse(n1)))))
% 11.21/1.80 = { by axiom 5 (multiply_add) }
% 11.21/1.80 add(inverse(X), add(inverse(n1), multiply(inverse(X), add(multiply(Y, inverse(n1)), inverse(n1)))))
% 11.21/1.80 = { by lemma 39 }
% 11.21/1.80 add(inverse(X), multiply(inverse(X), add(multiply(Y, inverse(n1)), inverse(n1))))
% 11.21/1.80 = { by lemma 15 }
% 11.21/1.80 add(inverse(X), multiply(inverse(X), multiply(inverse(n1), add(Y, n1))))
% 11.21/1.80 = { by lemma 22 }
% 11.21/1.80 add(inverse(X), multiply(inverse(X), inverse(n1)))
% 11.21/1.80 = { by lemma 42 R->L }
% 11.21/1.80 multiply(inverse(X), add(n1, inverse(n1)))
% 11.21/1.80 = { by lemma 38 }
% 11.21/1.80 multiply(inverse(X), n1)
% 11.21/1.80 = { by lemma 25 }
% 11.21/1.80 inverse(X)
% 11.21/1.80
% 11.21/1.80 Lemma 48: multiply(inverse(inverse(X)), add(inverse(X), Y)) = multiply(Y, inverse(inverse(X))).
% 11.21/1.80 Proof:
% 11.21/1.80 multiply(inverse(inverse(X)), add(inverse(X), Y))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) }
% 11.21/1.80 add(multiply(inverse(X), inverse(inverse(X))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 47 R->L }
% 11.21/1.80 add(multiply(inverse(X), inverse(add(inverse(X), inverse(n1)))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 47 R->L }
% 11.21/1.80 add(multiply(add(inverse(X), inverse(n1)), inverse(add(inverse(X), inverse(n1)))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 27 R->L }
% 11.21/1.80 add(multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 39 R->L }
% 11.21/1.80 add(add(inverse(n1), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 25 R->L }
% 11.21/1.80 add(add(multiply(inverse(n1), n1), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by axiom 1 (additive_inverse) R->L }
% 11.21/1.80 add(add(multiply(inverse(n1), add(add(add(inverse(X), inverse(n1)), inverse(n1)), inverse(add(add(inverse(X), inverse(n1)), inverse(n1))))), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 17 }
% 11.21/1.80 add(add(add(inverse(n1), multiply(inverse(add(add(inverse(X), inverse(n1)), inverse(n1))), inverse(n1))), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 46 }
% 11.21/1.80 add(add(add(inverse(n1), multiply(inverse(add(inverse(X), inverse(n1))), inverse(n1))), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 39 }
% 11.21/1.80 add(add(multiply(inverse(add(inverse(X), inverse(n1))), inverse(n1)), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 add(add(multiply(inverse(n1), inverse(add(inverse(X), inverse(n1)))), multiply(add(inverse(X), inverse(n1)), add(inverse(n1), inverse(add(inverse(X), inverse(n1)))))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 9 }
% 11.21/1.80 add(pixley(inverse(n1), add(inverse(X), inverse(n1)), add(inverse(X), inverse(n1))), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by axiom 4 (pixley2) }
% 11.21/1.80 add(inverse(n1), multiply(Y, inverse(inverse(X))))
% 11.21/1.80 = { by lemma 39 }
% 11.21/1.80 multiply(Y, inverse(inverse(X)))
% 11.21/1.80
% 11.21/1.80 Lemma 49: inverse(inverse(inverse(X))) = inverse(X).
% 11.21/1.80 Proof:
% 11.21/1.80 inverse(inverse(inverse(X)))
% 11.21/1.80 = { by lemma 34 R->L }
% 11.21/1.80 add(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), inverse(inverse(inverse(X))))
% 11.21/1.80 = { by lemma 48 R->L }
% 11.21/1.80 add(multiply(inverse(inverse(X)), add(inverse(X), inverse(inverse(inverse(X))))), inverse(inverse(inverse(X))))
% 11.21/1.80 = { by lemma 24 R->L }
% 11.21/1.80 add(multiply(inverse(inverse(X)), add(inverse(X), inverse(inverse(inverse(X))))), multiply(inverse(inverse(inverse(X))), add(inverse(X), inverse(inverse(inverse(X))))))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.80 multiply(add(inverse(X), inverse(inverse(inverse(X)))), add(inverse(inverse(X)), inverse(inverse(inverse(X)))))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 multiply(add(inverse(inverse(X)), inverse(inverse(inverse(X)))), add(inverse(X), inverse(inverse(inverse(X)))))
% 11.21/1.80 = { by axiom 1 (additive_inverse) }
% 11.21/1.80 multiply(n1, add(inverse(X), inverse(inverse(inverse(X)))))
% 11.21/1.80 = { by lemma 37 }
% 11.21/1.80 add(multiply(inverse(X), n1), inverse(inverse(inverse(X))))
% 11.21/1.80 = { by lemma 35 }
% 11.21/1.80 add(inverse(X), inverse(inverse(inverse(X))))
% 11.21/1.80 = { by lemma 25 R->L }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), n1))
% 11.21/1.80 = { by axiom 4 (pixley2) R->L }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), pixley(n1, inverse(X), inverse(X))))
% 11.21/1.80 = { by lemma 9 R->L }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), add(multiply(n1, inverse(inverse(X))), multiply(inverse(X), add(n1, inverse(inverse(X)))))))
% 11.21/1.80 = { by lemma 40 }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), add(multiply(n1, inverse(inverse(X))), pixley(inverse(X), X, inverse(inverse(X))))))
% 11.21/1.80 = { by lemma 45 }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), add(multiply(n1, inverse(inverse(X))), inverse(X))))
% 11.21/1.80 = { by lemma 8 }
% 11.21/1.80 add(inverse(X), multiply(inverse(inverse(inverse(X))), add(inverse(inverse(X)), inverse(X))))
% 11.21/1.80 = { by lemma 48 }
% 11.21/1.80 add(inverse(X), multiply(inverse(X), inverse(inverse(inverse(X)))))
% 11.21/1.80 = { by lemma 42 R->L }
% 11.21/1.80 multiply(inverse(X), add(n1, inverse(inverse(inverse(X)))))
% 11.21/1.80 = { by lemma 38 }
% 11.21/1.80 multiply(inverse(X), n1)
% 11.21/1.80 = { by lemma 25 }
% 11.21/1.80 inverse(X)
% 11.21/1.80
% 11.21/1.80 Lemma 50: add(X, multiply(X, Y)) = X.
% 11.21/1.80 Proof:
% 11.21/1.80 add(X, multiply(X, Y))
% 11.21/1.80 = { by axiom 2 (pixley1) R->L }
% 11.21/1.80 add(X, multiply(X, pixley(inverse(inverse(Y)), inverse(inverse(Y)), Y)))
% 11.21/1.80 = { by lemma 9 R->L }
% 11.21/1.80 add(X, multiply(X, add(multiply(inverse(inverse(Y)), inverse(inverse(inverse(Y)))), multiply(Y, add(inverse(inverse(Y)), inverse(inverse(inverse(Y))))))))
% 11.21/1.80 = { by lemma 49 }
% 11.21/1.80 add(X, multiply(X, add(multiply(inverse(inverse(Y)), inverse(Y)), multiply(Y, add(inverse(inverse(Y)), inverse(inverse(inverse(Y))))))))
% 11.21/1.80 = { by lemma 49 }
% 11.21/1.80 add(X, multiply(X, add(multiply(inverse(inverse(Y)), inverse(Y)), multiply(Y, add(inverse(inverse(Y)), inverse(Y))))))
% 11.21/1.80 = { by lemma 9 }
% 11.21/1.80 add(X, multiply(X, pixley(inverse(inverse(Y)), Y, Y)))
% 11.21/1.80 = { by axiom 4 (pixley2) }
% 11.21/1.80 add(X, multiply(X, inverse(inverse(Y))))
% 11.21/1.80 = { by lemma 44 }
% 11.21/1.80 X
% 11.21/1.80
% 11.21/1.80 Lemma 51: multiply(X, add(add(Y, X), Z)) = X.
% 11.21/1.80 Proof:
% 11.21/1.80 multiply(X, add(add(Y, X), Z))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) }
% 11.21/1.80 add(multiply(add(Y, X), X), multiply(Z, X))
% 11.21/1.80 = { by axiom 5 (multiply_add) }
% 11.21/1.80 add(X, multiply(Z, X))
% 11.21/1.80 = { by lemma 33 }
% 11.21/1.80 add(X, multiply(X, Z))
% 11.21/1.80 = { by lemma 50 }
% 11.21/1.80 X
% 11.21/1.80
% 11.21/1.80 Lemma 52: multiply(add(X, Y), add(X, Z)) = add(X, multiply(Z, add(X, Y))).
% 11.21/1.80 Proof:
% 11.21/1.80 multiply(add(X, Y), add(X, Z))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) }
% 11.21/1.80 add(multiply(X, add(X, Y)), multiply(Z, add(X, Y)))
% 11.21/1.80 = { by lemma 41 }
% 11.21/1.80 add(add(X, multiply(X, Y)), multiply(Z, add(X, Y)))
% 11.21/1.80 = { by lemma 50 }
% 11.21/1.80 add(X, multiply(Z, add(X, Y)))
% 11.21/1.80
% 11.21/1.80 Goal 1 (prove_add_multiply_property): add(a, multiply(b, c)) = multiply(add(a, b), add(a, c)).
% 11.21/1.80 Proof:
% 11.21/1.80 add(a, multiply(b, c))
% 11.21/1.80 = { by lemma 33 }
% 11.21/1.80 add(a, multiply(c, b))
% 11.21/1.80 = { by lemma 34 R->L }
% 11.21/1.80 add(a, add(multiply(multiply(c, b), multiply(inverse(b), add(a, c))), multiply(c, b)))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 add(a, add(multiply(multiply(inverse(b), add(a, c)), multiply(c, b)), multiply(c, b)))
% 11.21/1.80 = { by lemma 18 R->L }
% 11.21/1.80 add(a, add(multiply(multiply(inverse(b), add(a, c)), multiply(c, b)), multiply(multiply(b, add(a, c)), multiply(c, b))))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.80 add(a, multiply(multiply(c, b), add(multiply(inverse(b), add(a, c)), multiply(b, add(a, c)))))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 add(a, multiply(multiply(b, c), add(multiply(inverse(b), add(a, c)), multiply(b, add(a, c)))))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), add(inverse(b), b))))
% 11.21/1.80 = { by lemma 35 R->L }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), add(inverse(b), multiply(b, n1)))))
% 11.21/1.80 = { by lemma 8 R->L }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), add(multiply(n1, inverse(b)), multiply(b, n1)))))
% 11.21/1.80 = { by lemma 38 R->L }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), add(multiply(n1, inverse(b)), multiply(b, add(n1, inverse(b)))))))
% 11.21/1.80 = { by lemma 9 }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), pixley(n1, b, b))))
% 11.21/1.80 = { by axiom 4 (pixley2) }
% 11.21/1.80 add(a, multiply(multiply(b, c), multiply(add(a, c), n1)))
% 11.21/1.80 = { by lemma 35 }
% 11.21/1.80 add(a, multiply(multiply(b, c), add(a, c)))
% 11.21/1.80 = { by lemma 52 R->L }
% 11.21/1.80 multiply(add(a, c), add(a, multiply(b, c)))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 multiply(add(a, multiply(b, c)), add(a, c))
% 11.21/1.80 = { by lemma 52 }
% 11.21/1.80 add(a, multiply(c, add(a, multiply(b, c))))
% 11.21/1.80 = { by lemma 33 }
% 11.21/1.80 add(a, multiply(add(a, multiply(b, c)), c))
% 11.21/1.80 = { by lemma 32 R->L }
% 11.21/1.80 add(a, multiply(add(a, multiply(b, c)), add(c, c)))
% 11.21/1.80 = { by lemma 33 }
% 11.21/1.80 add(a, multiply(add(c, c), add(a, multiply(b, c))))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(c, c))))
% 11.21/1.80 = { by lemma 51 R->L }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(multiply(c, add(add(multiply(add(X, c), inverse(Y)), c), b)), c))))
% 11.21/1.80 = { by lemma 51 R->L }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(multiply(c, add(add(multiply(add(X, c), inverse(Y)), multiply(c, add(add(X, c), inverse(Y)))), b)), c))))
% 11.21/1.80 = { by lemma 9 }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(multiply(c, add(pixley(add(X, c), Y, c), b)), c))))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(add(multiply(pixley(add(X, c), Y, c), c), multiply(b, c)), c))))
% 11.21/1.80 = { by lemma 33 R->L }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(multiply(b, c), add(add(multiply(c, pixley(add(X, c), Y, c)), multiply(b, c)), c))))
% 11.21/1.80 = { by lemma 51 }
% 11.21/1.80 add(a, add(multiply(a, add(c, c)), multiply(b, c)))
% 11.21/1.80 = { by lemma 32 }
% 11.21/1.80 add(a, add(multiply(a, c), multiply(b, c)))
% 11.21/1.80 = { by axiom 6 (multiply_add_property) R->L }
% 11.21/1.80 add(a, multiply(c, add(a, b)))
% 11.21/1.80 = { by lemma 52 R->L }
% 11.21/1.80 multiply(add(a, b), add(a, c))
% 11.21/1.80 % SZS output end Proof
% 11.21/1.80
% 11.21/1.80 RESULT: Unsatisfiable (the axioms are contradictory).
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