TSTP Solution File: RNG009-5 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : RNG009-5 : TPTP v8.1.2. Released v1.0.0.
% 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 : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:58:47 EDT 2023
% Result : Unsatisfiable 42.24s 5.70s
% Output : Proof 42.80s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : RNG009-5 : TPTP v8.1.2. Released v1.0.0.
% 0.03/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n009.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Sun Aug 27 02:32:05 EDT 2023
% 0.12/0.33 % CPUTime :
% 42.24/5.70 Command-line arguments: --ground-connectedness --complete-subsets
% 42.24/5.70
% 42.24/5.70 % SZS status Unsatisfiable
% 42.24/5.70
% 42.80/5.76 % SZS output start Proof
% 42.80/5.76 Axiom 1 (commutative_addition): add(X, Y) = add(Y, X).
% 42.80/5.76 Axiom 2 (right_identity): add(X, additive_identity) = X.
% 42.80/5.76 Axiom 3 (right_additive_inverse): add(X, additive_inverse(X)) = additive_identity.
% 42.80/5.76 Axiom 4 (associative_addition): add(add(X, Y), Z) = add(X, add(Y, Z)).
% 42.80/5.76 Axiom 5 (x_cubed_is_x): multiply(X, multiply(X, X)) = X.
% 42.80/5.76 Axiom 6 (associative_multiplication): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 42.80/5.76 Axiom 7 (distribute1): multiply(X, add(Y, Z)) = add(multiply(X, Y), multiply(X, Z)).
% 42.80/5.76 Axiom 8 (distribute2): multiply(add(X, Y), Z) = add(multiply(X, Z), multiply(Y, Z)).
% 42.80/5.76
% 42.80/5.76 Lemma 9: add(additive_identity, X) = X.
% 42.80/5.76 Proof:
% 42.80/5.76 add(additive_identity, X)
% 42.80/5.76 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.76 add(X, additive_identity)
% 42.80/5.76 = { by axiom 2 (right_identity) }
% 42.80/5.76 X
% 42.80/5.76
% 42.80/5.76 Lemma 10: add(X, add(Y, additive_inverse(X))) = Y.
% 42.80/5.76 Proof:
% 42.80/5.76 add(X, add(Y, additive_inverse(X)))
% 42.80/5.76 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.76 add(X, add(additive_inverse(X), Y))
% 42.80/5.76 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.76 add(add(X, additive_inverse(X)), Y)
% 42.80/5.76 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.76 add(additive_identity, Y)
% 42.80/5.76 = { by lemma 9 }
% 42.80/5.76 Y
% 42.80/5.76
% 42.80/5.76 Lemma 11: add(X, add(additive_inverse(X), Y)) = Y.
% 42.80/5.76 Proof:
% 42.80/5.76 add(X, add(additive_inverse(X), Y))
% 42.80/5.76 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.76 add(X, add(Y, additive_inverse(X)))
% 42.80/5.76 = { by lemma 10 }
% 42.80/5.76 Y
% 42.80/5.76
% 42.80/5.76 Lemma 12: additive_inverse(additive_inverse(X)) = X.
% 42.80/5.76 Proof:
% 42.80/5.76 additive_inverse(additive_inverse(X))
% 42.80/5.76 = { by lemma 11 R->L }
% 42.80/5.76 add(X, add(additive_inverse(X), additive_inverse(additive_inverse(X))))
% 42.80/5.76 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.76 add(X, additive_identity)
% 42.80/5.76 = { by axiom 2 (right_identity) }
% 42.80/5.76 X
% 42.80/5.76
% 42.80/5.76 Lemma 13: multiply(X, add(Y, multiply(X, X))) = add(X, multiply(X, Y)).
% 42.80/5.76 Proof:
% 42.80/5.76 multiply(X, add(Y, multiply(X, X)))
% 42.80/5.76 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.76 multiply(X, add(multiply(X, X), Y))
% 42.80/5.76 = { by axiom 7 (distribute1) }
% 42.80/5.76 add(multiply(X, multiply(X, X)), multiply(X, Y))
% 42.80/5.76 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.76 add(X, multiply(X, Y))
% 42.80/5.76
% 42.80/5.76 Lemma 14: multiply(X, additive_identity) = additive_identity.
% 42.80/5.76 Proof:
% 42.80/5.76 multiply(X, additive_identity)
% 42.80/5.76 = { by lemma 12 R->L }
% 42.80/5.76 multiply(additive_inverse(additive_inverse(X)), additive_identity)
% 42.80/5.76 = { by lemma 11 R->L }
% 42.80/5.76 add(additive_inverse(X), add(additive_inverse(additive_inverse(X)), multiply(additive_inverse(additive_inverse(X)), additive_identity)))
% 42.80/5.76 = { by lemma 13 R->L }
% 42.80/5.76 add(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), add(additive_identity, multiply(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X))))))
% 42.80/5.76 = { by lemma 9 }
% 42.80/5.76 add(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), multiply(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X)))))
% 42.80/5.76 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.76 add(additive_inverse(X), additive_inverse(additive_inverse(X)))
% 42.80/5.76 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.76 additive_identity
% 42.80/5.76
% 42.80/5.76 Lemma 15: multiply(X, multiply(additive_identity, Y)) = multiply(additive_identity, Y).
% 42.80/5.76 Proof:
% 42.80/5.76 multiply(X, multiply(additive_identity, Y))
% 42.80/5.76 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.76 multiply(multiply(X, additive_identity), Y)
% 42.80/5.76 = { by lemma 14 }
% 42.80/5.76 multiply(additive_identity, Y)
% 42.80/5.76
% 42.80/5.76 Lemma 16: multiply(add(X, Y), multiply(X, X)) = add(X, multiply(Y, multiply(X, X))).
% 42.80/5.76 Proof:
% 42.80/5.76 multiply(add(X, Y), multiply(X, X))
% 42.80/5.76 = { by axiom 8 (distribute2) }
% 42.80/5.76 add(multiply(X, multiply(X, X)), multiply(Y, multiply(X, X)))
% 42.80/5.76 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.76 add(X, multiply(Y, multiply(X, X)))
% 42.80/5.76
% 42.80/5.76 Lemma 17: add(X, add(Y, additive_inverse(add(X, Y)))) = additive_identity.
% 42.80/5.76 Proof:
% 42.80/5.76 add(X, add(Y, additive_inverse(add(X, Y))))
% 42.80/5.76 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.76 add(add(X, Y), additive_inverse(add(X, Y)))
% 42.80/5.76 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.76 additive_identity
% 42.80/5.76
% 42.80/5.76 Lemma 18: multiply(additive_identity, multiply(X, X)) = additive_identity.
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_identity, multiply(X, X))
% 42.80/5.77 = { by lemma 10 R->L }
% 42.80/5.77 add(X, add(multiply(additive_identity, multiply(X, X)), additive_inverse(X)))
% 42.80/5.77 = { by axiom 5 (x_cubed_is_x) R->L }
% 42.80/5.77 add(X, add(multiply(additive_identity, multiply(X, X)), additive_inverse(multiply(X, multiply(X, X)))))
% 42.80/5.77 = { by axiom 2 (right_identity) R->L }
% 42.80/5.77 add(X, add(multiply(additive_identity, multiply(X, X)), additive_inverse(multiply(add(X, additive_identity), multiply(X, X)))))
% 42.80/5.77 = { by lemma 16 }
% 42.80/5.77 add(X, add(multiply(additive_identity, multiply(X, X)), additive_inverse(add(X, multiply(additive_identity, multiply(X, X))))))
% 42.80/5.77 = { by lemma 17 }
% 42.80/5.77 additive_identity
% 42.80/5.77
% 42.80/5.77 Lemma 19: multiply(additive_identity, X) = additive_identity.
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_identity, X)
% 42.80/5.77 = { by lemma 15 R->L }
% 42.80/5.77 multiply(additive_identity, multiply(additive_identity, X))
% 42.80/5.77 = { by lemma 15 R->L }
% 42.80/5.77 multiply(additive_identity, multiply(multiply(additive_identity, X), multiply(additive_identity, X)))
% 42.80/5.77 = { by lemma 18 }
% 42.80/5.77 additive_identity
% 42.80/5.77
% 42.80/5.77 Lemma 20: add(X, additive_inverse(add(X, Y))) = additive_inverse(Y).
% 42.80/5.77 Proof:
% 42.80/5.77 add(X, additive_inverse(add(X, Y)))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 add(X, additive_inverse(add(X, additive_inverse(additive_inverse(Y)))))
% 42.80/5.77 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.77 add(X, additive_inverse(add(additive_inverse(additive_inverse(Y)), X)))
% 42.80/5.77 = { by lemma 11 R->L }
% 42.80/5.77 add(additive_inverse(Y), add(additive_inverse(additive_inverse(Y)), add(X, additive_inverse(add(additive_inverse(additive_inverse(Y)), X)))))
% 42.80/5.77 = { by lemma 17 }
% 42.80/5.77 add(additive_inverse(Y), additive_identity)
% 42.80/5.77 = { by axiom 2 (right_identity) }
% 42.80/5.77 additive_inverse(Y)
% 42.80/5.77
% 42.80/5.77 Lemma 21: multiply(X, multiply(add(X, Y), X)) = add(X, multiply(X, multiply(Y, X))).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(X, multiply(add(X, Y), X))
% 42.80/5.77 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.77 multiply(X, multiply(add(Y, X), X))
% 42.80/5.77 = { by axiom 8 (distribute2) }
% 42.80/5.77 multiply(X, add(multiply(Y, X), multiply(X, X)))
% 42.80/5.77 = { by lemma 13 }
% 42.80/5.77 add(X, multiply(X, multiply(Y, X)))
% 42.80/5.77
% 42.80/5.77 Lemma 22: multiply(add(X, X), Y) = multiply(X, add(Y, Y)).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(add(X, X), Y)
% 42.80/5.77 = { by axiom 8 (distribute2) }
% 42.80/5.77 add(multiply(X, Y), multiply(X, Y))
% 42.80/5.77 = { by axiom 7 (distribute1) R->L }
% 42.80/5.77 multiply(X, add(Y, Y))
% 42.80/5.77
% 42.80/5.77 Lemma 23: multiply(add(X, X), multiply(X, X)) = add(X, X).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(add(X, X), multiply(X, X))
% 42.80/5.77 = { by lemma 22 }
% 42.80/5.77 multiply(X, add(multiply(X, X), multiply(X, X)))
% 42.80/5.77 = { by lemma 13 }
% 42.80/5.77 add(X, multiply(X, multiply(X, X)))
% 42.80/5.77 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.77 add(X, X)
% 42.80/5.77
% 42.80/5.77 Lemma 24: multiply(add(X, X), multiply(Y, Z)) = multiply(X, multiply(add(Y, Y), Z)).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(add(X, X), multiply(Y, Z))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(multiply(add(X, X), Y), Z)
% 42.80/5.77 = { by lemma 22 }
% 42.80/5.77 multiply(multiply(X, add(Y, Y)), Z)
% 42.80/5.77 = { by axiom 6 (associative_multiplication) }
% 42.80/5.77 multiply(X, multiply(add(Y, Y), Z))
% 42.80/5.77
% 42.80/5.77 Lemma 25: multiply(additive_inverse(X), multiply(X, additive_inverse(X))) = X.
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_inverse(X), multiply(X, additive_inverse(X)))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 multiply(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), additive_inverse(X)))
% 42.80/5.77 = { by lemma 11 R->L }
% 42.80/5.77 add(X, add(additive_inverse(X), multiply(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), additive_inverse(X)))))
% 42.80/5.77 = { by lemma 21 R->L }
% 42.80/5.77 add(X, multiply(additive_inverse(X), multiply(add(additive_inverse(X), additive_inverse(additive_inverse(X))), additive_inverse(X))))
% 42.80/5.77 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.77 add(X, multiply(additive_inverse(X), multiply(additive_identity, additive_inverse(X))))
% 42.80/5.77 = { by lemma 15 }
% 42.80/5.77 add(X, multiply(additive_identity, additive_inverse(X)))
% 42.80/5.77 = { by lemma 19 }
% 42.80/5.77 add(X, additive_identity)
% 42.80/5.77 = { by axiom 2 (right_identity) }
% 42.80/5.77 X
% 42.80/5.77
% 42.80/5.77 Lemma 26: multiply(additive_inverse(X), multiply(X, multiply(additive_inverse(X), Y))) = multiply(X, Y).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_inverse(X), multiply(X, multiply(additive_inverse(X), Y)))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(additive_inverse(X), multiply(multiply(X, additive_inverse(X)), Y))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(multiply(additive_inverse(X), multiply(X, additive_inverse(X))), Y)
% 42.80/5.77 = { by lemma 25 }
% 42.80/5.77 multiply(X, Y)
% 42.80/5.77
% 42.80/5.77 Lemma 27: multiply(additive_inverse(X), multiply(additive_inverse(X), Y)) = multiply(X, multiply(X, Y)).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_inverse(X), multiply(additive_inverse(X), Y))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(multiply(additive_inverse(X), additive_inverse(X)), Y)
% 42.80/5.77 = { by lemma 25 R->L }
% 42.80/5.77 multiply(multiply(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))), Y)
% 42.80/5.77 = { by lemma 12 }
% 42.80/5.77 multiply(multiply(additive_inverse(X), multiply(X, multiply(additive_inverse(X), additive_inverse(additive_inverse(X))))), Y)
% 42.80/5.77 = { by lemma 12 }
% 42.80/5.77 multiply(multiply(additive_inverse(X), multiply(X, multiply(additive_inverse(X), X))), Y)
% 42.80/5.77 = { by lemma 26 }
% 42.80/5.77 multiply(multiply(X, X), Y)
% 42.80/5.77 = { by axiom 6 (associative_multiplication) }
% 42.80/5.77 multiply(X, multiply(X, Y))
% 42.80/5.77
% 42.80/5.77 Lemma 28: multiply(X, multiply(additive_inverse(X), additive_inverse(X))) = X.
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(X, multiply(additive_inverse(X), additive_inverse(X)))
% 42.80/5.77 = { by lemma 11 R->L }
% 42.80/5.77 add(X, add(additive_inverse(X), multiply(X, multiply(additive_inverse(X), additive_inverse(X)))))
% 42.80/5.77 = { by lemma 16 R->L }
% 42.80/5.77 add(X, multiply(add(additive_inverse(X), X), multiply(additive_inverse(X), additive_inverse(X))))
% 42.80/5.77 = { by axiom 1 (commutative_addition) }
% 42.80/5.77 add(X, multiply(add(X, additive_inverse(X)), multiply(additive_inverse(X), additive_inverse(X))))
% 42.80/5.77 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.77 add(X, multiply(additive_identity, multiply(additive_inverse(X), additive_inverse(X))))
% 42.80/5.77 = { by lemma 18 }
% 42.80/5.77 add(X, additive_identity)
% 42.80/5.77 = { by axiom 2 (right_identity) }
% 42.80/5.77 X
% 42.80/5.77
% 42.80/5.77 Lemma 29: multiply(additive_inverse(X), X) = multiply(X, additive_inverse(X)).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_inverse(X), X)
% 42.80/5.77 = { by lemma 28 R->L }
% 42.80/5.77 multiply(additive_inverse(X), multiply(X, multiply(additive_inverse(X), additive_inverse(X))))
% 42.80/5.77 = { by lemma 26 }
% 42.80/5.77 multiply(X, additive_inverse(X))
% 42.80/5.77
% 42.80/5.77 Lemma 30: add(X, multiply(X, multiply(additive_inverse(X), add(X, X)))) = additive_inverse(X).
% 42.80/5.77 Proof:
% 42.80/5.77 add(X, multiply(X, multiply(additive_inverse(X), add(X, X))))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 add(X, multiply(multiply(X, additive_inverse(X)), add(X, X)))
% 42.80/5.77 = { by lemma 29 R->L }
% 42.80/5.77 add(X, multiply(multiply(additive_inverse(X), X), add(X, X)))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) }
% 42.80/5.77 add(X, multiply(additive_inverse(X), multiply(X, add(X, X))))
% 42.80/5.77 = { by axiom 7 (distribute1) }
% 42.80/5.77 add(X, multiply(additive_inverse(X), add(multiply(X, X), multiply(X, X))))
% 42.80/5.77 = { by lemma 22 R->L }
% 42.80/5.77 add(X, multiply(add(additive_inverse(X), additive_inverse(X)), multiply(X, X)))
% 42.80/5.77 = { by lemma 16 R->L }
% 42.80/5.77 multiply(add(X, add(additive_inverse(X), additive_inverse(X))), multiply(X, X))
% 42.80/5.77 = { by lemma 10 }
% 42.80/5.77 multiply(additive_inverse(X), multiply(X, X))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 multiply(additive_inverse(X), multiply(X, additive_inverse(additive_inverse(X))))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 multiply(additive_inverse(X), multiply(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X))))
% 42.80/5.77 = { by lemma 28 }
% 42.80/5.77 additive_inverse(X)
% 42.80/5.77
% 42.80/5.77 Lemma 31: multiply(add(X, multiply(Y, Z)), multiply(Z, Z)) = multiply(add(Y, multiply(X, Z)), Z).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(add(X, multiply(Y, Z)), multiply(Z, Z))
% 42.80/5.77 = { by axiom 8 (distribute2) }
% 42.80/5.77 add(multiply(X, multiply(Z, Z)), multiply(multiply(Y, Z), multiply(Z, Z)))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) }
% 42.80/5.77 add(multiply(X, multiply(Z, Z)), multiply(Y, multiply(Z, multiply(Z, Z))))
% 42.80/5.77 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.77 add(multiply(X, multiply(Z, Z)), multiply(Y, Z))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 add(multiply(multiply(X, Z), Z), multiply(Y, Z))
% 42.80/5.77 = { by axiom 8 (distribute2) R->L }
% 42.80/5.77 multiply(add(multiply(X, Z), Y), Z)
% 42.80/5.77 = { by axiom 1 (commutative_addition) }
% 42.80/5.77 multiply(add(Y, multiply(X, Z)), Z)
% 42.80/5.77
% 42.80/5.77 Lemma 32: multiply(additive_inverse(X), additive_inverse(multiply(additive_inverse(X), additive_inverse(X)))) = X.
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(additive_inverse(X), additive_inverse(multiply(additive_inverse(X), additive_inverse(X))))
% 42.80/5.77 = { by lemma 11 R->L }
% 42.80/5.77 add(X, add(additive_inverse(X), multiply(additive_inverse(X), additive_inverse(multiply(additive_inverse(X), additive_inverse(X))))))
% 42.80/5.77 = { by lemma 13 R->L }
% 42.80/5.77 add(X, multiply(additive_inverse(X), add(additive_inverse(multiply(additive_inverse(X), additive_inverse(X))), multiply(additive_inverse(X), additive_inverse(X)))))
% 42.80/5.77 = { by axiom 1 (commutative_addition) }
% 42.80/5.77 add(X, multiply(additive_inverse(X), add(multiply(additive_inverse(X), additive_inverse(X)), additive_inverse(multiply(additive_inverse(X), additive_inverse(X))))))
% 42.80/5.77 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.77 add(X, multiply(additive_inverse(X), additive_identity))
% 42.80/5.77 = { by lemma 14 }
% 42.80/5.77 add(X, additive_identity)
% 42.80/5.77 = { by axiom 2 (right_identity) }
% 42.80/5.77 X
% 42.80/5.77
% 42.80/5.77 Lemma 33: multiply(X, multiply(X, multiply(X, Y))) = multiply(X, Y).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(X, multiply(X, multiply(X, Y)))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(X, multiply(multiply(X, X), Y))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(multiply(X, multiply(X, X)), Y)
% 42.80/5.77 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.77 multiply(X, Y)
% 42.80/5.77
% 42.80/5.77 Lemma 34: multiply(X, multiply(X, additive_inverse(X))) = additive_inverse(X).
% 42.80/5.77 Proof:
% 42.80/5.77 multiply(X, multiply(X, additive_inverse(X)))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 multiply(X, multiply(additive_inverse(additive_inverse(X)), additive_inverse(X)))
% 42.80/5.77 = { by lemma 12 R->L }
% 42.80/5.77 multiply(additive_inverse(additive_inverse(X)), multiply(additive_inverse(additive_inverse(X)), additive_inverse(X)))
% 42.80/5.77 = { by lemma 32 R->L }
% 42.80/5.77 multiply(additive_inverse(additive_inverse(X)), multiply(additive_inverse(additive_inverse(X)), multiply(additive_inverse(additive_inverse(X)), additive_inverse(multiply(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X)))))))
% 42.80/5.77 = { by lemma 33 }
% 42.80/5.77 multiply(additive_inverse(additive_inverse(X)), additive_inverse(multiply(additive_inverse(additive_inverse(X)), additive_inverse(additive_inverse(X)))))
% 42.80/5.77 = { by lemma 32 }
% 42.80/5.77 additive_inverse(X)
% 42.80/5.77
% 42.80/5.77 Lemma 35: add(X, add(X, add(X, X))) = additive_inverse(add(X, X)).
% 42.80/5.77 Proof:
% 42.80/5.77 add(X, add(X, add(X, X)))
% 42.80/5.77 = { by axiom 5 (x_cubed_is_x) R->L }
% 42.80/5.77 add(X, add(X, multiply(add(X, X), multiply(add(X, X), add(X, X)))))
% 42.80/5.77 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.77 add(add(X, X), multiply(add(X, X), multiply(add(X, X), add(X, X))))
% 42.80/5.77 = { by lemma 21 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), add(X, X)))
% 42.80/5.77 = { by lemma 23 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(X, X))))
% 42.80/5.77 = { by lemma 24 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(add(X, X), X))))
% 42.80/5.77 = { by lemma 22 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(X, add(X, X)))))
% 42.80/5.77 = { by lemma 27 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(additive_inverse(X), multiply(additive_inverse(X), add(X, X)))))
% 42.80/5.77 = { by lemma 30 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, multiply(X, multiply(additive_inverse(X), add(X, X)))), multiply(additive_inverse(X), add(X, X)))))
% 42.80/5.77 = { by lemma 31 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, multiply(X, multiply(additive_inverse(X), add(X, X)))), multiply(multiply(additive_inverse(X), add(X, X)), multiply(additive_inverse(X), add(X, X))))))
% 42.80/5.77 = { by lemma 30 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(additive_inverse(X), multiply(multiply(additive_inverse(X), add(X, X)), multiply(additive_inverse(X), add(X, X))))))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(additive_inverse(X), multiply(additive_inverse(X), multiply(add(X, X), multiply(additive_inverse(X), add(X, X)))))))
% 42.80/5.77 = { by lemma 27 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(X, multiply(add(X, X), multiply(additive_inverse(X), add(X, X)))))))
% 42.80/5.77 = { by lemma 24 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(add(X, X), multiply(X, multiply(additive_inverse(X), add(X, X)))))))
% 42.80/5.77 = { by lemma 24 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(X, multiply(X, multiply(additive_inverse(X), add(X, X)))))))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(multiply(X, X), multiply(additive_inverse(X), add(X, X))))))
% 42.80/5.77 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(multiply(add(X, X), multiply(X, X)), multiply(additive_inverse(X), add(X, X)))))
% 42.80/5.77 = { by lemma 23 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(additive_inverse(X), add(X, X)))))
% 42.80/5.77 = { by lemma 22 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(add(additive_inverse(X), additive_inverse(X)), X))))
% 42.80/5.77 = { by lemma 10 R->L }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(add(additive_inverse(X), additive_inverse(add(additive_inverse(X), add(X, additive_inverse(additive_inverse(X)))))), X))))
% 42.80/5.77 = { by lemma 20 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(additive_inverse(add(X, additive_inverse(additive_inverse(X)))), X))))
% 42.80/5.77 = { by lemma 12 }
% 42.80/5.77 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(add(X, X), multiply(additive_inverse(add(X, X)), X))))
% 42.80/5.77 = { by lemma 24 }
% 42.80/5.78 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(add(additive_inverse(add(X, X)), additive_inverse(add(X, X))), X))))
% 42.80/5.78 = { by lemma 22 }
% 42.80/5.78 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(additive_inverse(add(X, X)), add(X, X)))))
% 42.80/5.78 = { by lemma 29 }
% 42.80/5.78 multiply(add(X, X), multiply(add(add(X, X), add(X, X)), multiply(X, multiply(add(X, X), additive_inverse(add(X, X))))))
% 42.80/5.78 = { by lemma 24 }
% 42.80/5.78 multiply(add(X, X), multiply(add(X, X), multiply(add(X, X), multiply(add(X, X), additive_inverse(add(X, X))))))
% 42.80/5.78 = { by lemma 34 }
% 42.80/5.78 multiply(add(X, X), multiply(add(X, X), additive_inverse(add(X, X))))
% 42.80/5.78 = { by lemma 34 }
% 42.80/5.78 additive_inverse(add(X, X))
% 42.80/5.78
% 42.80/5.78 Lemma 36: add(multiply(X, Z), multiply(Y, add(Z, Z))) = multiply(add(X, add(Y, Y)), Z).
% 42.80/5.78 Proof:
% 42.80/5.78 add(multiply(X, Z), multiply(Y, add(Z, Z)))
% 42.80/5.78 = { by lemma 22 R->L }
% 42.80/5.78 add(multiply(X, Z), multiply(add(Y, Y), Z))
% 42.80/5.78 = { by axiom 8 (distribute2) R->L }
% 42.80/5.78 multiply(add(X, add(Y, Y)), Z)
% 42.80/5.78
% 42.80/5.78 Lemma 37: add(multiply(X, Y), multiply(add(X, X), Z)) = multiply(X, add(Z, add(Z, Y))).
% 42.80/5.78 Proof:
% 42.80/5.78 add(multiply(X, Y), multiply(add(X, X), Z))
% 42.80/5.78 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.78 add(multiply(add(X, X), Z), multiply(X, Y))
% 42.80/5.78 = { by lemma 22 }
% 42.80/5.78 add(multiply(X, add(Z, Z)), multiply(X, Y))
% 42.80/5.78 = { by axiom 7 (distribute1) R->L }
% 42.80/5.78 multiply(X, add(add(Z, Z), Y))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, add(Z, add(Z, Y)))
% 42.80/5.78
% 42.80/5.78 Lemma 38: multiply(additive_inverse(X), Y) = multiply(X, additive_inverse(Y)).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(additive_inverse(X), Y)
% 42.80/5.78 = { by lemma 20 R->L }
% 42.80/5.78 multiply(add(X, additive_inverse(add(X, X))), Y)
% 42.80/5.78 = { by lemma 35 R->L }
% 42.80/5.78 multiply(add(X, add(X, add(X, add(X, X)))), Y)
% 42.80/5.78 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.78 multiply(add(X, add(add(X, X), add(X, X))), Y)
% 42.80/5.78 = { by lemma 36 R->L }
% 42.80/5.78 add(multiply(X, Y), multiply(add(X, X), add(Y, Y)))
% 42.80/5.78 = { by lemma 37 }
% 42.80/5.78 multiply(X, add(add(Y, Y), add(add(Y, Y), Y)))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, add(Y, add(Y, add(add(Y, Y), Y))))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, add(Y, add(Y, add(Y, add(Y, Y)))))
% 42.80/5.78 = { by lemma 35 }
% 42.80/5.78 multiply(X, add(Y, additive_inverse(add(Y, Y))))
% 42.80/5.78 = { by lemma 20 }
% 42.80/5.78 multiply(X, additive_inverse(Y))
% 42.80/5.78
% 42.80/5.78 Lemma 39: multiply(X, multiply(Y, additive_inverse(Z))) = multiply(X, additive_inverse(multiply(Y, Z))).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, multiply(Y, additive_inverse(Z)))
% 42.80/5.78 = { by lemma 38 R->L }
% 42.80/5.78 multiply(X, multiply(additive_inverse(Y), Z))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.78 multiply(multiply(X, additive_inverse(Y)), Z)
% 42.80/5.78 = { by lemma 38 R->L }
% 42.80/5.78 multiply(multiply(additive_inverse(X), Y), Z)
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 multiply(additive_inverse(X), multiply(Y, Z))
% 42.80/5.78 = { by lemma 38 }
% 42.80/5.78 multiply(X, additive_inverse(multiply(Y, Z)))
% 42.80/5.78
% 42.80/5.78 Lemma 40: multiply(X, additive_inverse(multiply(Y, additive_inverse(Z)))) = multiply(X, multiply(Y, Z)).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, additive_inverse(multiply(Y, additive_inverse(Z))))
% 42.80/5.78 = { by lemma 39 R->L }
% 42.80/5.78 multiply(X, multiply(Y, additive_inverse(additive_inverse(Z))))
% 42.80/5.78 = { by lemma 12 }
% 42.80/5.78 multiply(X, multiply(Y, Z))
% 42.80/5.78
% 42.80/5.78 Lemma 41: multiply(X, multiply(Y, multiply(X, multiply(Y, multiply(X, Y))))) = multiply(X, Y).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, multiply(Y, multiply(X, multiply(Y, multiply(X, Y)))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.78 multiply(X, multiply(Y, multiply(multiply(X, Y), multiply(X, Y))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.78 multiply(multiply(X, Y), multiply(multiply(X, Y), multiply(X, Y)))
% 42.80/5.78 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.78 multiply(X, Y)
% 42.80/5.78
% 42.80/5.78 Lemma 42: multiply(X, additive_inverse(Y)) = additive_inverse(multiply(X, Y)).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, additive_inverse(Y))
% 42.80/5.78 = { by lemma 12 R->L }
% 42.80/5.78 additive_inverse(additive_inverse(multiply(X, additive_inverse(Y))))
% 42.80/5.78 = { by axiom 5 (x_cubed_is_x) R->L }
% 42.80/5.78 additive_inverse(multiply(additive_inverse(multiply(X, additive_inverse(Y))), multiply(additive_inverse(multiply(X, additive_inverse(Y))), additive_inverse(multiply(X, additive_inverse(Y))))))
% 42.80/5.78 = { by lemma 40 }
% 42.80/5.78 additive_inverse(multiply(additive_inverse(multiply(X, additive_inverse(Y))), multiply(additive_inverse(multiply(X, additive_inverse(Y))), multiply(X, Y))))
% 42.80/5.78 = { by lemma 27 }
% 42.80/5.78 additive_inverse(multiply(multiply(X, additive_inverse(Y)), multiply(multiply(X, additive_inverse(Y)), multiply(X, Y))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 additive_inverse(multiply(X, multiply(additive_inverse(Y), multiply(multiply(X, additive_inverse(Y)), multiply(X, Y)))))
% 42.80/5.78 = { by lemma 38 }
% 42.80/5.78 additive_inverse(multiply(X, multiply(Y, additive_inverse(multiply(multiply(X, additive_inverse(Y)), multiply(X, Y))))))
% 42.80/5.78 = { by lemma 39 }
% 42.80/5.78 additive_inverse(multiply(X, additive_inverse(multiply(Y, multiply(multiply(X, additive_inverse(Y)), multiply(X, Y))))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 additive_inverse(multiply(X, additive_inverse(multiply(Y, multiply(X, multiply(additive_inverse(Y), multiply(X, Y)))))))
% 42.80/5.78 = { by lemma 38 }
% 42.80/5.78 additive_inverse(multiply(X, additive_inverse(multiply(Y, multiply(X, multiply(Y, additive_inverse(multiply(X, Y))))))))
% 42.80/5.78 = { by lemma 39 }
% 42.80/5.78 additive_inverse(multiply(X, additive_inverse(multiply(Y, multiply(X, additive_inverse(multiply(Y, multiply(X, Y))))))))
% 42.80/5.78 = { by lemma 39 }
% 42.80/5.78 additive_inverse(multiply(X, additive_inverse(multiply(Y, additive_inverse(multiply(X, multiply(Y, multiply(X, Y))))))))
% 42.80/5.78 = { by lemma 40 }
% 42.80/5.78 additive_inverse(multiply(X, multiply(Y, multiply(X, multiply(Y, multiply(X, Y))))))
% 42.80/5.78 = { by lemma 41 }
% 42.80/5.78 additive_inverse(multiply(X, Y))
% 42.80/5.78
% 42.80/5.78 Lemma 43: multiply(additive_inverse(X), Y) = additive_inverse(multiply(X, Y)).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(additive_inverse(X), Y)
% 42.80/5.78 = { by lemma 20 R->L }
% 42.80/5.78 multiply(add(X, additive_inverse(add(X, X))), Y)
% 42.80/5.78 = { by lemma 35 R->L }
% 42.80/5.78 multiply(add(X, add(X, add(X, add(X, X)))), Y)
% 42.80/5.78 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.78 multiply(add(X, add(add(X, X), add(X, X))), Y)
% 42.80/5.78 = { by lemma 36 R->L }
% 42.80/5.78 add(multiply(X, Y), multiply(add(X, X), add(Y, Y)))
% 42.80/5.78 = { by lemma 37 }
% 42.80/5.78 multiply(X, add(add(Y, Y), add(add(Y, Y), Y)))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, add(Y, add(Y, add(add(Y, Y), Y))))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, add(Y, add(Y, add(Y, add(Y, Y)))))
% 42.80/5.78 = { by lemma 35 }
% 42.80/5.78 multiply(X, add(Y, additive_inverse(add(Y, Y))))
% 42.80/5.78 = { by lemma 20 }
% 42.80/5.78 multiply(X, additive_inverse(Y))
% 42.80/5.78 = { by lemma 42 }
% 42.80/5.78 additive_inverse(multiply(X, Y))
% 42.80/5.78
% 42.80/5.78 Lemma 44: add(multiply(X, Y), additive_inverse(multiply(Z, Y))) = multiply(add(X, additive_inverse(Z)), Y).
% 42.80/5.78 Proof:
% 42.80/5.78 add(multiply(X, Y), additive_inverse(multiply(Z, Y)))
% 42.80/5.78 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.78 add(additive_inverse(multiply(Z, Y)), multiply(X, Y))
% 42.80/5.78 = { by lemma 10 R->L }
% 42.80/5.78 add(additive_inverse(multiply(Z, Y)), multiply(add(Z, add(X, additive_inverse(Z))), Y))
% 42.80/5.78 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.78 add(multiply(add(Z, add(X, additive_inverse(Z))), Y), additive_inverse(multiply(Z, Y)))
% 42.80/5.78 = { by axiom 8 (distribute2) }
% 42.80/5.78 add(add(multiply(Z, Y), multiply(add(X, additive_inverse(Z)), Y)), additive_inverse(multiply(Z, Y)))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 add(multiply(Z, Y), add(multiply(add(X, additive_inverse(Z)), Y), additive_inverse(multiply(Z, Y))))
% 42.80/5.78 = { by axiom 1 (commutative_addition) }
% 42.80/5.78 add(multiply(Z, Y), add(additive_inverse(multiply(Z, Y)), multiply(add(X, additive_inverse(Z)), Y)))
% 42.80/5.78 = { by lemma 11 }
% 42.80/5.78 multiply(add(X, additive_inverse(Z)), Y)
% 42.80/5.78
% 42.80/5.78 Lemma 45: multiply(add(X, multiply(X, multiply(X, Y))), Z) = multiply(X, multiply(X, multiply(add(X, Y), Z))).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(add(X, multiply(X, multiply(X, Y))), Z)
% 42.80/5.78 = { by lemma 13 R->L }
% 42.80/5.78 multiply(multiply(X, add(multiply(X, Y), multiply(X, X))), Z)
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 multiply(X, multiply(add(multiply(X, Y), multiply(X, X)), Z))
% 42.80/5.78 = { by axiom 7 (distribute1) R->L }
% 42.80/5.78 multiply(X, multiply(multiply(X, add(Y, X)), Z))
% 42.80/5.78 = { by axiom 1 (commutative_addition) }
% 42.80/5.78 multiply(X, multiply(multiply(X, add(X, Y)), Z))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 multiply(X, multiply(X, multiply(add(X, Y), Z)))
% 42.80/5.78
% 42.80/5.78 Lemma 46: multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), Z)) = additive_identity.
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), Z))
% 42.80/5.78 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.78 multiply(X, multiply(add(additive_inverse(multiply(X, multiply(X, Y))), Y), Z))
% 42.80/5.78 = { by lemma 20 R->L }
% 42.80/5.78 multiply(X, multiply(add(add(X, additive_inverse(add(X, multiply(X, multiply(X, Y))))), Y), Z))
% 42.80/5.78 = { by axiom 4 (associative_addition) }
% 42.80/5.78 multiply(X, multiply(add(X, add(additive_inverse(add(X, multiply(X, multiply(X, Y)))), Y)), Z))
% 42.80/5.78 = { by axiom 1 (commutative_addition) }
% 42.80/5.78 multiply(X, multiply(add(X, add(Y, additive_inverse(add(X, multiply(X, multiply(X, Y)))))), Z))
% 42.80/5.78 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.78 multiply(X, multiply(add(add(X, Y), additive_inverse(add(X, multiply(X, multiply(X, Y))))), Z))
% 42.80/5.78 = { by lemma 44 R->L }
% 42.80/5.78 multiply(X, add(multiply(add(X, Y), Z), additive_inverse(multiply(add(X, multiply(X, multiply(X, Y))), Z))))
% 42.80/5.78 = { by lemma 45 }
% 42.80/5.78 multiply(X, add(multiply(add(X, Y), Z), additive_inverse(multiply(X, multiply(X, multiply(add(X, Y), Z))))))
% 42.80/5.78 = { by axiom 7 (distribute1) }
% 42.80/5.78 add(multiply(X, multiply(add(X, Y), Z)), multiply(X, additive_inverse(multiply(X, multiply(X, multiply(add(X, Y), Z))))))
% 42.80/5.78 = { by lemma 33 R->L }
% 42.80/5.78 add(multiply(X, multiply(X, multiply(X, multiply(add(X, Y), Z)))), multiply(X, additive_inverse(multiply(X, multiply(X, multiply(add(X, Y), Z))))))
% 42.80/5.78 = { by axiom 7 (distribute1) R->L }
% 42.80/5.78 multiply(X, add(multiply(X, multiply(X, multiply(add(X, Y), Z))), additive_inverse(multiply(X, multiply(X, multiply(add(X, Y), Z))))))
% 42.80/5.78 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.78 multiply(X, additive_identity)
% 42.80/5.78 = { by lemma 14 }
% 42.80/5.78 additive_identity
% 42.80/5.78
% 42.80/5.78 Lemma 47: multiply(X, multiply(multiply(X, Y), X)) = multiply(Y, X).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(X, multiply(multiply(X, Y), X))
% 42.80/5.78 = { by axiom 2 (right_identity) R->L }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), additive_identity)
% 42.80/5.78 = { by lemma 14 R->L }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), additive_identity))
% 42.80/5.78 = { by lemma 46 R->L }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), X))))))
% 42.80/5.78 = { by lemma 41 }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), X))
% 42.80/5.78 = { by axiom 8 (distribute2) }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), add(multiply(Y, X), multiply(additive_inverse(multiply(X, multiply(X, Y))), X)))
% 42.80/5.78 = { by lemma 43 }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), add(multiply(Y, X), additive_inverse(multiply(multiply(X, multiply(X, Y)), X))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 add(multiply(X, multiply(multiply(X, Y), X)), add(multiply(Y, X), additive_inverse(multiply(X, multiply(multiply(X, Y), X)))))
% 42.80/5.78 = { by lemma 10 }
% 42.80/5.78 multiply(Y, X)
% 42.80/5.78
% 42.80/5.78 Lemma 48: multiply(multiply(X, Y), multiply(Z, W)) = multiply(X, multiply(multiply(Y, Z), W)).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(multiply(X, Y), multiply(Z, W))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 multiply(X, multiply(Y, multiply(Z, W)))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.78 multiply(X, multiply(multiply(Y, Z), W))
% 42.80/5.78
% 42.80/5.78 Lemma 49: multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))) = multiply(Y, X).
% 42.80/5.78 Proof:
% 42.80/5.78 multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X)))
% 42.80/5.78 = { by axiom 2 (right_identity) R->L }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), additive_identity)
% 42.80/5.78 = { by lemma 14 R->L }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), additive_identity))
% 42.80/5.78 = { by lemma 46 R->L }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), multiply(X, multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), X))))))
% 42.80/5.78 = { by lemma 41 }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), multiply(add(Y, additive_inverse(multiply(X, multiply(X, Y)))), X))
% 42.80/5.78 = { by axiom 8 (distribute2) }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), multiply(additive_inverse(multiply(X, multiply(X, Y))), X)))
% 42.80/5.78 = { by lemma 43 }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(multiply(X, multiply(X, Y)), X))))
% 42.80/5.78 = { by axiom 6 (associative_multiplication) }
% 42.80/5.78 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(X, multiply(multiply(X, Y), X)))))
% 42.80/5.79 = { by lemma 47 R->L }
% 42.80/5.79 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(X, multiply(multiply(Y, multiply(multiply(Y, X), Y)), X)))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(X, multiply(Y, multiply(multiply(multiply(Y, X), Y), X))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(X, multiply(Y, multiply(multiply(Y, X), multiply(Y, X)))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 add(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))), add(multiply(Y, X), additive_inverse(multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X))))))
% 42.80/5.79 = { by lemma 10 }
% 42.80/5.79 multiply(Y, X)
% 42.80/5.79
% 42.80/5.79 Lemma 50: multiply(multiply(Y, X), multiply(Y, X)) = multiply(multiply(X, Y), multiply(Y, X)).
% 42.80/5.79 Proof:
% 42.80/5.79 multiply(multiply(Y, X), multiply(Y, X))
% 42.80/5.79 = { by lemma 9 R->L }
% 42.80/5.79 multiply(add(additive_identity, multiply(Y, X)), multiply(Y, X))
% 42.80/5.79 = { by lemma 49 R->L }
% 42.80/5.79 multiply(add(additive_identity, multiply(multiply(X, Y), multiply(multiply(Y, X), multiply(Y, X)))), multiply(Y, X))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 multiply(add(additive_identity, multiply(multiply(multiply(X, Y), multiply(Y, X)), multiply(Y, X))), multiply(Y, X))
% 42.80/5.79 = { by lemma 31 R->L }
% 42.80/5.79 multiply(add(multiply(multiply(X, Y), multiply(Y, X)), multiply(additive_identity, multiply(Y, X))), multiply(multiply(Y, X), multiply(Y, X)))
% 42.80/5.79 = { by axiom 8 (distribute2) R->L }
% 42.80/5.79 multiply(multiply(add(multiply(X, Y), additive_identity), multiply(Y, X)), multiply(multiply(Y, X), multiply(Y, X)))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 multiply(add(multiply(X, Y), additive_identity), multiply(multiply(Y, X), multiply(multiply(Y, X), multiply(Y, X))))
% 42.80/5.79 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.79 multiply(add(multiply(X, Y), additive_identity), multiply(Y, X))
% 42.80/5.79 = { by axiom 1 (commutative_addition) }
% 42.80/5.79 multiply(add(additive_identity, multiply(X, Y)), multiply(Y, X))
% 42.80/5.79 = { by lemma 9 }
% 42.80/5.79 multiply(multiply(X, Y), multiply(Y, X))
% 42.80/5.79
% 42.80/5.79 Lemma 51: multiply(multiply(Y, X), multiply(multiply(Y, X), Z)) = multiply(multiply(X, Y), multiply(multiply(Y, X), Z)).
% 42.80/5.79 Proof:
% 42.80/5.79 multiply(multiply(Y, X), multiply(multiply(Y, X), Z))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 multiply(multiply(multiply(Y, X), multiply(Y, X)), Z)
% 42.80/5.79 = { by lemma 50 }
% 42.80/5.79 multiply(multiply(multiply(X, Y), multiply(Y, X)), Z)
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 multiply(multiply(X, Y), multiply(multiply(Y, X), Z))
% 42.80/5.79
% 42.80/5.79 Lemma 52: multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))) = additive_identity.
% 42.80/5.79 Proof:
% 42.80/5.79 multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y))))
% 42.80/5.79 = { by lemma 41 R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y))))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(X, Y)), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 44 R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(multiply(Y, X), multiply(X, Y)), additive_inverse(multiply(multiply(X, Y), multiply(X, Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 48 }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(multiply(Y, X), multiply(X, Y)), additive_inverse(multiply(X, multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 42 R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(multiply(Y, X), multiply(X, Y)), multiply(X, additive_inverse(multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 50 R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(multiply(X, Y), multiply(X, Y)), multiply(X, additive_inverse(multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 multiply(multiply(X, Y), multiply(add(multiply(X, multiply(Y, multiply(X, Y))), multiply(X, additive_inverse(multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by axiom 7 (distribute1) R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(multiply(X, add(multiply(Y, multiply(X, Y)), additive_inverse(multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 multiply(multiply(X, Y), multiply(multiply(X, add(multiply(multiply(Y, X), Y), additive_inverse(multiply(multiply(Y, X), Y)))), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by axiom 3 (right_additive_inverse) }
% 42.80/5.79 multiply(multiply(X, Y), multiply(multiply(X, additive_identity), multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 14 }
% 42.80/5.79 multiply(multiply(X, Y), multiply(additive_identity, multiply(add(multiply(Y, X), additive_inverse(multiply(X, Y))), multiply(multiply(X, Y), add(multiply(Y, X), additive_inverse(multiply(X, Y)))))))
% 42.80/5.79 = { by lemma 19 }
% 42.80/5.79 multiply(multiply(X, Y), additive_identity)
% 42.80/5.79 = { by lemma 14 }
% 42.80/5.79 additive_identity
% 42.80/5.79
% 42.80/5.79 Goal 1 (prove_commutativity): multiply(a, b) = multiply(b, a).
% 42.80/5.79 Proof:
% 42.80/5.79 multiply(a, b)
% 42.80/5.79 = { by lemma 49 R->L }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(a, b), multiply(a, b)))
% 42.80/5.79 = { by lemma 50 }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(b, a), multiply(a, b)))
% 42.80/5.79 = { by lemma 11 R->L }
% 42.80/5.79 multiply(multiply(b, a), add(multiply(multiply(b, a), additive_inverse(multiply(b, a))), add(additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))), multiply(multiply(b, a), multiply(a, b)))))
% 42.80/5.79 = { by axiom 1 (commutative_addition) R->L }
% 42.80/5.79 multiply(multiply(b, a), add(multiply(multiply(b, a), additive_inverse(multiply(b, a))), add(multiply(multiply(b, a), multiply(a, b)), additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))))))
% 42.80/5.79 = { by axiom 4 (associative_addition) R->L }
% 42.80/5.79 multiply(multiply(b, a), add(add(multiply(multiply(b, a), additive_inverse(multiply(b, a))), multiply(multiply(b, a), multiply(a, b))), additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a))))))
% 42.80/5.79 = { by axiom 7 (distribute1) R->L }
% 42.80/5.79 multiply(multiply(b, a), add(multiply(multiply(b, a), add(additive_inverse(multiply(b, a)), multiply(a, b))), additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a))))))
% 42.80/5.79 = { by axiom 1 (commutative_addition) }
% 42.80/5.79 multiply(multiply(b, a), add(additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))), multiply(multiply(b, a), add(additive_inverse(multiply(b, a)), multiply(a, b)))))
% 42.80/5.79 = { by axiom 1 (commutative_addition) }
% 42.80/5.79 multiply(multiply(b, a), add(additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))), multiply(multiply(b, a), add(multiply(a, b), additive_inverse(multiply(b, a))))))
% 42.80/5.79 = { by lemma 52 }
% 42.80/5.79 multiply(multiply(b, a), add(additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))), additive_identity))
% 42.80/5.79 = { by axiom 2 (right_identity) }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(b, a), additive_inverse(multiply(b, a)))))
% 42.80/5.79 = { by axiom 2 (right_identity) R->L }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(add(multiply(b, a), additive_identity), additive_inverse(multiply(b, a)))))
% 42.80/5.79 = { by lemma 14 R->L }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(add(multiply(b, a), multiply(multiply(b, a), additive_identity)), additive_inverse(multiply(b, a)))))
% 42.80/5.79 = { by lemma 52 R->L }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(add(multiply(b, a), multiply(multiply(b, a), multiply(multiply(b, a), add(multiply(a, b), additive_inverse(multiply(b, a)))))), additive_inverse(multiply(b, a)))))
% 42.80/5.79 = { by lemma 45 }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(b, a), multiply(multiply(b, a), multiply(add(multiply(b, a), add(multiply(a, b), additive_inverse(multiply(b, a)))), additive_inverse(multiply(b, a)))))))
% 42.80/5.79 = { by lemma 10 }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(b, a), multiply(multiply(b, a), multiply(multiply(a, b), additive_inverse(multiply(b, a)))))))
% 42.80/5.79 = { by lemma 51 R->L }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(b, a), multiply(multiply(a, b), multiply(multiply(a, b), additive_inverse(multiply(b, a)))))))
% 42.80/5.79 = { by lemma 51 R->L }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(a, b), multiply(multiply(a, b), multiply(multiply(a, b), additive_inverse(multiply(b, a)))))))
% 42.80/5.79 = { by lemma 33 }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(multiply(multiply(a, b), additive_inverse(multiply(b, a)))))
% 42.80/5.79 = { by lemma 42 }
% 42.80/5.79 multiply(multiply(b, a), additive_inverse(additive_inverse(multiply(multiply(a, b), multiply(b, a)))))
% 42.80/5.79 = { by lemma 12 }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(a, b), multiply(b, a)))
% 42.80/5.79 = { by lemma 47 R->L }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(a, b), multiply(a, multiply(multiply(a, b), a))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(a, b), multiply(a, multiply(a, multiply(b, a)))))
% 42.80/5.79 = { by lemma 48 }
% 42.80/5.79 multiply(multiply(b, a), multiply(a, multiply(multiply(b, a), multiply(a, multiply(b, a)))))
% 42.80/5.79 = { by lemma 33 R->L }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(b, a), multiply(multiply(b, a), multiply(a, multiply(multiply(b, a), multiply(a, multiply(b, a)))))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) }
% 42.80/5.79 multiply(multiply(b, a), multiply(b, multiply(a, multiply(multiply(b, a), multiply(a, multiply(multiply(b, a), multiply(a, multiply(b, a))))))))
% 42.80/5.79 = { by lemma 41 }
% 42.80/5.79 multiply(multiply(b, a), multiply(b, multiply(a, multiply(b, a))))
% 42.80/5.79 = { by axiom 6 (associative_multiplication) R->L }
% 42.80/5.79 multiply(multiply(b, a), multiply(multiply(b, a), multiply(b, a)))
% 42.80/5.79 = { by axiom 5 (x_cubed_is_x) }
% 42.80/5.79 multiply(b, a)
% 42.80/5.79 % SZS output end Proof
% 42.80/5.79
% 42.80/5.79 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------