TSTP Solution File: GRP503-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP503-1 : TPTP v8.1.2. Released v2.6.0.
% 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 : n008.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 01:18:41 EDT 2023
% Result : Unsatisfiable 0.20s 0.52s
% Output : Proof 0.20s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP503-1 : TPTP v8.1.2. Released v2.6.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.17/0.35 % Computer : n008.cluster.edu
% 0.17/0.35 % Model : x86_64 x86_64
% 0.17/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.35 % Memory : 8042.1875MB
% 0.17/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.35 % CPULimit : 300
% 0.17/0.35 % WCLimit : 300
% 0.17/0.35 % DateTime : Tue Aug 29 02:42:32 EDT 2023
% 0.17/0.35 % CPUTime :
% 0.20/0.52 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.20/0.52
% 0.20/0.52 % SZS status Unsatisfiable
% 0.20/0.52
% 0.20/0.59 % SZS output start Proof
% 0.20/0.59 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 0.20/0.59 Axiom 2 (single_axiom): double_divide(double_divide(X, inverse(double_divide(Y, Z))), double_divide(inverse(Y), inverse(double_divide(W, double_divide(X, W))))) = Z.
% 0.20/0.59
% 0.20/0.59 Lemma 3: double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W))) = Y.
% 0.20/0.59 Proof:
% 0.20/0.59 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W)))
% 0.20/0.59 = { by axiom 1 (multiply) }
% 0.20/0.59 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 0.20/0.59 = { by axiom 1 (multiply) }
% 0.20/0.59 double_divide(double_divide(X, inverse(double_divide(Z, Y))), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 0.20/0.59 = { by axiom 2 (single_axiom) }
% 0.20/0.59 Y
% 0.20/0.59
% 0.20/0.59 Lemma 4: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X))) = inverse(W).
% 0.20/0.59 Proof:
% 0.20/0.59 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X)))
% 0.20/0.59 = { by axiom 1 (multiply) }
% 0.20/0.59 inverse(double_divide(double_divide(Y, multiply(W, X)), double_divide(inverse(X), multiply(double_divide(Y, Z), Z))))
% 0.20/0.59 = { by lemma 3 }
% 0.20/0.59 inverse(W)
% 0.20/0.59
% 0.20/0.59 Lemma 5: multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X)))) = inverse(V).
% 0.20/0.59 Proof:
% 0.20/0.59 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X))))
% 0.20/0.59 = { by axiom 1 (multiply) }
% 0.20/0.59 multiply(double_divide(inverse(double_divide(Y, X)), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X))))
% 0.20/0.59 = { by lemma 4 }
% 0.20/0.59 inverse(V)
% 0.20/0.59
% 0.20/0.59 Lemma 6: multiply(double_divide(X, Z), Z) = multiply(double_divide(X, Y), Y).
% 0.20/0.59 Proof:
% 0.20/0.59 multiply(double_divide(X, Z), Z)
% 0.20/0.60 = { by lemma 3 R->L }
% 0.20/0.60 double_divide(double_divide(W, multiply(multiply(double_divide(X, Z), Z), multiply(V, U))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by axiom 1 (multiply) }
% 0.20/0.60 double_divide(double_divide(W, inverse(double_divide(multiply(V, U), multiply(double_divide(X, Z), Z)))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by lemma 4 R->L }
% 0.20/0.60 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(S, double_divide(U, V)))), multiply(double_divide(X2, Y2), Y2)), double_divide(X2, multiply(double_divide(multiply(V, U), multiply(double_divide(X, Z), Z)), double_divide(X, multiply(S, double_divide(U, V))))))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by lemma 5 }
% 0.20/0.60 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(S, double_divide(U, V)))), multiply(double_divide(X2, Y2), Y2)), double_divide(X2, inverse(S)))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by lemma 5 R->L }
% 0.20/0.60 double_divide(double_divide(W, multiply(double_divide(inverse(double_divide(X, multiply(S, double_divide(U, V)))), multiply(double_divide(X2, Y2), Y2)), double_divide(X2, multiply(double_divide(multiply(V, U), multiply(double_divide(X, Y), Y)), double_divide(X, multiply(S, double_divide(U, V))))))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by lemma 4 }
% 0.20/0.60 double_divide(double_divide(W, inverse(double_divide(multiply(V, U), multiply(double_divide(X, Y), Y)))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by axiom 1 (multiply) R->L }
% 0.20/0.60 double_divide(double_divide(W, multiply(multiply(double_divide(X, Y), Y), multiply(V, U))), double_divide(inverse(multiply(V, U)), multiply(double_divide(W, T), T)))
% 0.20/0.60 = { by lemma 3 }
% 0.20/0.60 multiply(double_divide(X, Y), Y)
% 0.20/0.61
% 0.20/0.61 Lemma 7: multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, U), U))) = multiply(double_divide(Y, X), V).
% 0.20/0.61 Proof:
% 0.20/0.61 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, U), U)))
% 0.20/0.61 = { by lemma 6 }
% 0.20/0.61 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(double_divide(V, double_divide(Y, X)), double_divide(Y, X))))
% 0.20/0.61 = { by lemma 5 }
% 0.20/0.61 inverse(double_divide(V, double_divide(Y, X)))
% 0.20/0.61 = { by axiom 1 (multiply) R->L }
% 0.20/0.61 multiply(double_divide(Y, X), V)
% 0.20/0.61
% 0.20/0.61 Lemma 8: multiply(double_divide(W, double_divide(Y, W)), Z) = multiply(double_divide(X, double_divide(Y, X)), Z).
% 0.20/0.61 Proof:
% 0.20/0.61 multiply(double_divide(W, double_divide(Y, W)), Z)
% 0.20/0.61 = { by lemma 7 R->L }
% 0.20/0.61 multiply(double_divide(multiply(double_divide(Y, W), W), multiply(double_divide(V, U), U)), double_divide(V, multiply(double_divide(Z, T), T)))
% 0.20/0.61 = { by lemma 6 }
% 0.20/0.61 multiply(double_divide(multiply(double_divide(Y, X), X), multiply(double_divide(V, U), U)), double_divide(V, multiply(double_divide(Z, T), T)))
% 0.20/0.61 = { by lemma 7 }
% 0.20/0.61 multiply(double_divide(X, double_divide(Y, X)), Z)
% 0.20/0.61
% 0.20/0.61 Lemma 9: double_divide(Z, double_divide(Y, Z)) = double_divide(X, double_divide(Y, X)).
% 0.20/0.61 Proof:
% 0.20/0.61 double_divide(Z, double_divide(Y, Z))
% 0.20/0.61 = { by lemma 3 R->L }
% 0.20/0.61 double_divide(double_divide(W, multiply(double_divide(Z, double_divide(Y, Z)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 0.20/0.61 = { by lemma 8 }
% 0.20/0.61 double_divide(double_divide(W, multiply(double_divide(X, double_divide(Y, X)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 0.20/0.61 = { by lemma 3 }
% 0.20/0.62 double_divide(X, double_divide(Y, X))
% 0.20/0.62
% 0.20/0.62 Lemma 10: double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(X, V), V))) = Y.
% 0.20/0.62 Proof:
% 0.20/0.62 double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(X, V), V)))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(inverse(double_divide(Z, W)), multiply(double_divide(X, V), V)))
% 0.20/0.62 = { by lemma 3 }
% 0.20/0.62 Y
% 0.20/0.62
% 0.20/0.62 Lemma 11: multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, Z)) = multiply(double_divide(Z, W), W).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, Z))
% 0.20/0.62 = { by lemma 8 }
% 0.20/0.62 multiply(double_divide(Z, double_divide(Y, Z)), double_divide(Y, Z))
% 0.20/0.62 = { by lemma 6 R->L }
% 0.20/0.62 multiply(double_divide(Z, W), W)
% 0.20/0.62
% 0.20/0.62 Lemma 12: multiply(double_divide(X, multiply(double_divide(Y, Z), Z)), W) = multiply(multiply(double_divide(X, V), V), multiply(Y, W)).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(X, multiply(double_divide(Y, Z), Z)), W)
% 0.20/0.62 = { by lemma 7 R->L }
% 0.20/0.62 multiply(double_divide(multiply(multiply(double_divide(Y, Z), Z), X), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(W, S), S)))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 multiply(double_divide(multiply(multiply(double_divide(Y, Z), Z), X), multiply(double_divide(U, T), T)), double_divide(U, inverse(double_divide(S, double_divide(W, S)))))
% 0.20/0.62 = { by lemma 11 R->L }
% 0.20/0.62 multiply(double_divide(multiply(multiply(double_divide(S, double_divide(W, S)), double_divide(W, Y)), X), multiply(double_divide(U, T), T)), double_divide(U, inverse(double_divide(S, double_divide(W, S)))))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 multiply(double_divide(inverse(double_divide(X, multiply(double_divide(S, double_divide(W, S)), double_divide(W, Y)))), multiply(double_divide(U, T), T)), double_divide(U, inverse(double_divide(S, double_divide(W, S)))))
% 0.20/0.62 = { by lemma 5 R->L }
% 0.20/0.62 multiply(double_divide(inverse(double_divide(X, multiply(double_divide(S, double_divide(W, S)), double_divide(W, Y)))), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(multiply(Y, W), multiply(double_divide(X, V), V)), double_divide(X, multiply(double_divide(S, double_divide(W, S)), double_divide(W, Y))))))
% 0.20/0.62 = { by lemma 4 }
% 0.20/0.62 inverse(double_divide(multiply(Y, W), multiply(double_divide(X, V), V)))
% 0.20/0.62 = { by axiom 1 (multiply) R->L }
% 0.20/0.62 multiply(multiply(double_divide(X, V), V), multiply(Y, W))
% 0.20/0.62
% 0.20/0.62 Lemma 13: multiply(multiply(double_divide(inverse(X), Y), Y), multiply(Z, double_divide(Z, multiply(W, X)))) = inverse(W).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(multiply(double_divide(inverse(X), Y), Y), multiply(Z, double_divide(Z, multiply(W, X))))
% 0.20/0.62 = { by lemma 12 R->L }
% 0.20/0.62 multiply(double_divide(inverse(X), multiply(double_divide(Z, V), V)), double_divide(Z, multiply(W, X)))
% 0.20/0.62 = { by lemma 4 }
% 0.20/0.62 inverse(W)
% 0.20/0.62
% 0.20/0.62 Lemma 14: double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V))) = double_divide(Y, W).
% 0.20/0.62 Proof:
% 0.20/0.62 double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 0.20/0.62 = { by lemma 6 }
% 0.20/0.62 double_divide(double_divide(X, multiply(double_divide(Y, W), W)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 0.20/0.62 = { by lemma 3 }
% 0.20/0.62 double_divide(Y, W)
% 0.20/0.62
% 0.20/0.62 Lemma 15: double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V))) = double_divide(X, Z).
% 0.20/0.62 Proof:
% 0.20/0.62 double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V)))
% 0.20/0.62 = { by lemma 10 R->L }
% 0.20/0.62 double_divide(double_divide(U, multiply(double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V))), double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V)))), double_divide(multiply(multiply(double_divide(inverse(W), V), V), multiply(Y, double_divide(Y, multiply(Z, W)))), multiply(double_divide(U, T), T)))
% 0.20/0.62 = { by lemma 13 }
% 0.20/0.62 double_divide(double_divide(U, multiply(double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V))), double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V)))), double_divide(inverse(Z), multiply(double_divide(U, T), T)))
% 0.20/0.62 = { by lemma 14 }
% 0.20/0.62 double_divide(X, Z)
% 0.20/0.62
% 0.20/0.62 Lemma 16: double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), X)) = double_divide(multiply(double_divide(inverse(W), V), V), Z).
% 0.20/0.62 Proof:
% 0.20/0.62 double_divide(X, double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), X))
% 0.20/0.62 = { by lemma 9 }
% 0.20/0.62 double_divide(multiply(double_divide(inverse(W), V), V), double_divide(multiply(Y, double_divide(Y, multiply(Z, W))), multiply(double_divide(inverse(W), V), V)))
% 0.20/0.62 = { by lemma 15 }
% 0.20/0.62 double_divide(multiply(double_divide(inverse(W), V), V), Z)
% 0.20/0.62
% 0.20/0.62 Lemma 17: double_divide(double_divide(inverse(X), multiply(Y, double_divide(double_divide(Z, multiply(W, X)), Z))), W) = Y.
% 0.20/0.62 Proof:
% 0.20/0.62 double_divide(double_divide(inverse(X), multiply(Y, double_divide(double_divide(Z, multiply(W, X)), Z))), W)
% 0.20/0.62 = { by lemma 15 R->L }
% 0.20/0.62 double_divide(double_divide(inverse(X), multiply(Y, double_divide(double_divide(Z, multiply(W, X)), Z))), double_divide(multiply(Z, double_divide(Z, multiply(W, X))), multiply(double_divide(inverse(X), V), V)))
% 0.20/0.62 = { by lemma 10 }
% 0.20/0.62 Y
% 0.20/0.62
% 0.20/0.62 Lemma 18: multiply(double_divide(multiply(X, double_divide(multiply(double_divide(multiply(Y, Z), W), W), multiply(Y, Z))), V), V) = inverse(X).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(multiply(X, double_divide(multiply(double_divide(multiply(Y, Z), W), W), multiply(Y, Z))), V), V)
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 multiply(double_divide(multiply(X, double_divide(multiply(double_divide(inverse(double_divide(Z, Y)), W), W), multiply(Y, Z))), V), V)
% 0.20/0.62 = { by lemma 16 R->L }
% 0.20/0.62 multiply(double_divide(multiply(X, double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U))), V), V)
% 0.20/0.62 = { by lemma 11 R->L }
% 0.20/0.62 multiply(double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)), double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), multiply(X, double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)))))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 multiply(double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)), double_divide(multiply(T, double_divide(T, multiply(inverse(double_divide(Z, Y)), double_divide(Z, Y)))), multiply(X, double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)))))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 inverse(double_divide(double_divide(multiply(T, double_divide(T, multiply(inverse(double_divide(Z, Y)), double_divide(Z, Y)))), multiply(X, double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)))), double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U))))
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 inverse(double_divide(double_divide(inverse(double_divide(double_divide(T, multiply(inverse(double_divide(Z, Y)), double_divide(Z, Y))), T)), multiply(X, double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)))), double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U))))
% 0.20/0.62 = { by lemma 17 R->L }
% 0.20/0.62 inverse(double_divide(double_divide(inverse(double_divide(double_divide(T, multiply(inverse(double_divide(Z, Y)), double_divide(Z, Y))), T)), multiply(X, double_divide(double_divide(inverse(double_divide(Z, Y)), multiply(double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U)), double_divide(double_divide(T, multiply(inverse(double_divide(Z, Y)), double_divide(Z, Y))), T))), inverse(double_divide(Z, Y))))), double_divide(U, double_divide(multiply(T, double_divide(T, multiply(multiply(Y, Z), double_divide(Z, Y)))), U))))
% 0.20/0.62 = { by lemma 17 }
% 0.20/0.62 inverse(X)
% 0.20/0.62
% 0.20/0.62 Lemma 19: multiply(double_divide(multiply(X, double_divide(X, multiply(Y, Z))), W), W) = multiply(Y, multiply(double_divide(inverse(Z), V), V)).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(multiply(X, double_divide(X, multiply(Y, Z))), W), W)
% 0.20/0.62 = { by axiom 1 (multiply) }
% 0.20/0.62 inverse(double_divide(W, double_divide(multiply(X, double_divide(X, multiply(Y, Z))), W)))
% 0.20/0.62 = { by lemma 16 }
% 0.20/0.62 inverse(double_divide(multiply(double_divide(inverse(Z), V), V), Y))
% 0.20/0.62 = { by axiom 1 (multiply) R->L }
% 0.20/0.62 multiply(Y, multiply(double_divide(inverse(Z), V), V))
% 0.20/0.62
% 0.20/0.62 Lemma 20: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V))) = multiply(X, W).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V)))
% 0.20/0.62 = { by lemma 6 }
% 0.20/0.62 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, X), X)))
% 0.20/0.62 = { by lemma 4 }
% 0.20/0.62 inverse(double_divide(W, X))
% 0.20/0.62 = { by axiom 1 (multiply) R->L }
% 0.20/0.62 multiply(X, W)
% 0.20/0.62
% 0.20/0.62 Lemma 21: multiply(double_divide(multiply(double_divide(inverse(X), Y), Y), Z), W) = multiply(multiply(double_divide(multiply(Z, X), V), V), W).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(multiply(double_divide(inverse(X), Y), Y), Z), W)
% 0.20/0.62 = { by lemma 7 R->L }
% 0.20/0.62 multiply(double_divide(multiply(Z, multiply(double_divide(inverse(X), Y), Y)), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(W, S), S)))
% 0.20/0.62 = { by lemma 19 R->L }
% 0.20/0.62 multiply(double_divide(multiply(double_divide(multiply(multiply(double_divide(multiply(Z, X), V), V), double_divide(multiply(double_divide(multiply(Z, X), V), V), multiply(Z, X))), X2), X2), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(W, S), S)))
% 0.20/0.62 = { by lemma 18 }
% 0.20/0.62 multiply(double_divide(inverse(multiply(double_divide(multiply(Z, X), V), V)), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(W, S), S)))
% 0.20/0.62 = { by lemma 20 }
% 0.20/0.62 multiply(multiply(double_divide(multiply(Z, X), V), V), W)
% 0.20/0.62
% 0.20/0.62 Lemma 22: multiply(double_divide(Z, inverse(Z)), Y) = multiply(double_divide(X, inverse(X)), Y).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(Z, inverse(Z)), Y)
% 0.20/0.62 = { by lemma 7 R->L }
% 0.20/0.62 multiply(double_divide(multiply(inverse(Z), Z), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 18 R->L }
% 0.20/0.62 multiply(double_divide(multiply(multiply(double_divide(multiply(Z, double_divide(multiply(double_divide(multiply(T, S), X2), X2), multiply(T, S))), W2), W2), Z), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 21 R->L }
% 0.20/0.62 multiply(double_divide(multiply(double_divide(multiply(double_divide(inverse(double_divide(multiply(double_divide(multiply(T, S), X2), X2), multiply(T, S))), Z2), Z2), Z), Z), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 6 }
% 0.20/0.62 multiply(double_divide(multiply(double_divide(multiply(double_divide(inverse(double_divide(multiply(double_divide(multiply(T, S), X2), X2), multiply(T, S))), Z2), Z2), X), X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 21 }
% 0.20/0.62 multiply(double_divide(multiply(multiply(double_divide(multiply(X, double_divide(multiply(double_divide(multiply(T, S), X2), X2), multiply(T, S))), Y2), Y2), X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 18 }
% 0.20/0.62 multiply(double_divide(multiply(inverse(X), X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(Y, U), U)))
% 0.20/0.62 = { by lemma 7 }
% 0.20/0.62 multiply(double_divide(X, inverse(X)), Y)
% 0.20/0.62
% 0.20/0.62 Lemma 23: multiply(double_divide(X, inverse(X)), inverse(Y)) = multiply(double_divide(Y, Z), Z).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(double_divide(X, inverse(X)), inverse(Y))
% 0.20/0.62 = { by lemma 22 }
% 0.20/0.62 multiply(double_divide(Y, inverse(Y)), inverse(Y))
% 0.20/0.62 = { by lemma 6 R->L }
% 0.20/0.62 multiply(double_divide(Y, Z), Z)
% 0.20/0.62
% 0.20/0.62 Lemma 24: multiply(multiply(double_divide(multiply(X, double_divide(X, multiply(Y, Z))), W), W), multiply(inverse(Z), V)) = multiply(Y, V).
% 0.20/0.62 Proof:
% 0.20/0.62 multiply(multiply(double_divide(multiply(X, double_divide(X, multiply(Y, Z))), W), W), multiply(inverse(Z), V))
% 0.20/0.62 = { by lemma 12 R->L }
% 0.20/0.63 multiply(double_divide(multiply(X, double_divide(X, multiply(Y, Z))), multiply(double_divide(inverse(Z), U), U)), V)
% 0.20/0.63 = { by lemma 7 R->L }
% 0.20/0.63 multiply(double_divide(multiply(multiply(double_divide(inverse(Z), U), U), multiply(X, double_divide(X, multiply(Y, Z)))), multiply(double_divide(T, S), S)), double_divide(T, multiply(double_divide(V, X2), X2)))
% 0.20/0.63 = { by lemma 13 }
% 0.20/0.63 multiply(double_divide(inverse(Y), multiply(double_divide(T, S), S)), double_divide(T, multiply(double_divide(V, X2), X2)))
% 0.20/0.63 = { by lemma 20 }
% 0.20/0.63 multiply(Y, V)
% 0.20/0.63
% 0.20/0.63 Lemma 25: multiply(X, double_divide(inverse(Y), multiply(Z, double_divide(double_divide(W, multiply(X, Y)), W)))) = inverse(Z).
% 0.20/0.63 Proof:
% 0.20/0.63 multiply(X, double_divide(inverse(Y), multiply(Z, double_divide(double_divide(W, multiply(X, Y)), W))))
% 0.20/0.63 = { by lemma 24 R->L }
% 0.20/0.63 multiply(multiply(double_divide(multiply(W, double_divide(W, multiply(X, Y))), V), V), multiply(inverse(Y), double_divide(inverse(Y), multiply(Z, double_divide(double_divide(W, multiply(X, Y)), W)))))
% 0.20/0.63 = { by lemma 12 R->L }
% 0.20/0.63 multiply(double_divide(multiply(W, double_divide(W, multiply(X, Y))), multiply(double_divide(inverse(Y), U), U)), double_divide(inverse(Y), multiply(Z, double_divide(double_divide(W, multiply(X, Y)), W))))
% 0.20/0.63 = { by lemma 5 }
% 0.20/0.63 inverse(Z)
% 0.20/0.63
% 0.20/0.63 Lemma 26: multiply(double_divide(multiply(X, multiply(double_divide(multiply(inverse(Y), Y), Z), Z)), W), W) = inverse(X).
% 0.20/0.63 Proof:
% 0.20/0.63 multiply(double_divide(multiply(X, multiply(double_divide(multiply(inverse(Y), Y), Z), Z)), W), W)
% 0.20/0.63 = { by lemma 25 R->L }
% 0.20/0.63 multiply(double_divide(multiply(X, multiply(double_divide(multiply(multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T)))), Y), Z), Z)), W), W)
% 0.20/0.64 = { by lemma 3 R->L }
% 0.20/0.64 multiply(double_divide(multiply(X, double_divide(double_divide(S, multiply(multiply(double_divide(multiply(multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T)))), Y), Z), Z), X2)), double_divide(inverse(X2), multiply(double_divide(S, Y2), Y2)))), W), W)
% 0.20/0.64 = { by lemma 21 R->L }
% 0.20/0.64 multiply(double_divide(multiply(X, double_divide(double_divide(S, multiply(double_divide(multiply(double_divide(inverse(Y), Z2), Z2), multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T))))), X2)), double_divide(inverse(X2), multiply(double_divide(S, Y2), Y2)))), W), W)
% 0.20/0.64 = { by lemma 3 }
% 0.20/0.64 multiply(double_divide(multiply(X, double_divide(multiply(double_divide(inverse(Y), Z2), Z2), multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T)))))), W), W)
% 0.20/0.64 = { by lemma 25 R->L }
% 0.20/0.64 multiply(double_divide(multiply(X, double_divide(multiply(double_divide(multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T)))), Z2), Z2), multiply(V, double_divide(inverse(U), multiply(Y, double_divide(double_divide(T, multiply(V, U)), T)))))), W), W)
% 0.20/0.64 = { by lemma 18 }
% 0.20/0.64 inverse(X)
% 0.20/0.64
% 0.20/0.64 Lemma 27: multiply(multiply(double_divide(multiply(inverse(X), X), Y), Y), Z) = multiply(double_divide(multiply(double_divide(Z, W), W), V), V).
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(multiply(double_divide(multiply(inverse(X), X), Y), Y), Z)
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 inverse(double_divide(Z, multiply(double_divide(multiply(inverse(X), X), Y), Y)))
% 0.20/0.64 = { by lemma 26 R->L }
% 0.20/0.64 multiply(double_divide(multiply(double_divide(Z, multiply(double_divide(multiply(inverse(X), X), Y), Y)), multiply(double_divide(multiply(inverse(X), X), Y), Y)), V), V)
% 0.20/0.64 = { by lemma 6 R->L }
% 0.20/0.64 multiply(double_divide(multiply(double_divide(Z, W), W), V), V)
% 0.20/0.64
% 0.20/0.64 Lemma 28: double_divide(multiply(double_divide(Z, W), W), Z) = double_divide(multiply(double_divide(X, Y), Y), X).
% 0.20/0.64 Proof:
% 0.20/0.64 double_divide(multiply(double_divide(Z, W), W), Z)
% 0.20/0.64 = { by lemma 14 R->L }
% 0.20/0.64 double_divide(double_divide(Y2, multiply(double_divide(multiply(double_divide(Z, W), W), W2), W2)), double_divide(inverse(Z), multiply(double_divide(Y2, Z2), Z2)))
% 0.20/0.64 = { by lemma 27 R->L }
% 0.20/0.64 double_divide(double_divide(Y2, multiply(multiply(double_divide(multiply(inverse(S), S), X2), X2), Z)), double_divide(inverse(Z), multiply(double_divide(Y2, Z2), Z2)))
% 0.20/0.64 = { by lemma 3 }
% 0.20/0.64 multiply(double_divide(multiply(inverse(S), S), X2), X2)
% 0.20/0.64 = { by lemma 3 R->L }
% 0.20/0.64 double_divide(double_divide(V, multiply(multiply(double_divide(multiply(inverse(S), S), X2), X2), X)), double_divide(inverse(X), multiply(double_divide(V, T), T)))
% 0.20/0.64 = { by lemma 27 }
% 0.20/0.64 double_divide(double_divide(V, multiply(double_divide(multiply(double_divide(X, Y), Y), U), U)), double_divide(inverse(X), multiply(double_divide(V, T), T)))
% 0.20/0.64 = { by lemma 14 }
% 0.20/0.64 double_divide(multiply(double_divide(X, Y), Y), X)
% 0.20/0.64
% 0.20/0.64 Lemma 29: double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z))) = double_divide(W, double_divide(Y, W)).
% 0.20/0.64 Proof:
% 0.20/0.64 double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z)))
% 0.20/0.64 = { by lemma 9 }
% 0.20/0.64 double_divide(double_divide(X, Y), double_divide(Y, double_divide(X, Y)))
% 0.20/0.64 = { by lemma 9 R->L }
% 0.20/0.64 double_divide(W, double_divide(Y, W))
% 0.20/0.64
% 0.20/0.64 Lemma 30: double_divide(double_divide(multiply(double_divide(X, Y), Y), X), double_divide(Z, double_divide(inverse(W), Z))) = W.
% 0.20/0.64 Proof:
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(X, Y), Y), X), double_divide(Z, double_divide(inverse(W), Z)))
% 0.20/0.64 = { by lemma 28 }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(multiply(W, double_divide(multiply(double_divide(multiply(V, U), T), T), multiply(V, U))), S), S), multiply(W, double_divide(multiply(double_divide(multiply(V, U), T), T), multiply(V, U)))), double_divide(Z, double_divide(inverse(W), Z)))
% 0.20/0.64 = { by lemma 18 }
% 0.20/0.64 double_divide(double_divide(inverse(W), multiply(W, double_divide(multiply(double_divide(multiply(V, U), T), T), multiply(V, U)))), double_divide(Z, double_divide(inverse(W), Z)))
% 0.20/0.64 = { by lemma 29 }
% 0.20/0.64 double_divide(X2, double_divide(multiply(W, double_divide(multiply(double_divide(multiply(V, U), T), T), multiply(V, U))), X2))
% 0.20/0.64 = { by lemma 28 }
% 0.20/0.64 double_divide(X2, double_divide(multiply(W, double_divide(multiply(double_divide(inverse(Y2), Z2), Z2), inverse(Y2))), X2))
% 0.20/0.64 = { by lemma 16 R->L }
% 0.20/0.64 double_divide(X2, double_divide(multiply(W, double_divide(W2, double_divide(multiply(V2, double_divide(V2, multiply(inverse(Y2), Y2))), W2))), X2))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 double_divide(X2, double_divide(multiply(W, double_divide(W2, double_divide(inverse(double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2)), W2))), X2))
% 0.20/0.64 = { by lemma 17 R->L }
% 0.20/0.64 double_divide(X2, double_divide(multiply(W, double_divide(double_divide(inverse(Y2), multiply(double_divide(W2, double_divide(inverse(double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2)), W2)), double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2))), inverse(Y2))), X2))
% 0.20/0.64 = { by lemma 29 R->L }
% 0.20/0.64 double_divide(double_divide(inverse(double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2)), multiply(W, double_divide(double_divide(inverse(Y2), multiply(double_divide(W2, double_divide(inverse(double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2)), W2)), double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2))), inverse(Y2)))), double_divide(W2, double_divide(inverse(double_divide(double_divide(V2, multiply(inverse(Y2), Y2)), V2)), W2)))
% 0.20/0.64 = { by lemma 17 }
% 0.20/0.64 W
% 0.20/0.64
% 0.20/0.64 Lemma 31: double_divide(inverse(X), multiply(Y, multiply(double_divide(Y, Z), Z))) = X.
% 0.20/0.64 Proof:
% 0.20/0.64 double_divide(inverse(X), multiply(Y, multiply(double_divide(Y, Z), Z)))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 double_divide(inverse(X), inverse(double_divide(multiply(double_divide(Y, Z), Z), Y)))
% 0.20/0.64 = { by lemma 28 }
% 0.20/0.64 double_divide(inverse(X), inverse(double_divide(multiply(double_divide(inverse(W), V), V), inverse(W))))
% 0.20/0.64 = { by axiom 1 (multiply) R->L }
% 0.20/0.64 double_divide(inverse(X), multiply(inverse(W), multiply(double_divide(inverse(W), V), V)))
% 0.20/0.64 = { by lemma 19 R->L }
% 0.20/0.64 double_divide(inverse(X), multiply(double_divide(multiply(U, double_divide(U, multiply(inverse(W), W))), T), T))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 double_divide(inverse(X), multiply(double_divide(inverse(double_divide(double_divide(U, multiply(inverse(W), W)), U)), T), T))
% 0.20/0.64 = { by lemma 25 R->L }
% 0.20/0.64 double_divide(multiply(inverse(W), double_divide(inverse(W), multiply(X, double_divide(double_divide(U, multiply(inverse(W), W)), U)))), multiply(double_divide(inverse(double_divide(double_divide(U, multiply(inverse(W), W)), U)), T), T))
% 0.20/0.64 = { by lemma 30 R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(S, X2), X2), S), double_divide(Y2, double_divide(inverse(double_divide(multiply(inverse(W), double_divide(inverse(W), multiply(X, double_divide(double_divide(U, multiply(inverse(W), W)), U)))), multiply(double_divide(inverse(double_divide(double_divide(U, multiply(inverse(W), W)), U)), T), T))), Y2)))
% 0.20/0.64 = { by axiom 1 (multiply) R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(S, X2), X2), S), double_divide(Y2, double_divide(multiply(multiply(double_divide(inverse(double_divide(double_divide(U, multiply(inverse(W), W)), U)), T), T), multiply(inverse(W), double_divide(inverse(W), multiply(X, double_divide(double_divide(U, multiply(inverse(W), W)), U))))), Y2)))
% 0.20/0.64 = { by lemma 13 }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(S, X2), X2), S), double_divide(Y2, double_divide(inverse(X), Y2)))
% 0.20/0.64 = { by lemma 30 }
% 0.20/0.64 X
% 0.20/0.64
% 0.20/0.64 Lemma 32: multiply(double_divide(multiply(inverse(X), X), Y), Y) = double_divide(multiply(double_divide(Z, W), W), Z).
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(double_divide(multiply(inverse(X), X), Y), Y)
% 0.20/0.64 = { by lemma 3 R->L }
% 0.20/0.64 double_divide(double_divide(V, multiply(multiply(double_divide(multiply(inverse(X), X), Y), Y), Z)), double_divide(inverse(Z), multiply(double_divide(V, U), U)))
% 0.20/0.64 = { by lemma 27 }
% 0.20/0.64 double_divide(double_divide(V, multiply(double_divide(multiply(double_divide(Z, W), W), T), T)), double_divide(inverse(Z), multiply(double_divide(V, U), U)))
% 0.20/0.64 = { by lemma 14 }
% 0.20/0.64 double_divide(multiply(double_divide(Z, W), W), Z)
% 0.20/0.64
% 0.20/0.64 Lemma 33: inverse(multiply(X, multiply(double_divide(X, Y), Y))) = double_divide(multiply(double_divide(Z, W), W), Z).
% 0.20/0.64 Proof:
% 0.20/0.64 inverse(multiply(X, multiply(double_divide(X, Y), Y)))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 inverse(inverse(double_divide(multiply(double_divide(X, Y), Y), X)))
% 0.20/0.64 = { by lemma 26 R->L }
% 0.20/0.64 multiply(double_divide(multiply(inverse(double_divide(multiply(double_divide(X, Y), Y), X)), multiply(double_divide(multiply(inverse(V), V), U), U)), T), T)
% 0.20/0.64 = { by lemma 32 }
% 0.20/0.64 multiply(double_divide(multiply(inverse(double_divide(multiply(double_divide(X, Y), Y), X)), double_divide(multiply(double_divide(X, Y), Y), X)), T), T)
% 0.20/0.64 = { by lemma 32 }
% 0.20/0.64 double_divide(multiply(double_divide(Z, W), W), Z)
% 0.20/0.64
% 0.20/0.64 Lemma 34: multiply(X, multiply(double_divide(X, Y), Y)) = double_divide(Z, inverse(Z)).
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(X, multiply(double_divide(X, Y), Y))
% 0.20/0.64 = { by lemma 31 R->L }
% 0.20/0.64 double_divide(inverse(multiply(X, multiply(double_divide(X, Y), Y))), multiply(W, multiply(double_divide(W, V), V)))
% 0.20/0.64 = { by lemma 33 }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(W, V), V), W), multiply(W, multiply(double_divide(W, V), V)))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(W, V), V), W), inverse(double_divide(multiply(double_divide(W, V), V), W)))
% 0.20/0.64 = { by lemma 3 R->L }
% 0.20/0.64 double_divide(double_divide(U, multiply(double_divide(double_divide(multiply(double_divide(W, V), V), W), inverse(double_divide(multiply(double_divide(W, V), V), W))), T)), double_divide(inverse(T), multiply(double_divide(U, S), S)))
% 0.20/0.64 = { by lemma 22 R->L }
% 0.20/0.64 double_divide(double_divide(U, multiply(double_divide(Z, inverse(Z)), T)), double_divide(inverse(T), multiply(double_divide(U, S), S)))
% 0.20/0.64 = { by lemma 3 }
% 0.20/0.64 double_divide(Z, inverse(Z))
% 0.20/0.64
% 0.20/0.64 Lemma 35: multiply(double_divide(X, Y), Y) = inverse(X).
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(double_divide(X, Y), Y)
% 0.20/0.64 = { by lemma 23 R->L }
% 0.20/0.64 multiply(double_divide(Z, inverse(Z)), inverse(X))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 inverse(double_divide(inverse(X), double_divide(Z, inverse(Z))))
% 0.20/0.64 = { by lemma 34 R->L }
% 0.20/0.64 inverse(double_divide(inverse(X), multiply(W, multiply(double_divide(W, V), V))))
% 0.20/0.64 = { by lemma 31 }
% 0.20/0.64 inverse(X)
% 0.20/0.64
% 0.20/0.64 Lemma 36: multiply(X, multiply(inverse(Y), Y)) = X.
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(X, multiply(inverse(Y), Y))
% 0.20/0.64 = { by lemma 30 R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(multiply(X, multiply(inverse(Y), Y))), V)))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(multiply(X, inverse(double_divide(Y, inverse(Y))))), V)))
% 0.20/0.64 = { by lemma 34 R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(multiply(X, inverse(multiply(U, multiply(double_divide(U, T), T))))), V)))
% 0.20/0.64 = { by lemma 33 }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(multiply(X, double_divide(multiply(double_divide(S, X2), X2), S))), V)))
% 0.20/0.64 = { by lemma 32 R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(multiply(X, multiply(double_divide(multiply(inverse(Y2), Y2), Z2), Z2))), V)))
% 0.20/0.64 = { by lemma 35 R->L }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(multiply(double_divide(multiply(X, multiply(double_divide(multiply(inverse(Y2), Y2), Z2), Z2)), W2), W2), V)))
% 0.20/0.64 = { by lemma 26 }
% 0.20/0.64 double_divide(double_divide(multiply(double_divide(Z, W), W), Z), double_divide(V, double_divide(inverse(X), V)))
% 0.20/0.64 = { by lemma 30 }
% 0.20/0.64 X
% 0.20/0.64
% 0.20/0.64 Lemma 37: multiply(X, inverse(inverse(Y))) = multiply(X, Y).
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(X, inverse(inverse(Y)))
% 0.20/0.64 = { by lemma 35 R->L }
% 0.20/0.64 multiply(X, multiply(double_divide(inverse(Y), Z), Z))
% 0.20/0.64 = { by lemma 19 R->L }
% 0.20/0.64 multiply(double_divide(multiply(W, double_divide(W, multiply(X, Y))), V), V)
% 0.20/0.64 = { by lemma 36 R->L }
% 0.20/0.64 multiply(multiply(double_divide(multiply(W, double_divide(W, multiply(X, Y))), V), V), multiply(inverse(Y), Y))
% 0.20/0.64 = { by lemma 24 }
% 0.20/0.64 multiply(X, Y)
% 0.20/0.64
% 0.20/0.64 Lemma 38: multiply(double_divide(X, inverse(X)), Y) = Y.
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(double_divide(X, inverse(X)), Y)
% 0.20/0.64 = { by lemma 36 R->L }
% 0.20/0.64 multiply(double_divide(X, inverse(X)), multiply(Y, multiply(inverse(Z), Z)))
% 0.20/0.64 = { by axiom 1 (multiply) }
% 0.20/0.64 multiply(double_divide(X, inverse(X)), inverse(double_divide(multiply(inverse(Z), Z), Y)))
% 0.20/0.64 = { by lemma 23 }
% 0.20/0.64 multiply(double_divide(double_divide(multiply(inverse(Z), Z), Y), W), W)
% 0.20/0.64 = { by lemma 35 }
% 0.20/0.64 inverse(double_divide(multiply(inverse(Z), Z), Y))
% 0.20/0.64 = { by axiom 1 (multiply) R->L }
% 0.20/0.64 multiply(Y, multiply(inverse(Z), Z))
% 0.20/0.64 = { by lemma 36 }
% 0.20/0.64 Y
% 0.20/0.64
% 0.20/0.64 Lemma 39: inverse(inverse(X)) = X.
% 0.20/0.64 Proof:
% 0.20/0.64 inverse(inverse(X))
% 0.20/0.64 = { by lemma 35 R->L }
% 0.20/0.64 multiply(double_divide(inverse(X), inverse(inverse(X))), inverse(inverse(X)))
% 0.20/0.64 = { by lemma 37 }
% 0.20/0.64 multiply(double_divide(inverse(X), inverse(inverse(X))), X)
% 0.20/0.64 = { by lemma 38 }
% 0.20/0.64 X
% 0.20/0.64
% 0.20/0.64 Lemma 40: multiply(inverse(X), multiply(X, Y)) = Y.
% 0.20/0.64 Proof:
% 0.20/0.64 multiply(inverse(X), multiply(X, Y))
% 0.20/0.64 = { by lemma 35 R->L }
% 0.20/0.64 multiply(multiply(double_divide(X, Z), Z), multiply(X, Y))
% 0.20/0.64 = { by lemma 23 R->L }
% 0.20/0.64 multiply(multiply(double_divide(W, inverse(W)), inverse(X)), multiply(X, Y))
% 0.20/0.64 = { by lemma 39 R->L }
% 0.20/0.64 multiply(multiply(double_divide(W, inverse(W)), inverse(inverse(inverse(X)))), multiply(X, Y))
% 0.20/0.64 = { by lemma 35 R->L }
% 0.20/0.64 multiply(multiply(double_divide(W, inverse(W)), multiply(double_divide(inverse(inverse(X)), V), V)), multiply(X, Y))
% 0.20/0.64 = { by lemma 19 R->L }
% 0.20/0.64 multiply(multiply(double_divide(multiply(U, double_divide(U, multiply(double_divide(W, inverse(W)), inverse(X)))), T), T), multiply(X, Y))
% 0.20/0.64 = { by lemma 39 R->L }
% 0.20/0.64 multiply(multiply(double_divide(multiply(U, double_divide(U, multiply(double_divide(W, inverse(W)), inverse(X)))), T), T), multiply(inverse(inverse(X)), Y))
% 0.20/0.64 = { by lemma 24 }
% 0.20/0.64 multiply(double_divide(W, inverse(W)), Y)
% 0.20/0.64 = { by lemma 38 }
% 0.20/0.64 Y
% 0.20/0.64
% 0.20/0.64 Goal 1 (prove_these_axioms_2): multiply(multiply(inverse(b2), b2), a2) = a2.
% 0.20/0.65 Proof:
% 0.20/0.65 multiply(multiply(inverse(b2), b2), a2)
% 0.20/0.65 = { by lemma 37 R->L }
% 0.20/0.65 multiply(multiply(inverse(b2), inverse(inverse(b2))), a2)
% 0.20/0.65 = { by lemma 35 R->L }
% 0.20/0.65 multiply(multiply(inverse(b2), multiply(double_divide(inverse(b2), X), X)), a2)
% 0.20/0.65 = { by lemma 19 R->L }
% 0.20/0.65 multiply(multiply(double_divide(multiply(Y, double_divide(Y, multiply(inverse(b2), b2))), Z), Z), a2)
% 0.20/0.65 = { by lemma 40 R->L }
% 0.20/0.65 multiply(multiply(double_divide(multiply(Y, double_divide(Y, multiply(inverse(b2), b2))), Z), Z), multiply(inverse(b2), multiply(b2, a2)))
% 0.20/0.65 = { by lemma 24 }
% 0.20/0.65 multiply(inverse(b2), multiply(b2, a2))
% 0.20/0.65 = { by lemma 40 }
% 0.20/0.65 a2
% 0.20/0.65 % SZS output end Proof
% 0.20/0.65
% 0.20/0.65 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------