TSTP Solution File: BOO067-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : BOO067-1 : 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 : n014.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Aug 30 18:11:36 EDT 2023
% Result : Unsatisfiable 8.62s 1.52s
% Output : Proof 10.07s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : BOO067-1 : 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
% 0.18/0.34 % Computer : n014.cluster.edu
% 0.18/0.34 % Model : x86_64 x86_64
% 0.18/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.34 % Memory : 8042.1875MB
% 0.18/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.34 % CPULimit : 300
% 0.18/0.34 % WCLimit : 300
% 0.18/0.34 % DateTime : Sun Aug 27 07:47:16 EDT 2023
% 0.18/0.34 % CPUTime :
% 8.62/1.52 Command-line arguments: --no-flatten-goal
% 8.62/1.52
% 8.62/1.52 % SZS status Unsatisfiable
% 8.62/1.52
% 10.07/1.65 % SZS output start Proof
% 10.07/1.65 Axiom 1 (single_axiom): multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(Z, W, V), U, multiply(Z, W, T))), multiply(W, multiply(T, U, V), Z)) = Y.
% 10.07/1.65
% 10.07/1.65 Lemma 2: multiply(multiply(Z, X, Y), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V)) = multiply(X, multiply(Y, inverse(multiply(Z, X, W)), W), Z).
% 10.07/1.65 Proof:
% 10.07/1.65 multiply(multiply(Z, X, Y), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V))
% 10.07/1.65 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.65 multiply(multiply(multiply(multiply(Z, X, W), inverse(multiply(Z, X, W)), multiply(Z, X, Y)), inverse(multiply(multiply(Z, X, W), inverse(multiply(Z, X, W)), multiply(Z, X, Y))), multiply(X, multiply(Y, inverse(multiply(Z, X, W)), W), Z)), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V))
% 10.07/1.65 = { by axiom 1 (single_axiom) }
% 10.07/1.65 multiply(X, multiply(Y, inverse(multiply(Z, X, W)), W), Z)
% 10.07/1.65
% 10.07/1.65 Lemma 3: multiply(inverse(X), multiply(Y, inverse(multiply(X, inverse(X), Z)), Z), X) = Y.
% 10.07/1.65 Proof:
% 10.07/1.65 multiply(inverse(X), multiply(Y, inverse(multiply(X, inverse(X), Z)), Z), X)
% 10.07/1.65 = { by lemma 2 R->L }
% 10.07/1.65 multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(W, V, U), T, multiply(W, V, S))), multiply(V, multiply(S, T, U), W))
% 10.07/1.65 = { by axiom 1 (single_axiom) }
% 10.07/1.65 Y
% 10.07/1.65
% 10.07/1.65 Lemma 4: multiply(multiply(X, inverse(multiply(Y, inverse(Y), Z)), Z), multiply(Y, inverse(multiply(inverse(Y), multiply(X, inverse(multiply(Y, inverse(Y), Z)), Z), W)), W), inverse(Y)) = multiply(X, inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V)).
% 10.07/1.65 Proof:
% 10.07/1.66 multiply(multiply(X, inverse(multiply(Y, inverse(Y), Z)), Z), multiply(Y, inverse(multiply(inverse(Y), multiply(X, inverse(multiply(Y, inverse(Y), Z)), Z), W)), W), inverse(Y))
% 10.07/1.66 = { by lemma 2 R->L }
% 10.07/1.66 multiply(multiply(inverse(Y), multiply(X, inverse(multiply(Y, inverse(Y), Z)), Z), Y), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V))
% 10.07/1.66 = { by lemma 3 }
% 10.07/1.66 multiply(X, inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V))
% 10.07/1.66
% 10.07/1.66 Lemma 5: multiply(X, inverse(multiply(multiply(Y, Z, W), V, multiply(Y, Z, U))), multiply(Z, multiply(U, V, W), Y)) = multiply(multiply(X, inverse(multiply(T, inverse(T), S)), S), multiply(T, inverse(X), T), inverse(T)).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(X, inverse(multiply(multiply(Y, Z, W), V, multiply(Y, Z, U))), multiply(Z, multiply(U, V, W), Y))
% 10.07/1.66 = { by lemma 4 R->L }
% 10.07/1.66 multiply(multiply(X, inverse(multiply(T, inverse(T), S)), S), multiply(T, inverse(multiply(inverse(T), multiply(X, inverse(multiply(T, inverse(T), S)), S), T)), T), inverse(T))
% 10.07/1.66 = { by lemma 2 R->L }
% 10.07/1.66 multiply(multiply(X, inverse(multiply(T, inverse(T), S)), S), multiply(T, inverse(multiply(multiply(T, inverse(T), X), inverse(multiply(multiply(X2, Y2, Z2), W2, multiply(X2, Y2, V2))), multiply(Y2, multiply(V2, W2, Z2), X2))), T), inverse(T))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(multiply(X, inverse(multiply(T, inverse(T), S)), S), multiply(T, inverse(X), T), inverse(T))
% 10.07/1.66
% 10.07/1.66 Lemma 6: multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), V))), V) = Y.
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), V))), V)
% 10.07/1.66 = { by lemma 3 R->L }
% 10.07/1.66 multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), V))), multiply(inverse(Z), multiply(V, inverse(multiply(Z, inverse(Z), W)), W), Z))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 Y
% 10.07/1.66
% 10.07/1.66 Lemma 7: multiply(multiply(X, Y, Z), inverse(multiply(multiply(X2, Y2, Z2), W2, multiply(X2, Y2, V2))), multiply(Y2, multiply(V2, W2, Z2), X2)) = multiply(multiply(X, Y, Z), inverse(multiply(multiply(W, V, U), T, multiply(W, V, S))), multiply(V, multiply(S, T, U), W)).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(multiply(X, Y, Z), inverse(multiply(multiply(X2, Y2, Z2), W2, multiply(X2, Y2, V2))), multiply(Y2, multiply(V2, W2, Z2), X2))
% 10.07/1.66 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.66 multiply(multiply(multiply(multiply(X, Y, U2), inverse(multiply(X, Y, U2)), multiply(X, Y, Z)), inverse(multiply(multiply(X, Y, U2), inverse(multiply(X, Y, U2)), multiply(X, Y, Z))), multiply(Y, multiply(Z, inverse(multiply(X, Y, U2)), U2), X)), inverse(multiply(multiply(X2, Y2, Z2), W2, multiply(X2, Y2, V2))), multiply(Y2, multiply(V2, W2, Z2), X2))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(Y, multiply(Z, inverse(multiply(X, Y, U2)), U2), X)
% 10.07/1.66 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.66 multiply(multiply(multiply(multiply(X, Y, U2), inverse(multiply(X, Y, U2)), multiply(X, Y, Z)), inverse(multiply(multiply(X, Y, U2), inverse(multiply(X, Y, U2)), multiply(X, Y, Z))), multiply(Y, multiply(Z, inverse(multiply(X, Y, U2)), U2), X)), inverse(multiply(multiply(W, V, U), T, multiply(W, V, S))), multiply(V, multiply(S, T, U), W))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(multiply(X, Y, Z), inverse(multiply(multiply(W, V, U), T, multiply(W, V, S))), multiply(V, multiply(S, T, U), W))
% 10.07/1.66
% 10.07/1.66 Lemma 8: multiply(multiply(multiply(multiply(X, Y, Z), W, multiply(X, Y, V)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2)) = multiply(Y, multiply(V, W, Z), X).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(multiply(multiply(multiply(X, Y, Z), W, multiply(X, Y, V)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.66 = { by lemma 7 }
% 10.07/1.66 multiply(multiply(multiply(multiply(X, Y, Z), W, multiply(X, Y, V)), inverse(multiply(multiply(X, Y, Z), W, multiply(X, Y, V))), multiply(Y, multiply(V, W, Z), X)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(Y, multiply(V, W, Z), X)
% 10.07/1.66
% 10.07/1.66 Lemma 9: multiply(multiply(X, inverse(multiply(multiply(Y, Z, W), V, multiply(Y, Z, U))), multiply(Z, multiply(U, V, W), Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T)) = multiply(inverse(W2), multiply(V2, inverse(multiply(multiply(U2, inverse(U2), T2), inverse(multiply(U2, inverse(U2), T2)), multiply(U2, inverse(U2), multiply(W2, inverse(W2), V2)))), X), W2).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(multiply(X, inverse(multiply(multiply(Y, Z, W), V, multiply(Y, Z, U))), multiply(Z, multiply(U, V, W), Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T))
% 10.07/1.66 = { by lemma 6 R->L }
% 10.07/1.66 multiply(multiply(multiply(multiply(W2, inverse(W2), X), inverse(multiply(multiply(U2, inverse(U2), T2), inverse(multiply(U2, inverse(U2), T2)), multiply(U2, inverse(U2), multiply(W2, inverse(W2), V2)))), multiply(W2, inverse(W2), V2)), inverse(multiply(multiply(Y, Z, W), V, multiply(Y, Z, U))), multiply(Z, multiply(U, V, W), Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T))
% 10.07/1.66 = { by lemma 8 }
% 10.07/1.66 multiply(inverse(W2), multiply(V2, inverse(multiply(multiply(U2, inverse(U2), T2), inverse(multiply(U2, inverse(U2), T2)), multiply(U2, inverse(U2), multiply(W2, inverse(W2), V2)))), X), W2)
% 10.07/1.66
% 10.07/1.66 Lemma 10: multiply(multiply(multiply(Y, Z, X), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2)) = multiply(multiply(X, inverse(multiply(Y, Z, W)), W), multiply(Y, inverse(multiply(Z, multiply(X, inverse(multiply(Y, Z, W)), W), V)), V), Z).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(multiply(multiply(Y, Z, X), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.66 = { by lemma 2 }
% 10.07/1.66 multiply(multiply(Z, multiply(X, inverse(multiply(Y, Z, W)), W), Y), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.66 = { by lemma 2 }
% 10.07/1.66 multiply(multiply(X, inverse(multiply(Y, Z, W)), W), multiply(Y, inverse(multiply(Z, multiply(X, inverse(multiply(Y, Z, W)), W), V)), V), Z)
% 10.07/1.66
% 10.07/1.66 Lemma 11: multiply(inverse(X), multiply(Y, inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), multiply(X, inverse(X), Y)))), multiply(V, U, T)), X) = multiply(multiply(T, inverse(multiply(V, U, S)), S), multiply(V, inverse(multiply(U, multiply(T, inverse(multiply(V, U, S)), S), X2)), X2), U).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(inverse(X), multiply(Y, inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), multiply(X, inverse(X), Y)))), multiply(V, U, T)), X)
% 10.07/1.66 = { by lemma 9 R->L }
% 10.07/1.66 multiply(multiply(multiply(V, U, T), inverse(multiply(multiply(Y2, Z2, W2), V2, multiply(Y2, Z2, U2))), multiply(Z2, multiply(U2, V2, W2), Y2)), inverse(multiply(multiply(T2, S2, X3), Y3, multiply(T2, S2, Z3))), multiply(S2, multiply(Z3, Y3, X3), T2))
% 10.07/1.66 = { by lemma 10 }
% 10.07/1.66 multiply(multiply(T, inverse(multiply(V, U, S)), S), multiply(V, inverse(multiply(U, multiply(T, inverse(multiply(V, U, S)), S), X2)), X2), U)
% 10.07/1.66
% 10.07/1.66 Lemma 12: multiply(inverse(X), multiply(multiply(Y, inverse(multiply(Z, inverse(Z), W)), W), multiply(Z, inverse(Y), Z), inverse(Z)), X) = multiply(multiply(Y, inverse(multiply(V, inverse(V), U)), U), multiply(V, inverse(Y), V), inverse(V)).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(inverse(X), multiply(multiply(Y, inverse(multiply(Z, inverse(Z), W)), W), multiply(Z, inverse(Y), Z), inverse(Z)), X)
% 10.07/1.66 = { by lemma 5 R->L }
% 10.07/1.66 multiply(inverse(X), multiply(Y, inverse(multiply(multiply(T, inverse(T), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(T, inverse(T), W2))), multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T)), X)
% 10.07/1.66 = { by lemma 6 }
% 10.07/1.66 multiply(inverse(X), multiply(Y, inverse(multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T)), X)
% 10.07/1.66 = { by lemma 8 R->L }
% 10.07/1.66 multiply(multiply(multiply(multiply(X, inverse(X), multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T)), inverse(multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), multiply(X, inverse(X), Y)), inverse(multiply(multiply(V2, U2, T2), S2, multiply(V2, U2, X3))), multiply(U2, multiply(X3, S2, T2), V2)), inverse(multiply(multiply(Y3, Z3, W3), V3, multiply(Y3, Z3, U3))), multiply(Z3, multiply(U3, V3, W3), Y3))
% 10.07/1.66 = { by lemma 6 }
% 10.07/1.66 multiply(multiply(multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T), inverse(multiply(multiply(V2, U2, T2), S2, multiply(V2, U2, X3))), multiply(U2, multiply(X3, S2, T2), V2)), inverse(multiply(multiply(Y3, Z3, W3), V3, multiply(Y3, Z3, U3))), multiply(Z3, multiply(U3, V3, W3), Y3))
% 10.07/1.66 = { by lemma 6 R->L }
% 10.07/1.66 multiply(multiply(multiply(multiply(T3, inverse(T3), multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T)), inverse(multiply(multiply(S3, inverse(S3), X4), inverse(multiply(S3, inverse(S3), X4)), multiply(S3, inverse(S3), multiply(T3, inverse(T3), Y4)))), multiply(T3, inverse(T3), Y4)), inverse(multiply(multiply(V2, U2, T2), S2, multiply(V2, U2, X3))), multiply(U2, multiply(X3, S2, T2), V2)), inverse(multiply(multiply(Y3, Z3, W3), V3, multiply(Y3, Z3, U3))), multiply(Z3, multiply(U3, V3, W3), Y3))
% 10.07/1.66 = { by lemma 8 }
% 10.07/1.66 multiply(inverse(T3), multiply(Y4, inverse(multiply(multiply(S3, inverse(S3), X4), inverse(multiply(S3, inverse(S3), X4)), multiply(S3, inverse(S3), multiply(T3, inverse(T3), Y4)))), multiply(inverse(T), multiply(W2, inverse(multiply(multiply(Y2, inverse(Y2), Z2), inverse(multiply(Y2, inverse(Y2), Z2)), multiply(Y2, inverse(Y2), multiply(T, inverse(T), W2)))), multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), multiply(S, inverse(S), multiply(X, inverse(X), Y)))), T)), T3)
% 10.07/1.66 = { by lemma 11 }
% 10.07/1.66 multiply(inverse(T3), multiply(Y4, inverse(multiply(multiply(S3, inverse(S3), X4), inverse(multiply(S3, inverse(S3), X4)), multiply(S3, inverse(S3), multiply(T3, inverse(T3), Y4)))), multiply(multiply(multiply(S, inverse(S), multiply(X, inverse(X), Y)), inverse(multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), Z4)), Z4), multiply(multiply(S, inverse(S), X2), inverse(multiply(inverse(multiply(S, inverse(S), X2)), multiply(multiply(S, inverse(S), multiply(X, inverse(X), Y)), inverse(multiply(multiply(S, inverse(S), X2), inverse(multiply(S, inverse(S), X2)), Z4)), Z4), W4)), W4), inverse(multiply(S, inverse(S), X2)))), T3)
% 10.07/1.66 = { by lemma 4 }
% 10.07/1.66 multiply(inverse(T3), multiply(Y4, inverse(multiply(multiply(S3, inverse(S3), X4), inverse(multiply(S3, inverse(S3), X4)), multiply(S3, inverse(S3), multiply(T3, inverse(T3), Y4)))), multiply(multiply(S, inverse(S), multiply(X, inverse(X), Y)), inverse(multiply(multiply(V4, U4, T4), S4, multiply(V4, U4, X5))), multiply(U4, multiply(X5, S4, T4), V4))), T3)
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(inverse(T3), multiply(Y4, inverse(multiply(multiply(S3, inverse(S3), X4), inverse(multiply(S3, inverse(S3), X4)), multiply(S3, inverse(S3), multiply(T3, inverse(T3), Y4)))), multiply(X, inverse(X), Y)), T3)
% 10.07/1.66 = { by lemma 9 R->L }
% 10.07/1.66 multiply(multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(Y5, Z5, W5), V5, multiply(Y5, Z5, U5))), multiply(Z5, multiply(U5, V5, W5), Y5)), inverse(multiply(multiply(T5, S5, X6), Y6, multiply(T5, S5, Z6))), multiply(S5, multiply(Z6, Y6, X6), T5))
% 10.07/1.66 = { by lemma 10 }
% 10.07/1.66 multiply(multiply(Y, inverse(multiply(X, inverse(X), W6)), W6), multiply(X, inverse(multiply(inverse(X), multiply(Y, inverse(multiply(X, inverse(X), W6)), W6), V6)), V6), inverse(X))
% 10.07/1.66 = { by lemma 4 }
% 10.07/1.66 multiply(Y, inverse(multiply(multiply(U6, T6, S6), X7, multiply(U6, T6, Y7))), multiply(T6, multiply(Y7, X7, S6), U6))
% 10.07/1.66 = { by lemma 5 }
% 10.07/1.66 multiply(multiply(Y, inverse(multiply(V, inverse(V), U)), U), multiply(V, inverse(Y), V), inverse(V))
% 10.07/1.66
% 10.07/1.66 Lemma 13: multiply(inverse(X), Y, X) = Y.
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(inverse(X), Y, X)
% 10.07/1.66 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.66 multiply(inverse(X), multiply(multiply(Z, inverse(Z), Y), inverse(multiply(multiply(W, V, U), T, multiply(W, V, S))), multiply(V, multiply(S, T, U), W)), X)
% 10.07/1.66 = { by lemma 5 }
% 10.07/1.66 multiply(inverse(X), multiply(multiply(multiply(Z, inverse(Z), Y), inverse(multiply(X2, inverse(X2), Y2)), Y2), multiply(X2, inverse(multiply(Z, inverse(Z), Y)), X2), inverse(X2)), X)
% 10.07/1.66 = { by lemma 12 }
% 10.07/1.66 multiply(multiply(multiply(Z, inverse(Z), Y), inverse(multiply(Z2, inverse(Z2), W2)), W2), multiply(Z2, inverse(multiply(Z, inverse(Z), Y)), Z2), inverse(Z2))
% 10.07/1.66 = { by lemma 5 R->L }
% 10.07/1.66 multiply(multiply(Z, inverse(Z), Y), inverse(multiply(multiply(V2, U2, T2), S2, multiply(V2, U2, X3))), multiply(U2, multiply(X3, S2, T2), V2))
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 Y
% 10.07/1.66
% 10.07/1.66 Lemma 14: multiply(inverse(Z), Y, Z) = multiply(inverse(X), Y, X).
% 10.07/1.66 Proof:
% 10.07/1.66 multiply(inverse(Z), Y, Z)
% 10.07/1.66 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.66 multiply(inverse(Z), multiply(multiply(W, inverse(W), Y), inverse(multiply(multiply(S2, X3, Y3), Z3, multiply(S2, X3, W3))), multiply(X3, multiply(W3, Z3, Y3), S2)), Z)
% 10.07/1.66 = { by lemma 5 }
% 10.07/1.66 multiply(inverse(Z), multiply(multiply(multiply(W, inverse(W), Y), inverse(multiply(U2, inverse(U2), T2)), T2), multiply(U2, inverse(multiply(W, inverse(W), Y)), U2), inverse(U2)), Z)
% 10.07/1.66 = { by lemma 12 }
% 10.07/1.66 multiply(multiply(multiply(W, inverse(W), Y), inverse(multiply(W2, inverse(W2), V2)), V2), multiply(W2, inverse(multiply(W, inverse(W), Y)), W2), inverse(W2))
% 10.07/1.66 = { by lemma 12 R->L }
% 10.07/1.66 multiply(inverse(X), multiply(multiply(multiply(W, inverse(W), Y), inverse(multiply(Y2, inverse(Y2), Z2)), Z2), multiply(Y2, inverse(multiply(W, inverse(W), Y)), Y2), inverse(Y2)), X)
% 10.07/1.66 = { by lemma 5 R->L }
% 10.07/1.66 multiply(inverse(X), multiply(multiply(W, inverse(W), Y), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V)), X)
% 10.07/1.66 = { by axiom 1 (single_axiom) }
% 10.07/1.66 multiply(inverse(X), Y, X)
% 10.07/1.66
% 10.07/1.67 Lemma 15: multiply(inverse(inverse(X)), multiply(Y, inverse(multiply(inverse(Y), inverse(inverse(X)), Z)), Z), inverse(Y)) = X.
% 10.07/1.67 Proof:
% 10.07/1.67 multiply(inverse(inverse(X)), multiply(Y, inverse(multiply(inverse(Y), inverse(inverse(X)), Z)), Z), inverse(Y))
% 10.07/1.67 = { by lemma 2 R->L }
% 10.07/1.67 multiply(multiply(inverse(Y), inverse(inverse(X)), Y), inverse(multiply(multiply(W, inverse(W), V), U, multiply(W, inverse(W), T))), multiply(inverse(W), multiply(T, U, V), W))
% 10.07/1.67 = { by lemma 13 }
% 10.07/1.67 multiply(multiply(inverse(Y), inverse(inverse(X)), Y), inverse(multiply(multiply(W, inverse(W), V), U, multiply(W, inverse(W), T))), multiply(T, U, V))
% 10.07/1.67 = { by lemma 14 }
% 10.07/1.67 multiply(multiply(inverse(X), inverse(inverse(X)), X), inverse(multiply(multiply(W, inverse(W), V), U, multiply(W, inverse(W), T))), multiply(T, U, V))
% 10.07/1.67 = { by lemma 13 R->L }
% 10.07/1.67 multiply(multiply(inverse(X), inverse(inverse(X)), X), inverse(multiply(multiply(W, inverse(W), V), U, multiply(W, inverse(W), T))), multiply(inverse(W), multiply(T, U, V), W))
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 X
% 10.07/1.67
% 10.07/1.67 Lemma 16: multiply(X, multiply(inverse(V), inverse(multiply(Z, X, Y)), V), Z) = multiply(X, multiply(inverse(Y), inverse(multiply(Z, X, W)), W), Z).
% 10.07/1.67 Proof:
% 10.07/1.67 multiply(X, multiply(inverse(V), inverse(multiply(Z, X, Y)), V), Z)
% 10.07/1.67 = { by lemma 14 }
% 10.07/1.67 multiply(X, multiply(inverse(Y), inverse(multiply(Z, X, Y)), Y), Z)
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(multiply(multiply(multiply(Z, X, Y), inverse(multiply(Z, X, Y)), multiply(Z, X, inverse(Y))), inverse(multiply(multiply(Z, X, Y), inverse(multiply(Z, X, Y)), multiply(Z, X, inverse(Y)))), multiply(X, multiply(inverse(Y), inverse(multiply(Z, X, Y)), Y), Z)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 multiply(multiply(Z, X, inverse(Y)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(multiply(multiply(multiply(Z, X, W), inverse(multiply(Z, X, W)), multiply(Z, X, inverse(Y))), inverse(multiply(multiply(Z, X, W), inverse(multiply(Z, X, W)), multiply(Z, X, inverse(Y)))), multiply(X, multiply(inverse(Y), inverse(multiply(Z, X, W)), W), Z)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 multiply(X, multiply(inverse(Y), inverse(multiply(Z, X, W)), W), Z)
% 10.07/1.67
% 10.07/1.67 Lemma 17: multiply(inverse(X), multiply(inverse(Y), inverse(multiply(Z, inverse(Z), W)), Y), X) = inverse(W).
% 10.07/1.67 Proof:
% 10.07/1.67 multiply(inverse(X), multiply(inverse(Y), inverse(multiply(Z, inverse(Z), W)), Y), X)
% 10.07/1.67 = { by lemma 14 }
% 10.07/1.67 multiply(inverse(Z), multiply(inverse(Y), inverse(multiply(Z, inverse(Z), W)), Y), Z)
% 10.07/1.67 = { by lemma 16 }
% 10.07/1.67 multiply(inverse(Z), multiply(inverse(W), inverse(multiply(Z, inverse(Z), V)), V), Z)
% 10.07/1.67 = { by lemma 3 }
% 10.07/1.67 inverse(W)
% 10.07/1.67
% 10.07/1.67 Lemma 18: multiply(inverse(X), multiply(inverse(Y), inverse(multiply(Z, W, inverse(V))), Y), X) = inverse(multiply(W, multiply(inverse(U), inverse(multiply(Z, W, V)), U), Z)).
% 10.07/1.67 Proof:
% 10.07/1.67 multiply(inverse(X), multiply(inverse(Y), inverse(multiply(Z, W, inverse(V))), Y), X)
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(inverse(X), multiply(inverse(Y), inverse(multiply(multiply(multiply(Z, W, T), inverse(multiply(Z, W, T)), multiply(Z, W, inverse(V))), inverse(multiply(multiply(Z, W, T), inverse(multiply(Z, W, T)), multiply(Z, W, inverse(V)))), multiply(W, multiply(inverse(V), inverse(multiply(Z, W, T)), T), Z))), Y), X)
% 10.07/1.67 = { by lemma 17 }
% 10.07/1.67 inverse(multiply(W, multiply(inverse(V), inverse(multiply(Z, W, T)), T), Z))
% 10.07/1.67 = { by lemma 16 R->L }
% 10.07/1.67 inverse(multiply(W, multiply(inverse(U), inverse(multiply(Z, W, V)), U), Z))
% 10.07/1.67
% 10.07/1.67 Lemma 19: multiply(X, inverse(multiply(Y, inverse(Y), Y)), multiply(Y, inverse(multiply(multiply(Z, W, V), U, multiply(Z, W, T))), multiply(W, multiply(T, U, V), Z))) = multiply(X, inverse(multiply(multiply(S, X2, Y2), Z2, multiply(S, X2, W2))), multiply(X2, multiply(W2, Z2, Y2), S)).
% 10.07/1.67 Proof:
% 10.07/1.67 multiply(X, inverse(multiply(Y, inverse(Y), Y)), multiply(Y, inverse(multiply(multiply(Z, W, V), U, multiply(Z, W, T))), multiply(W, multiply(T, U, V), Z)))
% 10.07/1.67 = { by lemma 3 R->L }
% 10.07/1.67 multiply(X, inverse(multiply(multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2), inverse(Y), Y)), multiply(Y, inverse(multiply(multiply(Z, W, V), U, multiply(Z, W, T))), multiply(W, multiply(T, U, V), Z)))
% 10.07/1.67 = { by lemma 3 R->L }
% 10.07/1.67 multiply(X, inverse(multiply(multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2), inverse(Y), multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2))), multiply(Y, inverse(multiply(multiply(Z, W, V), U, multiply(Z, W, T))), multiply(W, multiply(T, U, V), Z)))
% 10.07/1.67 = { by lemma 5 }
% 10.07/1.67 multiply(X, inverse(multiply(multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2), inverse(Y), multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2))), multiply(multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), multiply(V2, inverse(Y), V2), inverse(V2)))
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(multiply(multiply(T2, inverse(T2), X), inverse(multiply(multiply(S2, X3, Y3), Z3, multiply(S2, X3, W3))), multiply(X3, multiply(W3, Z3, Y3), S2)), inverse(multiply(multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2), inverse(Y), multiply(inverse(V2), multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), V2))), multiply(multiply(Y, inverse(multiply(V2, inverse(V2), U2)), U2), multiply(V2, inverse(Y), V2), inverse(V2)))
% 10.07/1.67 = { by lemma 7 R->L }
% 10.07/1.67 multiply(multiply(multiply(T2, inverse(T2), X), inverse(multiply(multiply(S2, X3, Y3), Z3, multiply(S2, X3, W3))), multiply(X3, multiply(W3, Z3, Y3), S2)), inverse(multiply(multiply(S, X2, Y2), Z2, multiply(S, X2, W2))), multiply(X2, multiply(W2, Z2, Y2), S))
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 multiply(X, inverse(multiply(multiply(S, X2, Y2), Z2, multiply(S, X2, W2))), multiply(X2, multiply(W2, Z2, Y2), S))
% 10.07/1.67
% 10.07/1.67 Lemma 20: inverse(multiply(Z, inverse(Z), Y)) = inverse(multiply(X, inverse(X), Y)).
% 10.07/1.67 Proof:
% 10.07/1.67 inverse(multiply(Z, inverse(Z), Y))
% 10.07/1.67 = { by lemma 17 R->L }
% 10.07/1.67 multiply(inverse(W), multiply(inverse(Y), inverse(multiply(multiply(Z, inverse(Z), V2), inverse(multiply(Z, inverse(Z), V2)), multiply(Z, inverse(Z), Y))), Y), W)
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(inverse(W), multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T)), inverse(multiply(multiply(Z, inverse(Z), V2), inverse(multiply(Z, inverse(Z), V2)), multiply(Z, inverse(Z), Y))), Y), W)
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 multiply(inverse(W), multiply(multiply(multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T))), inverse(multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T)))), multiply(inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(multiply(S, multiply(Z2, Y2, X2), T), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), W2), multiply(U, inverse(U), inverse(Y)))), inverse(multiply(multiply(Z, inverse(Z), V2), inverse(multiply(Z, inverse(Z), V2)), multiply(Z, inverse(Z), Y))), Y), W)
% 10.07/1.67 = { by lemma 6 }
% 10.07/1.67 multiply(inverse(W), multiply(inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(multiply(S, multiply(Z2, Y2, X2), T), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), W2), multiply(U, inverse(U), inverse(Y))), W)
% 10.07/1.67 = { by lemma 6 R->L }
% 10.07/1.67 multiply(inverse(W), multiply(multiply(multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T))), inverse(multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T)))), multiply(inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(multiply(S, multiply(Z2, Y2, X2), T), inverse(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), W2)), W2), multiply(U, inverse(U), inverse(Y)))), inverse(multiply(multiply(X, inverse(X), V), inverse(multiply(X, inverse(X), V)), multiply(X, inverse(X), Y))), Y), W)
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 multiply(inverse(W), multiply(multiply(multiply(U, inverse(U), inverse(Y)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T)), inverse(multiply(multiply(X, inverse(X), V), inverse(multiply(X, inverse(X), V)), multiply(X, inverse(X), Y))), Y), W)
% 10.07/1.67 = { by axiom 1 (single_axiom) }
% 10.07/1.67 multiply(inverse(W), multiply(inverse(Y), inverse(multiply(multiply(X, inverse(X), V), inverse(multiply(X, inverse(X), V)), multiply(X, inverse(X), Y))), Y), W)
% 10.07/1.67 = { by lemma 17 }
% 10.07/1.67 inverse(multiply(X, inverse(X), Y))
% 10.07/1.67
% 10.07/1.67 Lemma 21: inverse(inverse(multiply(X, inverse(X), Y))) = multiply(Z, inverse(Z), Y).
% 10.07/1.67 Proof:
% 10.07/1.67 inverse(inverse(multiply(X, inverse(X), Y)))
% 10.07/1.67 = { by lemma 17 R->L }
% 10.07/1.67 multiply(inverse(W), multiply(inverse(V), inverse(multiply(multiply(X, inverse(X), Y), inverse(multiply(X, inverse(X), Y)), inverse(multiply(X, inverse(X), Y)))), V), W)
% 10.07/1.67 = { by lemma 18 }
% 10.07/1.67 inverse(multiply(inverse(multiply(X, inverse(X), Y)), multiply(inverse(Y), inverse(multiply(multiply(X, inverse(X), Y), inverse(multiply(X, inverse(X), Y)), multiply(X, inverse(X), Y))), Y), multiply(X, inverse(X), Y)))
% 10.07/1.67 = { by axiom 1 (single_axiom) R->L }
% 10.07/1.67 inverse(multiply(inverse(multiply(X, inverse(X), Y)), multiply(inverse(Y), inverse(multiply(multiply(X, inverse(X), Y), inverse(multiply(X, inverse(X), Y)), multiply(X, inverse(X), Y))), multiply(multiply(X, inverse(X), Y), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))), multiply(X, inverse(X), Y)))
% 10.07/1.67 = { by lemma 19 }
% 10.07/1.67 inverse(multiply(inverse(multiply(X, inverse(X), Y)), multiply(inverse(Y), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2)), multiply(X, inverse(X), Y)))
% 10.07/1.67 = { by lemma 5 }
% 10.07/1.67 inverse(multiply(inverse(multiply(X, inverse(X), Y)), multiply(multiply(inverse(Y), inverse(multiply(S2, inverse(S2), X3)), X3), multiply(S2, inverse(inverse(Y)), S2), inverse(S2)), multiply(X, inverse(X), Y)))
% 10.07/1.67 = { by lemma 12 }
% 10.07/1.67 inverse(multiply(multiply(inverse(Y), inverse(multiply(Y3, inverse(Y3), Z3)), Z3), multiply(Y3, inverse(inverse(Y)), Y3), inverse(Y3)))
% 10.07/1.67 = { by lemma 5 R->L }
% 10.07/1.67 inverse(multiply(inverse(Y), inverse(multiply(multiply(W3, V3, U3), T3, multiply(W3, V3, S3))), multiply(V3, multiply(S3, T3, U3), W3)))
% 10.07/1.67 = { by lemma 17 R->L }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(Y), inverse(multiply(multiply(inverse(Y), inverse(inverse(Y)), inverse(Y)), inverse(multiply(inverse(Y), inverse(inverse(Y)), inverse(Y))), multiply(inverse(Y), inverse(multiply(multiply(W3, V3, U3), T3, multiply(W3, V3, S3))), multiply(V3, multiply(S3, T3, U3), W3)))), Y), X4)
% 10.07/1.68 = { by lemma 19 }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(Y), inverse(multiply(multiply(inverse(Y), inverse(inverse(Y)), inverse(Y)), inverse(multiply(multiply(Y4, Z4, W4), V4, multiply(Y4, Z4, U4))), multiply(Z4, multiply(U4, V4, W4), Y4))), Y), X4)
% 10.07/1.68 = { by axiom 1 (single_axiom) }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(Y), inverse(inverse(Y)), Y), X4)
% 10.07/1.68 = { by lemma 15 R->L }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(inverse(multiply(inverse(Y), inverse(inverse(Y)), Y))), multiply(T4, inverse(multiply(inverse(T4), inverse(inverse(multiply(inverse(Y), inverse(inverse(Y)), Y))), S4)), S4), inverse(T4)), X4)
% 10.07/1.68 = { by lemma 20 R->L }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(inverse(multiply(Z, inverse(Z), Y))), multiply(T4, inverse(multiply(inverse(T4), inverse(inverse(multiply(inverse(Y), inverse(inverse(Y)), Y))), S4)), S4), inverse(T4)), X4)
% 10.07/1.68 = { by lemma 20 R->L }
% 10.07/1.68 multiply(inverse(X4), multiply(inverse(inverse(multiply(Z, inverse(Z), Y))), multiply(T4, inverse(multiply(inverse(T4), inverse(inverse(multiply(Z, inverse(Z), Y))), S4)), S4), inverse(T4)), X4)
% 10.07/1.68 = { by lemma 15 }
% 10.07/1.68 multiply(inverse(X4), multiply(Z, inverse(Z), Y), X4)
% 10.07/1.68 = { by lemma 13 }
% 10.07/1.68 multiply(Z, inverse(Z), Y)
% 10.07/1.68
% 10.07/1.68 Lemma 22: multiply(X, inverse(X), Y) = Y.
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(X, inverse(X), Y)
% 10.07/1.68 = { by lemma 15 R->L }
% 10.07/1.68 multiply(inverse(inverse(multiply(X, inverse(X), Y))), multiply(Z, inverse(multiply(inverse(Z), inverse(inverse(multiply(X, inverse(X), Y))), W)), W), inverse(Z))
% 10.07/1.68 = { by lemma 21 }
% 10.07/1.68 multiply(multiply(V, inverse(V), Y), multiply(Z, inverse(multiply(inverse(Z), inverse(inverse(multiply(X, inverse(X), Y))), W)), W), inverse(Z))
% 10.07/1.68 = { by lemma 21 }
% 10.07/1.68 multiply(multiply(V, inverse(V), Y), multiply(Z, inverse(multiply(inverse(Z), multiply(V, inverse(V), Y), W)), W), inverse(Z))
% 10.07/1.68 = { by lemma 2 R->L }
% 10.07/1.68 multiply(multiply(inverse(Z), multiply(V, inverse(V), Y), Z), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))
% 10.07/1.68 = { by lemma 13 }
% 10.07/1.68 multiply(multiply(V, inverse(V), Y), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U))
% 10.07/1.68 = { by lemma 2 }
% 10.07/1.68 multiply(inverse(V), multiply(Y, inverse(multiply(V, inverse(V), Z2)), Z2), V)
% 10.07/1.68 = { by lemma 3 }
% 10.07/1.68 Y
% 10.07/1.68
% 10.07/1.68 Lemma 23: inverse(inverse(X)) = X.
% 10.07/1.68 Proof:
% 10.07/1.68 inverse(inverse(X))
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 inverse(inverse(multiply(Y, inverse(Y), X)))
% 10.07/1.68 = { by lemma 21 }
% 10.07/1.68 multiply(Z, inverse(Z), X)
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 X
% 10.07/1.68
% 10.07/1.68 Lemma 24: multiply(X, inverse(Y), Y) = X.
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(X, inverse(Y), Y)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(X, inverse(multiply(Z, inverse(Z), Y)), Y)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(X, inverse(multiply(W, inverse(W), multiply(Z, inverse(Z), Y))), Y)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(X, inverse(multiply(W, inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), Y))), Y)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(V, inverse(V), X), inverse(multiply(W, inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), Y))), Y)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(V, inverse(V), X), inverse(multiply(multiply(Z, inverse(Z), W), inverse(multiply(Z, inverse(Z), W)), multiply(Z, inverse(Z), Y))), Y)
% 10.07/1.68 = { by lemma 6 }
% 10.07/1.68 X
% 10.07/1.68
% 10.07/1.68 Lemma 25: multiply(X, Y, inverse(X)) = Y.
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(X, Y, inverse(X))
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(X, inverse(X), X), Y, inverse(X))
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), Y, inverse(X))
% 10.07/1.68 = { by lemma 6 R->L }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(multiply(Z, inverse(Z), Y), inverse(multiply(multiply(W, inverse(W), V), inverse(multiply(W, inverse(W), V)), multiply(W, inverse(W), X))), X), inverse(X))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(multiply(Z, inverse(Z), Y), inverse(multiply(multiply(W, inverse(W), V), inverse(multiply(W, inverse(W), V)), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(multiply(Z, inverse(Z), Y), inverse(X), X), inverse(X))
% 10.07/1.68 = { by lemma 3 R->L }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(multiply(Z, inverse(Z), Y), inverse(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 6 R->L }
% 10.07/1.68 multiply(multiply(multiply(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X), inverse(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X)), multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(Z, inverse(Z), Y))), inverse(multiply(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X), inverse(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X)), multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(Z, inverse(Z), Y)))), multiply(multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(multiply(Z, inverse(Z), Y), inverse(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), X)), X), inverse(X))), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), S)
% 10.07/1.68 = { by axiom 1 (single_axiom) }
% 10.07/1.68 multiply(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(Z, inverse(Z), Y)), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), S)
% 10.07/1.68 = { by lemma 3 R->L }
% 10.07/1.68 multiply(multiply(inverse(X), multiply(X, inverse(multiply(X, inverse(X), X)), X), multiply(Z, inverse(Z), Y)), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), multiply(inverse(U), multiply(S, inverse(multiply(U, inverse(U), T)), T), U))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(inverse(X), multiply(X, inverse(X), X), multiply(Z, inverse(Z), Y)), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), multiply(inverse(U), multiply(S, inverse(multiply(U, inverse(U), T)), T), U))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(inverse(X), X, multiply(Z, inverse(Z), Y)), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), multiply(inverse(U), multiply(S, inverse(multiply(U, inverse(U), T)), T), U))
% 10.07/1.68 = { by lemma 23 R->L }
% 10.07/1.68 multiply(multiply(inverse(X), inverse(inverse(X)), multiply(Z, inverse(Z), Y)), inverse(multiply(multiply(U, inverse(U), T), inverse(multiply(U, inverse(U), T)), multiply(U, inverse(U), S))), multiply(inverse(U), multiply(S, inverse(multiply(U, inverse(U), T)), T), U))
% 10.07/1.68 = { by axiom 1 (single_axiom) }
% 10.07/1.68 multiply(Z, inverse(Z), Y)
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 Y
% 10.07/1.68
% 10.07/1.68 Lemma 26: multiply(inverse(X), Y, X) = multiply(Z, Y, inverse(Z)).
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(inverse(X), Y, X)
% 10.07/1.68 = { by lemma 14 }
% 10.07/1.68 multiply(inverse(inverse(Z)), Y, inverse(Z))
% 10.07/1.68 = { by lemma 23 }
% 10.07/1.68 multiply(Z, Y, inverse(Z))
% 10.07/1.68
% 10.07/1.68 Lemma 27: multiply(multiply(X, inverse(multiply(X, Y, Z)), Z), multiply(X, inverse(multiply(Y, multiply(X, inverse(multiply(X, Y, Z)), Z), W)), W), Y) = multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V)).
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(multiply(X, inverse(multiply(X, Y, Z)), Z), multiply(X, inverse(multiply(Y, multiply(X, inverse(multiply(X, Y, Z)), Z), W)), W), Y)
% 10.07/1.68 = { by lemma 11 R->L }
% 10.07/1.68 multiply(inverse(T), multiply(S, inverse(multiply(multiply(X2, inverse(X2), Y2), inverse(multiply(X2, inverse(X2), Y2)), multiply(X2, inverse(X2), multiply(T, inverse(T), S)))), multiply(X, Y, X)), T)
% 10.07/1.68 = { by lemma 9 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(multiply(inverse(V), multiply(X, inverse(multiply(V, inverse(V), U)), U), V), Y, multiply(inverse(V), multiply(X, inverse(multiply(V, inverse(V), U)), U), V))), multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V))), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.68 = { by lemma 3 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, Y, multiply(inverse(V), multiply(X, inverse(multiply(V, inverse(V), U)), U), V))), multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V))), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.68 = { by lemma 2 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, Y, multiply(multiply(V, inverse(V), X), inverse(multiply(multiply(S2, X3, Y3), Z3, multiply(S2, X3, W3))), multiply(X3, multiply(W3, Z3, Y3), S2)))), multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V))), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.68 = { by axiom 1 (single_axiom) }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V))), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.68 = { by axiom 1 (single_axiom) }
% 10.07/1.68 multiply(multiply(X, inverse(multiply(V, inverse(V), U)), U), multiply(V, Y, V), inverse(V))
% 10.07/1.68
% 10.07/1.68 Lemma 28: inverse(multiply(X, Y, X)) = inverse(X).
% 10.07/1.68 Proof:
% 10.07/1.68 inverse(multiply(X, Y, X))
% 10.07/1.68 = { by lemma 23 R->L }
% 10.07/1.68 inverse(multiply(X, Y, inverse(inverse(X))))
% 10.07/1.68 = { by lemma 25 R->L }
% 10.07/1.68 multiply(Z, inverse(multiply(X, Y, inverse(inverse(X)))), inverse(Z))
% 10.07/1.68 = { by lemma 26 R->L }
% 10.07/1.68 multiply(inverse(W), inverse(multiply(X, Y, inverse(inverse(X)))), W)
% 10.07/1.68 = { by lemma 25 R->L }
% 10.07/1.68 multiply(V, multiply(inverse(W), inverse(multiply(X, Y, inverse(inverse(X)))), W), inverse(V))
% 10.07/1.68 = { by lemma 26 R->L }
% 10.07/1.68 multiply(inverse(U), multiply(inverse(W), inverse(multiply(X, Y, inverse(inverse(X)))), W), U)
% 10.07/1.68 = { by lemma 18 }
% 10.07/1.68 inverse(multiply(Y, multiply(inverse(T), inverse(multiply(X, Y, inverse(X))), T), X))
% 10.07/1.68 = { by lemma 25 }
% 10.07/1.68 inverse(multiply(Y, multiply(inverse(T), inverse(Y), T), X))
% 10.07/1.68 = { by lemma 26 }
% 10.07/1.68 inverse(multiply(Y, multiply(S, inverse(Y), inverse(S)), X))
% 10.07/1.68 = { by lemma 25 }
% 10.07/1.68 inverse(multiply(Y, inverse(Y), X))
% 10.07/1.68 = { by lemma 20 R->L }
% 10.07/1.68 inverse(multiply(X2, inverse(X2), X))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 inverse(X)
% 10.07/1.68
% 10.07/1.68 Lemma 29: multiply(X, Y, X) = X.
% 10.07/1.68 Proof:
% 10.07/1.68 multiply(X, Y, X)
% 10.07/1.68 = { by lemma 24 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(X), X)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X)
% 10.07/1.68 = { by lemma 24 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X)), X)
% 10.07/1.68 = { by lemma 25 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(Z, multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), inverse(Z))), X)
% 10.07/1.68 = { by lemma 26 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), X)), X)
% 10.07/1.68 = { by lemma 25 R->L }
% 10.07/1.68 multiply(X, multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X), multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 24 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 27 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, inverse(X), W)), W), multiply(X, inverse(X), X), inverse(X))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, inverse(X), W)), W), X, inverse(X))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(W), W), X, inverse(X))
% 10.07/1.68 = { by lemma 24 }
% 10.07/1.68 multiply(multiply(X, Y, X), X, inverse(X))
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), multiply(X, inverse(X), X), inverse(X))
% 10.07/1.68 = { by lemma 28 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), multiply(X, inverse(multiply(X, Y, X)), X), inverse(X))
% 10.07/1.68 = { by lemma 24 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(X), X), multiply(X, inverse(multiply(X, Y, X)), X), inverse(X))
% 10.07/1.68 = { by lemma 28 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, inverse(X), X)), X), multiply(X, inverse(multiply(X, Y, X)), X), inverse(X))
% 10.07/1.68 = { by lemma 27 R->L }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X), multiply(multiply(X, Y, X), inverse(multiply(inverse(multiply(X, Y, X)), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 28 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(X), X)), X), multiply(multiply(X, Y, X), inverse(multiply(inverse(multiply(X, Y, X)), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 24 }
% 10.07/1.68 multiply(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X), multiply(multiply(X, Y, X), inverse(multiply(inverse(multiply(X, Y, X)), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(X, multiply(multiply(X, Y, X), inverse(multiply(inverse(multiply(X, Y, X)), multiply(multiply(X, Y, X), inverse(multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 22 }
% 10.07/1.68 multiply(X, multiply(multiply(X, Y, X), inverse(multiply(inverse(multiply(X, Y, X)), multiply(multiply(X, Y, X), inverse(X), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 28 }
% 10.07/1.68 multiply(X, multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(X), X), X)), X), inverse(multiply(X, Y, X)))
% 10.07/1.68 = { by lemma 28 }
% 10.07/1.68 multiply(X, multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(X), X), X)), X), inverse(X))
% 10.07/1.68 = { by lemma 25 }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(inverse(X), multiply(multiply(X, Y, X), inverse(X), X), X)), X)
% 10.07/1.68 = { by lemma 26 }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(V, multiply(multiply(X, Y, X), inverse(X), X), inverse(V))), X)
% 10.07/1.68 = { by lemma 26 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(inverse(U), multiply(multiply(X, Y, X), inverse(X), X), U)), X)
% 10.07/1.68 = { by lemma 22 R->L }
% 10.07/1.68 multiply(multiply(X, Y, X), inverse(multiply(inverse(U), multiply(multiply(X, Y, X), inverse(multiply(U, inverse(U), X)), X), U)), X)
% 10.07/1.69 = { by lemma 2 R->L }
% 10.07/1.69 multiply(multiply(X, Y, X), inverse(multiply(multiply(U, inverse(U), multiply(X, Y, X)), inverse(multiply(multiply(T, S, X2), Y2, multiply(T, S, Z2))), multiply(S, multiply(Z2, Y2, X2), T))), X)
% 10.07/1.69 = { by axiom 1 (single_axiom) }
% 10.07/1.69 multiply(multiply(X, Y, X), inverse(multiply(X, Y, X)), X)
% 10.07/1.69 = { by lemma 22 }
% 10.07/1.69 X
% 10.07/1.69
% 10.07/1.69 Lemma 30: multiply(Z, X, Y) = multiply(X, Y, Z).
% 10.07/1.69 Proof:
% 10.07/1.69 multiply(Z, X, Y)
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(T2, inverse(T2), multiply(Z, X, Y))
% 10.07/1.69 = { by lemma 28 R->L }
% 10.07/1.69 multiply(T2, inverse(multiply(T2, inverse(T2), T2)), multiply(Z, X, Y))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(T2, inverse(multiply(S2, inverse(S2), multiply(T2, inverse(T2), T2))), multiply(Z, X, Y))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(T2, inverse(multiply(multiply(S2, inverse(S2), X3), inverse(multiply(S2, inverse(S2), X3)), multiply(S2, inverse(S2), multiply(T2, inverse(T2), T2)))), multiply(Z, X, Y))
% 10.07/1.69 = { by lemma 25 R->L }
% 10.07/1.69 multiply(Y3, multiply(T2, inverse(multiply(multiply(S2, inverse(S2), X3), inverse(multiply(S2, inverse(S2), X3)), multiply(S2, inverse(S2), multiply(T2, inverse(T2), T2)))), multiply(Z, X, Y)), inverse(Y3))
% 10.07/1.69 = { by lemma 26 R->L }
% 10.07/1.69 multiply(inverse(T2), multiply(T2, inverse(multiply(multiply(S2, inverse(S2), X3), inverse(multiply(S2, inverse(S2), X3)), multiply(S2, inverse(S2), multiply(T2, inverse(T2), T2)))), multiply(Z, X, Y)), T2)
% 10.07/1.69 = { by lemma 9 R->L }
% 10.07/1.69 multiply(multiply(multiply(Z, X, Y), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V)), inverse(multiply(multiply(Y2, Z2, W2), V2, multiply(Y2, Z2, U2))), multiply(Z2, multiply(U2, V2, W2), Y2))
% 10.07/1.69 = { by lemma 29 R->L }
% 10.07/1.69 multiply(multiply(multiply(multiply(Z, X, Y), W, multiply(Z, X, Y)), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V)), inverse(multiply(multiply(Y2, Z2, W2), V2, multiply(Y2, Z2, U2))), multiply(Z2, multiply(U2, V2, W2), Y2))
% 10.07/1.69 = { by lemma 8 }
% 10.07/1.69 multiply(X, multiply(Y, W, Y), Z)
% 10.07/1.69 = { by lemma 29 }
% 10.07/1.69 multiply(X, Y, Z)
% 10.07/1.69
% 10.07/1.69 Lemma 31: multiply(X, Z, Y) = multiply(X, Y, Z).
% 10.07/1.69 Proof:
% 10.07/1.69 multiply(X, Z, Y)
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(X, Z, multiply(V, inverse(V), Y))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(S2, inverse(S2), multiply(X, Z, multiply(V, inverse(V), Y)))
% 10.07/1.69 = { by lemma 28 R->L }
% 10.07/1.69 multiply(S2, inverse(multiply(S2, inverse(S2), S2)), multiply(X, Z, multiply(V, inverse(V), Y)))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(S2, inverse(multiply(X3, inverse(X3), multiply(S2, inverse(S2), S2))), multiply(X, Z, multiply(V, inverse(V), Y)))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(S2, inverse(multiply(multiply(X3, inverse(X3), Y3), inverse(multiply(X3, inverse(X3), Y3)), multiply(X3, inverse(X3), multiply(S2, inverse(S2), S2)))), multiply(X, Z, multiply(V, inverse(V), Y)))
% 10.07/1.69 = { by lemma 25 R->L }
% 10.07/1.69 multiply(Z3, multiply(S2, inverse(multiply(multiply(X3, inverse(X3), Y3), inverse(multiply(X3, inverse(X3), Y3)), multiply(X3, inverse(X3), multiply(S2, inverse(S2), S2)))), multiply(X, Z, multiply(V, inverse(V), Y))), inverse(Z3))
% 10.07/1.69 = { by lemma 26 R->L }
% 10.07/1.69 multiply(inverse(S2), multiply(S2, inverse(multiply(multiply(X3, inverse(X3), Y3), inverse(multiply(X3, inverse(X3), Y3)), multiply(X3, inverse(X3), multiply(S2, inverse(S2), S2)))), multiply(X, Z, multiply(V, inverse(V), Y))), S2)
% 10.07/1.69 = { by lemma 9 R->L }
% 10.07/1.69 multiply(multiply(multiply(X, Z, multiply(V, inverse(V), Y)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(multiply(multiply(multiply(V, inverse(V), X), Z, multiply(V, inverse(V), Y)), inverse(multiply(multiply(U, T, S), X2, multiply(U, T, Y2))), multiply(T, multiply(Y2, X2, S), U)), inverse(multiply(multiply(Z2, W2, V2), U2, multiply(Z2, W2, T2))), multiply(W2, multiply(T2, U2, V2), Z2))
% 10.07/1.69 = { by lemma 8 }
% 10.07/1.69 multiply(inverse(V), multiply(Y, Z, X), V)
% 10.07/1.69 = { by lemma 26 }
% 10.07/1.69 multiply(W, multiply(Y, Z, X), inverse(W))
% 10.07/1.69 = { by lemma 25 }
% 10.07/1.69 multiply(Y, Z, X)
% 10.07/1.69 = { by lemma 30 }
% 10.07/1.69 multiply(Z, X, Y)
% 10.07/1.69 = { by lemma 30 }
% 10.07/1.69 multiply(X, Y, Z)
% 10.07/1.69
% 10.07/1.69 Lemma 32: multiply(Y, X, Z) = multiply(X, Y, Z).
% 10.07/1.69 Proof:
% 10.07/1.69 multiply(Y, X, Z)
% 10.07/1.69 = { by lemma 30 }
% 10.07/1.69 multiply(X, Z, Y)
% 10.07/1.69 = { by lemma 31 R->L }
% 10.07/1.69 multiply(X, Y, Z)
% 10.07/1.69
% 10.07/1.69 Goal 1 (prove_tba_axioms_1): multiply(multiply(d, e, a), b, multiply(d, e, c)) = multiply(d, e, multiply(a, b, c)).
% 10.07/1.69 Proof:
% 10.07/1.69 multiply(multiply(d, e, a), b, multiply(d, e, c))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(X, inverse(X), multiply(multiply(d, e, a), b, multiply(d, e, c)))
% 10.07/1.69 = { by lemma 28 R->L }
% 10.07/1.69 multiply(X, inverse(multiply(X, inverse(X), X)), multiply(multiply(d, e, a), b, multiply(d, e, c)))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(X, inverse(multiply(Y, inverse(Y), multiply(X, inverse(X), X))), multiply(multiply(d, e, a), b, multiply(d, e, c)))
% 10.07/1.69 = { by lemma 22 R->L }
% 10.07/1.69 multiply(X, inverse(multiply(multiply(Y, inverse(Y), Z), inverse(multiply(Y, inverse(Y), Z)), multiply(Y, inverse(Y), multiply(X, inverse(X), X)))), multiply(multiply(d, e, a), b, multiply(d, e, c)))
% 10.07/1.69 = { by lemma 25 R->L }
% 10.07/1.69 multiply(W, multiply(X, inverse(multiply(multiply(Y, inverse(Y), Z), inverse(multiply(Y, inverse(Y), Z)), multiply(Y, inverse(Y), multiply(X, inverse(X), X)))), multiply(multiply(d, e, a), b, multiply(d, e, c))), inverse(W))
% 10.07/1.69 = { by lemma 26 R->L }
% 10.07/1.69 multiply(inverse(X), multiply(X, inverse(multiply(multiply(Y, inverse(Y), Z), inverse(multiply(Y, inverse(Y), Z)), multiply(Y, inverse(Y), multiply(X, inverse(X), X)))), multiply(multiply(d, e, a), b, multiply(d, e, c))), X)
% 10.07/1.69 = { by lemma 9 R->L }
% 10.07/1.69 multiply(multiply(multiply(multiply(d, e, a), b, multiply(d, e, c)), inverse(multiply(multiply(d, e, a), b, multiply(d, e, c))), multiply(e, multiply(c, b, a), d)), inverse(multiply(multiply(V, U, T), S, multiply(V, U, X2))), multiply(U, multiply(X2, S, T), V))
% 10.07/1.69 = { by axiom 1 (single_axiom) }
% 10.07/1.69 multiply(e, multiply(c, b, a), d)
% 10.07/1.69 = { by lemma 31 R->L }
% 10.07/1.69 multiply(e, d, multiply(c, b, a))
% 10.07/1.69 = { by lemma 32 R->L }
% 10.07/1.69 multiply(d, e, multiply(c, b, a))
% 10.07/1.69 = { by lemma 32 R->L }
% 10.07/1.69 multiply(d, e, multiply(b, c, a))
% 10.07/1.69 = { by lemma 31 R->L }
% 10.07/1.69 multiply(d, e, multiply(b, a, c))
% 10.07/1.69 = { by lemma 32 R->L }
% 10.07/1.69 multiply(d, e, multiply(a, b, c))
% 10.07/1.69 % SZS output end Proof
% 10.07/1.69
% 10.07/1.69 RESULT: Unsatisfiable (the axioms are contradictory).
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