TSTP Solution File: GRP024-5 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP024-5 : TPTP v8.1.2. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n026.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 01:16:38 EDT 2023
% Result : Unsatisfiable 24.60s 3.70s
% Output : Proof 27.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : GRP024-5 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n026.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Mon Aug 28 23:38:05 EDT 2023
% 0.16/0.34 % CPUTime :
% 24.60/3.70 Command-line arguments: --no-flatten-goal
% 24.60/3.70
% 24.60/3.70 % SZS status Unsatisfiable
% 24.60/3.70
% 26.81/3.87 % SZS output start Proof
% 26.81/3.87 Axiom 1 (left_identity): multiply(identity, X) = X.
% 26.81/3.87 Axiom 2 (left_inverse): multiply(inverse(X), X) = identity.
% 26.81/3.87 Axiom 3 (associativity_of_commutator): commutator(commutator(X, Y), Z) = commutator(X, commutator(Y, Z)).
% 26.81/3.87 Axiom 4 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 26.81/3.87 Axiom 5 (name): commutator(X, Y) = multiply(inverse(X), multiply(inverse(Y), multiply(X, Y))).
% 26.81/3.87
% 26.81/3.87 Lemma 6: multiply(inverse(X), multiply(X, Y)) = Y.
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(inverse(X), multiply(X, Y))
% 26.81/3.87 = { by axiom 4 (associativity) R->L }
% 26.81/3.87 multiply(multiply(inverse(X), X), Y)
% 26.81/3.87 = { by axiom 2 (left_inverse) }
% 26.81/3.87 multiply(identity, Y)
% 26.81/3.87 = { by axiom 1 (left_identity) }
% 26.81/3.87 Y
% 26.81/3.87
% 26.81/3.87 Lemma 7: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(inverse(inverse(X)), Y)
% 26.81/3.87 = { by lemma 6 R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
% 26.81/3.87 = { by lemma 6 }
% 26.81/3.87 multiply(X, Y)
% 26.81/3.87
% 26.81/3.87 Lemma 8: multiply(inverse(inverse(X)), identity) = X.
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(inverse(inverse(X)), identity)
% 26.81/3.87 = { by axiom 2 (left_inverse) R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), multiply(inverse(X), X))
% 26.81/3.87 = { by lemma 6 }
% 26.81/3.87 X
% 26.81/3.87
% 26.81/3.87 Lemma 9: multiply(X, identity) = X.
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(X, identity)
% 26.81/3.87 = { by lemma 7 R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), identity)
% 26.81/3.87 = { by lemma 8 }
% 26.81/3.87 X
% 26.81/3.87
% 26.81/3.87 Lemma 10: inverse(inverse(X)) = X.
% 26.81/3.87 Proof:
% 26.81/3.87 inverse(inverse(X))
% 26.81/3.87 = { by lemma 9 R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), identity)
% 26.81/3.87 = { by lemma 8 }
% 26.81/3.87 X
% 26.81/3.87
% 26.81/3.87 Lemma 11: commutator(X, X) = identity.
% 26.81/3.87 Proof:
% 26.81/3.87 commutator(X, X)
% 26.81/3.87 = { by axiom 5 (name) }
% 26.81/3.87 multiply(inverse(X), multiply(inverse(X), multiply(X, X)))
% 26.81/3.87 = { by lemma 6 }
% 26.81/3.87 multiply(inverse(X), X)
% 26.81/3.87 = { by axiom 2 (left_inverse) }
% 26.81/3.87 identity
% 26.81/3.87
% 26.81/3.87 Lemma 12: commutator(identity, X) = identity.
% 26.81/3.87 Proof:
% 26.81/3.87 commutator(identity, X)
% 26.81/3.87 = { by axiom 5 (name) }
% 26.81/3.87 multiply(inverse(identity), multiply(inverse(X), multiply(identity, X)))
% 26.81/3.87 = { by axiom 1 (left_identity) }
% 26.81/3.87 multiply(inverse(identity), multiply(inverse(X), X))
% 26.81/3.87 = { by axiom 2 (left_inverse) }
% 26.81/3.87 multiply(inverse(identity), identity)
% 26.81/3.87 = { by axiom 2 (left_inverse) }
% 26.81/3.87 identity
% 26.81/3.87
% 26.81/3.87 Lemma 13: multiply(X, inverse(X)) = identity.
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(X, inverse(X))
% 26.81/3.87 = { by lemma 7 R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), inverse(X))
% 26.81/3.87 = { by axiom 2 (left_inverse) }
% 26.81/3.87 identity
% 26.81/3.87
% 26.81/3.87 Lemma 14: multiply(X, multiply(inverse(X), Y)) = Y.
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(X, multiply(inverse(X), Y))
% 26.81/3.87 = { by lemma 7 R->L }
% 26.81/3.87 multiply(inverse(inverse(X)), multiply(inverse(X), Y))
% 26.81/3.87 = { by lemma 6 }
% 26.81/3.87 Y
% 26.81/3.87
% 26.81/3.87 Lemma 15: multiply(inverse(multiply(X, Y)), multiply(X, Z)) = multiply(inverse(Y), Z).
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(inverse(multiply(X, Y)), multiply(X, Z))
% 26.81/3.87 = { by lemma 14 R->L }
% 26.81/3.87 multiply(inverse(multiply(X, Y)), multiply(X, multiply(Y, multiply(inverse(Y), Z))))
% 26.81/3.87 = { by axiom 4 (associativity) R->L }
% 26.81/3.87 multiply(inverse(multiply(X, Y)), multiply(multiply(X, Y), multiply(inverse(Y), Z)))
% 26.81/3.87 = { by lemma 6 }
% 26.81/3.87 multiply(inverse(Y), Z)
% 26.81/3.87
% 26.81/3.87 Lemma 16: multiply(inverse(X), multiply(inverse(Y), Z)) = multiply(inverse(multiply(Y, X)), Z).
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(inverse(X), multiply(inverse(Y), Z))
% 26.81/3.87 = { by lemma 6 R->L }
% 26.81/3.87 multiply(inverse(multiply(inverse(Y), multiply(Y, X))), multiply(inverse(Y), Z))
% 26.81/3.87 = { by lemma 15 }
% 26.81/3.87 multiply(inverse(multiply(Y, X)), Z)
% 26.81/3.87
% 26.81/3.87 Lemma 17: multiply(X, multiply(inverse(multiply(X, Y)), Y)) = commutator(inverse(X), Y).
% 26.81/3.87 Proof:
% 26.81/3.87 multiply(X, multiply(inverse(multiply(X, Y)), Y))
% 26.81/3.87 = { by lemma 16 R->L }
% 26.81/3.88 multiply(X, multiply(inverse(Y), multiply(inverse(X), Y)))
% 26.81/3.88 = { by lemma 7 R->L }
% 26.81/3.88 multiply(inverse(inverse(X)), multiply(inverse(Y), multiply(inverse(X), Y)))
% 26.81/3.88 = { by axiom 5 (name) R->L }
% 26.81/3.88 commutator(inverse(X), Y)
% 26.81/3.88
% 26.81/3.88 Lemma 18: multiply(X, inverse(multiply(Y, X))) = inverse(Y).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(X, inverse(multiply(Y, X)))
% 26.81/3.88 = { by lemma 6 R->L }
% 26.81/3.88 multiply(inverse(Y), multiply(Y, multiply(X, inverse(multiply(Y, X)))))
% 26.81/3.88 = { by axiom 4 (associativity) R->L }
% 26.81/3.88 multiply(inverse(Y), multiply(multiply(Y, X), inverse(multiply(Y, X))))
% 26.81/3.88 = { by lemma 13 }
% 26.81/3.88 multiply(inverse(Y), identity)
% 26.81/3.88 = { by lemma 9 }
% 26.81/3.88 inverse(Y)
% 26.81/3.88
% 26.81/3.88 Lemma 19: multiply(inverse(X), inverse(Y)) = inverse(multiply(Y, X)).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(inverse(X), inverse(Y))
% 26.81/3.88 = { by lemma 18 R->L }
% 26.81/3.88 multiply(inverse(X), multiply(X, inverse(multiply(Y, X))))
% 26.81/3.88 = { by lemma 6 }
% 26.81/3.88 inverse(multiply(Y, X))
% 26.81/3.88
% 26.81/3.88 Lemma 20: inverse(multiply(inverse(X), Y)) = multiply(inverse(Y), X).
% 26.81/3.88 Proof:
% 26.81/3.88 inverse(multiply(inverse(X), Y))
% 26.81/3.88 = { by lemma 19 R->L }
% 26.81/3.88 multiply(inverse(Y), inverse(inverse(X)))
% 26.81/3.88 = { by lemma 10 }
% 26.81/3.88 multiply(inverse(Y), X)
% 26.81/3.88
% 26.81/3.88 Lemma 21: inverse(commutator(X, Y)) = commutator(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 inverse(commutator(X, Y))
% 26.81/3.88 = { by lemma 10 R->L }
% 26.81/3.88 inverse(commutator(inverse(inverse(X)), Y))
% 26.81/3.88 = { by lemma 17 R->L }
% 26.81/3.88 inverse(multiply(inverse(X), multiply(inverse(multiply(inverse(X), Y)), Y)))
% 26.81/3.88 = { by lemma 20 }
% 26.81/3.88 multiply(inverse(multiply(inverse(multiply(inverse(X), Y)), Y)), X)
% 26.81/3.88 = { by lemma 20 }
% 26.81/3.88 multiply(multiply(inverse(Y), multiply(inverse(X), Y)), X)
% 26.81/3.88 = { by axiom 4 (associativity) }
% 26.81/3.88 multiply(inverse(Y), multiply(multiply(inverse(X), Y), X))
% 26.81/3.88 = { by axiom 4 (associativity) }
% 26.81/3.88 multiply(inverse(Y), multiply(inverse(X), multiply(Y, X)))
% 26.81/3.88 = { by axiom 5 (name) R->L }
% 26.81/3.88 commutator(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 22: multiply(inverse(multiply(X, Y)), X) = inverse(Y).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(inverse(multiply(X, Y)), X)
% 26.81/3.88 = { by lemma 20 R->L }
% 26.81/3.88 inverse(multiply(inverse(X), multiply(X, Y)))
% 26.81/3.88 = { by lemma 6 }
% 26.81/3.88 inverse(Y)
% 26.81/3.88
% 26.81/3.88 Lemma 23: multiply(inverse(X), multiply(Y, X)) = multiply(Y, commutator(Y, X)).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(inverse(X), multiply(Y, X))
% 26.81/3.88 = { by lemma 14 R->L }
% 26.81/3.88 multiply(Y, multiply(inverse(Y), multiply(inverse(X), multiply(Y, X))))
% 26.81/3.88 = { by axiom 5 (name) R->L }
% 26.81/3.88 multiply(Y, commutator(Y, X))
% 26.81/3.88
% 26.81/3.88 Lemma 24: commutator(X, commutator(X, Y)) = identity.
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, commutator(X, Y))
% 26.81/3.88 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.88 commutator(commutator(X, X), Y)
% 26.81/3.88 = { by lemma 11 }
% 26.81/3.88 commutator(identity, Y)
% 26.81/3.88 = { by lemma 12 }
% 26.81/3.88 identity
% 26.81/3.88
% 26.81/3.88 Lemma 25: multiply(X, multiply(Y, commutator(Y, X))) = multiply(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(X, multiply(Y, commutator(Y, X)))
% 26.81/3.88 = { by lemma 7 R->L }
% 26.81/3.88 multiply(inverse(inverse(X)), multiply(Y, commutator(Y, X)))
% 26.81/3.88 = { by lemma 23 R->L }
% 26.81/3.88 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(Y, X)))
% 26.81/3.88 = { by lemma 6 }
% 26.81/3.88 multiply(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 26: multiply(commutator(X, Y), X) = multiply(X, commutator(X, Y)).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(commutator(X, Y), X)
% 26.81/3.88 = { by lemma 9 R->L }
% 26.81/3.88 multiply(commutator(X, Y), multiply(X, identity))
% 26.81/3.88 = { by lemma 24 R->L }
% 26.81/3.88 multiply(commutator(X, Y), multiply(X, commutator(X, commutator(X, Y))))
% 26.81/3.88 = { by lemma 25 }
% 26.81/3.88 multiply(X, commutator(X, Y))
% 26.81/3.88
% 26.81/3.88 Lemma 27: commutator(X, commutator(Y, X)) = identity.
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, commutator(Y, X))
% 26.81/3.88 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.88 commutator(commutator(X, Y), X)
% 26.81/3.88 = { by axiom 5 (name) }
% 26.81/3.88 multiply(inverse(commutator(X, Y)), multiply(inverse(X), multiply(commutator(X, Y), X)))
% 26.81/3.88 = { by lemma 26 }
% 26.81/3.88 multiply(inverse(commutator(X, Y)), multiply(inverse(X), multiply(X, commutator(X, Y))))
% 26.81/3.88 = { by lemma 6 }
% 26.81/3.88 multiply(inverse(commutator(X, Y)), commutator(X, Y))
% 26.81/3.88 = { by axiom 2 (left_inverse) }
% 26.81/3.88 identity
% 26.81/3.88
% 26.81/3.88 Lemma 28: multiply(commutator(X, Y), Y) = multiply(Y, commutator(X, Y)).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(commutator(X, Y), Y)
% 26.81/3.88 = { by lemma 9 R->L }
% 26.81/3.88 multiply(commutator(X, Y), multiply(Y, identity))
% 26.81/3.88 = { by lemma 27 R->L }
% 26.81/3.88 multiply(commutator(X, Y), multiply(Y, commutator(Y, commutator(X, Y))))
% 26.81/3.88 = { by lemma 25 }
% 26.81/3.88 multiply(Y, commutator(X, Y))
% 26.81/3.88
% 26.81/3.88 Lemma 29: commutator(X, multiply(Y, X)) = inverse(commutator(Y, X)).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, multiply(Y, X))
% 26.81/3.88 = { by lemma 10 R->L }
% 26.81/3.88 commutator(inverse(inverse(X)), multiply(Y, X))
% 26.81/3.88 = { by lemma 17 R->L }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(multiply(inverse(X), multiply(Y, X))), multiply(Y, X)))
% 26.81/3.88 = { by lemma 23 }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(multiply(Y, commutator(Y, X))), multiply(Y, X)))
% 26.81/3.88 = { by lemma 16 }
% 26.81/3.88 multiply(inverse(multiply(multiply(Y, commutator(Y, X)), X)), multiply(Y, X))
% 26.81/3.88 = { by axiom 4 (associativity) }
% 26.81/3.88 multiply(inverse(multiply(Y, multiply(commutator(Y, X), X))), multiply(Y, X))
% 26.81/3.88 = { by lemma 15 }
% 26.81/3.88 multiply(inverse(multiply(commutator(Y, X), X)), X)
% 26.81/3.88 = { by lemma 28 }
% 26.81/3.88 multiply(inverse(multiply(X, commutator(Y, X))), X)
% 26.81/3.88 = { by lemma 22 }
% 26.81/3.88 inverse(commutator(Y, X))
% 26.81/3.88
% 26.81/3.88 Lemma 30: commutator(multiply(inverse(X), Y), X) = commutator(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(multiply(inverse(X), Y), X)
% 26.81/3.88 = { by axiom 5 (name) }
% 26.81/3.88 multiply(inverse(multiply(inverse(X), Y)), multiply(inverse(X), multiply(multiply(inverse(X), Y), X)))
% 26.81/3.88 = { by lemma 15 }
% 26.81/3.88 multiply(inverse(Y), multiply(multiply(inverse(X), Y), X))
% 26.81/3.88 = { by axiom 4 (associativity) }
% 26.81/3.88 multiply(inverse(Y), multiply(inverse(X), multiply(Y, X)))
% 26.81/3.88 = { by axiom 5 (name) R->L }
% 26.81/3.88 commutator(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 31: commutator(multiply(X, Y), inverse(X)) = commutator(Y, inverse(X)).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(multiply(X, Y), inverse(X))
% 26.81/3.88 = { by lemma 7 R->L }
% 26.81/3.88 commutator(multiply(inverse(inverse(X)), Y), inverse(X))
% 26.81/3.88 = { by lemma 30 }
% 26.81/3.88 commutator(Y, inverse(X))
% 26.81/3.88
% 26.81/3.88 Lemma 32: commutator(X, multiply(X, Y)) = commutator(X, Y).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, multiply(X, Y))
% 26.81/3.88 = { by axiom 5 (name) }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(multiply(X, Y)), multiply(X, multiply(X, Y))))
% 26.81/3.88 = { by lemma 15 }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(Y), multiply(X, Y)))
% 26.81/3.88 = { by axiom 5 (name) R->L }
% 26.81/3.88 commutator(X, Y)
% 26.81/3.88
% 26.81/3.88 Lemma 33: commutator(inverse(X), Y) = commutator(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(inverse(X), Y)
% 26.81/3.88 = { by lemma 22 R->L }
% 26.81/3.88 commutator(multiply(inverse(multiply(Y, X)), Y), Y)
% 26.81/3.88 = { by lemma 6 R->L }
% 26.81/3.88 commutator(multiply(inverse(multiply(Y, X)), Y), multiply(inverse(inverse(multiply(Y, X))), multiply(inverse(multiply(Y, X)), Y)))
% 26.81/3.88 = { by lemma 29 }
% 26.81/3.88 inverse(commutator(inverse(inverse(multiply(Y, X))), multiply(inverse(multiply(Y, X)), Y)))
% 26.81/3.88 = { by lemma 21 }
% 26.81/3.88 commutator(multiply(inverse(multiply(Y, X)), Y), inverse(inverse(multiply(Y, X))))
% 26.81/3.88 = { by lemma 31 }
% 26.81/3.88 commutator(Y, inverse(inverse(multiply(Y, X))))
% 26.81/3.88 = { by lemma 10 }
% 26.81/3.88 commutator(Y, multiply(Y, X))
% 26.81/3.88 = { by lemma 32 }
% 26.81/3.88 commutator(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 34: commutator(X, inverse(Y)) = commutator(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, inverse(Y))
% 26.81/3.88 = { by lemma 21 R->L }
% 26.81/3.88 inverse(commutator(inverse(Y), X))
% 26.81/3.88 = { by lemma 17 R->L }
% 26.81/3.88 inverse(multiply(Y, multiply(inverse(multiply(Y, X)), X)))
% 26.81/3.88 = { by lemma 19 R->L }
% 26.81/3.88 multiply(inverse(multiply(inverse(multiply(Y, X)), X)), inverse(Y))
% 26.81/3.88 = { by lemma 16 R->L }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(inverse(multiply(Y, X))), inverse(Y)))
% 26.81/3.88 = { by lemma 19 R->L }
% 26.81/3.88 multiply(inverse(X), multiply(inverse(multiply(inverse(X), inverse(Y))), inverse(Y)))
% 26.81/3.88 = { by lemma 17 }
% 26.81/3.88 commutator(inverse(inverse(X)), inverse(Y))
% 26.81/3.88 = { by lemma 33 }
% 26.81/3.88 commutator(inverse(Y), inverse(X))
% 26.81/3.88 = { by lemma 33 }
% 26.81/3.88 commutator(inverse(X), Y)
% 26.81/3.88 = { by lemma 33 }
% 26.81/3.88 commutator(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 35: inverse(multiply(X, inverse(Y))) = multiply(Y, inverse(X)).
% 26.81/3.88 Proof:
% 26.81/3.88 inverse(multiply(X, inverse(Y)))
% 26.81/3.88 = { by lemma 14 R->L }
% 26.81/3.88 multiply(Y, multiply(inverse(Y), inverse(multiply(X, inverse(Y)))))
% 26.81/3.88 = { by lemma 18 }
% 26.81/3.88 multiply(Y, inverse(X))
% 26.81/3.88
% 26.81/3.88 Lemma 36: commutator(Y, commutator(X, Z)) = commutator(X, commutator(Y, Z)).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(Y, commutator(X, Z))
% 26.81/3.88 = { by lemma 34 R->L }
% 26.81/3.88 commutator(Y, commutator(Z, inverse(X)))
% 26.81/3.88 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.88 commutator(commutator(Y, Z), inverse(X))
% 26.81/3.88 = { by lemma 34 }
% 26.81/3.88 commutator(X, commutator(Y, Z))
% 26.81/3.88
% 26.81/3.88 Lemma 37: commutator(X, commutator(Z, Y)) = commutator(X, commutator(Y, Z)).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, commutator(Z, Y))
% 26.81/3.88 = { by lemma 36 }
% 26.81/3.88 commutator(Z, commutator(X, Y))
% 26.81/3.88 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.88 commutator(commutator(Z, X), Y)
% 26.81/3.88 = { by lemma 34 R->L }
% 26.81/3.88 commutator(commutator(X, inverse(Z)), Y)
% 26.81/3.88 = { by axiom 3 (associativity_of_commutator) }
% 26.81/3.88 commutator(X, commutator(inverse(Z), Y))
% 26.81/3.88 = { by lemma 33 }
% 26.81/3.88 commutator(X, commutator(Y, Z))
% 26.81/3.88
% 26.81/3.88 Lemma 38: commutator(multiply(X, Y), Y) = commutator(X, Y).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(multiply(X, Y), Y)
% 26.81/3.88 = { by lemma 6 R->L }
% 26.81/3.88 commutator(multiply(X, Y), multiply(inverse(X), multiply(X, Y)))
% 26.81/3.88 = { by lemma 29 }
% 26.81/3.88 inverse(commutator(inverse(X), multiply(X, Y)))
% 26.81/3.88 = { by lemma 21 }
% 26.81/3.88 commutator(multiply(X, Y), inverse(X))
% 26.81/3.88 = { by lemma 31 }
% 26.81/3.88 commutator(Y, inverse(X))
% 26.81/3.88 = { by lemma 34 }
% 26.81/3.88 commutator(X, Y)
% 26.81/3.88
% 26.81/3.88 Lemma 39: commutator(X, multiply(Y, multiply(Z, X))) = commutator(X, multiply(Y, Z)).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(X, multiply(Y, multiply(Z, X)))
% 26.81/3.88 = { by axiom 4 (associativity) R->L }
% 26.81/3.88 commutator(X, multiply(multiply(Y, Z), X))
% 26.81/3.88 = { by lemma 29 }
% 26.81/3.88 inverse(commutator(multiply(Y, Z), X))
% 26.81/3.88 = { by lemma 21 }
% 26.81/3.88 commutator(X, multiply(Y, Z))
% 26.81/3.88
% 26.81/3.88 Lemma 40: commutator(multiply(X, Y), X) = commutator(Y, X).
% 26.81/3.88 Proof:
% 26.81/3.88 commutator(multiply(X, Y), X)
% 26.81/3.88 = { by lemma 30 R->L }
% 26.81/3.88 commutator(multiply(inverse(X), multiply(X, Y)), X)
% 26.81/3.88 = { by lemma 6 }
% 26.81/3.88 commutator(Y, X)
% 26.81/3.88
% 26.81/3.88 Lemma 41: multiply(inverse(multiply(X, Y)), multiply(Y, multiply(X, Z))) = multiply(commutator(Y, X), Z).
% 26.81/3.88 Proof:
% 26.81/3.88 multiply(inverse(multiply(X, Y)), multiply(Y, multiply(X, Z)))
% 26.81/3.88 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(inverse(multiply(X, Y)), multiply(multiply(Y, X), Z))
% 26.81/3.89 = { by lemma 16 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(inverse(X), multiply(multiply(Y, X), Z)))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(multiply(inverse(X), multiply(Y, X)), Z))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(multiply(inverse(Y), multiply(inverse(X), multiply(Y, X))), Z)
% 26.81/3.89 = { by axiom 5 (name) R->L }
% 26.81/3.89 multiply(commutator(Y, X), Z)
% 26.81/3.89
% 26.81/3.89 Lemma 42: multiply(X, multiply(Y, multiply(commutator(Y, X), Z))) = multiply(Y, multiply(X, Z)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(X, multiply(Y, multiply(commutator(Y, X), Z)))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(multiply(X, Y), multiply(commutator(Y, X), Z))
% 26.81/3.89 = { by lemma 7 R->L }
% 26.81/3.89 multiply(inverse(inverse(multiply(X, Y))), multiply(commutator(Y, X), Z))
% 26.81/3.89 = { by lemma 41 R->L }
% 26.81/3.89 multiply(inverse(inverse(multiply(X, Y))), multiply(inverse(multiply(X, Y)), multiply(Y, multiply(X, Z))))
% 26.81/3.89 = { by lemma 6 }
% 26.81/3.89 multiply(Y, multiply(X, Z))
% 26.81/3.89
% 26.81/3.89 Lemma 43: multiply(inverse(X), multiply(commutator(X, Y), Z)) = multiply(Y, multiply(inverse(multiply(Y, X)), Z)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(inverse(X), multiply(commutator(X, Y), Z))
% 26.81/3.89 = { by lemma 40 R->L }
% 26.81/3.89 multiply(inverse(X), multiply(commutator(multiply(Y, X), Y), Z))
% 26.81/3.89 = { by lemma 34 R->L }
% 26.81/3.89 multiply(inverse(X), multiply(commutator(Y, inverse(multiply(Y, X))), Z))
% 26.81/3.89 = { by lemma 15 R->L }
% 26.81/3.89 multiply(inverse(multiply(Y, X)), multiply(Y, multiply(commutator(Y, inverse(multiply(Y, X))), Z)))
% 26.81/3.89 = { by lemma 42 }
% 26.81/3.89 multiply(Y, multiply(inverse(multiply(Y, X)), Z))
% 26.81/3.89
% 26.81/3.89 Lemma 44: multiply(inverse(X), multiply(Y, multiply(X, inverse(Y)))) = commutator(Y, X).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(inverse(X), multiply(Y, multiply(X, inverse(Y))))
% 26.81/3.89 = { by lemma 7 R->L }
% 26.81/3.89 multiply(inverse(X), multiply(inverse(inverse(Y)), multiply(X, inverse(Y))))
% 26.81/3.89 = { by axiom 5 (name) R->L }
% 26.81/3.89 commutator(X, inverse(Y))
% 26.81/3.89 = { by lemma 34 }
% 26.81/3.89 commutator(Y, X)
% 26.81/3.89
% 26.81/3.89 Lemma 45: multiply(X, multiply(Y, inverse(multiply(Y, X)))) = commutator(X, Y).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(X, multiply(Y, inverse(multiply(Y, X))))
% 26.81/3.89 = { by lemma 19 R->L }
% 26.81/3.89 multiply(X, multiply(Y, multiply(inverse(X), inverse(Y))))
% 26.81/3.89 = { by lemma 7 R->L }
% 26.81/3.89 multiply(inverse(inverse(X)), multiply(Y, multiply(inverse(X), inverse(Y))))
% 26.81/3.89 = { by lemma 44 }
% 26.81/3.89 commutator(Y, inverse(X))
% 26.81/3.89 = { by lemma 34 }
% 26.81/3.89 commutator(X, Y)
% 26.81/3.89
% 26.81/3.89 Lemma 46: multiply(commutator(X, Y), commutator(Y, X)) = identity.
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(commutator(X, Y), commutator(Y, X))
% 26.81/3.89 = { by lemma 33 R->L }
% 26.81/3.89 multiply(commutator(X, Y), commutator(inverse(X), Y))
% 26.81/3.89 = { by lemma 17 R->L }
% 26.81/3.89 multiply(commutator(X, Y), multiply(X, multiply(inverse(multiply(X, Y)), Y)))
% 26.81/3.89 = { by lemma 20 R->L }
% 26.81/3.89 multiply(commutator(X, Y), multiply(X, inverse(multiply(inverse(Y), multiply(X, Y)))))
% 26.81/3.89 = { by lemma 23 }
% 26.81/3.89 multiply(commutator(X, Y), multiply(X, inverse(multiply(X, commutator(X, Y)))))
% 26.81/3.89 = { by lemma 45 }
% 26.81/3.89 commutator(commutator(X, Y), X)
% 26.81/3.89 = { by axiom 3 (associativity_of_commutator) }
% 26.81/3.89 commutator(X, commutator(Y, X))
% 26.81/3.89 = { by lemma 27 }
% 26.81/3.89 identity
% 26.81/3.89
% 26.81/3.89 Lemma 47: multiply(inverse(X), commutator(Y, Z)) = inverse(multiply(commutator(Z, Y), X)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(inverse(X), commutator(Y, Z))
% 26.81/3.89 = { by lemma 15 R->L }
% 26.81/3.89 multiply(inverse(multiply(commutator(Z, Y), X)), multiply(commutator(Z, Y), commutator(Y, Z)))
% 26.81/3.89 = { by lemma 46 }
% 26.81/3.89 multiply(inverse(multiply(commutator(Z, Y), X)), identity)
% 26.81/3.89 = { by lemma 9 }
% 26.81/3.89 inverse(multiply(commutator(Z, Y), X))
% 26.81/3.89
% 26.81/3.89 Lemma 48: commutator(Y, commutator(Y, X)) = commutator(X, commutator(Y, X)).
% 26.81/3.89 Proof:
% 26.81/3.89 commutator(Y, commutator(Y, X))
% 26.81/3.89 = { by lemma 38 R->L }
% 26.81/3.89 commutator(Y, commutator(multiply(Y, X), X))
% 26.81/3.89 = { by lemma 34 R->L }
% 26.81/3.89 commutator(Y, commutator(X, inverse(multiply(Y, X))))
% 26.81/3.89 = { by lemma 36 }
% 26.81/3.89 commutator(X, commutator(Y, inverse(multiply(Y, X))))
% 26.81/3.89 = { by lemma 32 R->L }
% 26.81/3.89 commutator(X, commutator(Y, multiply(Y, inverse(multiply(Y, X)))))
% 26.81/3.89 = { by lemma 36 R->L }
% 26.81/3.89 commutator(Y, commutator(X, multiply(Y, inverse(multiply(Y, X)))))
% 26.81/3.89 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.89 commutator(commutator(Y, X), multiply(Y, inverse(multiply(Y, X))))
% 26.81/3.89 = { by lemma 39 R->L }
% 26.81/3.89 commutator(commutator(Y, X), multiply(Y, multiply(inverse(multiply(Y, X)), commutator(Y, X))))
% 26.81/3.89 = { by lemma 43 R->L }
% 26.81/3.89 commutator(commutator(Y, X), multiply(inverse(X), multiply(commutator(X, Y), commutator(Y, X))))
% 26.81/3.89 = { by lemma 39 }
% 26.81/3.89 commutator(commutator(Y, X), multiply(inverse(X), commutator(X, Y)))
% 26.81/3.89 = { by lemma 47 }
% 26.81/3.89 commutator(commutator(Y, X), inverse(multiply(commutator(Y, X), X)))
% 26.81/3.89 = { by lemma 34 }
% 26.81/3.89 commutator(multiply(commutator(Y, X), X), commutator(Y, X))
% 26.81/3.89 = { by lemma 28 }
% 26.81/3.89 commutator(multiply(X, commutator(Y, X)), commutator(Y, X))
% 26.81/3.89 = { by lemma 38 }
% 26.81/3.89 commutator(X, commutator(Y, X))
% 26.81/3.89
% 26.81/3.89 Lemma 49: multiply(commutator(X, Y), multiply(Y, Z)) = multiply(Y, multiply(commutator(X, Y), Z)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(commutator(X, Y), multiply(Y, Z))
% 26.81/3.89 = { by lemma 33 R->L }
% 26.81/3.89 multiply(commutator(inverse(Y), X), multiply(Y, Z))
% 26.81/3.89 = { by lemma 30 R->L }
% 26.81/3.89 multiply(commutator(multiply(inverse(X), inverse(Y)), X), multiply(Y, Z))
% 26.81/3.89 = { by lemma 19 }
% 26.81/3.89 multiply(commutator(inverse(multiply(Y, X)), X), multiply(Y, Z))
% 26.81/3.89 = { by lemma 41 R->L }
% 26.81/3.89 multiply(inverse(multiply(X, inverse(multiply(Y, X)))), multiply(inverse(multiply(Y, X)), multiply(X, multiply(Y, Z))))
% 26.81/3.89 = { by lemma 41 }
% 26.81/3.89 multiply(inverse(multiply(X, inverse(multiply(Y, X)))), multiply(commutator(X, Y), Z))
% 26.81/3.89 = { by lemma 35 }
% 26.81/3.89 multiply(multiply(multiply(Y, X), inverse(X)), multiply(commutator(X, Y), Z))
% 26.81/3.89 = { by axiom 4 (associativity) }
% 26.81/3.89 multiply(multiply(Y, X), multiply(inverse(X), multiply(commutator(X, Y), Z)))
% 26.81/3.89 = { by axiom 4 (associativity) }
% 26.81/3.89 multiply(Y, multiply(X, multiply(inverse(X), multiply(commutator(X, Y), Z))))
% 26.81/3.89 = { by lemma 14 }
% 26.81/3.89 multiply(Y, multiply(commutator(X, Y), Z))
% 26.81/3.89
% 26.81/3.89 Lemma 50: multiply(inverse(X), multiply(Y, inverse(Z))) = inverse(multiply(Z, multiply(inverse(Y), X))).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(inverse(X), multiply(Y, inverse(Z)))
% 26.81/3.89 = { by lemma 35 R->L }
% 26.81/3.89 multiply(inverse(X), inverse(multiply(Z, inverse(Y))))
% 26.81/3.89 = { by lemma 19 }
% 26.81/3.89 inverse(multiply(multiply(Z, inverse(Y)), X))
% 26.81/3.89 = { by axiom 4 (associativity) }
% 26.81/3.89 inverse(multiply(Z, multiply(inverse(Y), X)))
% 26.81/3.89
% 26.81/3.89 Lemma 51: multiply(commutator(X, Y), multiply(inverse(Y), Z)) = multiply(inverse(Y), multiply(commutator(X, Y), Z)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(commutator(X, Y), multiply(inverse(Y), Z))
% 26.81/3.89 = { by lemma 6 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(Y, multiply(commutator(X, Y), multiply(inverse(Y), Z))))
% 26.81/3.89 = { by lemma 49 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(commutator(X, Y), multiply(Y, multiply(inverse(Y), Z))))
% 26.81/3.89 = { by lemma 14 }
% 26.81/3.89 multiply(inverse(Y), multiply(commutator(X, Y), Z))
% 26.81/3.89
% 26.81/3.89 Lemma 52: multiply(X, inverse(multiply(X, commutator(X, Y)))) = commutator(Y, X).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(X, inverse(multiply(X, commutator(X, Y))))
% 26.81/3.89 = { by lemma 14 R->L }
% 26.81/3.89 multiply(X, multiply(Y, multiply(inverse(Y), inverse(multiply(X, commutator(X, Y))))))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(multiply(X, Y), multiply(inverse(Y), inverse(multiply(X, commutator(X, Y)))))
% 26.81/3.89 = { by lemma 23 R->L }
% 26.81/3.89 multiply(multiply(X, Y), multiply(inverse(Y), inverse(multiply(inverse(Y), multiply(X, Y)))))
% 26.81/3.89 = { by lemma 45 }
% 26.81/3.89 commutator(multiply(X, Y), inverse(Y))
% 26.81/3.89 = { by lemma 6 R->L }
% 26.81/3.89 commutator(multiply(X, Y), inverse(multiply(inverse(X), multiply(X, Y))))
% 26.81/3.89 = { by lemma 32 R->L }
% 26.81/3.89 commutator(multiply(X, Y), multiply(multiply(X, Y), inverse(multiply(inverse(X), multiply(X, Y)))))
% 26.81/3.89 = { by lemma 18 }
% 26.81/3.89 commutator(multiply(X, Y), inverse(inverse(X)))
% 26.81/3.89 = { by lemma 10 }
% 26.81/3.89 commutator(multiply(X, Y), X)
% 26.81/3.89 = { by lemma 40 }
% 26.81/3.89 commutator(Y, X)
% 26.81/3.89
% 26.81/3.89 Lemma 53: inverse(multiply(commutator(X, Y), commutator(Y, Z))) = multiply(commutator(Z, Y), commutator(Y, X)).
% 26.81/3.89 Proof:
% 26.81/3.89 inverse(multiply(commutator(X, Y), commutator(Y, Z)))
% 26.81/3.89 = { by lemma 22 R->L }
% 26.81/3.89 multiply(inverse(multiply(Y, multiply(commutator(X, Y), commutator(Y, Z)))), Y)
% 26.81/3.89 = { by lemma 49 R->L }
% 26.81/3.89 multiply(inverse(multiply(commutator(X, Y), multiply(Y, commutator(Y, Z)))), Y)
% 26.81/3.89 = { by lemma 26 R->L }
% 26.81/3.89 multiply(inverse(multiply(commutator(X, Y), multiply(commutator(Y, Z), Y))), Y)
% 26.81/3.89 = { by lemma 21 R->L }
% 26.81/3.89 multiply(inverse(multiply(commutator(X, Y), multiply(inverse(commutator(Z, Y)), Y))), Y)
% 26.81/3.89 = { by lemma 33 R->L }
% 26.81/3.89 multiply(inverse(multiply(commutator(inverse(Y), X), multiply(inverse(commutator(Z, Y)), Y))), Y)
% 26.81/3.89 = { by lemma 20 R->L }
% 26.81/3.89 inverse(multiply(inverse(Y), multiply(commutator(inverse(Y), X), multiply(inverse(commutator(Z, Y)), Y))))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 inverse(multiply(multiply(inverse(Y), commutator(inverse(Y), X)), multiply(inverse(commutator(Z, Y)), Y)))
% 26.81/3.89 = { by lemma 50 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(commutator(Z, Y), inverse(multiply(inverse(Y), commutator(inverse(Y), X)))))
% 26.81/3.89 = { by lemma 51 R->L }
% 26.81/3.89 multiply(commutator(Z, Y), multiply(inverse(Y), inverse(multiply(inverse(Y), commutator(inverse(Y), X)))))
% 26.81/3.89 = { by lemma 52 }
% 26.81/3.89 multiply(commutator(Z, Y), commutator(X, inverse(Y)))
% 26.81/3.89 = { by lemma 34 }
% 26.81/3.89 multiply(commutator(Z, Y), commutator(Y, X))
% 26.81/3.89
% 26.81/3.89 Lemma 54: multiply(X, multiply(Y, multiply(inverse(multiply(X, Y)), Z))) = Z.
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(X, multiply(Y, multiply(inverse(multiply(X, Y)), Z)))
% 26.81/3.89 = { by axiom 4 (associativity) R->L }
% 26.81/3.89 multiply(multiply(X, Y), multiply(inverse(multiply(X, Y)), Z))
% 26.81/3.89 = { by lemma 14 }
% 26.81/3.89 Z
% 26.81/3.89
% 26.81/3.89 Lemma 55: multiply(X, multiply(inverse(multiply(Y, X)), Z)) = multiply(inverse(Y), Z).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(X, multiply(inverse(multiply(Y, X)), Z))
% 26.81/3.89 = { by lemma 6 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(Y, multiply(X, multiply(inverse(multiply(Y, X)), Z))))
% 26.81/3.89 = { by lemma 54 }
% 26.81/3.89 multiply(inverse(Y), Z)
% 26.81/3.89
% 26.81/3.89 Lemma 56: commutator(inverse(X), commutator(X, Y)) = identity.
% 26.81/3.89 Proof:
% 26.81/3.89 commutator(inverse(X), commutator(X, Y))
% 26.81/3.89 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.89 commutator(commutator(inverse(X), X), Y)
% 26.81/3.89 = { by axiom 5 (name) }
% 26.81/3.89 commutator(multiply(inverse(inverse(X)), multiply(inverse(X), multiply(inverse(X), X))), Y)
% 26.81/3.89 = { by lemma 6 }
% 26.81/3.89 commutator(multiply(inverse(X), X), Y)
% 26.81/3.89 = { by axiom 2 (left_inverse) }
% 26.81/3.89 commutator(identity, Y)
% 26.81/3.89 = { by lemma 12 }
% 26.81/3.89 identity
% 26.81/3.89
% 26.81/3.89 Lemma 57: multiply(inverse(multiply(X, Y)), multiply(Y, Z)) = multiply(inverse(X), multiply(commutator(Y, X), Z)).
% 26.81/3.89 Proof:
% 26.81/3.89 multiply(inverse(multiply(X, Y)), multiply(Y, Z))
% 26.81/3.89 = { by lemma 16 R->L }
% 26.81/3.89 multiply(inverse(Y), multiply(inverse(X), multiply(Y, Z)))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 multiply(multiply(inverse(Y), inverse(X)), multiply(Y, Z))
% 26.81/3.90 = { by lemma 35 R->L }
% 26.81/3.90 multiply(inverse(multiply(X, inverse(inverse(Y)))), multiply(Y, Z))
% 26.81/3.90 = { by lemma 7 R->L }
% 26.81/3.90 multiply(inverse(multiply(X, inverse(inverse(Y)))), multiply(inverse(inverse(Y)), Z))
% 26.81/3.90 = { by lemma 14 R->L }
% 26.81/3.90 multiply(inverse(multiply(X, inverse(inverse(Y)))), multiply(inverse(inverse(Y)), multiply(X, multiply(inverse(X), Z))))
% 26.81/3.90 = { by lemma 41 }
% 26.81/3.90 multiply(commutator(inverse(inverse(Y)), X), multiply(inverse(X), Z))
% 26.81/3.90 = { by lemma 33 }
% 26.81/3.90 multiply(commutator(X, inverse(Y)), multiply(inverse(X), Z))
% 26.81/3.90 = { by axiom 1 (left_identity) R->L }
% 26.81/3.90 multiply(commutator(X, inverse(Y)), multiply(inverse(X), multiply(identity, Z)))
% 26.81/3.90 = { by lemma 56 R->L }
% 26.81/3.90 multiply(commutator(X, inverse(Y)), multiply(inverse(X), multiply(commutator(inverse(X), commutator(X, inverse(Y))), Z)))
% 26.81/3.90 = { by lemma 42 }
% 26.81/3.90 multiply(inverse(X), multiply(commutator(X, inverse(Y)), Z))
% 26.81/3.90 = { by lemma 34 }
% 26.81/3.90 multiply(inverse(X), multiply(commutator(Y, X), Z))
% 26.81/3.90
% 26.81/3.90 Lemma 58: multiply(inverse(multiply(X, Y)), multiply(Y, X)) = commutator(Y, X).
% 26.81/3.90 Proof:
% 26.81/3.90 multiply(inverse(multiply(X, Y)), multiply(Y, X))
% 26.81/3.90 = { by lemma 20 R->L }
% 26.81/3.90 inverse(multiply(inverse(multiply(Y, X)), multiply(X, Y)))
% 26.81/3.90 = { by lemma 19 R->L }
% 26.81/3.90 multiply(inverse(multiply(X, Y)), inverse(inverse(multiply(Y, X))))
% 26.81/3.90 = { by lemma 16 R->L }
% 26.81/3.90 multiply(inverse(Y), multiply(inverse(X), inverse(inverse(multiply(Y, X)))))
% 26.81/3.90 = { by lemma 15 R->L }
% 26.81/3.90 multiply(inverse(Y), multiply(inverse(multiply(Y, X)), multiply(Y, inverse(inverse(multiply(Y, X))))))
% 26.81/3.90 = { by lemma 44 }
% 26.81/3.90 commutator(inverse(multiply(Y, X)), Y)
% 26.81/3.90 = { by lemma 33 }
% 26.81/3.90 commutator(Y, multiply(Y, X))
% 26.81/3.90 = { by lemma 32 }
% 26.81/3.90 commutator(Y, X)
% 26.81/3.90
% 26.81/3.90 Lemma 59: commutator(X, multiply(commutator(Y, X), Z)) = commutator(X, Z).
% 26.81/3.90 Proof:
% 26.81/3.90 commutator(X, multiply(commutator(Y, X), Z))
% 26.81/3.90 = { by lemma 34 R->L }
% 26.81/3.90 commutator(multiply(commutator(Y, X), Z), inverse(X))
% 26.81/3.90 = { by lemma 33 R->L }
% 26.81/3.90 commutator(inverse(inverse(X)), multiply(commutator(Y, X), Z))
% 26.81/3.90 = { by lemma 17 R->L }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(multiply(inverse(X), multiply(commutator(Y, X), Z))), multiply(commutator(Y, X), Z)))
% 26.81/3.90 = { by lemma 57 R->L }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(multiply(inverse(multiply(X, Y)), multiply(Y, Z))), multiply(commutator(Y, X), Z)))
% 26.81/3.90 = { by lemma 16 }
% 26.81/3.90 multiply(inverse(multiply(multiply(inverse(multiply(X, Y)), multiply(Y, Z)), X)), multiply(commutator(Y, X), Z))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 multiply(inverse(multiply(inverse(multiply(X, Y)), multiply(multiply(Y, Z), X))), multiply(commutator(Y, X), Z))
% 26.81/3.90 = { by lemma 20 }
% 26.81/3.90 multiply(multiply(inverse(multiply(multiply(Y, Z), X)), multiply(X, Y)), multiply(commutator(Y, X), Z))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 multiply(inverse(multiply(multiply(Y, Z), X)), multiply(multiply(X, Y), multiply(commutator(Y, X), Z)))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 multiply(inverse(multiply(Y, multiply(Z, X))), multiply(multiply(X, Y), multiply(commutator(Y, X), Z)))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 multiply(inverse(multiply(Y, multiply(Z, X))), multiply(X, multiply(Y, multiply(commutator(Y, X), Z))))
% 26.81/3.90 = { by lemma 42 }
% 26.81/3.90 multiply(inverse(multiply(Y, multiply(Z, X))), multiply(Y, multiply(X, Z)))
% 26.81/3.90 = { by lemma 15 }
% 26.81/3.90 multiply(inverse(multiply(Z, X)), multiply(X, Z))
% 26.81/3.90 = { by lemma 58 }
% 26.81/3.90 commutator(X, Z)
% 26.81/3.90
% 26.81/3.90 Lemma 60: commutator(X, multiply(Y, multiply(X, Z))) = commutator(X, multiply(Y, Z)).
% 26.81/3.90 Proof:
% 26.81/3.90 commutator(X, multiply(Y, multiply(X, Z)))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 commutator(X, multiply(multiply(Y, X), Z))
% 26.81/3.90 = { by lemma 7 R->L }
% 26.81/3.90 commutator(X, multiply(inverse(inverse(multiply(Y, X))), Z))
% 26.81/3.90 = { by lemma 55 R->L }
% 26.81/3.90 commutator(X, multiply(X, multiply(inverse(multiply(inverse(multiply(Y, X)), X)), Z)))
% 26.81/3.90 = { by lemma 32 }
% 26.81/3.90 commutator(X, multiply(inverse(multiply(inverse(multiply(Y, X)), X)), Z))
% 26.81/3.90 = { by lemma 16 R->L }
% 26.81/3.90 commutator(X, multiply(inverse(multiply(inverse(X), multiply(inverse(Y), X))), Z))
% 26.81/3.90 = { by lemma 23 }
% 26.81/3.90 commutator(X, multiply(inverse(multiply(inverse(Y), commutator(inverse(Y), X))), Z))
% 26.81/3.90 = { by lemma 20 }
% 26.81/3.90 commutator(X, multiply(multiply(inverse(commutator(inverse(Y), X)), Y), Z))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 commutator(X, multiply(inverse(commutator(inverse(Y), X)), multiply(Y, Z)))
% 26.81/3.90 = { by lemma 59 R->L }
% 26.81/3.90 commutator(X, multiply(commutator(inverse(Y), X), multiply(inverse(commutator(inverse(Y), X)), multiply(Y, Z))))
% 26.81/3.90 = { by lemma 14 }
% 26.81/3.90 commutator(X, multiply(Y, Z))
% 26.81/3.90
% 26.81/3.90 Lemma 61: multiply(inverse(X), multiply(Y, multiply(commutator(Y, Z), X))) = multiply(Y, commutator(Y, multiply(Z, X))).
% 26.81/3.90 Proof:
% 26.81/3.90 multiply(inverse(X), multiply(Y, multiply(commutator(Y, Z), X)))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 multiply(inverse(X), multiply(multiply(Y, commutator(Y, Z)), X))
% 26.81/3.90 = { by lemma 7 R->L }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(inverse(multiply(Y, commutator(Y, Z)))), X))
% 26.81/3.90 = { by lemma 15 R->L }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(multiply(Y, inverse(multiply(Y, commutator(Y, Z))))), multiply(Y, X)))
% 26.81/3.90 = { by lemma 52 }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(commutator(Z, Y)), multiply(Y, X)))
% 26.81/3.90 = { by lemma 21 }
% 26.81/3.90 multiply(inverse(X), multiply(commutator(Y, Z), multiply(Y, X)))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 multiply(multiply(inverse(X), commutator(Y, Z)), multiply(Y, X))
% 26.81/3.90 = { by lemma 47 }
% 26.81/3.90 multiply(inverse(multiply(commutator(Z, Y), X)), multiply(Y, X))
% 26.81/3.90 = { by lemma 15 R->L }
% 26.81/3.90 multiply(inverse(multiply(Y, multiply(commutator(Z, Y), X))), multiply(Y, multiply(Y, X)))
% 26.81/3.90 = { by lemma 15 R->L }
% 26.81/3.90 multiply(inverse(multiply(Z, multiply(Y, multiply(commutator(Z, Y), X)))), multiply(Z, multiply(Y, multiply(Y, X))))
% 26.81/3.90 = { by lemma 42 R->L }
% 26.81/3.90 multiply(inverse(multiply(Z, multiply(Y, multiply(commutator(Z, Y), X)))), multiply(Y, multiply(Z, multiply(commutator(Z, Y), multiply(Y, X)))))
% 26.81/3.90 = { by lemma 49 }
% 26.81/3.90 multiply(inverse(multiply(Z, multiply(Y, multiply(commutator(Z, Y), X)))), multiply(Y, multiply(Z, multiply(Y, multiply(commutator(Z, Y), X)))))
% 26.81/3.90 = { by lemma 23 }
% 26.81/3.90 multiply(Y, commutator(Y, multiply(Z, multiply(Y, multiply(commutator(Z, Y), X)))))
% 26.81/3.90 = { by lemma 60 }
% 26.81/3.90 multiply(Y, commutator(Y, multiply(Z, multiply(commutator(Z, Y), X))))
% 26.81/3.90 = { by lemma 32 R->L }
% 26.81/3.90 multiply(Y, commutator(Y, multiply(Y, multiply(Z, multiply(commutator(Z, Y), X)))))
% 26.81/3.90 = { by lemma 42 }
% 26.81/3.90 multiply(Y, commutator(Y, multiply(Z, multiply(Y, X))))
% 26.81/3.90 = { by lemma 60 }
% 26.81/3.90 multiply(Y, commutator(Y, multiply(Z, X)))
% 26.81/3.90
% 26.81/3.90 Lemma 62: commutator(multiply(X, multiply(Y, Z)), Z) = commutator(multiply(X, Y), Z).
% 26.81/3.90 Proof:
% 26.81/3.90 commutator(multiply(X, multiply(Y, Z)), Z)
% 26.81/3.90 = { by lemma 52 R->L }
% 26.81/3.90 multiply(Z, inverse(multiply(Z, commutator(Z, multiply(X, multiply(Y, Z))))))
% 26.81/3.90 = { by lemma 39 }
% 26.81/3.90 multiply(Z, inverse(multiply(Z, commutator(Z, multiply(X, Y)))))
% 26.81/3.90 = { by lemma 52 }
% 26.81/3.90 commutator(multiply(X, Y), Z)
% 26.81/3.90
% 26.81/3.90 Lemma 63: multiply(commutator(X, Y), commutator(Y, multiply(X, Z))) = commutator(multiply(Y, commutator(Y, X)), Z).
% 26.81/3.90 Proof:
% 26.81/3.90 multiply(commutator(X, Y), commutator(Y, multiply(X, Z)))
% 26.81/3.90 = { by lemma 53 R->L }
% 26.81/3.90 inverse(multiply(commutator(multiply(X, Z), Y), commutator(Y, X)))
% 26.81/3.90 = { by lemma 47 R->L }
% 26.81/3.90 multiply(inverse(commutator(Y, X)), commutator(Y, multiply(X, Z)))
% 26.81/3.90 = { by lemma 55 R->L }
% 26.81/3.90 multiply(Z, multiply(inverse(multiply(commutator(Y, X), Z)), commutator(Y, multiply(X, Z))))
% 26.81/3.90 = { by lemma 15 R->L }
% 26.81/3.90 multiply(Z, multiply(inverse(multiply(Y, multiply(commutator(Y, X), Z))), multiply(Y, commutator(Y, multiply(X, Z)))))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 multiply(multiply(Z, inverse(multiply(Y, multiply(commutator(Y, X), Z)))), multiply(Y, commutator(Y, multiply(X, Z))))
% 26.81/3.90 = { by lemma 35 R->L }
% 26.81/3.90 multiply(inverse(multiply(multiply(Y, multiply(commutator(Y, X), Z)), inverse(Z))), multiply(Y, commutator(Y, multiply(X, Z))))
% 26.81/3.90 = { by lemma 61 R->L }
% 26.81/3.90 multiply(inverse(multiply(multiply(Y, multiply(commutator(Y, X), Z)), inverse(Z))), multiply(inverse(Z), multiply(Y, multiply(commutator(Y, X), Z))))
% 26.81/3.90 = { by lemma 58 }
% 26.81/3.90 commutator(inverse(Z), multiply(Y, multiply(commutator(Y, X), Z)))
% 26.81/3.90 = { by lemma 33 }
% 26.81/3.90 commutator(multiply(Y, multiply(commutator(Y, X), Z)), Z)
% 26.81/3.90 = { by lemma 62 }
% 26.81/3.90 commutator(multiply(Y, commutator(Y, X)), Z)
% 26.81/3.90
% 26.81/3.90 Lemma 64: multiply(commutator(X, Y), commutator(multiply(X, commutator(X, Y)), Z)) = commutator(X, multiply(Y, Z)).
% 26.81/3.90 Proof:
% 26.81/3.90 multiply(commutator(X, Y), commutator(multiply(X, commutator(X, Y)), Z))
% 26.81/3.90 = { by lemma 62 R->L }
% 26.81/3.90 multiply(commutator(X, Y), commutator(multiply(X, multiply(commutator(X, Y), Z)), Z))
% 26.81/3.90 = { by lemma 33 R->L }
% 26.81/3.90 multiply(commutator(X, Y), commutator(inverse(Z), multiply(X, multiply(commutator(X, Y), Z))))
% 26.81/3.90 = { by lemma 14 R->L }
% 26.81/3.90 multiply(commutator(X, Y), multiply(Z, multiply(inverse(Z), commutator(inverse(Z), multiply(X, multiply(commutator(X, Y), Z))))))
% 26.81/3.90 = { by axiom 4 (associativity) R->L }
% 26.81/3.90 multiply(multiply(commutator(X, Y), Z), multiply(inverse(Z), commutator(inverse(Z), multiply(X, multiply(commutator(X, Y), Z)))))
% 26.81/3.90 = { by lemma 23 R->L }
% 26.81/3.90 multiply(multiply(commutator(X, Y), Z), multiply(inverse(multiply(X, multiply(commutator(X, Y), Z))), multiply(inverse(Z), multiply(X, multiply(commutator(X, Y), Z)))))
% 26.81/3.90 = { by lemma 55 }
% 26.81/3.90 multiply(inverse(X), multiply(inverse(Z), multiply(X, multiply(commutator(X, Y), Z))))
% 26.81/3.90 = { by lemma 61 }
% 26.81/3.90 multiply(inverse(X), multiply(X, commutator(X, multiply(Y, Z))))
% 26.81/3.90 = { by lemma 6 }
% 26.81/3.90 commutator(X, multiply(Y, Z))
% 26.81/3.90
% 26.81/3.90 Lemma 65: commutator(multiply(commutator(X, Y), Z), Y) = commutator(Z, Y).
% 26.81/3.90 Proof:
% 26.81/3.90 commutator(multiply(commutator(X, Y), Z), Y)
% 26.81/3.90 = { by lemma 33 R->L }
% 26.81/3.90 commutator(inverse(Y), multiply(commutator(X, Y), Z))
% 26.81/3.90 = { by axiom 5 (name) }
% 26.81/3.90 multiply(inverse(inverse(Y)), multiply(inverse(multiply(commutator(X, Y), Z)), multiply(inverse(Y), multiply(commutator(X, Y), Z))))
% 26.81/3.90 = { by lemma 57 R->L }
% 26.81/3.90 multiply(inverse(inverse(Y)), multiply(inverse(multiply(commutator(X, Y), Z)), multiply(inverse(multiply(Y, X)), multiply(X, Z))))
% 26.81/3.90 = { by lemma 7 }
% 26.81/3.90 multiply(Y, multiply(inverse(multiply(commutator(X, Y), Z)), multiply(inverse(multiply(Y, X)), multiply(X, Z))))
% 26.81/3.90 = { by lemma 16 }
% 26.81/3.90 multiply(Y, multiply(inverse(multiply(multiply(Y, X), multiply(commutator(X, Y), Z))), multiply(X, Z)))
% 26.81/3.90 = { by axiom 4 (associativity) }
% 26.81/3.90 multiply(Y, multiply(inverse(multiply(Y, multiply(X, multiply(commutator(X, Y), Z)))), multiply(X, Z)))
% 26.81/3.90 = { by lemma 42 }
% 26.81/3.90 multiply(Y, multiply(inverse(multiply(X, multiply(Y, Z))), multiply(X, Z)))
% 26.81/3.90 = { by lemma 15 }
% 26.81/3.90 multiply(Y, multiply(inverse(multiply(Y, Z)), Z))
% 26.81/3.90 = { by lemma 17 }
% 26.81/3.90 commutator(inverse(Y), Z)
% 26.81/3.90 = { by lemma 33 }
% 26.81/3.90 commutator(Z, Y)
% 26.81/3.90
% 26.81/3.90 Lemma 66: commutator(X, multiply(Y, commutator(Z, X))) = commutator(X, Y).
% 26.81/3.90 Proof:
% 26.81/3.90 commutator(X, multiply(Y, commutator(Z, X)))
% 26.81/3.90 = { by lemma 33 R->L }
% 26.81/3.90 commutator(inverse(multiply(Y, commutator(Z, X))), X)
% 26.81/3.90 = { by lemma 65 R->L }
% 26.81/3.90 commutator(multiply(commutator(Z, X), inverse(multiply(Y, commutator(Z, X)))), X)
% 26.81/3.90 = { by lemma 18 }
% 26.81/3.90 commutator(inverse(Y), X)
% 26.81/3.90 = { by lemma 33 }
% 26.81/3.90 commutator(X, Y)
% 26.81/3.90
% 26.81/3.90 Lemma 67: multiply(commutator(X, Y), inverse(Z)) = inverse(multiply(Z, commutator(Y, X))).
% 26.81/3.90 Proof:
% 26.81/3.90 multiply(commutator(X, Y), inverse(Z))
% 26.81/3.90 = { by lemma 9 R->L }
% 26.81/3.90 multiply(commutator(X, Y), inverse(multiply(Z, identity)))
% 26.81/3.90 = { by lemma 46 R->L }
% 26.81/3.91 multiply(commutator(X, Y), inverse(multiply(Z, multiply(commutator(Y, X), commutator(X, Y)))))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(commutator(X, Y), inverse(multiply(multiply(Z, commutator(Y, X)), commutator(X, Y))))
% 26.81/3.91 = { by lemma 18 }
% 26.81/3.91 inverse(multiply(Z, commutator(Y, X)))
% 26.81/3.91
% 26.81/3.91 Lemma 68: multiply(inverse(Y), commutator(Y, X)) = multiply(X, inverse(multiply(X, Y))).
% 26.81/3.91 Proof:
% 26.81/3.91 multiply(inverse(Y), commutator(Y, X))
% 26.81/3.91 = { by lemma 45 R->L }
% 26.81/3.91 multiply(inverse(Y), multiply(Y, multiply(X, inverse(multiply(X, Y)))))
% 26.81/3.91 = { by lemma 6 }
% 26.81/3.91 multiply(X, inverse(multiply(X, Y)))
% 26.81/3.91
% 26.81/3.91 Lemma 69: multiply(commutator(Y, Z), commutator(Z, commutator(Y, X))) = multiply(inverse(X), multiply(commutator(Y, Z), X)).
% 26.81/3.91 Proof:
% 26.81/3.91 multiply(commutator(Y, Z), commutator(Z, commutator(Y, X)))
% 26.81/3.91 = { by lemma 36 }
% 26.81/3.91 multiply(commutator(Y, Z), commutator(Y, commutator(Z, X)))
% 26.81/3.91 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.91 multiply(commutator(Y, Z), commutator(commutator(Y, Z), X))
% 26.81/3.91 = { by lemma 23 R->L }
% 26.81/3.91 multiply(inverse(X), multiply(commutator(Y, Z), X))
% 26.81/3.91
% 26.81/3.91 Lemma 70: commutator(X, commutator(Y, multiply(inverse(X), Z))) = commutator(X, commutator(Z, Y)).
% 26.81/3.91 Proof:
% 26.81/3.91 commutator(X, commutator(Y, multiply(inverse(X), Z)))
% 26.81/3.91 = { by lemma 37 }
% 26.81/3.91 commutator(X, commutator(multiply(inverse(X), Z), Y))
% 26.81/3.91 = { by lemma 36 }
% 26.81/3.91 commutator(multiply(inverse(X), Z), commutator(X, Y))
% 26.81/3.91 = { by axiom 3 (associativity_of_commutator) R->L }
% 26.81/3.91 commutator(commutator(multiply(inverse(X), Z), X), Y)
% 26.81/3.91 = { by lemma 30 }
% 26.81/3.91 commutator(commutator(Z, X), Y)
% 26.81/3.91 = { by axiom 3 (associativity_of_commutator) }
% 26.81/3.91 commutator(Z, commutator(X, Y))
% 26.81/3.91 = { by lemma 36 R->L }
% 26.81/3.91 commutator(X, commutator(Z, Y))
% 26.81/3.91
% 26.81/3.91 Lemma 71: multiply(commutator(X, Y), commutator(Z, commutator(X, Y))) = commutator(X, multiply(Y, commutator(Z, Y))).
% 26.81/3.91 Proof:
% 26.81/3.91 multiply(commutator(X, Y), commutator(Z, commutator(X, Y)))
% 26.81/3.91 = { by lemma 41 R->L }
% 26.81/3.91 multiply(inverse(multiply(Y, X)), multiply(X, multiply(Y, commutator(Z, commutator(X, Y)))))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(inverse(multiply(Y, X)), multiply(multiply(X, Y), commutator(Z, commutator(X, Y))))
% 26.81/3.91 = { by lemma 16 R->L }
% 26.81/3.91 multiply(inverse(X), multiply(inverse(Y), multiply(multiply(X, Y), commutator(Z, commutator(X, Y)))))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(inverse(X), multiply(multiply(inverse(Y), multiply(X, Y)), commutator(Z, commutator(X, Y))))
% 26.81/3.91 = { by axiom 5 (name) }
% 26.81/3.91 multiply(inverse(X), multiply(multiply(inverse(Y), multiply(X, Y)), commutator(Z, multiply(inverse(X), multiply(inverse(Y), multiply(X, Y))))))
% 26.81/3.91 = { by lemma 44 R->L }
% 26.81/3.91 multiply(inverse(X), multiply(multiply(inverse(Y), multiply(X, Y)), multiply(inverse(multiply(inverse(X), multiply(inverse(Y), multiply(X, Y)))), multiply(Z, multiply(multiply(inverse(X), multiply(inverse(Y), multiply(X, Y))), inverse(Z))))))
% 26.81/3.91 = { by lemma 54 }
% 26.81/3.91 multiply(Z, multiply(multiply(inverse(X), multiply(inverse(Y), multiply(X, Y))), inverse(Z)))
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 multiply(Z, multiply(inverse(X), multiply(multiply(inverse(Y), multiply(X, Y)), inverse(Z))))
% 26.81/3.91 = { by lemma 50 }
% 26.81/3.91 multiply(Z, inverse(multiply(Z, multiply(inverse(multiply(inverse(Y), multiply(X, Y))), X))))
% 26.81/3.91 = { by lemma 20 }
% 26.81/3.91 multiply(Z, inverse(multiply(Z, multiply(multiply(inverse(multiply(X, Y)), Y), X))))
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 multiply(Z, inverse(multiply(Z, multiply(inverse(multiply(X, Y)), multiply(Y, X)))))
% 26.81/3.91 = { by lemma 58 }
% 26.81/3.91 multiply(Z, inverse(multiply(Z, commutator(Y, X))))
% 26.81/3.91 = { by lemma 66 R->L }
% 26.81/3.91 multiply(Z, inverse(multiply(Z, commutator(Y, multiply(X, commutator(X, Y))))))
% 26.81/3.91 = { by lemma 67 R->L }
% 26.81/3.91 multiply(Z, multiply(commutator(multiply(X, commutator(X, Y)), Y), inverse(Z)))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(multiply(Z, commutator(multiply(X, commutator(X, Y)), Y)), inverse(Z))
% 26.81/3.91 = { by lemma 35 R->L }
% 26.81/3.91 inverse(multiply(Z, inverse(multiply(Z, commutator(multiply(X, commutator(X, Y)), Y)))))
% 26.81/3.91 = { by lemma 68 R->L }
% 26.81/3.91 inverse(multiply(inverse(commutator(multiply(X, commutator(X, Y)), Y)), commutator(commutator(multiply(X, commutator(X, Y)), Y), Z)))
% 26.81/3.91 = { by lemma 25 R->L }
% 26.81/3.91 inverse(multiply(commutator(commutator(multiply(X, commutator(X, Y)), Y), Z), multiply(inverse(commutator(multiply(X, commutator(X, Y)), Y)), commutator(inverse(commutator(multiply(X, commutator(X, Y)), Y)), commutator(commutator(multiply(X, commutator(X, Y)), Y), Z)))))
% 26.81/3.91 = { by lemma 56 }
% 26.81/3.91 inverse(multiply(commutator(commutator(multiply(X, commutator(X, Y)), Y), Z), multiply(inverse(commutator(multiply(X, commutator(X, Y)), Y)), identity)))
% 26.81/3.91 = { by lemma 9 }
% 26.81/3.91 inverse(multiply(commutator(commutator(multiply(X, commutator(X, Y)), Y), Z), inverse(commutator(multiply(X, commutator(X, Y)), Y))))
% 26.81/3.91 = { by lemma 35 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), inverse(commutator(commutator(multiply(X, commutator(X, Y)), Y), Z)))
% 26.81/3.91 = { by lemma 21 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Z, commutator(multiply(X, commutator(X, Y)), Y)))
% 26.81/3.91 = { by lemma 14 R->L }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), multiply(Z, multiply(inverse(Z), commutator(Z, commutator(multiply(X, commutator(X, Y)), Y)))))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z), multiply(inverse(Z), commutator(Z, commutator(multiply(X, commutator(X, Y)), Y))))
% 26.81/3.91 = { by lemma 7 R->L }
% 26.81/3.91 multiply(inverse(inverse(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z))), multiply(inverse(Z), commutator(Z, commutator(multiply(X, commutator(X, Y)), Y))))
% 26.81/3.91 = { by lemma 68 }
% 26.81/3.91 multiply(inverse(inverse(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z))), multiply(commutator(multiply(X, commutator(X, Y)), Y), inverse(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z))))
% 26.81/3.91 = { by lemma 69 R->L }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(multiply(X, commutator(X, Y)), inverse(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z)))))
% 26.81/3.91 = { by lemma 34 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(multiply(commutator(multiply(X, commutator(X, Y)), Y), Z), multiply(X, commutator(X, Y)))))
% 26.81/3.91 = { by lemma 21 R->L }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(multiply(inverse(commutator(Y, multiply(X, commutator(X, Y)))), Z), multiply(X, commutator(X, Y)))))
% 26.81/3.91 = { by lemma 65 R->L }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(multiply(commutator(Y, multiply(X, commutator(X, Y))), multiply(inverse(commutator(Y, multiply(X, commutator(X, Y)))), Z)), multiply(X, commutator(X, Y)))))
% 26.81/3.91 = { by lemma 14 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(Z, multiply(X, commutator(X, Y)))))
% 26.81/3.91 = { by lemma 37 R->L }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(Y, commutator(multiply(X, commutator(X, Y)), Z)))
% 26.81/3.91 = { by lemma 69 }
% 26.81/3.91 multiply(inverse(Z), multiply(commutator(multiply(X, commutator(X, Y)), Y), Z))
% 26.81/3.91 = { by lemma 45 R->L }
% 26.81/3.91 multiply(inverse(Z), multiply(multiply(multiply(X, commutator(X, Y)), multiply(Y, inverse(multiply(Y, multiply(X, commutator(X, Y)))))), Z))
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 multiply(inverse(Z), multiply(multiply(X, commutator(X, Y)), multiply(multiply(Y, inverse(multiply(Y, multiply(X, commutator(X, Y))))), Z)))
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 multiply(inverse(Z), multiply(multiply(X, commutator(X, Y)), multiply(Y, multiply(inverse(multiply(Y, multiply(X, commutator(X, Y)))), Z))))
% 26.81/3.91 = { by axiom 4 (associativity) R->L }
% 26.81/3.91 multiply(multiply(inverse(Z), multiply(X, commutator(X, Y))), multiply(Y, multiply(inverse(multiply(Y, multiply(X, commutator(X, Y)))), Z)))
% 26.81/3.91 = { by lemma 20 R->L }
% 26.81/3.91 multiply(inverse(multiply(inverse(multiply(X, commutator(X, Y))), Z)), multiply(Y, multiply(inverse(multiply(Y, multiply(X, commutator(X, Y)))), Z)))
% 26.81/3.91 = { by lemma 43 R->L }
% 26.81/3.91 multiply(inverse(multiply(inverse(multiply(X, commutator(X, Y))), Z)), multiply(inverse(multiply(X, commutator(X, Y))), multiply(commutator(multiply(X, commutator(X, Y)), Y), Z)))
% 26.81/3.91 = { by lemma 42 R->L }
% 26.81/3.91 multiply(inverse(multiply(inverse(multiply(X, commutator(X, Y))), Z)), multiply(commutator(multiply(X, commutator(X, Y)), Y), multiply(inverse(multiply(X, commutator(X, Y))), multiply(commutator(inverse(multiply(X, commutator(X, Y))), commutator(multiply(X, commutator(X, Y)), Y)), Z))))
% 26.81/3.91 = { by lemma 56 }
% 26.81/3.91 multiply(inverse(multiply(inverse(multiply(X, commutator(X, Y))), Z)), multiply(commutator(multiply(X, commutator(X, Y)), Y), multiply(inverse(multiply(X, commutator(X, Y))), multiply(identity, Z))))
% 26.81/3.91 = { by axiom 1 (left_identity) }
% 26.81/3.91 multiply(inverse(multiply(inverse(multiply(X, commutator(X, Y))), Z)), multiply(commutator(multiply(X, commutator(X, Y)), Y), multiply(inverse(multiply(X, commutator(X, Y))), Z)))
% 26.81/3.91 = { by lemma 23 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(commutator(multiply(X, commutator(X, Y)), Y), multiply(inverse(multiply(X, commutator(X, Y))), Z)))
% 26.81/3.91 = { by axiom 3 (associativity_of_commutator) }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(multiply(X, commutator(X, Y)), commutator(Y, multiply(inverse(multiply(X, commutator(X, Y))), Z))))
% 26.81/3.91 = { by lemma 70 }
% 26.81/3.91 multiply(commutator(multiply(X, commutator(X, Y)), Y), commutator(multiply(X, commutator(X, Y)), commutator(Z, Y)))
% 26.81/3.91 = { by lemma 34 R->L }
% 26.81/3.91 multiply(commutator(Y, inverse(multiply(X, commutator(X, Y)))), commutator(multiply(X, commutator(X, Y)), commutator(Z, Y)))
% 26.81/3.91 = { by lemma 59 R->L }
% 26.81/3.91 multiply(commutator(Y, multiply(commutator(X, Y), inverse(multiply(X, commutator(X, Y))))), commutator(multiply(X, commutator(X, Y)), commutator(Z, Y)))
% 26.81/3.91 = { by lemma 18 }
% 26.81/3.91 multiply(commutator(Y, inverse(X)), commutator(multiply(X, commutator(X, Y)), commutator(Z, Y)))
% 26.81/3.91 = { by lemma 34 }
% 26.81/3.91 multiply(commutator(X, Y), commutator(multiply(X, commutator(X, Y)), commutator(Z, Y)))
% 26.81/3.91 = { by lemma 64 }
% 26.81/3.91 commutator(X, multiply(Y, commutator(Z, Y)))
% 26.81/3.91
% 26.81/3.91 Lemma 72: multiply(commutator(X, Y), commutator(Z, Y)) = commutator(multiply(X, Z), Y).
% 26.81/3.91 Proof:
% 26.81/3.91 multiply(commutator(X, Y), commutator(Z, Y))
% 26.81/3.91 = { by lemma 34 R->L }
% 26.81/3.91 multiply(commutator(X, Y), commutator(Y, inverse(Z)))
% 26.81/3.91 = { by lemma 18 R->L }
% 26.81/3.91 multiply(commutator(X, Y), commutator(Y, multiply(X, inverse(multiply(Z, X)))))
% 26.81/3.91 = { by lemma 63 }
% 26.81/3.91 commutator(multiply(Y, commutator(Y, X)), inverse(multiply(Z, X)))
% 26.81/3.91 = { by lemma 34 }
% 26.81/3.91 commutator(multiply(Z, X), multiply(Y, commutator(Y, X)))
% 26.81/3.91 = { by lemma 64 R->L }
% 26.81/3.91 multiply(commutator(multiply(Z, X), Y), commutator(multiply(multiply(Z, X), commutator(multiply(Z, X), Y)), commutator(Y, X)))
% 26.81/3.91 = { by lemma 37 R->L }
% 26.81/3.91 multiply(commutator(multiply(Z, X), Y), commutator(multiply(multiply(Z, X), commutator(multiply(Z, X), Y)), commutator(X, Y)))
% 26.81/3.91 = { by lemma 64 }
% 26.81/3.91 commutator(multiply(Z, X), multiply(Y, commutator(X, Y)))
% 26.81/3.91 = { by lemma 71 R->L }
% 26.81/3.91 multiply(commutator(multiply(Z, X), Y), commutator(X, commutator(multiply(Z, X), Y)))
% 26.81/3.91 = { by lemma 33 R->L }
% 26.81/3.91 multiply(commutator(multiply(Z, X), Y), commutator(X, commutator(inverse(Y), multiply(Z, X))))
% 26.81/3.91 = { by lemma 33 R->L }
% 26.81/3.91 multiply(commutator(inverse(Y), multiply(Z, X)), commutator(X, commutator(inverse(Y), multiply(Z, X))))
% 26.81/3.91 = { by lemma 71 }
% 26.81/3.91 commutator(inverse(Y), multiply(multiply(Z, X), commutator(X, multiply(Z, X))))
% 26.81/3.91 = { by lemma 33 }
% 26.81/3.91 commutator(multiply(multiply(Z, X), commutator(X, multiply(Z, X))), Y)
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 commutator(multiply(Z, multiply(X, commutator(X, multiply(Z, X)))), Y)
% 26.81/3.91 = { by lemma 10 R->L }
% 26.81/3.91 commutator(multiply(Z, multiply(X, commutator(inverse(inverse(X)), multiply(Z, X)))), Y)
% 26.81/3.91 = { by lemma 17 R->L }
% 26.81/3.91 commutator(multiply(Z, multiply(X, multiply(inverse(X), multiply(inverse(multiply(inverse(X), multiply(Z, X))), multiply(Z, X))))), Y)
% 26.81/3.91 = { by lemma 23 }
% 26.81/3.91 commutator(multiply(Z, multiply(X, multiply(inverse(X), multiply(inverse(multiply(Z, commutator(Z, X))), multiply(Z, X))))), Y)
% 26.81/3.91 = { by lemma 16 }
% 26.81/3.91 commutator(multiply(Z, multiply(X, multiply(inverse(multiply(multiply(Z, commutator(Z, X)), X)), multiply(Z, X)))), Y)
% 26.81/3.91 = { by axiom 4 (associativity) }
% 26.81/3.91 commutator(multiply(Z, multiply(X, multiply(inverse(multiply(Z, multiply(commutator(Z, X), X))), multiply(Z, X)))), Y)
% 26.81/3.92 = { by lemma 15 }
% 26.81/3.92 commutator(multiply(Z, multiply(X, multiply(inverse(multiply(commutator(Z, X), X)), X))), Y)
% 26.81/3.92 = { by lemma 28 }
% 26.81/3.92 commutator(multiply(Z, multiply(X, multiply(inverse(multiply(X, commutator(Z, X))), X))), Y)
% 26.81/3.92 = { by lemma 22 }
% 26.81/3.92 commutator(multiply(Z, multiply(X, inverse(commutator(Z, X)))), Y)
% 26.81/3.92 = { by lemma 21 }
% 26.81/3.92 commutator(multiply(Z, multiply(X, commutator(X, Z))), Y)
% 26.81/3.92 = { by lemma 25 }
% 27.56/3.92 commutator(multiply(X, Z), Y)
% 27.56/3.92
% 27.56/3.92 Lemma 73: multiply(commutator(X, Y), multiply(commutator(X, Z), Y)) = multiply(Y, commutator(X, multiply(Z, Y))).
% 27.56/3.92 Proof:
% 27.56/3.92 multiply(commutator(X, Y), multiply(commutator(X, Z), Y))
% 27.56/3.92 = { by lemma 33 R->L }
% 27.56/3.92 multiply(commutator(inverse(Y), X), multiply(commutator(X, Z), Y))
% 27.56/3.92 = { by lemma 41 R->L }
% 27.56/3.92 multiply(inverse(multiply(X, inverse(Y))), multiply(inverse(Y), multiply(X, multiply(commutator(X, Z), Y))))
% 27.56/3.92 = { by lemma 61 }
% 27.56/3.92 multiply(inverse(multiply(X, inverse(Y))), multiply(X, commutator(X, multiply(Z, Y))))
% 27.56/3.92 = { by lemma 15 }
% 27.56/3.92 multiply(inverse(inverse(Y)), commutator(X, multiply(Z, Y)))
% 27.56/3.92 = { by lemma 47 }
% 27.56/3.92 inverse(multiply(commutator(multiply(Z, Y), X), inverse(Y)))
% 27.56/3.92 = { by lemma 35 }
% 27.56/3.92 multiply(Y, inverse(commutator(multiply(Z, Y), X)))
% 27.56/3.93 = { by lemma 21 }
% 27.56/3.93 multiply(Y, commutator(X, multiply(Z, Y)))
% 27.56/3.93
% 27.56/3.93 Lemma 74: multiply(commutator(Y, Z), commutator(X, Y)) = multiply(commutator(X, Y), commutator(Y, Z)).
% 27.56/3.93 Proof:
% 27.56/3.93 multiply(commutator(Y, Z), commutator(X, Y))
% 27.56/3.93 = { by axiom 1 (left_identity) R->L }
% 27.56/3.93 multiply(identity, multiply(commutator(Y, Z), commutator(X, Y)))
% 27.56/3.93 = { by lemma 24 R->L }
% 27.56/3.93 multiply(commutator(X, commutator(X, Y)), multiply(commutator(Y, Z), commutator(X, Y)))
% 27.56/3.93 = { by lemma 48 }
% 27.56/3.93 multiply(commutator(Y, commutator(X, Y)), multiply(commutator(Y, Z), commutator(X, Y)))
% 27.56/3.93 = { by lemma 73 }
% 27.56/3.93 multiply(commutator(X, Y), commutator(Y, multiply(Z, commutator(X, Y))))
% 27.56/3.93 = { by lemma 66 }
% 27.56/3.93 multiply(commutator(X, Y), commutator(Y, Z))
% 27.56/3.93
% 27.56/3.93 Lemma 75: commutator(multiply(Y, X), Z) = commutator(multiply(X, Y), Z).
% 27.56/3.93 Proof:
% 27.56/3.93 commutator(multiply(Y, X), Z)
% 27.56/3.93 = { by lemma 72 R->L }
% 27.56/3.93 multiply(commutator(Y, Z), commutator(X, Z))
% 27.56/3.93 = { by lemma 34 R->L }
% 27.56/3.93 multiply(commutator(Z, inverse(Y)), commutator(X, Z))
% 27.56/3.93 = { by lemma 74 }
% 27.56/3.93 multiply(commutator(X, Z), commutator(Z, inverse(Y)))
% 27.56/3.93 = { by lemma 34 }
% 27.56/3.93 multiply(commutator(X, Z), commutator(Y, Z))
% 27.56/3.93 = { by lemma 72 }
% 27.56/3.93 commutator(multiply(X, Y), Z)
% 27.56/3.93
% 27.56/3.93 Lemma 76: multiply(commutator(X, Y), commutator(multiply(Y, Z), X)) = commutator(Z, multiply(X, commutator(X, Y))).
% 27.56/3.93 Proof:
% 27.56/3.93 multiply(commutator(X, Y), commutator(multiply(Y, Z), X))
% 27.56/3.93 = { by lemma 74 }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), commutator(X, Y))
% 27.56/3.93 = { by lemma 9 R->L }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), multiply(commutator(X, Y), identity))
% 27.56/3.93 = { by lemma 13 R->L }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), multiply(commutator(X, Y), multiply(Z, inverse(Z))))
% 27.56/3.93 = { by axiom 4 (associativity) R->L }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), multiply(multiply(commutator(X, Y), Z), inverse(Z)))
% 27.56/3.93 = { by lemma 6 R->L }
% 27.56/3.93 multiply(inverse(X), multiply(X, multiply(commutator(multiply(Y, Z), X), multiply(multiply(commutator(X, Y), Z), inverse(Z)))))
% 27.56/3.93 = { by lemma 49 R->L }
% 27.56/3.93 multiply(inverse(X), multiply(commutator(multiply(Y, Z), X), multiply(X, multiply(multiply(commutator(X, Y), Z), inverse(Z)))))
% 27.56/3.93 = { by axiom 4 (associativity) R->L }
% 27.56/3.93 multiply(inverse(X), multiply(commutator(multiply(Y, Z), X), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z))))
% 27.56/3.93 = { by lemma 51 R->L }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), multiply(inverse(X), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z))))
% 27.56/3.93 = { by lemma 55 R->L }
% 27.56/3.93 multiply(commutator(multiply(Y, Z), X), multiply(commutator(X, multiply(Y, Z)), multiply(inverse(multiply(X, commutator(X, multiply(Y, Z)))), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z)))))
% 27.56/3.93 = { by axiom 4 (associativity) R->L }
% 27.56/3.94 multiply(multiply(commutator(multiply(Y, Z), X), commutator(X, multiply(Y, Z))), multiply(inverse(multiply(X, commutator(X, multiply(Y, Z)))), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z))))
% 27.56/3.94 = { by lemma 46 }
% 27.56/3.94 multiply(identity, multiply(inverse(multiply(X, commutator(X, multiply(Y, Z)))), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z))))
% 27.56/3.94 = { by axiom 1 (left_identity) }
% 27.56/3.94 multiply(inverse(multiply(X, commutator(X, multiply(Y, Z)))), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z)))
% 27.56/3.94 = { by lemma 61 R->L }
% 27.56/3.94 multiply(inverse(multiply(inverse(Z), multiply(X, multiply(commutator(X, Y), Z)))), multiply(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z)))
% 27.56/3.94 = { by lemma 58 }
% 27.56/3.94 commutator(multiply(X, multiply(commutator(X, Y), Z)), inverse(Z))
% 27.56/3.94 = { by lemma 34 }
% 27.56/3.94 commutator(Z, multiply(X, multiply(commutator(X, Y), Z)))
% 27.56/3.94 = { by lemma 39 }
% 27.56/3.94 commutator(Z, multiply(X, commutator(X, Y)))
% 27.56/3.94
% 27.56/3.94 Goal 1 (prove_center): multiply(a, commutator(b, c)) = multiply(commutator(b, c), a).
% 27.56/3.94 Proof:
% 27.56/3.94 multiply(a, commutator(b, c))
% 27.56/3.94 = { by lemma 25 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(a, commutator(b, c))))
% 27.56/3.94 = { by lemma 37 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(a, commutator(c, b))))
% 27.56/3.94 = { by lemma 36 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(c, commutator(a, b))))
% 27.56/3.94 = { by lemma 37 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(c, commutator(b, a))))
% 27.56/3.94 = { by lemma 33 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(inverse(commutator(b, a)), c)))
% 27.56/3.94 = { by lemma 29 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(commutator(a, multiply(b, a)), c)))
% 27.56/3.94 = { by axiom 3 (associativity_of_commutator) }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(a, commutator(multiply(b, a), c))))
% 27.56/3.94 = { by lemma 70 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(a, commutator(c, multiply(inverse(a), multiply(b, a))))))
% 27.56/3.94 = { by lemma 23 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(a, commutator(c, multiply(b, commutator(b, a))))))
% 27.56/3.94 = { by axiom 3 (associativity_of_commutator) R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, commutator(commutator(a, c), multiply(b, commutator(b, a)))))
% 27.56/3.94 = { by lemma 76 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(commutator(b, a), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 21 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(inverse(commutator(a, b)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 22 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, c), commutator(a, b))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by axiom 1 (left_identity) R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(identity, multiply(commutator(a, c), commutator(a, b)))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 27 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(b, commutator(a, b)), multiply(commutator(a, c), commutator(a, b)))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 48 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, commutator(a, b)), multiply(commutator(a, c), commutator(a, b)))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 73 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, b), commutator(a, multiply(c, commutator(a, b))))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 33 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, b), commutator(inverse(multiply(c, commutator(a, b))), a))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 67 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, b), commutator(multiply(commutator(b, a), inverse(c)), a))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 65 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, b), commutator(inverse(c), a))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 33 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(a, b), commutator(a, c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 40 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(multiply(b, a), b), commutator(a, c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 33 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(inverse(b), multiply(b, a)), commutator(a, c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 6 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(inverse(b), multiply(b, a)), commutator(multiply(inverse(b), multiply(b, a)), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 32 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(inverse(b), multiply(inverse(b), multiply(b, a))), commutator(multiply(inverse(b), multiply(b, a)), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 32 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(inverse(b), multiply(inverse(b), multiply(b, a))), commutator(multiply(inverse(b), multiply(b, a)), multiply(multiply(inverse(b), multiply(b, a)), c)))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by axiom 4 (associativity) }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(multiply(commutator(inverse(b), multiply(inverse(b), multiply(b, a))), commutator(multiply(inverse(b), multiply(b, a)), multiply(inverse(b), multiply(multiply(b, a), c))))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 63 }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(multiply(inverse(b), multiply(b, a)), commutator(multiply(inverse(b), multiply(b, a)), inverse(b))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by axiom 4 (associativity) }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(b), multiply(multiply(b, a), commutator(multiply(inverse(b), multiply(b, a)), inverse(b)))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 7 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(inverse(inverse(b))), multiply(multiply(b, a), commutator(multiply(inverse(b), multiply(b, a)), inverse(b)))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by lemma 15 R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(multiply(inverse(b), inverse(inverse(b)))), multiply(inverse(b), multiply(multiply(b, a), commutator(multiply(inverse(b), multiply(b, a)), inverse(b))))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.94 = { by axiom 4 (associativity) R->L }
% 27.56/3.94 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(multiply(inverse(b), inverse(inverse(b)))), multiply(multiply(inverse(b), multiply(b, a)), commutator(multiply(inverse(b), multiply(b, a)), inverse(b)))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 23 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(multiply(inverse(b), inverse(inverse(b)))), multiply(inverse(inverse(b)), multiply(multiply(inverse(b), multiply(b, a)), inverse(b)))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by axiom 4 (associativity) }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(multiply(inverse(b), inverse(inverse(b)))), multiply(inverse(inverse(b)), multiply(inverse(b), multiply(multiply(b, a), inverse(b))))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 41 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(commutator(inverse(inverse(b)), inverse(b)), multiply(multiply(b, a), inverse(b))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 33 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(commutator(inverse(b), inverse(b)), multiply(multiply(b, a), inverse(b))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 11 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(identity, multiply(multiply(b, a), inverse(b))), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by axiom 1 (left_identity) }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(multiply(b, a), inverse(b)), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 75 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(b), multiply(b, a)), multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 6 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(a, multiply(multiply(b, a), c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by axiom 4 (associativity) }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(a, multiply(b, multiply(a, c)))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 60 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(a, multiply(b, c))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 33 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(inverse(multiply(b, c)), a)), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 19 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(c), inverse(b)), a)), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 75 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(multiply(inverse(b), inverse(c)), a)), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 19 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(inverse(multiply(c, b)), a)), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 33 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(inverse(commutator(a, multiply(c, b))), commutator(a, c)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 47 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(inverse(multiply(commutator(c, a), commutator(a, multiply(c, b)))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 74 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(inverse(multiply(commutator(a, multiply(c, b)), commutator(c, a))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 33 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(inverse(multiply(commutator(a, multiply(c, b)), commutator(inverse(a), c))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 34 R->L }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(inverse(multiply(commutator(multiply(c, b), inverse(a)), commutator(inverse(a), c))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 53 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(commutator(c, inverse(a)), commutator(inverse(a), multiply(c, b))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 34 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(commutator(a, c), commutator(inverse(a), multiply(c, b))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 33 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(multiply(commutator(a, c), commutator(multiply(c, b), a)), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 76 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, multiply(commutator(b, multiply(a, commutator(a, c))), commutator(multiply(a, commutator(a, c)), b))))
% 27.56/3.95 = { by lemma 46 }
% 27.56/3.95 multiply(commutator(b, c), multiply(a, identity))
% 27.56/3.95 = { by lemma 9 }
% 27.56/3.95 multiply(commutator(b, c), a)
% 27.56/3.95 % SZS output end Proof
% 27.56/3.95
% 27.56/3.95 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------