TSTP Solution File: GRP400-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP400-1 : TPTP v8.1.2. Released v2.5.0.
% 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 : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:18:16 EDT 2023
% Result : Unsatisfiable 114.04s 15.02s
% Output : Proof 122.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP400-1 : TPTP v8.1.2. Released v2.5.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n031.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Mon Aug 28 20:04:10 EDT 2023
% 0.14/0.35 % CPUTime :
% 114.04/15.02 Command-line arguments: --flatten
% 114.04/15.02
% 114.04/15.02 % SZS status Unsatisfiable
% 114.04/15.02
% 121.92/15.94 % SZS output start Proof
% 121.92/15.94 Take the following subset of the input axioms:
% 121.92/15.94 fof(associativity_of_commutator, axiom, ![A, B, C]: commutator(commutator(A, B), C)=commutator(A, commutator(B, C))).
% 121.92/15.94 fof(associativity_of_multiply, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
% 121.92/15.94 fof(commutator, axiom, ![B2, A3]: multiply(A3, B2)=multiply(B2, multiply(A3, commutator(A3, B2)))).
% 121.92/15.94 fof(left_cancellation, axiom, ![B2, C2, A3]: (multiply(A3, B2)!=multiply(A3, C2) | B2=C2)).
% 121.92/15.94 fof(prove_commutator_distributes_over_product, negated_conjecture, commutator(multiply(a, b), c)!=multiply(commutator(a, c), commutator(b, c))).
% 121.92/15.94 fof(right_cancellation, axiom, ![A2, B2, C2]: (multiply(A2, B2)!=multiply(C2, B2) | A2=C2)).
% 121.92/15.94
% 121.92/15.94 Now clausify the problem and encode Horn clauses using encoding 3 of
% 121.92/15.94 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 121.92/15.94 We repeatedly replace C & s=t => u=v by the two clauses:
% 121.92/15.94 fresh(y, y, x1...xn) = u
% 121.92/15.94 C => fresh(s, t, x1...xn) = v
% 121.92/15.94 where fresh is a fresh function symbol and x1..xn are the free
% 121.92/15.94 variables of u and v.
% 121.92/15.94 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 121.92/15.94 input problem has no model of domain size 1).
% 121.92/15.94
% 121.92/15.94 The encoding turns the above axioms into the following unit equations and goals:
% 121.92/15.94
% 121.92/15.94 Axiom 1 (left_cancellation): fresh(X, X, Y, Z) = Z.
% 121.92/15.94 Axiom 2 (right_cancellation): fresh2(X, X, Y, Z) = Z.
% 121.92/15.94 Axiom 3 (associativity_of_commutator): commutator(commutator(X, Y), Z) = commutator(X, commutator(Y, Z)).
% 121.92/15.94 Axiom 4 (associativity_of_multiply): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 121.92/15.95 Axiom 5 (commutator): multiply(X, Y) = multiply(Y, multiply(X, commutator(X, Y))).
% 121.92/15.95 Axiom 6 (left_cancellation): fresh(multiply(X, Y), multiply(X, Z), Y, Z) = Y.
% 121.92/15.95 Axiom 7 (right_cancellation): fresh2(multiply(X, Y), multiply(Z, Y), X, Z) = X.
% 121.92/15.95
% 121.92/15.95 Lemma 8: multiply(X, commutator(X, X)) = X.
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(X, commutator(X, X))
% 121.92/15.95 = { by axiom 1 (left_cancellation) R->L }
% 121.92/15.95 fresh(multiply(X, X), multiply(X, X), X, multiply(X, commutator(X, X)))
% 121.92/15.95 = { by axiom 5 (commutator) }
% 121.92/15.95 fresh(multiply(X, X), multiply(X, multiply(X, commutator(X, X))), X, multiply(X, commutator(X, X)))
% 121.92/15.95 = { by axiom 6 (left_cancellation) }
% 121.92/15.95 X
% 121.92/15.95
% 121.92/15.95 Lemma 9: multiply(X, multiply(commutator(X, X), Y)) = multiply(X, Y).
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(X, multiply(commutator(X, X), Y))
% 121.92/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.92/15.95 multiply(multiply(X, commutator(X, X)), Y)
% 121.92/15.95 = { by lemma 8 }
% 121.92/15.95 multiply(X, Y)
% 121.92/15.95
% 121.92/15.95 Lemma 10: multiply(X, multiply(Y, multiply(commutator(Y, X), Z))) = multiply(Y, multiply(X, Z)).
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(X, multiply(Y, multiply(commutator(Y, X), Z)))
% 121.92/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.92/15.95 multiply(X, multiply(multiply(Y, commutator(Y, X)), Z))
% 121.92/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.92/15.95 multiply(multiply(X, multiply(Y, commutator(Y, X))), Z)
% 121.92/15.95 = { by axiom 5 (commutator) R->L }
% 121.92/15.95 multiply(multiply(Y, X), Z)
% 121.92/15.95 = { by axiom 4 (associativity_of_multiply) }
% 121.92/15.95 multiply(Y, multiply(X, Z))
% 121.92/15.95
% 121.92/15.95 Lemma 11: multiply(X, commutator(X, commutator(X, commutator(X, X)))) = X.
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(X, commutator(X, commutator(X, commutator(X, X))))
% 121.92/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.92/15.95 multiply(X, commutator(commutator(X, X), commutator(X, X)))
% 121.92/15.95 = { by lemma 9 R->L }
% 121.92/15.95 multiply(X, multiply(commutator(X, X), commutator(commutator(X, X), commutator(X, X))))
% 121.92/15.95 = { by lemma 8 }
% 121.92/15.95 multiply(X, commutator(X, X))
% 121.92/15.95 = { by lemma 8 }
% 121.92/15.95 X
% 121.92/15.95
% 121.92/15.95 Lemma 12: fresh(multiply(X, Y), X, Y, commutator(X, X)) = Y.
% 121.92/15.95 Proof:
% 121.92/15.95 fresh(multiply(X, Y), X, Y, commutator(X, X))
% 121.92/15.95 = { by lemma 8 R->L }
% 121.92/15.95 fresh(multiply(X, Y), multiply(X, commutator(X, X)), Y, commutator(X, X))
% 121.92/15.95 = { by axiom 6 (left_cancellation) }
% 121.92/15.95 Y
% 121.92/15.95
% 121.92/15.95 Lemma 13: multiply(commutator(X, X), commutator(X, X)) = commutator(X, X).
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(commutator(X, X), commutator(X, X))
% 121.92/15.95 = { by axiom 1 (left_cancellation) R->L }
% 121.92/15.95 multiply(commutator(X, X), fresh(X, X, commutator(X, commutator(X, commutator(X, X))), commutator(X, X)))
% 121.92/15.95 = { by lemma 11 R->L }
% 121.92/15.95 multiply(commutator(X, X), fresh(multiply(X, commutator(X, commutator(X, commutator(X, X)))), X, commutator(X, commutator(X, commutator(X, X))), commutator(X, X)))
% 121.92/15.95 = { by lemma 12 }
% 121.92/15.95 multiply(commutator(X, X), commutator(X, commutator(X, commutator(X, X))))
% 121.92/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.92/15.95 multiply(commutator(X, X), commutator(commutator(X, X), commutator(X, X)))
% 121.92/15.95 = { by lemma 8 }
% 121.92/15.95 commutator(X, X)
% 121.92/15.95
% 121.92/15.95 Lemma 14: multiply(commutator(X, X), multiply(commutator(X, X), Y)) = multiply(commutator(X, X), Y).
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(commutator(X, X), multiply(commutator(X, X), Y))
% 121.92/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.92/15.95 multiply(multiply(commutator(X, X), commutator(X, X)), Y)
% 121.92/15.95 = { by lemma 13 }
% 121.92/15.95 multiply(commutator(X, X), Y)
% 121.92/15.95
% 121.92/15.95 Lemma 15: multiply(X, commutator(X, commutator(X, X))) = multiply(commutator(X, X), X).
% 121.92/15.95 Proof:
% 121.92/15.95 multiply(X, commutator(X, commutator(X, X)))
% 121.92/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.92/15.95 multiply(X, commutator(commutator(X, X), X))
% 121.92/15.95 = { by lemma 9 R->L }
% 121.92/15.95 multiply(X, multiply(commutator(X, X), commutator(commutator(X, X), X)))
% 121.92/15.95 = { by axiom 5 (commutator) R->L }
% 121.92/15.95 multiply(commutator(X, X), X)
% 121.92/15.95
% 121.92/15.95 Lemma 16: multiply(commutator(X, X), multiply(commutator(X, X), X)) = X.
% 121.92/15.95 Proof:
% 121.98/15.95 multiply(commutator(X, X), multiply(commutator(X, X), X))
% 121.98/15.95 = { by lemma 15 R->L }
% 121.98/15.95 multiply(commutator(X, X), multiply(X, commutator(X, commutator(X, X))))
% 121.98/15.95 = { by axiom 5 (commutator) R->L }
% 121.98/15.95 multiply(X, commutator(X, X))
% 121.98/15.95 = { by lemma 8 }
% 121.98/15.95 X
% 121.98/15.95
% 121.98/15.95 Lemma 17: fresh2(multiply(X, Y), Y, X, commutator(Y, Y)) = X.
% 121.98/15.95 Proof:
% 121.98/15.95 fresh2(multiply(X, Y), Y, X, commutator(Y, Y))
% 121.98/15.95 = { by lemma 13 R->L }
% 121.98/15.95 fresh2(multiply(X, Y), Y, X, multiply(commutator(Y, Y), commutator(Y, Y)))
% 121.98/15.95 = { by lemma 16 R->L }
% 121.98/15.95 fresh2(multiply(X, Y), multiply(commutator(Y, Y), multiply(commutator(Y, Y), Y)), X, multiply(commutator(Y, Y), commutator(Y, Y)))
% 121.98/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.95 fresh2(multiply(X, Y), multiply(multiply(commutator(Y, Y), commutator(Y, Y)), Y), X, multiply(commutator(Y, Y), commutator(Y, Y)))
% 121.98/15.95 = { by axiom 7 (right_cancellation) }
% 121.98/15.95 X
% 121.98/15.95
% 121.98/15.95 Lemma 18: commutator(multiply(X, Y), multiply(X, Y)) = commutator(X, X).
% 121.98/15.95 Proof:
% 121.98/15.95 commutator(multiply(X, Y), multiply(X, Y))
% 121.98/15.95 = { by axiom 2 (right_cancellation) R->L }
% 121.98/15.95 fresh2(multiply(X, Y), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 9 R->L }
% 121.98/15.95 fresh2(multiply(X, multiply(commutator(X, X), Y)), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 10 R->L }
% 121.98/15.95 fresh2(multiply(commutator(X, X), multiply(X, multiply(commutator(X, commutator(X, X)), Y))), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.95 fresh2(multiply(commutator(X, X), multiply(X, multiply(commutator(commutator(X, X), X), Y))), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 9 R->L }
% 121.98/15.95 fresh2(multiply(commutator(X, X), multiply(X, multiply(commutator(X, X), multiply(commutator(commutator(X, X), X), Y)))), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 10 }
% 121.98/15.95 fresh2(multiply(commutator(X, X), multiply(commutator(X, X), multiply(X, Y))), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 14 }
% 121.98/15.95 fresh2(multiply(commutator(X, X), multiply(X, Y)), multiply(X, Y), commutator(X, X), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.95 = { by lemma 17 }
% 121.98/15.95 commutator(X, X)
% 121.98/15.95
% 121.98/15.95 Lemma 19: commutator(Y, Y) = commutator(X, X).
% 121.98/15.95 Proof:
% 121.98/15.95 commutator(Y, Y)
% 121.98/15.95 = { by lemma 18 R->L }
% 121.98/15.95 commutator(multiply(Y, X), multiply(Y, X))
% 121.98/15.95 = { by axiom 5 (commutator) }
% 121.98/15.95 commutator(multiply(Y, X), multiply(X, multiply(Y, commutator(Y, X))))
% 121.98/15.95 = { by axiom 5 (commutator) }
% 121.98/15.95 commutator(multiply(X, multiply(Y, commutator(Y, X))), multiply(X, multiply(Y, commutator(Y, X))))
% 121.98/15.95 = { by lemma 18 }
% 121.98/15.95 commutator(X, X)
% 121.98/15.95
% 121.98/15.95 Lemma 20: multiply(X, commutator(Y, Y)) = X.
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(X, commutator(Y, Y))
% 121.98/15.95 = { by lemma 19 }
% 121.98/15.95 multiply(X, commutator(X, X))
% 121.98/15.95 = { by lemma 8 }
% 121.98/15.95 X
% 121.98/15.95
% 121.98/15.95 Lemma 21: multiply(commutator(X, X), X) = X.
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(commutator(X, X), X)
% 121.98/15.95 = { by lemma 14 R->L }
% 121.98/15.95 multiply(commutator(X, X), multiply(commutator(X, X), X))
% 121.98/15.95 = { by lemma 16 }
% 121.98/15.95 X
% 121.98/15.95
% 121.98/15.95 Lemma 22: multiply(commutator(X, X), Y) = Y.
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(commutator(X, X), Y)
% 121.98/15.95 = { by lemma 21 R->L }
% 121.98/15.95 multiply(commutator(X, X), multiply(commutator(Y, Y), Y))
% 121.98/15.95 = { by lemma 19 }
% 121.98/15.95 multiply(commutator(Y, Y), multiply(commutator(Y, Y), Y))
% 121.98/15.95 = { by lemma 16 }
% 121.98/15.95 Y
% 121.98/15.95
% 121.98/15.95 Lemma 23: commutator(Z, commutator(Z, Y)) = commutator(X, commutator(X, Y)).
% 121.98/15.95 Proof:
% 121.98/15.95 commutator(Z, commutator(Z, Y))
% 121.98/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.95 commutator(commutator(Z, Z), Y)
% 121.98/15.95 = { by lemma 19 }
% 121.98/15.95 commutator(commutator(X, X), Y)
% 121.98/15.95 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.95 commutator(X, commutator(X, Y))
% 121.98/15.95
% 121.98/15.95 Lemma 24: commutator(X, commutator(X, X)) = commutator(X, X).
% 121.98/15.95 Proof:
% 121.98/15.95 commutator(X, commutator(X, X))
% 121.98/15.95 = { by lemma 17 R->L }
% 121.98/15.95 fresh2(multiply(commutator(X, commutator(X, X)), X), X, commutator(X, commutator(X, X)), commutator(X, X))
% 121.98/15.95 = { by lemma 11 R->L }
% 121.98/15.95 fresh2(multiply(commutator(X, commutator(X, X)), multiply(X, commutator(X, commutator(X, commutator(X, X))))), X, commutator(X, commutator(X, X)), commutator(X, X))
% 121.98/15.95 = { by axiom 5 (commutator) R->L }
% 121.98/15.95 fresh2(multiply(X, commutator(X, commutator(X, X))), X, commutator(X, commutator(X, X)), commutator(X, X))
% 121.98/15.95 = { by lemma 15 }
% 121.98/15.95 fresh2(multiply(commutator(X, X), X), X, commutator(X, commutator(X, X)), commutator(X, X))
% 121.98/15.95 = { by lemma 21 }
% 121.98/15.95 fresh2(X, X, commutator(X, commutator(X, X)), commutator(X, X))
% 121.98/15.95 = { by axiom 2 (right_cancellation) }
% 121.98/15.95 commutator(X, X)
% 121.98/15.95
% 121.98/15.95 Lemma 25: commutator(X, commutator(X, Y)) = commutator(Z, Z).
% 121.98/15.95 Proof:
% 121.98/15.95 commutator(X, commutator(X, Y))
% 121.98/15.95 = { by lemma 23 }
% 121.98/15.95 commutator(Y, commutator(Y, Y))
% 121.98/15.95 = { by lemma 24 }
% 121.98/15.95 commutator(Y, Y)
% 121.98/15.95 = { by lemma 19 R->L }
% 121.98/15.95 commutator(Z, Z)
% 121.98/15.95
% 121.98/15.95 Lemma 26: multiply(X, multiply(Y, multiply(Z, commutator(Z, multiply(X, Y))))) = multiply(Z, multiply(X, Y)).
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(X, multiply(Y, multiply(Z, commutator(Z, multiply(X, Y)))))
% 121.98/15.95 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.95 multiply(multiply(X, Y), multiply(Z, commutator(Z, multiply(X, Y))))
% 121.98/15.95 = { by axiom 5 (commutator) R->L }
% 121.98/15.95 multiply(Z, multiply(X, Y))
% 121.98/15.95
% 121.98/15.95 Lemma 27: multiply(X, multiply(Y, commutator(X, commutator(Y, multiply(Y, X))))) = multiply(commutator(X, Y), multiply(Y, X)).
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(X, multiply(Y, commutator(X, commutator(Y, multiply(Y, X)))))
% 121.98/15.95 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.95 multiply(X, multiply(Y, commutator(commutator(X, Y), multiply(Y, X))))
% 121.98/15.95 = { by lemma 10 R->L }
% 121.98/15.95 multiply(Y, multiply(X, multiply(commutator(X, Y), commutator(commutator(X, Y), multiply(Y, X)))))
% 121.98/15.95 = { by lemma 26 }
% 121.98/15.95 multiply(commutator(X, Y), multiply(Y, X))
% 121.98/15.95
% 121.98/15.95 Lemma 28: multiply(X, multiply(commutator(Y, X), Y)) = multiply(Y, X).
% 121.98/15.95 Proof:
% 121.98/15.95 multiply(X, multiply(commutator(Y, X), Y))
% 121.98/15.95 = { by lemma 22 R->L }
% 121.98/15.95 multiply(X, multiply(commutator(Z, Z), multiply(commutator(Y, X), Y)))
% 121.98/15.96 = { by lemma 25 R->L }
% 121.98/15.96 multiply(X, multiply(commutator(Y, commutator(Y, X)), multiply(commutator(Y, X), Y)))
% 121.98/15.96 = { by lemma 27 R->L }
% 121.98/15.96 multiply(X, multiply(Y, multiply(commutator(Y, X), commutator(Y, commutator(commutator(Y, X), multiply(commutator(Y, X), Y))))))
% 121.98/15.96 = { by lemma 10 }
% 121.98/15.96 multiply(Y, multiply(X, commutator(Y, commutator(commutator(Y, X), multiply(commutator(Y, X), Y)))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.96 multiply(Y, multiply(X, commutator(Y, commutator(Y, commutator(X, multiply(commutator(Y, X), Y))))))
% 121.98/15.96 = { by lemma 23 R->L }
% 121.98/15.96 multiply(Y, multiply(X, commutator(W, commutator(W, commutator(X, multiply(commutator(Y, X), Y))))))
% 121.98/15.96 = { by lemma 25 }
% 121.98/15.96 multiply(Y, multiply(X, commutator(V, V)))
% 121.98/15.96 = { by lemma 20 }
% 121.98/15.96 multiply(Y, X)
% 121.98/15.96
% 121.98/15.96 Lemma 29: multiply(commutator(X, Y), X) = multiply(X, commutator(X, Y)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(X, Y), X)
% 121.98/15.96 = { by lemma 22 R->L }
% 121.98/15.96 multiply(commutator(X, Y), multiply(commutator(Z, Z), X))
% 121.98/15.96 = { by lemma 25 R->L }
% 121.98/15.96 multiply(commutator(X, Y), multiply(commutator(X, commutator(X, Y)), X))
% 121.98/15.96 = { by lemma 28 }
% 121.98/15.96 multiply(X, commutator(X, Y))
% 121.98/15.96
% 121.98/15.96 Lemma 30: multiply(commutator(Y, X), X) = multiply(X, commutator(Y, X)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(Y, X), X)
% 121.98/15.96 = { by axiom 5 (commutator) }
% 121.98/15.96 multiply(X, multiply(commutator(Y, X), commutator(commutator(Y, X), X)))
% 121.98/15.96 = { by lemma 22 R->L }
% 121.98/15.96 multiply(X, multiply(commutator(X, X), multiply(commutator(Y, X), commutator(commutator(Y, X), X))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.96 multiply(X, multiply(commutator(X, X), multiply(commutator(Y, X), commutator(Y, commutator(X, X)))))
% 121.98/15.96 = { by lemma 24 R->L }
% 121.98/15.96 multiply(X, multiply(commutator(X, X), multiply(commutator(Y, X), commutator(Y, commutator(X, commutator(X, X))))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.96 multiply(X, multiply(commutator(X, X), multiply(commutator(Y, X), commutator(commutator(Y, X), commutator(X, X)))))
% 121.98/15.96 = { by axiom 5 (commutator) R->L }
% 121.98/15.96 multiply(X, multiply(commutator(Y, X), commutator(X, X)))
% 121.98/15.96 = { by lemma 20 }
% 121.98/15.96 multiply(X, commutator(Y, X))
% 121.98/15.96
% 121.98/15.96 Lemma 31: fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, W)), Z, W) = Z.
% 121.98/15.96 Proof:
% 121.98/15.96 fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, W)), Z, W)
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 fresh(multiply(X, multiply(Y, Z)), multiply(multiply(X, Y), W), Z, W)
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), W), Z, W)
% 121.98/15.96 = { by axiom 6 (left_cancellation) }
% 121.98/15.96 Z
% 121.98/15.96
% 121.98/15.96 Lemma 32: fresh(multiply(X, multiply(Y, Z)), multiply(Y, X), Z, commutator(Y, X)) = Z.
% 121.98/15.96 Proof:
% 121.98/15.96 fresh(multiply(X, multiply(Y, Z)), multiply(Y, X), Z, commutator(Y, X))
% 121.98/15.96 = { by axiom 5 (commutator) }
% 121.98/15.96 fresh(multiply(X, multiply(Y, Z)), multiply(X, multiply(Y, commutator(Y, X))), Z, commutator(Y, X))
% 121.98/15.96 = { by lemma 31 }
% 121.98/15.96 Z
% 121.98/15.96
% 121.98/15.96 Lemma 33: commutator(X, multiply(Y, X)) = commutator(X, Y).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(X, multiply(Y, X))
% 121.98/15.96 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.96 fresh(multiply(X, multiply(Y, X)), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 fresh(multiply(multiply(X, Y), X), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by axiom 5 (commutator) }
% 121.98/15.96 fresh(multiply(multiply(Y, multiply(X, commutator(X, Y))), X), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 fresh(multiply(multiply(multiply(Y, X), commutator(X, Y)), X), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 fresh(multiply(multiply(Y, X), multiply(commutator(X, Y), X)), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by lemma 29 }
% 121.98/15.96 fresh(multiply(multiply(Y, X), multiply(X, commutator(X, Y))), multiply(X, multiply(Y, X)), commutator(X, Y), commutator(X, multiply(Y, X)))
% 121.98/15.96 = { by lemma 32 }
% 121.98/15.96 commutator(X, Y)
% 121.98/15.96
% 121.98/15.96 Lemma 34: multiply(X, multiply(Y, commutator(X, commutator(Y, X)))) = multiply(Y, multiply(commutator(X, Y), X)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(X, multiply(Y, commutator(X, commutator(Y, X))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.96 multiply(X, multiply(Y, commutator(commutator(X, Y), X)))
% 121.98/15.96 = { by lemma 10 R->L }
% 121.98/15.96 multiply(Y, multiply(X, multiply(commutator(X, Y), commutator(commutator(X, Y), X))))
% 121.98/15.96 = { by axiom 5 (commutator) R->L }
% 121.98/15.96 multiply(Y, multiply(commutator(X, Y), X))
% 121.98/15.96
% 121.98/15.96 Lemma 35: commutator(X, commutator(Y, X)) = commutator(Z, Z).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(X, commutator(Y, X))
% 121.98/15.96 = { by lemma 12 R->L }
% 121.98/15.96 fresh(multiply(multiply(X, Y), commutator(X, commutator(Y, X))), multiply(X, Y), commutator(X, commutator(Y, X)), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 fresh(multiply(X, multiply(Y, commutator(X, commutator(Y, X)))), multiply(X, Y), commutator(X, commutator(Y, X)), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.96 = { by lemma 34 }
% 121.98/15.96 fresh(multiply(Y, multiply(commutator(X, Y), X)), multiply(X, Y), commutator(X, commutator(Y, X)), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.96 = { by lemma 28 }
% 121.98/15.96 fresh(multiply(X, Y), multiply(X, Y), commutator(X, commutator(Y, X)), commutator(multiply(X, Y), multiply(X, Y)))
% 121.98/15.96 = { by axiom 1 (left_cancellation) }
% 121.98/15.96 commutator(multiply(X, Y), multiply(X, Y))
% 121.98/15.96 = { by lemma 19 R->L }
% 121.98/15.96 commutator(Z, Z)
% 121.98/15.96
% 121.98/15.96 Lemma 36: commutator(X, commutator(Y, Y)) = commutator(Z, Z).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(X, commutator(Y, Y))
% 121.98/15.96 = { by lemma 19 }
% 121.98/15.96 commutator(X, commutator(X, X))
% 121.98/15.96 = { by lemma 24 }
% 121.98/15.96 commutator(X, X)
% 121.98/15.96 = { by lemma 19 R->L }
% 121.98/15.96 commutator(Z, Z)
% 121.98/15.96
% 121.98/15.96 Lemma 37: multiply(X, multiply(commutator(Y, Z), commutator(Y, commutator(Z, X)))) = multiply(commutator(Y, Z), X).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(X, multiply(commutator(Y, Z), commutator(Y, commutator(Z, X))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.96 multiply(X, multiply(commutator(Y, Z), commutator(commutator(Y, Z), X)))
% 121.98/15.96 = { by axiom 5 (commutator) R->L }
% 121.98/15.96 multiply(commutator(Y, Z), X)
% 121.98/15.96
% 121.98/15.96 Lemma 38: multiply(multiply(X, Z), commutator(Y, Z)) = multiply(X, multiply(commutator(Y, Z), Z)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(multiply(X, Z), commutator(Y, Z))
% 121.98/15.96 = { by lemma 20 R->L }
% 121.98/15.96 multiply(multiply(X, Z), multiply(commutator(Y, Z), commutator(V, V)))
% 121.98/15.96 = { by lemma 36 R->L }
% 121.98/15.96 multiply(multiply(X, Z), multiply(commutator(Y, Z), commutator(Y, commutator(W, W))))
% 121.98/15.96 = { by lemma 19 }
% 121.98/15.96 multiply(multiply(X, Z), multiply(commutator(Y, Z), commutator(Y, commutator(Z, Z))))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(X, multiply(Z, multiply(commutator(Y, Z), commutator(Y, commutator(Z, Z)))))
% 121.98/15.96 = { by lemma 37 }
% 121.98/15.96 multiply(X, multiply(commutator(Y, Z), Z))
% 121.98/15.96
% 121.98/15.96 Lemma 39: commutator(multiply(X, Y), Y) = commutator(X, Y).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(multiply(X, Y), Y)
% 121.98/15.96 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.96 fresh(multiply(multiply(X, Y), Y), multiply(multiply(X, Y), Y), commutator(X, Y), commutator(multiply(X, Y), Y))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 fresh(multiply(X, multiply(Y, Y)), multiply(multiply(X, Y), Y), commutator(X, Y), commutator(multiply(X, Y), Y))
% 121.98/15.96 = { by lemma 10 R->L }
% 121.98/15.96 fresh(multiply(Y, multiply(X, multiply(commutator(X, Y), Y))), multiply(multiply(X, Y), Y), commutator(X, Y), commutator(multiply(X, Y), Y))
% 121.98/15.96 = { by lemma 38 R->L }
% 121.98/15.96 fresh(multiply(Y, multiply(multiply(X, Y), commutator(X, Y))), multiply(multiply(X, Y), Y), commutator(X, Y), commutator(multiply(X, Y), Y))
% 121.98/15.96 = { by lemma 32 }
% 121.98/15.96 commutator(X, Y)
% 121.98/15.96
% 121.98/15.96 Lemma 40: multiply(multiply(X, Y), multiply(Z, commutator(Z, Y))) = multiply(X, multiply(Z, Y)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(multiply(X, Y), multiply(Z, commutator(Z, Y)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(X, multiply(Y, multiply(Z, commutator(Z, Y))))
% 121.98/15.96 = { by axiom 5 (commutator) R->L }
% 121.98/15.96 multiply(X, multiply(Z, Y))
% 121.98/15.96
% 121.98/15.96 Lemma 41: commutator(multiply(X, Y), Y) = commutator(X, multiply(X, Y)).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(multiply(X, Y), Y)
% 121.98/15.96 = { by lemma 32 R->L }
% 121.98/15.96 fresh(multiply(multiply(X, Y), multiply(X, commutator(multiply(X, Y), Y))), multiply(X, multiply(X, Y)), commutator(multiply(X, Y), Y), commutator(X, multiply(X, Y)))
% 121.98/15.96 = { by lemma 39 }
% 121.98/15.96 fresh(multiply(multiply(X, Y), multiply(X, commutator(X, Y))), multiply(X, multiply(X, Y)), commutator(multiply(X, Y), Y), commutator(X, multiply(X, Y)))
% 121.98/15.96 = { by lemma 40 }
% 121.98/15.96 fresh(multiply(X, multiply(X, Y)), multiply(X, multiply(X, Y)), commutator(multiply(X, Y), Y), commutator(X, multiply(X, Y)))
% 121.98/15.96 = { by axiom 1 (left_cancellation) }
% 121.98/15.96 commutator(X, multiply(X, Y))
% 121.98/15.96
% 121.98/15.96 Lemma 42: multiply(X, multiply(commutator(Y, X), multiply(Y, Z))) = multiply(Y, multiply(X, Z)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(X, multiply(commutator(Y, X), multiply(Y, Z)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(X, multiply(multiply(commutator(Y, X), Y), Z))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(multiply(X, multiply(commutator(Y, X), Y)), Z)
% 121.98/15.96 = { by lemma 28 }
% 121.98/15.96 multiply(multiply(Y, X), Z)
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(Y, multiply(X, Z))
% 121.98/15.96
% 121.98/15.96 Lemma 43: multiply(commutator(Y, X), multiply(X, Z)) = multiply(X, multiply(commutator(Y, X), Z)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(Y, X), multiply(X, Z))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(multiply(commutator(Y, X), X), Z)
% 121.98/15.96 = { by lemma 30 }
% 121.98/15.96 multiply(multiply(X, commutator(Y, X)), Z)
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(X, multiply(commutator(Y, X), Z))
% 121.98/15.96
% 121.98/15.96 Lemma 44: commutator(Z, commutator(commutator(Z, W), Y)) = commutator(commutator(X, X), Y).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(Z, commutator(commutator(Z, W), Y))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.96 commutator(commutator(Z, commutator(Z, W)), Y)
% 121.98/15.96 = { by lemma 25 }
% 121.98/15.96 commutator(commutator(X, X), Y)
% 121.98/15.96
% 121.98/15.96 Lemma 45: commutator(X, commutator(commutator(X, Y), Z)) = commutator(W, W).
% 121.98/15.96 Proof:
% 121.98/15.96 commutator(X, commutator(commutator(X, Y), Z))
% 121.98/15.96 = { by lemma 44 }
% 121.98/15.96 commutator(commutator(V, V), Z)
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.96 commutator(V, commutator(V, Z))
% 121.98/15.96 = { by lemma 25 }
% 121.98/15.96 commutator(W, W)
% 121.98/15.96
% 121.98/15.96 Lemma 46: multiply(commutator(X, Y), multiply(Y, X)) = multiply(X, Y).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(X, Y), multiply(Y, X))
% 121.98/15.96 = { by lemma 43 }
% 121.98/15.96 multiply(Y, multiply(commutator(X, Y), X))
% 121.98/15.96 = { by lemma 22 R->L }
% 121.98/15.96 multiply(Y, multiply(commutator(Z, Z), multiply(commutator(X, Y), X)))
% 121.98/15.96 = { by lemma 25 R->L }
% 121.98/15.96 multiply(Y, multiply(commutator(X, commutator(X, Y)), multiply(commutator(X, Y), X)))
% 121.98/15.96 = { by lemma 27 R->L }
% 121.98/15.96 multiply(Y, multiply(X, multiply(commutator(X, Y), commutator(X, commutator(commutator(X, Y), multiply(commutator(X, Y), X))))))
% 121.98/15.96 = { by lemma 10 }
% 121.98/15.96 multiply(X, multiply(Y, commutator(X, commutator(commutator(X, Y), multiply(commutator(X, Y), X)))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.96 multiply(X, multiply(Y, commutator(X, commutator(X, commutator(Y, multiply(commutator(X, Y), X))))))
% 121.98/15.96 = { by lemma 23 R->L }
% 121.98/15.96 multiply(X, multiply(Y, commutator(W, commutator(W, commutator(Y, multiply(commutator(X, Y), X))))))
% 121.98/15.96 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.96 multiply(X, multiply(Y, commutator(W, commutator(commutator(W, Y), multiply(commutator(X, Y), X)))))
% 121.98/15.96 = { by lemma 45 }
% 121.98/15.96 multiply(X, multiply(Y, commutator(V, V)))
% 121.98/15.96 = { by lemma 20 }
% 121.98/15.96 multiply(X, Y)
% 121.98/15.96
% 121.98/15.96 Lemma 47: multiply(commutator(X, Y), multiply(Y, multiply(X, Z))) = multiply(X, multiply(Y, Z)).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(X, Y), multiply(Y, multiply(X, Z)))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(commutator(X, Y), multiply(multiply(Y, X), Z))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(multiply(commutator(X, Y), multiply(Y, X)), Z)
% 121.98/15.96 = { by lemma 46 }
% 121.98/15.96 multiply(multiply(X, Y), Z)
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(X, multiply(Y, Z))
% 121.98/15.96
% 121.98/15.96 Lemma 48: multiply(commutator(X, Y), multiply(Y, multiply(multiply(X, Z), commutator(Z, Y)))) = multiply(multiply(X, Z), Y).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(X, Y), multiply(Y, multiply(multiply(X, Z), commutator(Z, Y))))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.96 multiply(commutator(X, Y), multiply(Y, multiply(X, multiply(Z, commutator(Z, Y)))))
% 121.98/15.96 = { by lemma 47 }
% 121.98/15.96 multiply(X, multiply(Y, multiply(Z, commutator(Z, Y))))
% 121.98/15.96 = { by axiom 5 (commutator) R->L }
% 121.98/15.96 multiply(X, multiply(Z, Y))
% 121.98/15.96 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.96 multiply(multiply(X, Z), Y)
% 121.98/15.96
% 121.98/15.96 Lemma 49: multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))) = commutator(multiply(X, Z), Y).
% 121.98/15.96 Proof:
% 121.98/15.96 multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y)))
% 121.98/15.96 = { by lemma 32 R->L }
% 121.98/15.96 fresh(multiply(Y, multiply(multiply(X, Z), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))))), multiply(multiply(X, Z), Y), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(X, Z), Y))
% 121.98/15.96 = { by lemma 42 }
% 121.98/15.96 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y)))), multiply(multiply(X, Z), Y), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(X, Z), Y))
% 121.98/15.97 = { by lemma 43 R->L }
% 121.98/15.97 fresh(multiply(commutator(X, Y), multiply(Y, multiply(multiply(X, Z), commutator(Z, Y)))), multiply(multiply(X, Z), Y), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(X, Z), Y))
% 121.98/15.97 = { by lemma 48 }
% 121.98/15.97 fresh(multiply(multiply(X, Z), Y), multiply(multiply(X, Z), Y), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(X, Z), Y))
% 121.98/15.97 = { by axiom 1 (left_cancellation) }
% 121.98/15.97 commutator(multiply(X, Z), Y)
% 121.98/15.97
% 121.98/15.97 Lemma 50: multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y))) = multiply(multiply(X, Z), commutator(multiply(X, Z), Y)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y)))
% 121.98/15.97 = { by lemma 42 R->L }
% 121.98/15.97 multiply(multiply(X, Z), multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.97 = { by lemma 49 }
% 121.98/15.97 multiply(multiply(X, Z), commutator(multiply(X, Z), Y))
% 121.98/15.97
% 121.98/15.97 Lemma 51: multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), W))) = multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), Y), W)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), W)))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.97 multiply(commutator(X, Y), multiply(multiply(multiply(X, Z), commutator(Z, Y)), W))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.97 multiply(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y))), W)
% 121.98/15.97 = { by lemma 50 }
% 121.98/15.97 multiply(multiply(multiply(X, Z), commutator(multiply(X, Z), Y)), W)
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), Y), W))
% 121.98/15.97
% 121.98/15.97 Lemma 52: multiply(multiply(X, Y), commutator(Y, Z)) = multiply(X, multiply(commutator(Y, Z), Y)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(multiply(X, Y), commutator(Y, Z))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 multiply(X, multiply(Y, commutator(Y, Z)))
% 121.98/15.97 = { by lemma 29 R->L }
% 121.98/15.97 multiply(X, multiply(commutator(Y, Z), Y))
% 121.98/15.97
% 121.98/15.97 Lemma 53: commutator(commutator(X, Y), X) = commutator(Z, Z).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(commutator(X, Y), X)
% 121.98/15.97 = { by lemma 12 R->L }
% 121.98/15.97 fresh(multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by lemma 32 R->L }
% 121.98/15.97 fresh(fresh(multiply(Y, multiply(multiply(W, X), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)))), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by lemma 51 R->L }
% 121.98/15.97 fresh(fresh(multiply(Y, multiply(commutator(W, Y), multiply(multiply(W, X), multiply(commutator(X, Y), commutator(commutator(X, Y), X))))), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by lemma 40 }
% 121.98/15.97 fresh(fresh(multiply(Y, multiply(commutator(W, Y), multiply(W, multiply(commutator(X, Y), X)))), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by lemma 52 R->L }
% 121.98/15.97 fresh(fresh(multiply(Y, multiply(commutator(W, Y), multiply(multiply(W, X), commutator(X, Y)))), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by lemma 50 }
% 121.98/15.97 fresh(fresh(multiply(Y, multiply(multiply(W, X), commutator(multiply(W, X), Y))), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by axiom 5 (commutator) R->L }
% 121.98/15.97 fresh(fresh(multiply(multiply(W, X), Y), multiply(multiply(W, X), Y), multiply(commutator(multiply(W, X), Y), commutator(commutator(X, Y), X)), commutator(multiply(W, X), Y)), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by axiom 1 (left_cancellation) }
% 121.98/15.97 fresh(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y), commutator(commutator(X, Y), X), commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y)))
% 121.98/15.97 = { by axiom 1 (left_cancellation) }
% 121.98/15.97 commutator(commutator(multiply(W, X), Y), commutator(multiply(W, X), Y))
% 121.98/15.97 = { by lemma 19 R->L }
% 121.98/15.97 commutator(Z, Z)
% 121.98/15.97
% 121.98/15.97 Lemma 54: multiply(commutator(X, Y), multiply(X, Z)) = multiply(X, multiply(commutator(X, Y), Z)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(commutator(X, Y), multiply(X, Z))
% 121.98/15.97 = { by lemma 22 R->L }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(V, V), multiply(X, Z)))
% 121.98/15.97 = { by lemma 25 R->L }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(W, commutator(W, Y)), multiply(X, Z)))
% 121.98/15.97 = { by lemma 23 }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(X, commutator(X, Y)), multiply(X, Z)))
% 121.98/15.97 = { by lemma 42 }
% 121.98/15.97 multiply(X, multiply(commutator(X, Y), Z))
% 121.98/15.97
% 121.98/15.97 Lemma 55: fresh(multiply(X, Y), multiply(X, multiply(Y, Z)), commutator(X, Y), multiply(commutator(X, Y), Z)) = commutator(X, Y).
% 121.98/15.97 Proof:
% 121.98/15.97 fresh(multiply(X, Y), multiply(X, multiply(Y, Z)), commutator(X, Y), multiply(commutator(X, Y), Z))
% 121.98/15.97 = { by lemma 10 R->L }
% 121.98/15.97 fresh(multiply(X, Y), multiply(Y, multiply(X, multiply(commutator(X, Y), Z))), commutator(X, Y), multiply(commutator(X, Y), Z))
% 121.98/15.97 = { by axiom 5 (commutator) }
% 121.98/15.97 fresh(multiply(Y, multiply(X, commutator(X, Y))), multiply(Y, multiply(X, multiply(commutator(X, Y), Z))), commutator(X, Y), multiply(commutator(X, Y), Z))
% 121.98/15.97 = { by lemma 31 }
% 121.98/15.97 commutator(X, Y)
% 121.98/15.97
% 121.98/15.97 Lemma 56: multiply(commutator(X, Y), commutator(Y, X)) = commutator(Z, Z).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(commutator(X, Y), commutator(Y, X))
% 121.98/15.97 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.97 fresh(commutator(X, Y), commutator(X, Y), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 55 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, multiply(Y, multiply(commutator(X, Y), commutator(Y, X)))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 20 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, multiply(Y, multiply(commutator(X, Y), multiply(commutator(Y, X), commutator(W, W))))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 36 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, multiply(Y, multiply(commutator(X, Y), multiply(commutator(Y, X), commutator(Y, commutator(V, V)))))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 25 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, multiply(Y, multiply(commutator(X, Y), multiply(commutator(Y, X), commutator(Y, commutator(X, commutator(X, Y))))))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 37 }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, multiply(Y, multiply(commutator(Y, X), commutator(X, Y)))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 10 }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(Y, multiply(X, commutator(X, Y))), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 29 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(Y, multiply(commutator(X, Y), X)), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 43 R->L }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(commutator(X, Y), multiply(Y, X)), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 46 }
% 121.98/15.97 fresh(commutator(X, Y), fresh(multiply(X, Y), multiply(X, Y), commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by axiom 1 (left_cancellation) }
% 121.98/15.97 fresh(commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))
% 121.98/15.97 = { by lemma 22 R->L }
% 121.98/15.97 fresh(commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Y, X))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X))))
% 121.98/15.97 = { by lemma 9 R->L }
% 121.98/15.97 fresh(commutator(X, Y), multiply(commutator(X, Y), multiply(commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X))))
% 121.98/15.97 = { by lemma 8 R->L }
% 121.98/15.97 fresh(multiply(commutator(X, Y), commutator(commutator(X, Y), commutator(X, Y))), multiply(commutator(X, Y), multiply(commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X)))), commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(commutator(X, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Y, X))))
% 121.98/15.97 = { by axiom 6 (left_cancellation) }
% 121.98/15.97 commutator(commutator(X, Y), commutator(X, Y))
% 121.98/15.97 = { by lemma 19 R->L }
% 121.98/15.97 commutator(Z, Z)
% 121.98/15.97
% 121.98/15.97 Lemma 57: commutator(commutator(X, Y), commutator(Z, Y)) = commutator(W, W).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(commutator(X, Y), commutator(Z, Y))
% 121.98/15.97 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.97 commutator(X, commutator(Y, commutator(Z, Y)))
% 121.98/15.97 = { by lemma 35 }
% 121.98/15.97 commutator(X, commutator(V, V))
% 121.98/15.97 = { by lemma 36 }
% 121.98/15.97 commutator(W, W)
% 121.98/15.97
% 121.98/15.97 Lemma 58: multiply(commutator(Z, Y), commutator(X, Y)) = multiply(commutator(X, Y), commutator(Z, Y)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(commutator(Z, Y), commutator(X, Y))
% 121.98/15.97 = { by axiom 5 (commutator) }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(Z, Y), commutator(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by lemma 57 }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(Z, Y), commutator(W, W)))
% 121.98/15.97 = { by lemma 20 }
% 121.98/15.97 multiply(commutator(X, Y), commutator(Z, Y))
% 121.98/15.97
% 121.98/15.97 Lemma 59: commutator(X, commutator(Y, commutator(X, Z))) = commutator(W, W).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(X, commutator(Y, commutator(X, Z)))
% 121.98/15.97 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.97 commutator(X, commutator(commutator(Y, X), Z))
% 121.98/15.97 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.97 commutator(commutator(X, commutator(Y, X)), Z)
% 121.98/15.97 = { by lemma 35 }
% 121.98/15.97 commutator(commutator(V, V), Z)
% 121.98/15.97 = { by lemma 44 R->L }
% 121.98/15.97 commutator(U, commutator(commutator(U, T), Z))
% 121.98/15.97 = { by lemma 45 }
% 121.98/15.97 commutator(W, W)
% 121.98/15.97
% 121.98/15.97 Lemma 60: commutator(X, commutator(multiply(Y, X), Z)) = commutator(X, commutator(Y, Z)).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(X, commutator(multiply(Y, X), Z))
% 121.98/15.97 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.97 commutator(commutator(X, multiply(Y, X)), Z)
% 121.98/15.97 = { by lemma 33 }
% 121.98/15.97 commutator(commutator(X, Y), Z)
% 121.98/15.97 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.97 commutator(X, commutator(Y, Z))
% 121.98/15.97
% 121.98/15.97 Lemma 61: multiply(multiply(commutator(Z, Y), commutator(X, Y)), W) = multiply(commutator(X, Y), multiply(commutator(Z, Y), W)).
% 121.98/15.97 Proof:
% 121.98/15.97 multiply(multiply(commutator(Z, Y), commutator(X, Y)), W)
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 multiply(commutator(Z, Y), multiply(commutator(X, Y), W))
% 121.98/15.97 = { by lemma 10 R->L }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(commutator(Z, Y), commutator(X, Y)), W)))
% 121.98/15.97 = { by lemma 57 }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(V, V), W)))
% 121.98/15.97 = { by lemma 22 }
% 121.98/15.97 multiply(commutator(X, Y), multiply(commutator(Z, Y), W))
% 121.98/15.97
% 121.98/15.97 Lemma 62: commutator(commutator(X, Y), multiply(commutator(Z, Y), commutator(X, Y))) = commutator(W, W).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(commutator(X, Y), multiply(commutator(Z, Y), commutator(X, Y)))
% 121.98/15.97 = { by lemma 58 }
% 121.98/15.97 commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y)))
% 121.98/15.97 = { by lemma 32 R->L }
% 121.98/15.97 fresh(multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(multiply(commutator(Z, Y), commutator(X, Y)), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 fresh(multiply(commutator(Z, Y), multiply(commutator(X, Y), multiply(multiply(commutator(Z, Y), commutator(X, Y)), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y)))))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 fresh(multiply(commutator(Z, Y), multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))))))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by lemma 26 }
% 121.98/15.97 fresh(multiply(commutator(Z, Y), multiply(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y)))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.97 fresh(multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(X, Y), commutator(Z, Y))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by lemma 58 R->L }
% 121.98/15.97 fresh(multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by lemma 58 R->L }
% 121.98/15.97 fresh(multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))), commutator(commutator(X, Y), multiply(commutator(Z, Y), commutator(X, Y))), commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y))))
% 121.98/15.97 = { by axiom 1 (left_cancellation) }
% 121.98/15.97 commutator(multiply(commutator(Z, Y), commutator(X, Y)), multiply(commutator(Z, Y), commutator(X, Y)))
% 121.98/15.97 = { by lemma 19 R->L }
% 121.98/15.97 commutator(W, W)
% 121.98/15.97
% 121.98/15.97 Lemma 63: commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)) = commutator(W, W).
% 121.98/15.97 Proof:
% 121.98/15.97 commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y))
% 121.98/15.97 = { by lemma 32 R->L }
% 121.98/15.97 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Z, Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.97 fresh(multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.97 = { by lemma 61 }
% 121.98/15.97 fresh(multiply(multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.97 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.97 fresh(multiply(commutator(Z, Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.97 = { by lemma 20 R->L }
% 121.98/15.97 fresh(multiply(commutator(Z, Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(V, V)))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by lemma 36 R->L }
% 121.98/15.98 fresh(multiply(commutator(Z, Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(U, U))))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by lemma 62 R->L }
% 121.98/15.98 fresh(multiply(commutator(Z, Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by lemma 37 }
% 121.98/15.98 fresh(multiply(commutator(Z, Y), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), multiply(commutator(X, Y), commutator(Z, Y)))), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by lemma 28 }
% 121.98/15.98 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), multiply(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by lemma 61 R->L }
% 121.98/15.98 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Z, Y)), commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))))
% 121.98/15.98 = { by axiom 1 (left_cancellation) }
% 121.98/15.98 commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y)))
% 121.98/15.98 = { by lemma 62 }
% 121.98/15.98 commutator(W, W)
% 121.98/15.98
% 121.98/15.98 Lemma 64: commutator(multiply(commutator(Z, W), commutator(V, W)), commutator(commutator(V, W), Y)) = commutator(commutator(X, X), Y).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(multiply(commutator(Z, W), commutator(V, W)), commutator(commutator(V, W), Y))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 commutator(commutator(multiply(commutator(Z, W), commutator(V, W)), commutator(V, W)), Y)
% 121.98/15.98 = { by lemma 63 }
% 121.98/15.98 commutator(commutator(X, X), Y)
% 121.98/15.98
% 121.98/15.98 Lemma 65: commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(Z, Y), W)) = commutator(V, V).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(Z, Y), W))
% 121.98/15.98 = { by lemma 64 }
% 121.98/15.98 commutator(commutator(U, U), W)
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.98 commutator(U, commutator(U, W))
% 121.98/15.98 = { by lemma 25 }
% 121.98/15.98 commutator(V, V)
% 121.98/15.98
% 121.98/15.98 Lemma 66: commutator(X, commutator(commutator(Y, X), Z)) = commutator(W, W).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(X, commutator(commutator(Y, X), Z))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 commutator(commutator(X, commutator(Y, X)), Z)
% 121.98/15.98 = { by lemma 35 }
% 121.98/15.98 commutator(commutator(V, V), Z)
% 121.98/15.98 = { by lemma 64 R->L }
% 121.98/15.98 commutator(multiply(commutator(U, T), commutator(S, T)), commutator(commutator(S, T), Z))
% 121.98/15.98 = { by lemma 65 }
% 121.98/15.98 commutator(W, W)
% 121.98/15.98
% 121.98/15.98 Lemma 67: commutator(multiply(X, Y), commutator(Y, Z)) = commutator(X, commutator(Y, Z)).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(multiply(X, Y), commutator(Y, Z))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 commutator(commutator(multiply(X, Y), Y), Z)
% 121.98/15.98 = { by lemma 39 }
% 121.98/15.98 commutator(commutator(X, Y), Z)
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.98 commutator(X, commutator(Y, Z))
% 121.98/15.98
% 121.98/15.98 Lemma 68: commutator(multiply(X, Y), commutator(Y, Z)) = commutator(X, commutator(multiply(X, Y), Z)).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(multiply(X, Y), commutator(Y, Z))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 commutator(commutator(multiply(X, Y), Y), Z)
% 121.98/15.98 = { by lemma 41 }
% 121.98/15.98 commutator(commutator(X, multiply(X, Y)), Z)
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.98 commutator(X, commutator(multiply(X, Y), Z))
% 121.98/15.98
% 121.98/15.98 Lemma 69: multiply(X, multiply(Y, multiply(Z, commutator(Z, commutator(X, Y))))) = multiply(Y, multiply(X, multiply(Z, commutator(X, Y)))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(X, multiply(Y, multiply(Z, commutator(Z, commutator(X, Y)))))
% 121.98/15.98 = { by lemma 10 R->L }
% 121.98/15.98 multiply(Y, multiply(X, multiply(commutator(X, Y), multiply(Z, commutator(Z, commutator(X, Y))))))
% 121.98/15.98 = { by axiom 5 (commutator) R->L }
% 121.98/15.98 multiply(Y, multiply(X, multiply(Z, commutator(X, Y))))
% 121.98/15.98
% 121.98/15.98 Lemma 70: multiply(Y, multiply(multiply(X, Z), commutator(Z, commutator(Y, X)))) = multiply(X, multiply(Y, multiply(Z, commutator(Y, X)))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(Y, multiply(multiply(X, Z), commutator(Z, commutator(Y, X))))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(Y, multiply(X, multiply(Z, commutator(Z, commutator(Y, X)))))
% 121.98/15.98 = { by lemma 69 }
% 121.98/15.98 multiply(X, multiply(Y, multiply(Z, commutator(Y, X))))
% 121.98/15.98
% 121.98/15.98 Lemma 71: commutator(commutator(X, Y), multiply(Z, X)) = commutator(commutator(X, Y), Z).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(commutator(X, Y), multiply(Z, X))
% 121.98/15.98 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.98 fresh(multiply(commutator(X, Y), multiply(Z, X)), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by lemma 20 R->L }
% 121.98/15.98 fresh(multiply(commutator(X, Y), multiply(multiply(Z, X), commutator(W, W))), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by lemma 45 R->L }
% 121.98/15.98 fresh(multiply(commutator(X, Y), multiply(multiply(Z, X), commutator(X, commutator(commutator(X, Y), Z)))), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by lemma 70 }
% 121.98/15.98 fresh(multiply(Z, multiply(commutator(X, Y), multiply(X, commutator(commutator(X, Y), Z)))), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by lemma 54 }
% 121.98/15.98 fresh(multiply(Z, multiply(X, multiply(commutator(X, Y), commutator(commutator(X, Y), Z)))), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 fresh(multiply(multiply(Z, X), multiply(commutator(X, Y), commutator(commutator(X, Y), Z))), multiply(commutator(X, Y), multiply(Z, X)), commutator(commutator(X, Y), Z), commutator(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98 = { by lemma 32 }
% 121.98/15.98 commutator(commutator(X, Y), Z)
% 121.98/15.98
% 121.98/15.98 Lemma 72: commutator(commutator(X, Y), commutator(commutator(Z, Y), W)) = commutator(V, V).
% 121.98/15.98 Proof:
% 121.98/15.98 commutator(commutator(X, Y), commutator(commutator(Z, Y), W))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 commutator(commutator(commutator(X, Y), commutator(Z, Y)), W)
% 121.98/15.98 = { by lemma 57 }
% 121.98/15.98 commutator(commutator(U, U), W)
% 121.98/15.98 = { by lemma 44 R->L }
% 121.98/15.98 commutator(T, commutator(commutator(T, S), W))
% 121.98/15.98 = { by lemma 45 }
% 121.98/15.98 commutator(V, V)
% 121.98/15.98
% 121.98/15.98 Lemma 73: multiply(X, multiply(Y, multiply(Z, commutator(Z, X)))) = multiply(Y, multiply(commutator(X, Y), multiply(Z, X))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(X, multiply(Y, multiply(Z, commutator(Z, X))))
% 121.98/15.98 = { by lemma 42 R->L }
% 121.98/15.98 multiply(Y, multiply(commutator(X, Y), multiply(X, multiply(Z, commutator(Z, X)))))
% 121.98/15.98 = { by axiom 5 (commutator) R->L }
% 121.98/15.98 multiply(Y, multiply(commutator(X, Y), multiply(Z, X)))
% 121.98/15.98
% 121.98/15.98 Lemma 74: multiply(X, multiply(multiply(commutator(Y, Z), commutator(W, Z)), commutator(commutator(W, Z), X))) = multiply(commutator(W, Z), multiply(X, commutator(Y, Z))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(X, multiply(multiply(commutator(Y, Z), commutator(W, Z)), commutator(commutator(W, Z), X)))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(X, multiply(commutator(Y, Z), multiply(commutator(W, Z), commutator(commutator(W, Z), X))))
% 121.98/15.98 = { by lemma 29 R->L }
% 121.98/15.98 multiply(X, multiply(commutator(Y, Z), multiply(commutator(commutator(W, Z), X), commutator(W, Z))))
% 121.98/15.98 = { by lemma 42 R->L }
% 121.98/15.98 multiply(X, multiply(commutator(commutator(W, Z), X), multiply(commutator(commutator(Y, Z), commutator(commutator(W, Z), X)), multiply(commutator(Y, Z), commutator(W, Z)))))
% 121.98/15.98 = { by lemma 72 }
% 121.98/15.98 multiply(X, multiply(commutator(commutator(W, Z), X), multiply(commutator(V, V), multiply(commutator(Y, Z), commutator(W, Z)))))
% 121.98/15.98 = { by lemma 22 }
% 121.98/15.98 multiply(X, multiply(commutator(commutator(W, Z), X), multiply(commutator(Y, Z), commutator(W, Z))))
% 121.98/15.98 = { by lemma 73 R->L }
% 121.98/15.98 multiply(commutator(W, Z), multiply(X, multiply(commutator(Y, Z), commutator(commutator(Y, Z), commutator(W, Z)))))
% 121.98/15.98 = { by lemma 57 }
% 121.98/15.98 multiply(commutator(W, Z), multiply(X, multiply(commutator(Y, Z), commutator(U, U))))
% 121.98/15.98 = { by lemma 20 }
% 121.98/15.98 multiply(commutator(W, Z), multiply(X, commutator(Y, Z)))
% 121.98/15.98
% 121.98/15.98 Lemma 75: multiply(X, multiply(commutator(Y, X), multiply(multiply(Y, Z), W))) = multiply(Y, multiply(X, multiply(Z, W))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(X, multiply(commutator(Y, X), multiply(multiply(Y, Z), W)))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(X, multiply(commutator(Y, X), multiply(Y, multiply(Z, W))))
% 121.98/15.98 = { by lemma 42 }
% 121.98/15.98 multiply(Y, multiply(X, multiply(Z, W)))
% 121.98/15.98
% 121.98/15.98 Lemma 76: multiply(X, multiply(multiply(commutator(Y, Z), commutator(W, Z)), commutator(commutator(Y, Z), X))) = multiply(commutator(Y, Z), multiply(X, commutator(W, Z))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(X, multiply(multiply(commutator(Y, Z), commutator(W, Z)), commutator(commutator(Y, Z), X)))
% 121.98/15.98 = { by lemma 61 }
% 121.98/15.98 multiply(X, multiply(commutator(W, Z), multiply(commutator(Y, Z), commutator(commutator(Y, Z), X))))
% 121.98/15.98 = { by lemma 29 R->L }
% 121.98/15.98 multiply(X, multiply(commutator(W, Z), multiply(commutator(commutator(Y, Z), X), commutator(Y, Z))))
% 121.98/15.98 = { by lemma 74 R->L }
% 121.98/15.98 multiply(X, multiply(commutator(commutator(Y, Z), X), multiply(multiply(commutator(Y, Z), commutator(W, Z)), commutator(commutator(W, Z), commutator(commutator(Y, Z), X)))))
% 121.98/15.98 = { by lemma 75 }
% 121.98/15.98 multiply(commutator(Y, Z), multiply(X, multiply(commutator(W, Z), commutator(commutator(W, Z), commutator(commutator(Y, Z), X)))))
% 121.98/15.98 = { by lemma 72 }
% 121.98/15.98 multiply(commutator(Y, Z), multiply(X, multiply(commutator(W, Z), commutator(V, V))))
% 121.98/15.98 = { by lemma 20 }
% 121.98/15.98 multiply(commutator(Y, Z), multiply(X, commutator(W, Z)))
% 121.98/15.98
% 121.98/15.98 Lemma 77: multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)) = commutator(multiply(X, Z), Y).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z))
% 121.98/15.98 = { by lemma 32 R->L }
% 121.98/15.98 fresh(multiply(Y, multiply(multiply(X, Z), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 fresh(multiply(Y, multiply(X, multiply(Z, multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by lemma 76 }
% 121.98/15.98 fresh(multiply(Y, multiply(X, multiply(commutator(X, Y), multiply(Z, commutator(Z, Y))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by lemma 54 R->L }
% 121.98/15.98 fresh(multiply(Y, multiply(commutator(X, Y), multiply(X, multiply(Z, commutator(Z, Y))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y)))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by lemma 50 }
% 121.98/15.98 fresh(multiply(Y, multiply(multiply(X, Z), commutator(multiply(X, Z), Y))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by axiom 5 (commutator) R->L }
% 121.98/15.98 fresh(multiply(multiply(X, Z), Y), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(commutator(X, Y), Z)), commutator(multiply(X, Z), Y))
% 121.98/15.98 = { by axiom 1 (left_cancellation) }
% 121.98/15.98 commutator(multiply(X, Z), Y)
% 121.98/15.98
% 121.98/15.98 Lemma 78: multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), W)) = multiply(commutator(multiply(X, Z), Y), W).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), W))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 multiply(multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(commutator(X, Y), commutator(Z, Y))), W)
% 121.98/15.98 = { by lemma 49 }
% 121.98/15.98 multiply(commutator(multiply(X, Z), Y), W)
% 121.98/15.98
% 121.98/15.98 Lemma 79: multiply(multiply(commutator(Y, Z), commutator(X, Z)), X) = multiply(X, commutator(multiply(Y, X), Z)).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(multiply(commutator(Y, Z), commutator(X, Z)), X)
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(commutator(Y, Z), multiply(commutator(X, Z), X))
% 121.98/15.98 = { by lemma 29 }
% 121.98/15.98 multiply(commutator(Y, Z), multiply(X, commutator(X, Z)))
% 121.98/15.98 = { by lemma 76 R->L }
% 121.98/15.98 multiply(X, multiply(multiply(commutator(Y, Z), commutator(X, Z)), commutator(commutator(Y, Z), X)))
% 121.98/15.98 = { by lemma 77 }
% 121.98/15.98 multiply(X, commutator(multiply(Y, X), Z))
% 121.98/15.98
% 121.98/15.98 Lemma 80: multiply(multiply(commutator(Y, Z), commutator(X, Z)), multiply(X, W)) = multiply(X, multiply(commutator(multiply(Y, X), Z), W)).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(multiply(commutator(Y, Z), commutator(X, Z)), multiply(X, W))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 multiply(multiply(multiply(commutator(Y, Z), commutator(X, Z)), X), W)
% 121.98/15.98 = { by lemma 79 }
% 121.98/15.98 multiply(multiply(X, commutator(multiply(Y, X), Z)), W)
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(X, multiply(commutator(multiply(Y, X), Z), W))
% 121.98/15.98
% 121.98/15.98 Lemma 81: multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)) = multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y))
% 121.98/15.98 = { by axiom 5 (commutator) }
% 121.98/15.98 multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(Y, multiply(X, commutator(X, Y))))
% 121.98/15.98 = { by lemma 80 }
% 121.98/15.98 multiply(Y, multiply(commutator(multiply(X, Y), Z), multiply(X, commutator(X, Y))))
% 121.98/15.98 = { by lemma 73 }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, commutator(multiply(X, Y), Z)), multiply(X, Y)))
% 121.98/15.98 = { by lemma 60 }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, commutator(X, Z)), multiply(X, Y)))
% 121.98/15.98 = { by lemma 22 R->L }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, commutator(X, Z)), multiply(X, multiply(commutator(S, S), Y))))
% 121.98/15.98 = { by lemma 59 R->L }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, commutator(X, Z)), multiply(X, multiply(commutator(X, commutator(Y, commutator(X, Z))), Y))))
% 121.98/15.98 = { by lemma 10 }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(X, multiply(commutator(Y, commutator(X, Z)), Y)))
% 121.98/15.98 = { by lemma 60 R->L }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(X, multiply(commutator(Y, commutator(multiply(X, Y), Z)), Y)))
% 121.98/15.98 = { by lemma 29 }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(X, multiply(Y, commutator(Y, commutator(multiply(X, Y), Z)))))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 multiply(commutator(multiply(X, Y), Z), multiply(multiply(X, Y), commutator(Y, commutator(multiply(X, Y), Z))))
% 121.98/15.98 = { by lemma 54 }
% 121.98/15.98 multiply(multiply(X, Y), multiply(commutator(multiply(X, Y), Z), commutator(Y, commutator(multiply(X, Y), Z))))
% 121.98/15.98 = { by lemma 60 }
% 121.98/15.98 multiply(multiply(X, Y), multiply(commutator(multiply(X, Y), Z), commutator(Y, commutator(X, Z))))
% 121.98/15.98 = { by lemma 51 R->L }
% 121.98/15.98 multiply(commutator(X, Z), multiply(multiply(X, Y), multiply(commutator(Y, Z), commutator(Y, commutator(X, Z)))))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(commutator(X, Z), multiply(X, multiply(Y, multiply(commutator(Y, Z), commutator(Y, commutator(X, Z))))))
% 121.98/15.98 = { by lemma 54 }
% 121.98/15.98 multiply(X, multiply(commutator(X, Z), multiply(Y, multiply(commutator(Y, Z), commutator(Y, commutator(X, Z))))))
% 121.98/15.98 = { by lemma 70 R->L }
% 121.98/15.98 multiply(X, multiply(Y, multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(commutator(Y, Z), commutator(Y, commutator(X, Z))))))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(commutator(Y, Z), commutator(Y, commutator(X, Z)))))
% 121.98/15.98 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(commutator(commutator(Y, Z), Y), commutator(X, Z))))
% 121.98/15.98 = { by lemma 53 }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(commutator(T, T), commutator(X, Z))))
% 121.98/15.98 = { by lemma 44 R->L }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(V, commutator(commutator(V, U), commutator(X, Z)))))
% 121.98/15.98 = { by lemma 45 }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(W, W)))
% 121.98/15.98 = { by lemma 20 }
% 121.98/15.98 multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z)))
% 121.98/15.98
% 121.98/15.98 Lemma 82: multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(multiply(X, Y), W)) = multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), W)).
% 121.98/15.98 Proof:
% 121.98/15.98 multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(multiply(X, Y), W))
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.98 multiply(multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), W)
% 121.98/15.98 = { by lemma 81 }
% 121.98/15.98 multiply(multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))), W)
% 121.98/15.98 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.98 multiply(multiply(X, Y), multiply(multiply(commutator(X, Z), commutator(Y, Z)), W))
% 121.98/15.98
% 121.98/15.98 Lemma 83: multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))) = multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)).
% 121.98/15.98 Proof:
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))
% 121.98/15.99 = { by lemma 82 }
% 121.98/15.99 multiply(multiply(X, Z), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))
% 121.98/15.99 = { by axiom 5 (commutator) R->L }
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))
% 121.98/15.99
% 121.98/15.99 Lemma 84: commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)) = commutator(W, W).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))
% 121.98/15.99 = { by lemma 32 R->L }
% 121.98/15.99 fresh(multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 54 R->L }
% 121.98/15.99 fresh(multiply(commutator(multiply(X, Z), Y), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 78 R->L }
% 121.98/15.99 fresh(multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 83 }
% 121.98/15.99 fresh(multiply(commutator(commutator(X, Y), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 78 }
% 121.98/15.99 fresh(multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 29 }
% 121.98/15.99 fresh(multiply(multiply(X, Z), commutator(multiply(X, Z), Y)), multiply(commutator(multiply(X, Z), Y), multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by lemma 29 }
% 121.98/15.99 fresh(multiply(multiply(X, Z), commutator(multiply(X, Z), Y)), multiply(multiply(X, Z), commutator(multiply(X, Z), Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), commutator(commutator(multiply(X, Z), Y), multiply(X, Z)))
% 121.98/15.99 = { by axiom 1 (left_cancellation) }
% 121.98/15.99 commutator(commutator(multiply(X, Z), Y), multiply(X, Z))
% 121.98/15.99 = { by lemma 53 }
% 121.98/15.99 commutator(W, W)
% 121.98/15.99
% 121.98/15.99 Lemma 85: multiply(multiply(X, Y), multiply(Z, multiply(commutator(Z, Y), W))) = multiply(X, multiply(Z, multiply(Y, W))).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(multiply(X, Y), multiply(Z, multiply(commutator(Z, Y), W)))
% 121.98/15.99 = { by axiom 4 (associativity_of_multiply) }
% 121.98/15.99 multiply(X, multiply(Y, multiply(Z, multiply(commutator(Z, Y), W))))
% 121.98/15.99 = { by lemma 10 }
% 121.98/15.99 multiply(X, multiply(Z, multiply(Y, W)))
% 121.98/15.99
% 121.98/15.99 Lemma 86: multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), commutator(Z, Y))) = commutator(multiply(X, Z), Y).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), commutator(Z, Y)))
% 121.98/15.99 = { by lemma 20 R->L }
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(W, W)))
% 121.98/15.99 = { by lemma 84 R->L }
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))
% 121.98/15.99 = { by lemma 32 R->L }
% 121.98/15.99 fresh(multiply(Y, multiply(multiply(X, Z), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 69 R->L }
% 121.98/15.99 fresh(multiply(Y, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 85 }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(commutator(Z, Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(commutator(commutator(Z, Y), multiply(commutator(X, Y), commutator(Z, Y))), multiply(X, Z))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 62 }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(commutator(V, V), multiply(X, Z))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 44 R->L }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(U, commutator(commutator(U, T), multiply(X, Z)))))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 45 }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(S, S)))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 36 }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(X2, X2))))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 20 }
% 121.98/15.99 fresh(multiply(Y, multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y)))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 43 R->L }
% 121.98/15.99 fresh(multiply(commutator(X, Y), multiply(Y, multiply(multiply(X, Z), commutator(Z, Y)))), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by lemma 48 }
% 121.98/15.99 fresh(multiply(multiply(X, Z), Y), multiply(multiply(X, Z), Y), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)))), commutator(multiply(X, Z), Y))
% 121.98/15.99 = { by axiom 1 (left_cancellation) }
% 121.98/15.99 commutator(multiply(X, Z), Y)
% 121.98/15.99
% 121.98/15.99 Lemma 87: fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), W), commutator(multiply(X, Z), Y), W, commutator(multiply(X, Z), commutator(Z, Y))) = W.
% 121.98/15.99 Proof:
% 121.98/15.99 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), W), commutator(multiply(X, Z), Y), W, commutator(multiply(X, Z), commutator(Z, Y)))
% 121.98/15.99 = { by lemma 86 R->L }
% 121.98/15.99 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), commutator(Z, Y))), W, commutator(multiply(X, Z), commutator(Z, Y)))
% 121.98/15.99 = { by axiom 6 (left_cancellation) }
% 121.98/15.99 W
% 121.98/15.99
% 121.98/15.99 Lemma 88: multiply(commutator(X, Y), multiply(Z, multiply(commutator(commutator(Z, X), Y), W))) = multiply(Z, multiply(commutator(X, Y), W)).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(X, Y), multiply(Z, multiply(commutator(commutator(Z, X), Y), W)))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 multiply(commutator(X, Y), multiply(Z, multiply(commutator(Z, commutator(X, Y)), W)))
% 121.98/15.99 = { by lemma 10 }
% 121.98/15.99 multiply(Z, multiply(commutator(X, Y), W))
% 121.98/15.99
% 121.98/15.99 Lemma 89: multiply(commutator(X, Y), multiply(multiply(Z, Y), commutator(multiply(Z, Y), commutator(Y, X)))) = multiply(multiply(Z, Y), commutator(X, Y)).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(X, Y), multiply(multiply(Z, Y), commutator(multiply(Z, Y), commutator(Y, X))))
% 121.98/15.99 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.99 multiply(commutator(X, Y), multiply(multiply(Z, Y), fresh(commutator(multiply(Z, Y), X), commutator(multiply(Z, Y), X), commutator(commutator(Z, X), Y), commutator(multiply(Z, Y), commutator(Y, X)))))
% 121.98/15.99 = { by lemma 77 R->L }
% 121.98/15.99 multiply(commutator(X, Y), multiply(multiply(Z, Y), fresh(multiply(multiply(commutator(Z, X), commutator(Y, X)), commutator(commutator(Z, X), Y)), commutator(multiply(Z, Y), X), commutator(commutator(Z, X), Y), commutator(multiply(Z, Y), commutator(Y, X)))))
% 121.98/15.99 = { by lemma 87 }
% 121.98/15.99 multiply(commutator(X, Y), multiply(multiply(Z, Y), commutator(commutator(Z, X), Y)))
% 121.98/15.99 = { by lemma 38 }
% 121.98/15.99 multiply(commutator(X, Y), multiply(Z, multiply(commutator(commutator(Z, X), Y), Y)))
% 121.98/15.99 = { by lemma 88 }
% 121.98/15.99 multiply(Z, multiply(commutator(X, Y), Y))
% 121.98/15.99 = { by lemma 38 R->L }
% 121.98/15.99 multiply(multiply(Z, Y), commutator(X, Y))
% 121.98/15.99
% 121.98/15.99 Lemma 90: commutator(commutator(multiply(X, Y), Z), Y) = commutator(multiply(X, Y), commutator(Y, Z)).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(commutator(multiply(X, Y), Z), Y)
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 commutator(multiply(X, Y), commutator(Z, Y))
% 121.98/15.99 = { by axiom 1 (left_cancellation) R->L }
% 121.98/15.99 fresh(multiply(multiply(X, Y), commutator(Z, Y)), multiply(multiply(X, Y), commutator(Z, Y)), commutator(multiply(X, Y), commutator(Y, Z)), commutator(multiply(X, Y), commutator(Z, Y)))
% 121.98/15.99 = { by lemma 89 R->L }
% 121.98/15.99 fresh(multiply(commutator(Z, Y), multiply(multiply(X, Y), commutator(multiply(X, Y), commutator(Y, Z)))), multiply(multiply(X, Y), commutator(Z, Y)), commutator(multiply(X, Y), commutator(Y, Z)), commutator(multiply(X, Y), commutator(Z, Y)))
% 121.98/15.99 = { by lemma 32 }
% 121.98/15.99 commutator(multiply(X, Y), commutator(Y, Z))
% 121.98/15.99
% 121.98/15.99 Lemma 91: multiply(commutator(X, Y), multiply(X, multiply(Z, W))) = multiply(X, multiply(multiply(commutator(X, Y), Z), W)).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(X, Y), multiply(X, multiply(Z, W)))
% 121.98/15.99 = { by lemma 75 R->L }
% 121.98/15.99 multiply(X, multiply(commutator(commutator(X, Y), X), multiply(multiply(commutator(X, Y), Z), W)))
% 121.98/15.99 = { by lemma 53 }
% 121.98/15.99 multiply(X, multiply(commutator(V, V), multiply(multiply(commutator(X, Y), Z), W)))
% 121.98/15.99 = { by lemma 22 }
% 121.98/15.99 multiply(X, multiply(multiply(commutator(X, Y), Z), W))
% 121.98/15.99
% 121.98/15.99 Lemma 92: multiply(commutator(X, Y), multiply(commutator(Y, X), Z)) = Z.
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(X, Y), multiply(commutator(Y, X), Z))
% 121.98/15.99 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Y, X)), Z)
% 121.98/15.99 = { by lemma 56 }
% 121.98/15.99 multiply(commutator(W, W), Z)
% 121.98/15.99 = { by lemma 22 }
% 121.98/15.99 Z
% 121.98/15.99
% 121.98/15.99 Lemma 93: multiply(commutator(Z, Y), multiply(commutator(X, Y), W)) = multiply(commutator(X, Y), multiply(commutator(Z, Y), W)).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(Z, Y), multiply(commutator(X, Y), W))
% 121.98/15.99 = { by lemma 10 R->L }
% 121.98/15.99 multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(commutator(Z, Y), commutator(X, Y)), W)))
% 121.98/15.99 = { by lemma 57 }
% 121.98/15.99 multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(V, V), W)))
% 121.98/15.99 = { by lemma 22 }
% 121.98/15.99 multiply(commutator(X, Y), multiply(commutator(Z, Y), W))
% 121.98/15.99
% 121.98/15.99 Lemma 94: commutator(X, commutator(Y, commutator(commutator(X, Z), W))) = commutator(V, V).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(X, commutator(Y, commutator(commutator(X, Z), W)))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 commutator(X, commutator(Y, commutator(X, commutator(Z, W))))
% 121.98/15.99 = { by lemma 59 }
% 121.98/15.99 commutator(V, V)
% 121.98/15.99
% 121.98/15.99 Lemma 95: multiply(commutator(X, commutator(commutator(Y, Z), W)), Y) = multiply(Y, commutator(X, commutator(commutator(Y, Z), W))).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(X, commutator(commutator(Y, Z), W)), Y)
% 121.98/15.99 = { by lemma 20 R->L }
% 121.98/15.99 multiply(commutator(X, commutator(commutator(Y, Z), W)), multiply(Y, commutator(V, V)))
% 121.98/15.99 = { by lemma 94 R->L }
% 121.98/15.99 multiply(commutator(X, commutator(commutator(Y, Z), W)), multiply(Y, commutator(Y, commutator(X, commutator(commutator(Y, Z), W)))))
% 121.98/15.99 = { by axiom 5 (commutator) R->L }
% 121.98/15.99 multiply(Y, commutator(X, commutator(commutator(Y, Z), W)))
% 121.98/15.99
% 121.98/15.99 Lemma 96: multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), W)) = multiply(commutator(multiply(X, Z), Y), W).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), W))
% 121.98/15.99 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/15.99 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), commutator(Z, Y))), W)
% 121.98/15.99 = { by lemma 86 }
% 121.98/15.99 multiply(commutator(multiply(X, Z), Y), W)
% 121.98/15.99
% 121.98/15.99 Lemma 97: commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)) = commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y))
% 121.98/15.99 = { by lemma 32 R->L }
% 121.98/15.99 fresh(multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(multiply(X, Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)))), multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))))
% 121.98/15.99 = { by lemma 83 }
% 121.98/15.99 fresh(multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))))
% 121.98/15.99 = { by lemma 81 R->L }
% 121.98/15.99 fresh(multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))))
% 121.98/15.99 = { by axiom 1 (left_cancellation) }
% 121.98/15.99 commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z)))
% 121.98/15.99
% 121.98/15.99 Lemma 98: commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(X, Y), W)) = commutator(multiply(X, Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), W)).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(X, Y), W))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.99 commutator(commutator(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), W)
% 121.98/15.99 = { by lemma 97 }
% 121.98/15.99 commutator(commutator(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))), W)
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 commutator(multiply(X, Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), W))
% 121.98/15.99
% 121.98/15.99 Lemma 99: commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), W)) = commutator(V, V).
% 121.98/15.99 Proof:
% 121.98/15.99 commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), W))
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) R->L }
% 121.98/15.99 commutator(commutator(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), W)
% 121.98/15.99 = { by lemma 84 }
% 121.98/15.99 commutator(commutator(U, U), W)
% 121.98/15.99 = { by lemma 63 R->L }
% 121.98/15.99 commutator(commutator(multiply(commutator(T, S), commutator(X2, S)), commutator(X2, S)), W)
% 121.98/15.99 = { by axiom 3 (associativity_of_commutator) }
% 121.98/15.99 commutator(multiply(commutator(T, S), commutator(X2, S)), commutator(commutator(X2, S), W))
% 121.98/15.99 = { by lemma 65 }
% 121.98/15.99 commutator(V, V)
% 121.98/15.99
% 121.98/15.99 Lemma 100: multiply(commutator(multiply(X, Y), Z), multiply(Y, commutator(Y, multiply(commutator(X, Z), commutator(Y, Z))))) = multiply(multiply(commutator(X, Z), commutator(Y, Z)), Y).
% 121.98/15.99 Proof:
% 121.98/15.99 multiply(commutator(multiply(X, Y), Z), multiply(Y, commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/15.99 = { by lemma 96 R->L }
% 121.98/15.99 multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(Y, commutator(Y, multiply(commutator(X, Z), commutator(Y, Z))))))
% 121.98/15.99 = { by lemma 73 }
% 121.98/15.99 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(X, Y), commutator(Y, Z))), multiply(Y, multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 98 }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(commutator(multiply(X, Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(Y, Z))), multiply(Y, multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 63 }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(commutator(multiply(X, Y), commutator(W, W)), multiply(Y, multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 36 }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(commutator(V, V), multiply(Y, multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 22 }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(Y, multiply(commutator(X, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 22 R->L }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(Y, multiply(commutator(U, U), multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 59 R->L }
% 121.98/16.00 multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(Y, multiply(commutator(Y, commutator(multiply(X, Y), commutator(Y, Z))), multiply(commutator(X, Z), commutator(Y, Z)))))
% 121.98/16.00 = { by lemma 10 }
% 121.98/16.00 multiply(Y, multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(commutator(X, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 20 R->L }
% 121.98/16.00 multiply(Y, multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(T, T))))
% 121.98/16.00 = { by lemma 99 R->L }
% 121.98/16.00 multiply(Y, multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(X, Y), commutator(Y, Z))))))
% 121.98/16.00 = { by axiom 5 (commutator) R->L }
% 121.98/16.00 multiply(Y, multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(X, Y), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 86 }
% 121.98/16.00 multiply(Y, commutator(multiply(X, Y), Z))
% 121.98/16.00 = { by lemma 79 R->L }
% 121.98/16.00 multiply(multiply(commutator(X, Z), commutator(Y, Z)), Y)
% 121.98/16.00
% 121.98/16.00 Lemma 101: commutator(X, multiply(commutator(Y, Z), commutator(X, Z))) = commutator(X, commutator(multiply(Y, X), Z)).
% 121.98/16.00 Proof:
% 121.98/16.00 commutator(X, multiply(commutator(Y, Z), commutator(X, Z)))
% 121.98/16.00 = { by lemma 32 R->L }
% 121.98/16.00 fresh(multiply(commutator(multiply(Y, X), Z), multiply(X, commutator(X, multiply(commutator(Y, Z), commutator(X, Z))))), multiply(X, commutator(multiply(Y, X), Z)), commutator(X, multiply(commutator(Y, Z), commutator(X, Z))), commutator(X, commutator(multiply(Y, X), Z)))
% 121.98/16.00 = { by lemma 100 }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, Z), commutator(X, Z)), X), multiply(X, commutator(multiply(Y, X), Z)), commutator(X, multiply(commutator(Y, Z), commutator(X, Z))), commutator(X, commutator(multiply(Y, X), Z)))
% 121.98/16.00 = { by lemma 79 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, Z), commutator(X, Z)), X), multiply(multiply(commutator(Y, Z), commutator(X, Z)), X), commutator(X, multiply(commutator(Y, Z), commutator(X, Z))), commutator(X, commutator(multiply(Y, X), Z)))
% 121.98/16.00 = { by axiom 1 (left_cancellation) }
% 121.98/16.00 commutator(X, commutator(multiply(Y, X), Z))
% 121.98/16.00
% 121.98/16.00 Lemma 102: commutator(multiply(commutator(Y, X), commutator(Z, X)), X) = commutator(X, multiply(commutator(Y, X), commutator(Z, X))).
% 121.98/16.00 Proof:
% 121.98/16.00 commutator(multiply(commutator(Y, X), commutator(Z, X)), X)
% 121.98/16.00 = { by lemma 32 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(multiply(commutator(Y, X), commutator(Z, X)), X))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by axiom 4 (associativity_of_multiply) }
% 121.98/16.00 fresh(multiply(commutator(Y, X), multiply(commutator(Z, X), multiply(X, commutator(multiply(commutator(Y, X), commutator(Z, X)), X)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 43 }
% 121.98/16.00 fresh(multiply(commutator(Y, X), multiply(X, multiply(commutator(Z, X), commutator(multiply(commutator(Y, X), commutator(Z, X)), X)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 43 }
% 121.98/16.00 fresh(multiply(X, multiply(commutator(Y, X), multiply(commutator(Z, X), commutator(multiply(commutator(Y, X), commutator(Z, X)), X)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 58 }
% 121.98/16.00 fresh(multiply(X, multiply(commutator(Y, X), multiply(commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(Z, X)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 40 R->L }
% 121.98/16.00 fresh(multiply(X, multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(Z, X))))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 10 }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(Z, X)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by axiom 3 (associativity_of_commutator) }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(multiply(commutator(Y, X), commutator(Z, X)), commutator(X, commutator(Z, X))))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 35 }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(multiply(commutator(Y, X), commutator(Z, X)), commutator(V, V)))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 36 }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(W, W))), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 20 }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), X), multiply(X, multiply(commutator(Y, X), commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 43 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), X), multiply(commutator(Y, X), multiply(X, commutator(Z, X))), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by lemma 30 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), X), multiply(commutator(Y, X), multiply(commutator(Z, X), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by axiom 4 (associativity_of_multiply) R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(Y, X), commutator(Z, X)), X), multiply(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, multiply(commutator(Y, X), commutator(Z, X))))
% 121.98/16.00 = { by axiom 1 (left_cancellation) }
% 121.98/16.00 commutator(X, multiply(commutator(Y, X), commutator(Z, X)))
% 121.98/16.00
% 121.98/16.00 Lemma 103: commutator(multiply(commutator(Y, X), commutator(Z, X)), X) = commutator(X, commutator(multiply(Y, Z), X)).
% 121.98/16.00 Proof:
% 121.98/16.00 commutator(multiply(commutator(Y, X), commutator(Z, X)), X)
% 121.98/16.00 = { by lemma 32 R->L }
% 121.98/16.00 fresh(multiply(commutator(multiply(Y, Z), X), multiply(X, commutator(multiply(commutator(Y, X), commutator(Z, X)), X))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 102 }
% 121.98/16.00 fresh(multiply(commutator(multiply(Y, Z), X), multiply(X, commutator(X, multiply(commutator(Y, X), commutator(Z, X))))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 78 R->L }
% 121.98/16.00 fresh(multiply(commutator(commutator(Y, X), multiply(Y, Z)), multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, commutator(X, multiply(commutator(Y, X), commutator(Z, X)))))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by axiom 5 (commutator) R->L }
% 121.98/16.00 fresh(multiply(commutator(commutator(Y, X), multiply(Y, Z)), multiply(X, multiply(commutator(Y, X), commutator(Z, X)))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 22 R->L }
% 121.98/16.00 fresh(multiply(commutator(commutator(Y, X), multiply(Y, Z)), multiply(X, multiply(commutator(W, W), multiply(commutator(Y, X), commutator(Z, X))))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 66 R->L }
% 121.98/16.00 fresh(multiply(commutator(commutator(Y, X), multiply(Y, Z)), multiply(X, multiply(commutator(X, commutator(commutator(Y, X), multiply(Y, Z))), multiply(commutator(Y, X), commutator(Z, X))))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 10 }
% 121.98/16.00 fresh(multiply(X, multiply(commutator(commutator(Y, X), multiply(Y, Z)), multiply(commutator(Y, X), commutator(Z, X)))), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 49 }
% 121.98/16.00 fresh(multiply(X, commutator(multiply(Y, Z), X)), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 30 R->L }
% 121.98/16.00 fresh(multiply(commutator(multiply(Y, Z), X), X), multiply(X, commutator(multiply(Y, Z), X)), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by lemma 30 R->L }
% 121.98/16.00 fresh(multiply(commutator(multiply(Y, Z), X), X), multiply(commutator(multiply(Y, Z), X), X), commutator(multiply(commutator(Y, X), commutator(Z, X)), X), commutator(X, commutator(multiply(Y, Z), X)))
% 121.98/16.00 = { by axiom 1 (left_cancellation) }
% 121.98/16.00 commutator(X, commutator(multiply(Y, Z), X))
% 121.98/16.00
% 121.98/16.00 Lemma 104: commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))) = commutator(V, V).
% 121.98/16.00 Proof:
% 121.98/16.00 commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z)))
% 121.98/16.00 = { by axiom 1 (left_cancellation) R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 61 }
% 121.98/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), multiply(commutator(Y, Z), multiply(commutator(W, Z), commutator(multiply(X, Y), Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 58 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), multiply(commutator(Y, Z), multiply(commutator(multiply(X, Y), Z), commutator(W, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 93 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, Z), commutator(W, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 58 R->L }
% 121.98/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(multiply(X, Y), Z)), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by axiom 4 (associativity_of_multiply) }
% 121.98/16.00 fresh(multiply(commutator(W, Z), multiply(commutator(Y, Z), commutator(multiply(X, Y), Z))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 28 R->L }
% 121.98/16.00 fresh(multiply(commutator(W, Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(Y, Z)))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 93 R->L }
% 121.98/16.00 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), multiply(commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(Y, Z)))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by lemma 29 }
% 121.98/16.00 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), multiply(commutator(Y, Z), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z))))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 121.98/16.00 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.00 fresh(multiply(commutator(multiply(X, Y), Z), multiply(multiply(commutator(W, Z), commutator(Y, Z)), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 122.39/16.00 = { by lemma 61 }
% 122.39/16.00 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, Z), multiply(commutator(W, Z), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z))))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 122.39/16.00 = { by lemma 93 }
% 122.39/16.00 fresh(multiply(commutator(Y, Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z))))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 122.39/16.00 = { by lemma 93 }
% 122.39/16.00 fresh(multiply(commutator(Y, Z), multiply(commutator(W, Z), multiply(commutator(multiply(X, Y), Z), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z))))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 122.39/16.00 = { by lemma 61 R->L }
% 122.39/16.00 fresh(multiply(multiply(commutator(W, Z), commutator(Y, Z)), multiply(commutator(multiply(X, Y), Z), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)))), multiply(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))), commutator(commutator(Y, Z), commutator(multiply(X, Y), Z)), commutator(commutator(multiply(X, Y), Z), multiply(commutator(W, Z), commutator(Y, Z))))
% 122.39/16.01 = { by lemma 32 }
% 122.39/16.01 commutator(commutator(Y, Z), commutator(multiply(X, Y), Z))
% 122.39/16.01 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.01 commutator(Y, commutator(Z, commutator(multiply(X, Y), Z)))
% 122.39/16.01 = { by lemma 103 R->L }
% 122.39/16.01 commutator(Y, commutator(multiply(commutator(X, Z), commutator(Y, Z)), Z))
% 122.39/16.01 = { by lemma 102 }
% 122.39/16.01 commutator(Y, commutator(Z, multiply(commutator(X, Z), commutator(Y, Z))))
% 122.39/16.01 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.39/16.01 commutator(commutator(Y, Z), multiply(commutator(X, Z), commutator(Y, Z)))
% 122.39/16.01 = { by lemma 62 }
% 122.39/16.01 commutator(V, V)
% 122.39/16.01
% 122.39/16.01 Lemma 105: commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y) = commutator(W, W).
% 122.39/16.01 Proof:
% 122.39/16.01 commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y)
% 122.39/16.01 = { by lemma 17 R->L }
% 122.39/16.01 fresh2(multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), multiply(Y, multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 20 R->L }
% 122.39/16.01 fresh2(multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), multiply(Y, multiply(multiply(X, Z), commutator(V, V)))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 104 R->L }
% 122.39/16.01 fresh2(multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), multiply(Y, multiply(multiply(X, Z), commutator(commutator(multiply(X, Z), Y), multiply(commutator(X, Y), commutator(Z, Y)))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.01 fresh2(multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), multiply(Y, multiply(multiply(X, Z), commutator(multiply(X, Z), commutator(Y, multiply(commutator(X, Y), commutator(Z, Y))))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 102 R->L }
% 122.39/16.01 fresh2(multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), multiply(Y, multiply(multiply(X, Z), commutator(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 103 }
% 122.39/16.01 fresh2(multiply(commutator(Y, commutator(multiply(X, Z), Y)), multiply(Y, multiply(multiply(X, Z), commutator(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 103 }
% 122.39/16.01 fresh2(multiply(commutator(Y, commutator(multiply(X, Z), Y)), multiply(Y, multiply(multiply(X, Z), commutator(multiply(X, Z), commutator(Y, commutator(multiply(X, Z), Y)))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 73 }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y), multiply(multiply(X, Z), commutator(Y, commutator(multiply(X, Z), Y))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 42 R->L }
% 122.39/16.01 fresh2(multiply(commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y), multiply(commutator(Y, commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y)), multiply(Y, multiply(multiply(X, Z), commutator(Y, commutator(multiply(X, Z), Y)))))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 34 }
% 122.39/16.01 fresh2(multiply(commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y), multiply(commutator(Y, commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y)), multiply(multiply(X, Z), multiply(commutator(Y, multiply(X, Z)), Y)))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 28 }
% 122.39/16.01 fresh2(multiply(commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y), multiply(commutator(Y, commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y)), multiply(Y, multiply(X, Z)))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 42 }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(commutator(Y, commutator(multiply(X, Z), Y)), Y), multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 103 R->L }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), Y), multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Y, Y)), multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 19 R->L }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), commutator(U, U)), multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 36 }
% 122.39/16.01 fresh2(multiply(Y, multiply(commutator(T, T), multiply(X, Z))), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by lemma 22 }
% 122.39/16.01 fresh2(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Y), commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z))))
% 122.39/16.01 = { by axiom 2 (right_cancellation) }
% 122.39/16.01 commutator(multiply(Y, multiply(X, Z)), multiply(Y, multiply(X, Z)))
% 122.39/16.01 = { by lemma 18 }
% 122.39/16.01 commutator(Y, Y)
% 122.39/16.01 = { by lemma 19 R->L }
% 122.39/16.01 commutator(W, W)
% 122.39/16.01
% 122.39/16.01 Lemma 106: multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, W))) = multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), Y), W)).
% 122.39/16.01 Proof:
% 122.39/16.01 multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, W)))
% 122.39/16.01 = { by lemma 91 R->L }
% 122.39/16.01 multiply(commutator(X, Y), multiply(X, multiply(commutator(Z, Y), multiply(Z, W))))
% 122.39/16.01 = { by lemma 75 R->L }
% 122.39/16.01 multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(X, commutator(Z, Y)), multiply(multiply(X, Z), W))))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(X, commutator(Z, Y)), multiply(multiply(X, Z), W)))
% 122.39/16.01 = { by lemma 67 R->L }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(multiply(X, Z), W)))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z)), W))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.01 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z))), W)
% 122.39/16.01 = { by lemma 22 R->L }
% 122.39/16.01 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(V, V), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z)))), W)
% 122.39/16.01 = { by lemma 25 R->L }
% 122.39/16.01 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(multiply(X, Z), commutator(Z, Y))), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z)))), W)
% 122.39/16.01 = { by lemma 27 R->L }
% 122.39/16.01 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), commutator(Z, Y)), commutator(multiply(X, Z), commutator(commutator(multiply(X, Z), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z))))))), W)
% 122.39/16.01 = { by lemma 85 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(commutator(multiply(X, Z), commutator(Z, Y)), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z))))))), W)
% 122.39/16.01 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.01 multiply(multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(multiply(X, Z), commutator(multiply(X, Z), commutator(commutator(Z, Y), multiply(commutator(multiply(X, Z), commutator(Z, Y)), multiply(X, Z)))))))), W)
% 122.39/16.01 = { by lemma 25 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), commutator(U, U)))), W)
% 122.39/16.01 = { by lemma 20 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, Y))), W)
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.01 multiply(commutator(X, Y), multiply(multiply(multiply(X, Z), commutator(Z, Y)), W))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.01 multiply(commutator(X, Y), multiply(multiply(X, Z), multiply(commutator(Z, Y), W)))
% 122.39/16.01 = { by lemma 51 }
% 122.39/16.01 multiply(multiply(X, Z), multiply(commutator(multiply(X, Z), Y), W))
% 122.39/16.01
% 122.39/16.01 Lemma 107: fresh(multiply(multiply(X, Y), Z), multiply(multiply(commutator(X, W), commutator(Y, W)), multiply(X, Y)), Z, multiply(commutator(X, W), commutator(Y, W))) = Z.
% 122.39/16.01 Proof:
% 122.39/16.01 fresh(multiply(multiply(X, Y), Z), multiply(multiply(commutator(X, W), commutator(Y, W)), multiply(X, Y)), Z, multiply(commutator(X, W), commutator(Y, W)))
% 122.39/16.01 = { by lemma 81 }
% 122.39/16.01 fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(commutator(X, W), commutator(Y, W))), Z, multiply(commutator(X, W), commutator(Y, W)))
% 122.39/16.01 = { by axiom 6 (left_cancellation) }
% 122.39/16.01 Z
% 122.39/16.01
% 122.39/16.01 Lemma 108: commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z) = commutator(commutator(X, Y), multiply(Z, X)).
% 122.39/16.01 Proof:
% 122.39/16.01 commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z)
% 122.39/16.01 = { by axiom 1 (left_cancellation) R->L }
% 122.39/16.01 fresh(multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), commutator(commutator(X, Y), multiply(Z, X)), commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z))
% 122.39/16.01 = { by lemma 61 }
% 122.39/16.01 fresh(multiply(commutator(X, Y), multiply(commutator(Z, Y), Z)), multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), commutator(commutator(X, Y), multiply(Z, X)), commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z))
% 122.39/16.01 = { by lemma 29 }
% 122.39/16.01 fresh(multiply(commutator(X, Y), multiply(Z, commutator(Z, Y))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), commutator(commutator(X, Y), multiply(Z, X)), commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z))
% 122.39/16.01 = { by lemma 74 R->L }
% 122.39/16.01 fresh(multiply(Z, multiply(multiply(commutator(Z, Y), commutator(X, Y)), commutator(commutator(X, Y), Z))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), commutator(commutator(X, Y), multiply(Z, X)), commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z))
% 122.39/16.01 = { by lemma 71 R->L }
% 122.39/16.01 fresh(multiply(Z, multiply(multiply(commutator(Z, Y), commutator(X, Y)), commutator(commutator(X, Y), multiply(Z, X)))), multiply(multiply(commutator(Z, Y), commutator(X, Y)), Z), commutator(commutator(X, Y), multiply(Z, X)), commutator(multiply(commutator(Z, Y), commutator(X, Y)), Z))
% 122.39/16.01 = { by lemma 32 }
% 122.39/16.01 commutator(commutator(X, Y), multiply(Z, X))
% 122.39/16.01
% 122.39/16.01 Lemma 109: multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(Z, commutator(commutator(Z, W), multiply(X, Z))))) = multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(X, Z)).
% 122.39/16.01 Proof:
% 122.39/16.01 multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(Z, commutator(commutator(Z, W), multiply(X, Z)))))
% 122.39/16.01 = { by lemma 71 }
% 122.39/16.01 multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(Z, commutator(commutator(Z, W), X))))
% 122.39/16.01 = { by lemma 91 R->L }
% 122.39/16.01 multiply(commutator(X, Y), multiply(X, multiply(commutator(Z, W), multiply(Z, commutator(commutator(Z, W), X)))))
% 122.39/16.01 = { by lemma 70 R->L }
% 122.39/16.01 multiply(commutator(X, Y), multiply(commutator(Z, W), multiply(multiply(X, Z), commutator(Z, commutator(commutator(Z, W), X)))))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(multiply(X, Z), commutator(Z, commutator(commutator(Z, W), X))))
% 122.39/16.01 = { by lemma 45 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(multiply(X, Z), commutator(V, V)))
% 122.39/16.01 = { by lemma 20 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, W)), multiply(X, Z))
% 122.39/16.01
% 122.39/16.01 Lemma 110: multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(multiply(commutator(X, Y), commutator(Z, Y)), X))) = multiply(Z, multiply(commutator(X, Y), commutator(Z, Y))).
% 122.39/16.01 Proof:
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(multiply(commutator(X, Y), commutator(Z, Y)), X)))
% 122.39/16.01 = { by lemma 108 }
% 122.39/16.01 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z))))
% 122.39/16.01 = { by axiom 6 (left_cancellation) R->L }
% 122.39/16.01 fresh(multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z))))), multiply(X, multiply(Z, multiply(commutator(X, Y), commutator(Z, Y)))), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z)))), multiply(Z, multiply(commutator(X, Y), commutator(Z, Y))))
% 122.39/16.01 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.01 fresh(multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z))))), multiply(multiply(X, Z), multiply(commutator(X, Y), commutator(Z, Y))), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z)))), multiply(Z, multiply(commutator(X, Y), commutator(Z, Y))))
% 122.39/16.01 = { by lemma 81 R->L }
% 122.39/16.01 fresh(multiply(X, multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z))))), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z)))), multiply(Z, multiply(commutator(X, Y), commutator(Z, Y))))
% 122.39/16.01 = { by lemma 109 }
% 122.39/16.01 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(X, Z)), multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(Z, commutator(commutator(Z, Y), multiply(X, Z)))), multiply(Z, multiply(commutator(X, Y), commutator(Z, Y))))
% 122.39/16.01 = { by axiom 1 (left_cancellation) }
% 122.39/16.01 multiply(Z, multiply(commutator(X, Y), commutator(Z, Y)))
% 122.39/16.01
% 122.39/16.01 Lemma 111: commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y) = commutator(X, multiply(commutator(Y, Z), commutator(X, Z))).
% 122.39/16.01 Proof:
% 122.39/16.01 commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y)
% 122.39/16.01 = { by lemma 32 R->L }
% 122.39/16.01 fresh(multiply(multiply(commutator(Y, Z), commutator(X, Z)), multiply(X, commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y))), multiply(X, multiply(commutator(Y, Z), commutator(X, Z))), commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y), commutator(X, multiply(commutator(Y, Z), commutator(X, Z))))
% 122.39/16.01 = { by lemma 110 }
% 122.39/16.02 fresh(multiply(X, multiply(commutator(Y, Z), commutator(X, Z))), multiply(X, multiply(commutator(Y, Z), commutator(X, Z))), commutator(multiply(commutator(Y, Z), commutator(X, Z)), Y), commutator(X, multiply(commutator(Y, Z), commutator(X, Z))))
% 122.39/16.02 = { by axiom 1 (left_cancellation) }
% 122.39/16.02 commutator(X, multiply(commutator(Y, Z), commutator(X, Z)))
% 122.39/16.02
% 122.39/16.02 Lemma 112: multiply(multiply(Y, Z), multiply(X, multiply(commutator(multiply(Y, Z), commutator(Z, X)), W))) = multiply(X, multiply(Y, multiply(commutator(multiply(Y, Z), X), multiply(Z, W)))).
% 122.39/16.02 Proof:
% 122.39/16.02 multiply(multiply(Y, Z), multiply(X, multiply(commutator(multiply(Y, Z), commutator(Z, X)), W)))
% 122.39/16.02 = { by lemma 90 R->L }
% 122.39/16.02 multiply(multiply(Y, Z), multiply(X, multiply(commutator(commutator(multiply(Y, Z), X), Z), W)))
% 122.39/16.02 = { by lemma 10 R->L }
% 122.39/16.02 multiply(X, multiply(multiply(Y, Z), multiply(commutator(multiply(Y, Z), X), multiply(commutator(commutator(multiply(Y, Z), X), Z), W))))
% 122.39/16.02 = { by lemma 85 }
% 122.39/16.02 multiply(X, multiply(Y, multiply(commutator(multiply(Y, Z), X), multiply(Z, W))))
% 122.39/16.02
% 122.39/16.02 Lemma 113: multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), commutator(multiply(V, Z), Y)) = multiply(commutator(multiply(V, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W)).
% 122.39/16.02 Proof:
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), commutator(multiply(V, Z), Y))
% 122.39/16.02 = { by lemma 20 R->L }
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(commutator(multiply(V, Z), Y), commutator(U, U)))
% 122.39/16.02 = { by lemma 65 R->L }
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(commutator(multiply(V, Z), Y), commutator(multiply(commutator(T, S), commutator(X2, S)), commutator(commutator(X2, S), W))))
% 122.39/16.02 = { by lemma 64 }
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(commutator(multiply(V, Z), Y), commutator(commutator(Y2, Y2), W)))
% 122.39/16.02 = { by lemma 104 R->L }
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(commutator(multiply(V, Z), Y), commutator(commutator(commutator(multiply(V, Z), Y), multiply(commutator(X, Y), commutator(Z, Y))), W)))
% 122.39/16.02 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.02 multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(commutator(multiply(V, Z), Y), commutator(commutator(multiply(V, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W))))
% 122.39/16.02 = { by axiom 5 (commutator) R->L }
% 122.39/16.02 multiply(commutator(multiply(V, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W))
% 122.39/16.02
% 122.39/16.02 Lemma 114: multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(W, commutator(multiply(X, Z), commutator(Z, Y)))) = multiply(W, multiply(commutator(multiply(X, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W))).
% 122.39/16.02 Proof:
% 122.39/16.02 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(W, commutator(multiply(X, Z), commutator(Z, Y))))
% 122.39/16.02 = { by lemma 42 R->L }
% 122.39/16.02 multiply(W, multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(multiply(X, Z), commutator(Z, Y)))))
% 122.39/16.02 = { by lemma 86 }
% 122.39/16.02 multiply(W, multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), commutator(multiply(X, Z), Y)))
% 122.39/16.02 = { by lemma 113 }
% 122.39/16.02 multiply(W, multiply(commutator(multiply(X, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W)))
% 122.39/16.02
% 122.39/16.02 Lemma 115: multiply(multiply(Y, Z), multiply(X, multiply(commutator(commutator(multiply(Y, Z), X), Z), W))) = multiply(X, multiply(Y, multiply(commutator(multiply(Y, Z), X), multiply(Z, W)))).
% 122.39/16.02 Proof:
% 122.39/16.02 multiply(multiply(Y, Z), multiply(X, multiply(commutator(commutator(multiply(Y, Z), X), Z), W)))
% 122.39/16.02 = { by lemma 10 R->L }
% 122.39/16.02 multiply(X, multiply(multiply(Y, Z), multiply(commutator(multiply(Y, Z), X), multiply(commutator(commutator(multiply(Y, Z), X), Z), W))))
% 122.39/16.02 = { by lemma 85 }
% 122.39/16.02 multiply(X, multiply(Y, multiply(commutator(multiply(Y, Z), X), multiply(Z, W))))
% 122.39/16.02
% 122.39/16.02 Lemma 116: multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), W))) = multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(X, Y), commutator(Y, Z)), W))).
% 122.39/16.02 Proof:
% 122.39/16.02 multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), W)))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.02 multiply(multiply(X, Y), multiply(multiply(Z, commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), W))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))), W)
% 122.39/16.02 = { by lemma 10 R->L }
% 122.39/16.02 multiply(multiply(Z, multiply(multiply(X, Y), multiply(commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)))), W)
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(Y, multiply(commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))))), W)
% 122.39/16.02 = { by lemma 114 R->L }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(Y, commutator(multiply(X, Y), commutator(Y, Z)))))), W)
% 122.39/16.02 = { by axiom 5 (commutator) }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(commutator(multiply(X, Y), commutator(Y, Z)), multiply(Y, commutator(Y, commutator(multiply(X, Y), commutator(Y, Z)))))))), W)
% 122.39/16.02 = { by lemma 96 }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(commutator(multiply(X, Y), Z), multiply(Y, commutator(Y, commutator(multiply(X, Y), commutator(Y, Z))))))), W)
% 122.39/16.02 = { by lemma 115 R->L }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(commutator(multiply(X, Y), Z), Y), commutator(Y, commutator(multiply(X, Y), commutator(Y, Z)))))), W)
% 122.39/16.02 = { by lemma 90 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(X, Y), commutator(Y, Z)), commutator(Y, commutator(multiply(X, Y), commutator(Y, Z)))))), W)
% 122.39/16.02 = { by lemma 59 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(X, Y), commutator(Y, Z)), commutator(T, T)))), W)
% 122.39/16.02 = { by lemma 20 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, commutator(multiply(X, Y), commutator(Y, Z)))), W)
% 122.39/16.02 = { by lemma 68 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, commutator(X, commutator(multiply(X, Y), Z)))), W)
% 122.39/16.02 = { by lemma 47 R->L }
% 122.39/16.02 multiply(multiply(commutator(multiply(X, Y), Z), multiply(Z, multiply(multiply(X, Y), commutator(X, commutator(multiply(X, Y), Z))))), W)
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 multiply(multiply(commutator(multiply(X, Y), Z), multiply(Z, multiply(X, multiply(Y, commutator(X, commutator(multiply(X, Y), Z)))))), W)
% 122.39/16.02 = { by lemma 43 }
% 122.39/16.02 multiply(multiply(Z, multiply(commutator(multiply(X, Y), Z), multiply(X, multiply(Y, commutator(X, commutator(multiply(X, Y), Z)))))), W)
% 122.39/16.02 = { by lemma 69 R->L }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(commutator(multiply(X, Y), Z), multiply(Y, commutator(Y, commutator(X, commutator(multiply(X, Y), Z))))))), W)
% 122.39/16.02 = { by lemma 115 R->L }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(commutator(multiply(X, Y), Z), Y), commutator(Y, commutator(X, commutator(multiply(X, Y), Z)))))), W)
% 122.39/16.02 = { by lemma 60 R->L }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(commutator(multiply(X, Y), Z), Y), commutator(Y, commutator(multiply(X, Y), commutator(multiply(X, Y), Z)))))), W)
% 122.39/16.02 = { by lemma 25 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(commutator(multiply(X, Y), Z), Y), commutator(Y, commutator(U, U))))), W)
% 122.39/16.02 = { by lemma 36 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, multiply(commutator(commutator(multiply(X, Y), Z), Y), commutator(V, V)))), W)
% 122.39/16.02 = { by lemma 20 }
% 122.39/16.02 multiply(multiply(multiply(X, Y), multiply(Z, commutator(commutator(multiply(X, Y), Z), Y))), W)
% 122.39/16.02 = { by lemma 10 R->L }
% 122.39/16.02 multiply(multiply(Z, multiply(multiply(X, Y), multiply(commutator(multiply(X, Y), Z), commutator(commutator(multiply(X, Y), Z), Y)))), W)
% 122.39/16.02 = { by lemma 40 }
% 122.39/16.02 multiply(multiply(Z, multiply(X, multiply(commutator(multiply(X, Y), Z), Y))), W)
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 multiply(Z, multiply(multiply(X, multiply(commutator(multiply(X, Y), Z), Y)), W))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 multiply(Z, multiply(X, multiply(multiply(commutator(multiply(X, Y), Z), Y), W)))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 multiply(Z, multiply(X, multiply(commutator(multiply(X, Y), Z), multiply(Y, W))))
% 122.39/16.02 = { by lemma 112 R->L }
% 122.39/16.02 multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(X, Y), commutator(Y, Z)), W)))
% 122.39/16.02
% 122.39/16.02 Lemma 117: commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y) = commutator(multiply(X, Y), commutator(Y, Z)).
% 122.39/16.02 Proof:
% 122.39/16.02 commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)
% 122.39/16.02 = { by lemma 87 R->L }
% 122.39/16.02 fresh(multiply(multiply(commutator(X, Z), commutator(Y, Z)), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by axiom 1 (left_cancellation) R->L }
% 122.39/16.02 fresh(multiply(fresh(multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 81 }
% 122.39/16.02 fresh(multiply(fresh(multiply(multiply(X, Y), multiply(commutator(X, Z), commutator(Y, Z))), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 fresh(multiply(fresh(multiply(X, multiply(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by axiom 5 (commutator) }
% 122.39/16.02 fresh(multiply(fresh(multiply(X, multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(Y, commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))))), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 106 }
% 122.39/16.02 fresh(multiply(fresh(multiply(multiply(X, Y), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z))))), multiply(multiply(commutator(X, Z), commutator(Y, Z)), multiply(X, Y)), multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 107 }
% 122.39/16.02 fresh(multiply(multiply(commutator(multiply(X, Y), Z), commutator(Y, multiply(commutator(X, Z), commutator(Y, Z)))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y)), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(Y, multiply(commutator(X, Z), commutator(Y, Z))), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 111 R->L }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), X), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 108 }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), multiply(X, Y)), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 71 }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), X), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 20 R->L }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), X), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(S, S)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 36 R->L }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), X), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(T, T))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 45 R->L }
% 122.39/16.02 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(commutator(Y, Z), X), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(Y, commutator(commutator(Y, Z), X)))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.02 = { by lemma 37 }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(Y, Z), X))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(Y, commutator(Z, X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 32 R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(commutator(Z, X), multiply(multiply(X, Y), commutator(Y, commutator(Z, X)))), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 22 R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(commutator(Z, X), multiply(commutator(U, U), multiply(multiply(X, Y), commutator(Y, commutator(Z, X))))), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 35 R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(commutator(Z, X), multiply(commutator(X, commutator(Z, X)), multiply(multiply(X, Y), commutator(Y, commutator(Z, X))))), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 75 }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(X, multiply(commutator(Z, X), multiply(Y, commutator(Y, commutator(Z, X))))), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 5 (commutator) R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(X, multiply(Y, commutator(Z, X))), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), fresh(multiply(multiply(X, Y), commutator(Z, X)), multiply(multiply(X, Y), commutator(Z, X)), commutator(Y, commutator(Z, X)), commutator(multiply(X, Y), commutator(Z, X))))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 1 (left_cancellation) }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Z, X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.39/16.03 fresh(multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 1 (left_cancellation) R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), Z), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 20 R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(V, V))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 36 R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(W, W)))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 19 }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(multiply(commutator(X, Z), commutator(Y, Z)), commutator(X, X)))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(commutator(multiply(commutator(X, Z), commutator(Y, Z)), X), X))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 111 }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(commutator(Y, multiply(commutator(X, Z), commutator(Y, Z))), X))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 101 }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(commutator(Y, commutator(multiply(X, Y), Z)), X))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, commutator(Y, commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 47 R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(Z, multiply(multiply(X, Y), commutator(Y, commutator(commutator(multiply(X, Y), Z), X))))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 43 }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(Z, multiply(commutator(multiply(X, Y), Z), multiply(multiply(X, Y), commutator(Y, commutator(commutator(multiply(X, Y), Z), X))))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 70 }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(Z, multiply(X, multiply(commutator(multiply(X, Y), Z), multiply(Y, commutator(commutator(multiply(X, Y), Z), X))))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 112 R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(X, Y), commutator(Y, Z)), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 116 R->L }
% 122.39/16.03 fresh(fresh(multiply(multiply(X, Y), Z), multiply(multiply(X, Y), multiply(Z, multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), multiply(commutator(multiply(X, Y), Z), multiply(commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(commutator(multiply(X, Y), Z), X)))), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by lemma 55 }
% 122.39/16.03 fresh(commutator(multiply(X, Y), Z), commutator(multiply(X, Y), Z), commutator(multiply(commutator(X, Z), commutator(Y, Z)), Y), commutator(multiply(X, Y), commutator(Y, Z)))
% 122.39/16.03 = { by axiom 1 (left_cancellation) }
% 122.39/16.03 commutator(multiply(X, Y), commutator(Y, Z))
% 122.39/16.03
% 122.39/16.03 Lemma 118: commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z) = commutator(multiply(commutator(X, Y), commutator(Z, Y)), X).
% 122.39/16.03 Proof:
% 122.39/16.03 commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)
% 122.39/16.03 = { by lemma 32 R->L }
% 122.39/16.03 fresh(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 20 R->L }
% 122.39/16.03 fresh(multiply(commutator(X, Y), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(W, W))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 56 R->L }
% 122.39/16.03 fresh(multiply(commutator(X, Y), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), multiply(commutator(Z, Y), commutator(Y, Z)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 26 R->L }
% 122.39/16.03 fresh(multiply(commutator(X, Y), multiply(commutator(Z, Y), multiply(commutator(Y, Z), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), multiply(commutator(Z, Y), commutator(Y, Z))))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.03 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Y, Z), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), multiply(commutator(Z, Y), commutator(Y, Z)))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 56 }
% 122.39/16.03 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Y, Z), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(V, V))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 36 }
% 122.39/16.03 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Y, Z), multiply(multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)), commutator(U, U)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 20 }
% 122.39/16.03 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Y, Z), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 117 }
% 122.39/16.03 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(commutator(Y, Z), multiply(multiply(X, Z), commutator(multiply(X, Z), commutator(Z, Y))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.03 = { by lemma 89 }
% 122.39/16.04 fresh(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(X, Z), commutator(Y, Z))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 82 }
% 122.39/16.04 fresh(multiply(multiply(X, Z), multiply(multiply(commutator(X, Y), commutator(Z, Y)), commutator(Y, Z))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 fresh(multiply(multiply(X, Z), multiply(commutator(X, Y), multiply(commutator(Z, Y), commutator(Y, Z)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 56 }
% 122.39/16.04 fresh(multiply(multiply(X, Z), multiply(commutator(X, Y), commutator(T, T))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 20 }
% 122.39/16.04 fresh(multiply(multiply(X, Z), commutator(X, Y)), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), Z), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by axiom 1 (left_cancellation) }
% 122.39/16.04 commutator(multiply(X, Z), commutator(X, Y))
% 122.39/16.04 = { by axiom 1 (left_cancellation) R->L }
% 122.39/16.04 fresh(multiply(multiply(X, Z), commutator(X, Y)), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 fresh(multiply(X, multiply(Z, commutator(X, Y))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 22 R->L }
% 122.39/16.04 fresh(multiply(X, multiply(commutator(S, S), multiply(Z, commutator(X, Y)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 53 R->L }
% 122.39/16.04 fresh(multiply(X, multiply(commutator(commutator(X, Y), X), multiply(Z, commutator(X, Y)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 73 R->L }
% 122.39/16.04 fresh(multiply(commutator(X, Y), multiply(X, multiply(Z, commutator(Z, commutator(X, Y))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 fresh(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, commutator(X, Y)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 60 R->L }
% 122.39/16.04 fresh(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, commutator(multiply(X, Z), Y)))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 101 R->L }
% 122.39/16.04 fresh(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(Z, multiply(commutator(X, Y), commutator(Z, Y))))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 111 R->L }
% 122.39/16.04 fresh(multiply(commutator(X, Y), multiply(multiply(X, Z), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X))), multiply(multiply(X, Z), commutator(X, Y)), commutator(multiply(commutator(X, Y), commutator(Z, Y)), X), commutator(multiply(X, Z), commutator(X, Y)))
% 122.39/16.04 = { by lemma 32 }
% 122.39/16.04 commutator(multiply(commutator(X, Y), commutator(Z, Y)), X)
% 122.39/16.04
% 122.39/16.04 Lemma 119: multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, W)) = multiply(X, multiply(multiply(commutator(Y, X), commutator(Z, X)), W)).
% 122.39/16.04 Proof:
% 122.39/16.04 multiply(multiply(commutator(Y, X), commutator(Z, X)), multiply(X, W))
% 122.39/16.04 = { by lemma 61 }
% 122.39/16.04 multiply(commutator(Z, X), multiply(commutator(Y, X), multiply(X, W)))
% 122.39/16.04 = { by lemma 43 }
% 122.39/16.04 multiply(commutator(Z, X), multiply(X, multiply(commutator(Y, X), W)))
% 122.39/16.04 = { by lemma 43 }
% 122.39/16.04 multiply(X, multiply(commutator(Z, X), multiply(commutator(Y, X), W)))
% 122.39/16.04 = { by lemma 61 R->L }
% 122.39/16.04 multiply(X, multiply(multiply(commutator(Y, X), commutator(Z, X)), W))
% 122.39/16.04
% 122.39/16.04 Lemma 120: multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), V)) = multiply(commutator(X, multiply(Y, Z)), multiply(multiply(commutator(Y, W), commutator(Z, W)), V)).
% 122.39/16.04 Proof:
% 122.39/16.04 multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), V))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), commutator(X, multiply(Y, Z))), V)
% 122.39/16.04 = { by lemma 20 R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(T, T))), V)
% 122.39/16.04 = { by lemma 36 R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, commutator(U, U)))), V)
% 122.39/16.04 = { by lemma 84 R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, commutator(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z))))), V)
% 122.39/16.04 = { by lemma 32 R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, fresh(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(multiply(Y, Z), commutator(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)))), multiply(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W))), commutator(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), commutator(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W))))))), V)
% 122.39/16.04 = { by lemma 83 }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, fresh(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), multiply(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W))), commutator(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), commutator(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W))))))), V)
% 122.39/16.04 = { by lemma 81 R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, fresh(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), commutator(multiply(commutator(Y, W), commutator(Z, W)), multiply(Y, Z)), commutator(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W))))))), V)
% 122.39/16.04 = { by axiom 1 (left_cancellation) }
% 122.39/16.04 multiply(multiply(multiply(commutator(Y, W), commutator(Z, W)), multiply(commutator(X, multiply(Y, Z)), commutator(X, commutator(multiply(Y, Z), multiply(commutator(Y, W), commutator(Z, W)))))), V)
% 122.39/16.04 = { by lemma 37 }
% 122.39/16.04 multiply(multiply(commutator(X, multiply(Y, Z)), multiply(commutator(Y, W), commutator(Z, W))), V)
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 multiply(commutator(X, multiply(Y, Z)), multiply(multiply(commutator(Y, W), commutator(Z, W)), V))
% 122.39/16.04
% 122.39/16.04 Lemma 121: multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(W, multiply(commutator(multiply(X, Z), commutator(Z, Y)), V))) = multiply(W, multiply(commutator(multiply(X, Z), Y), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), V))).
% 122.39/16.04 Proof:
% 122.39/16.04 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(W, multiply(commutator(multiply(X, Z), commutator(Z, Y)), V)))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(multiply(W, commutator(multiply(X, Z), commutator(Z, Y))), V))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 multiply(multiply(multiply(commutator(X, Y), commutator(Z, Y)), multiply(W, commutator(multiply(X, Z), commutator(Z, Y)))), V)
% 122.39/16.04 = { by lemma 114 }
% 122.39/16.04 multiply(multiply(W, multiply(commutator(multiply(X, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W))), V)
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 multiply(W, multiply(multiply(commutator(multiply(X, Z), Y), commutator(multiply(commutator(X, Y), commutator(Z, Y)), W)), V))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 multiply(W, multiply(commutator(multiply(X, Z), Y), multiply(commutator(multiply(commutator(X, Y), commutator(Z, Y)), W), V)))
% 122.39/16.04
% 122.39/16.04 Goal 1 (prove_commutator_distributes_over_product): commutator(multiply(a, b), c) = multiply(commutator(a, c), commutator(b, c)).
% 122.39/16.04 Proof:
% 122.39/16.04 commutator(multiply(a, b), c)
% 122.39/16.04 = { by axiom 1 (left_cancellation) R->L }
% 122.39/16.04 fresh(multiply(multiply(a, b), c), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.04 fresh(multiply(a, multiply(b, c)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 10 R->L }
% 122.39/16.04 fresh(multiply(b, multiply(a, multiply(commutator(a, b), c))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 54 R->L }
% 122.39/16.04 fresh(multiply(b, multiply(commutator(a, b), multiply(a, c))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 39 R->L }
% 122.39/16.04 fresh(multiply(b, multiply(commutator(multiply(a, b), b), multiply(a, c))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 43 R->L }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(b, multiply(a, c))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 46 R->L }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(b, multiply(commutator(a, c), multiply(c, a)))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(b, multiply(multiply(commutator(a, c), c), a))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(b, multiply(commutator(a, c), c)), a)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 20 R->L }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(b, multiply(commutator(a, c), multiply(c, commutator(X, X)))), a)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 43 }
% 122.39/16.04 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(b, multiply(c, multiply(commutator(a, c), commutator(X, X)))), a)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.04 = { by lemma 57 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(b, multiply(c, multiply(commutator(a, c), commutator(commutator(a, c), commutator(b, c))))), a)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 69 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(c, multiply(b, multiply(commutator(a, c), commutator(b, c)))), a)), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(b, multiply(commutator(a, c), commutator(b, c))), a))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 110 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(b, commutator(multiply(commutator(a, c), commutator(b, c)), a))), a))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(b, commutator(multiply(commutator(a, c), commutator(b, c)), a)), a)))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by axiom 4 (associativity_of_multiply) }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(b, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 80 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(b, multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 118 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(b, multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), b), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 121 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(b, multiply(commutator(multiply(a, b), commutator(b, c)), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 80 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(b, multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(a, b), commutator(b, c)), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 90 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(b, multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), b), a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 10 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(b, a)))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 43 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(b, a)))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 22 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(Y, Y), multiply(b, a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 99 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), a)), multiply(b, a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(commutator(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), a), multiply(b, a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 97 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(commutator(multiply(a, b), multiply(commutator(a, c), commutator(b, c))), a), multiply(b, a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by axiom 3 (associativity_of_commutator) }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 22 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(Z, Z), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 43 }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(Z, Z), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 105 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), c), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 121 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(a, b), commutator(b, c)), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.39/16.05 = { by lemma 117 R->L }
% 122.39/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), b), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.05 = { by lemma 118 }
% 122.77/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.05 = { by lemma 22 R->L }
% 122.77/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, multiply(commutator(W, W), a))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.05 = { by lemma 25 R->L }
% 122.77/16.05 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(b, multiply(commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a))), a))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 5 (commutator) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(a, multiply(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), commutator(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(a, multiply(b, multiply(commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a))), commutator(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), multiply(multiply(a, b), multiply(commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a))), commutator(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(multiply(a, b), multiply(commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)), commutator(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), commutator(multiply(b, commutator(multiply(a, b), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), a)))), a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 25 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), commutator(multiply(b, commutator(V, V)), a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 20 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), commutator(b, a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 33 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 111 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(b, multiply(commutator(a, c), commutator(b, c))), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 32 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(fresh(multiply(commutator(multiply(a, b), c), multiply(b, commutator(b, multiply(commutator(a, c), commutator(b, c))))), multiply(b, commutator(multiply(a, b), c)), commutator(b, multiply(commutator(a, c), commutator(b, c))), commutator(b, commutator(multiply(a, b), c))), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 100 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(fresh(multiply(multiply(commutator(a, c), commutator(b, c)), b), multiply(b, commutator(multiply(a, b), c)), commutator(b, multiply(commutator(a, c), commutator(b, c))), commutator(b, commutator(multiply(a, b), c))), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 79 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(fresh(multiply(multiply(commutator(a, c), commutator(b, c)), b), multiply(multiply(commutator(a, c), commutator(b, c)), b), commutator(b, multiply(commutator(a, c), commutator(b, c))), commutator(b, commutator(multiply(a, b), c))), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 1 (left_cancellation) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(multiply(a, b), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(c, multiply(a, b)), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(a, b))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 119 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 109 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(c, multiply(a, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(b, commutator(commutator(b, c), multiply(a, b)))))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(c, multiply(a, multiply(commutator(a, c), multiply(commutator(b, c), multiply(b, commutator(commutator(b, c), multiply(a, b))))))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, multiply(c, multiply(commutator(b, c), multiply(b, commutator(commutator(b, c), multiply(a, b)))))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 43 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, multiply(commutator(b, c), multiply(c, multiply(b, commutator(commutator(b, c), multiply(a, b)))))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 47 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, multiply(b, multiply(c, commutator(commutator(b, c), multiply(a, b))))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(multiply(a, b), multiply(c, commutator(commutator(b, c), multiply(a, b)))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(multiply(c, commutator(commutator(b, c), multiply(a, b))), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, multiply(commutator(commutator(b, c), multiply(a, b)), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 108 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), a), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 118 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(commutator(a, c), commutator(b, c)), b), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 116 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(a, b), commutator(b, c)), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 90 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, multiply(commutator(commutator(multiply(a, b), c), b), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), b), multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 85 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(a, multiply(commutator(multiply(a, b), c), multiply(b, multiply(commutator(b, commutator(multiply(a, b), c)), commutator(b, multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(a, multiply(b, multiply(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 10 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(a, b), multiply(c, commutator(b, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 47 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(multiply(a, b), commutator(b, multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 30 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(multiply(a, b), c), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 43 }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(b, multiply(a, b)), multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 20 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), commutator(U, U)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 66 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), commutator(T, commutator(commutator(multiply(a, b), T), S))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.06 = { by lemma 95 R->L }
% 122.77/16.06 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(b, multiply(a, b)), multiply(commutator(T, commutator(commutator(multiply(a, b), T), S)), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 10 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(T, commutator(commutator(multiply(a, b), T), S)), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 95 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(multiply(a, b), commutator(T, commutator(commutator(multiply(a, b), T), S)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 66 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(multiply(a, b), commutator(X2, X2))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 20 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(a, b), commutator(c, commutator(b, multiply(a, b)))), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 29 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(multiply(a, b), commutator(multiply(a, b), commutator(c, commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(commutator(multiply(a, b), c), multiply(multiply(a, b), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 54 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 106 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(a, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(b, commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 85 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), b), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 82 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), b), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 117 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(multiply(a, b), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 37 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(a, b), commutator(b, c)), commutator(multiply(a, b), commutator(commutator(b, c), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 94 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(a, b), commutator(b, c)), commutator(Y2, Y2)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 20 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(b, c)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 117 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 118 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), a))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 82 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), a))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 118 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 117 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(b, c)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 42 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))), multiply(multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), commutator(b, c))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 86 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))), commutator(multiply(a, b), c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 113 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(c, multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(a, b), c), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 10 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), multiply(multiply(a, b), multiply(commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))), multiply(commutator(multiply(a, b), c), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 88 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(c, commutator(b, multiply(a, b))), multiply(commutator(multiply(a, b), c), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 10 R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(c, commutator(b, multiply(a, b))), multiply(commutator(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), c)), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(c, commutator(b, multiply(a, b))), multiply(commutator(c, commutator(commutator(b, multiply(a, b)), commutator(multiply(a, b), c))), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(c, commutator(b, multiply(a, b))), multiply(commutator(c, commutator(commutator(commutator(b, multiply(a, b)), multiply(a, b)), c)), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 35 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(c, commutator(b, multiply(a, b))), multiply(commutator(Z2, Z2), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 22 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 10 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(commutator(multiply(a, b), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), commutator(c, commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by lemma 98 }
% 122.77/16.07 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), commutator(c, commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.07 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(commutator(multiply(commutator(a, c), commutator(b, c)), c), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 105 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(commutator(W2, W2), commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 64 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(multiply(commutator(V2, U2), commutator(T2, U2)), commutator(commutator(T2, U2), commutator(b, multiply(a, b)))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 65 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(multiply(a, b), commutator(S2, S2))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 36 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(c, commutator(b, multiply(a, b))), commutator(X3, X3)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 20 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, commutator(c, commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 92 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, commutator(c, commutator(b, multiply(a, b))))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 5 (commutator) R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(commutator(multiply(a, b), b), multiply(c, commutator(b, multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 42 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(commutator(multiply(a, b), b), c), multiply(commutator(multiply(a, b), b), commutator(b, multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 56 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(commutator(multiply(a, b), b), c), commutator(Y3, Y3)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 20 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, commutator(commutator(multiply(a, b), b), c))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 3 (associativity_of_commutator) }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, commutator(multiply(a, b), commutator(b, c)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 20 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(a, b), commutator(b, c)), commutator(Z3, Z3)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 35 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(a, b), commutator(b, c)), commutator(multiply(a, b), commutator(commutator(b, c), multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 3 (associativity_of_commutator) R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(multiply(a, b), commutator(b, c)), commutator(commutator(multiply(a, b), commutator(b, c)), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 68 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(a, commutator(multiply(a, b), c)), commutator(commutator(multiply(a, b), commutator(b, c)), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 68 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(a, b), multiply(c, multiply(commutator(a, commutator(multiply(a, b), c)), commutator(commutator(a, commutator(multiply(a, b), c)), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 73 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(a, commutator(multiply(a, b), c)), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 42 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(a, multiply(commutator(multiply(a, b), c), b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 40 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(commutator(multiply(a, b), c), b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 90 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(multiply(a, b), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 54 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(multiply(a, b), c), multiply(multiply(a, b), commutator(multiply(a, b), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 29 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(multiply(a, b), commutator(b, c)), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 67 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(multiply(a, b), c), multiply(commutator(a, commutator(b, c)), multiply(a, b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 78 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(a, commutator(b, c)), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(commutator(a, c), multiply(commutator(b, c), multiply(commutator(a, commutator(b, c)), multiply(a, b)))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 42 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(commutator(a, c), multiply(a, multiply(commutator(b, c), b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 52 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(commutator(a, c), multiply(multiply(a, b), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 50 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(multiply(a, b), commutator(multiply(a, b), c)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 4 (associativity_of_multiply) }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(a, multiply(b, commutator(multiply(a, b), c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 79 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(a, multiply(multiply(commutator(a, c), commutator(b, c)), b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 40 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(multiply(a, b), multiply(multiply(commutator(a, c), commutator(b, c)), commutator(multiply(commutator(a, c), commutator(b, c)), b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 82 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(commutator(a, c), multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 78 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(commutator(multiply(a, b), c), multiply(multiply(a, b), commutator(multiply(commutator(a, c), commutator(b, c)), b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 54 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(multiply(commutator(a, c), commutator(b, c)), b)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 118 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(multiply(commutator(a, c), commutator(b, c)), a)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 108 }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 107 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), fresh(multiply(multiply(a, b), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b)))), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b))), multiply(commutator(a, c), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 51 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), fresh(multiply(commutator(a, c), multiply(multiply(a, b), multiply(commutator(b, c), commutator(commutator(b, c), multiply(a, b))))), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b))), multiply(commutator(a, c), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 5 (commutator) R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), fresh(multiply(commutator(a, c), multiply(commutator(b, c), multiply(a, b))), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b))), multiply(commutator(a, c), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 4 (associativity_of_multiply) R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), fresh(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b)), multiply(commutator(multiply(a, b), c), commutator(commutator(b, c), multiply(a, b))), multiply(commutator(a, c), commutator(b, c))))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by axiom 1 (left_cancellation) }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(a, b), multiply(commutator(a, c), commutator(b, c)))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 81 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 119 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 120 R->L }
% 122.77/16.08 fresh(multiply(commutator(multiply(a, b), b), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 41 }
% 122.77/16.08 fresh(multiply(commutator(a, multiply(a, b)), multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 120 R->L }
% 122.77/16.08 fresh(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(a, multiply(a, b)), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 41 R->L }
% 122.77/16.08 fresh(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(commutator(multiply(a, b), b), multiply(commutator(b, multiply(a, b)), multiply(c, multiply(a, b))))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.08 = { by lemma 92 }
% 122.77/16.09 fresh(multiply(multiply(commutator(a, c), commutator(b, c)), multiply(c, multiply(a, b))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.09 = { by lemma 119 }
% 122.77/16.09 fresh(multiply(c, multiply(multiply(commutator(a, c), commutator(b, c)), multiply(a, b))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.09 = { by lemma 81 }
% 122.77/16.09 fresh(multiply(c, multiply(multiply(a, b), multiply(commutator(a, c), commutator(b, c)))), multiply(multiply(a, b), c), multiply(commutator(a, c), commutator(b, c)), commutator(multiply(a, b), c))
% 122.77/16.09 = { by lemma 32 }
% 122.77/16.09 multiply(commutator(a, c), commutator(b, c))
% 122.77/16.09 % SZS output end Proof
% 122.77/16.09
% 122.77/16.09 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------