%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP083-1 : TPTP v9.3.1. Bugfixed v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 10:09:13 AM UTC 2026
% Result : Unsatisfiable 2.32s 0.73s
% Output : Proof 3.78s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRP083-1 : TPTP v9.3.1. Bugfixed v2.3.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.07/0.35 % Computer : n003.cluster.edu
% 0.07/0.35 % Model : x86_64 x86_64
% 0.07/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.35 % Memory : 8046.5625MB
% 0.07/0.35 % OS : Linux 6.8.0-71-generic
% 0.07/0.35 % CPULimit : 300
% 0.07/0.35 % WCLimit : 300
% 0.07/0.35 % DateTime : Sun Sep 27 09:20:10 UTC 2026
% 0.07/0.36 % CPUTime :
% 0.07/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 2.32/0.73 Command-line arguments: --no-flatten-goal
% 2.32/0.73
% 2.32/0.73 % SZS status Unsatisfiable
% 2.32/0.73
% 3.78/0.96 % SZS output start Proof
% 3.78/0.96 Axiom 1 (multiply): multiply(X, Y) = inverse(double_divide(Y, X)).
% 3.78/0.96 Axiom 2 (single_axiom): double_divide(double_divide(X, inverse(double_divide(Y, Z))), double_divide(inverse(Y), inverse(double_divide(W, double_divide(X, W))))) = Z.
% 3.78/0.96
% 3.78/0.96 Lemma 3: double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W))) = Y.
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, W), W)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(X, inverse(double_divide(Z, Y))), double_divide(inverse(Z), inverse(double_divide(W, double_divide(X, W)))))
% 3.78/0.96 = { by axiom 2 (single_axiom) }
% 3.78/0.96 Y
% 3.78/0.96
% 3.78/0.96 Lemma 4: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X))) = inverse(W).
% 3.78/0.96 Proof:
% 3.78/0.96 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(W, X)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 inverse(double_divide(double_divide(Y, multiply(W, X)), double_divide(inverse(X), multiply(double_divide(Y, Z), Z))))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 inverse(W)
% 3.78/0.96
% 3.78/0.96 Lemma 5: double_divide(inverse(X), multiply(double_divide(Y, W), W)) = double_divide(inverse(X), multiply(double_divide(Y, Z), Z)).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(inverse(X), multiply(double_divide(Y, W), W))
% 3.78/0.96 = { by lemma 3 R->L }
% 3.78/0.96 double_divide(double_divide(V, multiply(double_divide(inverse(X), multiply(double_divide(Y, W), W)), double_divide(Y, multiply(U, X)))), double_divide(inverse(double_divide(Y, multiply(U, X))), multiply(double_divide(V, T), T)))
% 3.78/0.96 = { by lemma 4 }
% 3.78/0.96 double_divide(double_divide(V, inverse(U)), double_divide(inverse(double_divide(Y, multiply(U, X))), multiply(double_divide(V, T), T)))
% 3.78/0.96 = { by lemma 4 R->L }
% 3.78/0.96 double_divide(double_divide(V, multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(U, X)))), double_divide(inverse(double_divide(Y, multiply(U, X))), multiply(double_divide(V, T), T)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 double_divide(inverse(X), multiply(double_divide(Y, Z), Z))
% 3.78/0.96
% 3.78/0.96 Lemma 6: multiply(double_divide(X, Z), Z) = multiply(double_divide(X, Y), Y).
% 3.78/0.96 Proof:
% 3.78/0.96 multiply(double_divide(X, Z), Z)
% 3.78/0.96 = { by lemma 3 R->L }
% 3.78/0.96 double_divide(double_divide(W, multiply(multiply(double_divide(X, Z), Z), inverse(V))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(W, inverse(double_divide(inverse(V), multiply(double_divide(X, Z), Z)))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 3.78/0.96 = { by lemma 5 }
% 3.78/0.96 double_divide(double_divide(W, inverse(double_divide(inverse(V), multiply(double_divide(X, Y), Y)))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 3.78/0.96 = { by axiom 1 (multiply) R->L }
% 3.78/0.96 double_divide(double_divide(W, multiply(multiply(double_divide(X, Y), Y), inverse(V))), double_divide(inverse(inverse(V)), multiply(double_divide(W, U), U)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 multiply(double_divide(X, Y), Y)
% 3.78/0.96
% 3.78/0.96 Lemma 7: multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V))) = multiply(X, W).
% 3.78/0.96 Proof:
% 3.78/0.96 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, V), V)))
% 3.78/0.96 = { by lemma 6 }
% 3.78/0.96 multiply(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(Y, multiply(double_divide(W, X), X)))
% 3.78/0.96 = { by lemma 4 }
% 3.78/0.96 inverse(double_divide(W, X))
% 3.78/0.96 = { by axiom 1 (multiply) R->L }
% 3.78/0.96 multiply(X, W)
% 3.78/0.96
% 3.78/0.96 Lemma 8: double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V))) = double_divide(Y, W).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(double_divide(X, multiply(double_divide(Y, Z), Z)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 3.78/0.96 = { by lemma 6 }
% 3.78/0.96 double_divide(double_divide(X, multiply(double_divide(Y, W), W)), double_divide(inverse(W), multiply(double_divide(X, V), V)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 double_divide(Y, W)
% 3.78/0.96
% 3.78/0.96 Lemma 9: double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(X, V), V))) = Y.
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(multiply(W, Z), multiply(double_divide(X, V), V)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(X, multiply(Y, double_divide(Z, W))), double_divide(inverse(double_divide(Z, W)), multiply(double_divide(X, V), V)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 Y
% 3.78/0.96
% 3.78/0.96 Lemma 10: double_divide(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(multiply(W, V), multiply(double_divide(inverse(X), U), U))) = double_divide(Y, double_divide(V, W)).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(double_divide(inverse(X), multiply(double_divide(Y, Z), Z)), double_divide(multiply(W, V), multiply(double_divide(inverse(X), U), U)))
% 3.78/0.96 = { by lemma 5 }
% 3.78/0.96 double_divide(double_divide(inverse(X), multiply(double_divide(Y, double_divide(V, W)), double_divide(V, W))), double_divide(multiply(W, V), multiply(double_divide(inverse(X), U), U)))
% 3.78/0.96 = { by lemma 9 }
% 3.78/0.96 double_divide(Y, double_divide(V, W))
% 3.78/0.96
% 3.78/0.96 Lemma 11: double_divide(X, double_divide(Y, multiply(double_divide(Z, double_divide(X, Z)), W))) = double_divide(inverse(W), multiply(double_divide(Y, V), V)).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(X, double_divide(Y, multiply(double_divide(Z, double_divide(X, Z)), W)))
% 3.78/0.96 = { by lemma 10 R->L }
% 3.78/0.96 double_divide(double_divide(inverse(U), multiply(double_divide(X, Z), Z)), double_divide(multiply(multiply(double_divide(Z, double_divide(X, Z)), W), Y), multiply(double_divide(inverse(U), T), T)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(inverse(U), inverse(double_divide(Z, double_divide(X, Z)))), double_divide(multiply(multiply(double_divide(Z, double_divide(X, Z)), W), Y), multiply(double_divide(inverse(U), T), T)))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(double_divide(inverse(U), inverse(double_divide(Z, double_divide(X, Z)))), double_divide(inverse(double_divide(Y, multiply(double_divide(Z, double_divide(X, Z)), W))), multiply(double_divide(inverse(U), T), T)))
% 3.78/0.96 = { by lemma 4 R->L }
% 3.78/0.96 double_divide(double_divide(inverse(U), multiply(double_divide(inverse(W), multiply(double_divide(Y, V), V)), double_divide(Y, multiply(double_divide(Z, double_divide(X, Z)), W)))), double_divide(inverse(double_divide(Y, multiply(double_divide(Z, double_divide(X, Z)), W))), multiply(double_divide(inverse(U), T), T)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 double_divide(inverse(W), multiply(double_divide(Y, V), V))
% 3.78/0.96
% 3.78/0.96 Lemma 12: double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)) = double_divide(Y, double_divide(Z, multiply(double_divide(X, V), V))).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(multiply(X, Y), multiply(double_divide(Z, W), W))
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 double_divide(inverse(double_divide(Y, X)), multiply(double_divide(Z, W), W))
% 3.78/0.96 = { by lemma 11 R->L }
% 3.78/0.96 double_divide(Y, double_divide(Z, multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, X))))
% 3.78/0.96 = { by lemma 6 R->L }
% 3.78/0.96 double_divide(Y, double_divide(Z, multiply(double_divide(X, V), V)))
% 3.78/0.96
% 3.78/0.96 Lemma 13: multiply(double_divide(X, multiply(double_divide(Y, Z), Z)), W) = multiply(multiply(double_divide(X, V), V), multiply(Y, W)).
% 3.78/0.96 Proof:
% 3.78/0.96 multiply(double_divide(X, multiply(double_divide(Y, Z), Z)), W)
% 3.78/0.96 = { by axiom 1 (multiply) }
% 3.78/0.96 inverse(double_divide(W, double_divide(X, multiply(double_divide(Y, Z), Z))))
% 3.78/0.96 = { by lemma 12 R->L }
% 3.78/0.96 inverse(double_divide(multiply(Y, W), multiply(double_divide(X, V), V)))
% 3.78/0.96 = { by axiom 1 (multiply) R->L }
% 3.78/0.96 multiply(multiply(double_divide(X, V), V), multiply(Y, W))
% 3.78/0.96
% 3.78/0.96 Lemma 14: multiply(multiply(double_divide(X, Y), Y), multiply(Z, multiply(double_divide(Z, W), W))) = multiply(double_divide(X, V), V).
% 3.78/0.96 Proof:
% 3.78/0.96 multiply(multiply(double_divide(X, Y), Y), multiply(Z, multiply(double_divide(Z, W), W)))
% 3.78/0.96 = { by lemma 13 R->L }
% 3.78/0.96 multiply(double_divide(X, multiply(double_divide(Z, W), W)), multiply(double_divide(Z, W), W))
% 3.78/0.96 = { by lemma 6 R->L }
% 3.78/0.96 multiply(double_divide(X, V), V)
% 3.78/0.96
% 3.78/0.96 Lemma 15: double_divide(X, multiply(Y, multiply(double_divide(Y, Z), Z))) = multiply(double_divide(X, W), W).
% 3.78/0.96 Proof:
% 3.78/0.96 double_divide(X, multiply(Y, multiply(double_divide(Y, Z), Z)))
% 3.78/0.96 = { by lemma 8 R->L }
% 3.78/0.96 double_divide(double_divide(V, multiply(double_divide(X, U), U)), double_divide(inverse(multiply(Y, multiply(double_divide(Y, Z), Z))), multiply(double_divide(V, T), T)))
% 3.78/0.96 = { by lemma 14 R->L }
% 3.78/0.96 double_divide(double_divide(V, multiply(multiply(double_divide(X, W), W), multiply(Y, multiply(double_divide(Y, Z), Z)))), double_divide(inverse(multiply(Y, multiply(double_divide(Y, Z), Z))), multiply(double_divide(V, T), T)))
% 3.78/0.96 = { by lemma 3 }
% 3.78/0.96 multiply(double_divide(X, W), W)
% 3.78/0.96
% 3.78/0.96 Lemma 16: multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), Z), Z) = inverse(X).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), Z), Z)
% 3.78/0.97 = { by lemma 14 R->L }
% 3.78/0.97 multiply(multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), W), W), multiply(V, multiply(double_divide(V, U), U)))
% 3.78/0.97 = { by lemma 15 R->L }
% 3.78/0.97 multiply(multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), W), W), multiply(V, double_divide(V, multiply(X, multiply(double_divide(X, Y), Y)))))
% 3.78/0.97 = { by axiom 1 (multiply) }
% 3.78/0.97 multiply(multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), W), W), inverse(double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V)))
% 3.78/0.97 = { by axiom 1 (multiply) }
% 3.78/0.97 inverse(double_divide(inverse(double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V)), multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), W), W)))
% 3.78/0.97 = { by lemma 11 R->L }
% 3.78/0.97 inverse(double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), double_divide(inverse(multiply(double_divide(X, Y), Y)), multiply(double_divide(V, double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V)), double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V)))))
% 3.78/0.97 = { by axiom 1 (multiply) R->L }
% 3.78/0.97 multiply(double_divide(inverse(multiply(double_divide(X, Y), Y)), multiply(double_divide(V, double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V)), double_divide(double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))), V))), double_divide(V, multiply(X, multiply(double_divide(X, Y), Y))))
% 3.78/0.97 = { by lemma 4 }
% 3.78/0.97 inverse(X)
% 3.78/0.97
% 3.78/0.97 Lemma 17: double_divide(X, double_divide(W, double_divide(Z, W))) = double_divide(X, double_divide(Y, double_divide(Z, Y))).
% 3.78/0.97 Proof:
% 3.78/0.97 double_divide(X, double_divide(W, double_divide(Z, W)))
% 3.78/0.97 = { by lemma 10 R->L }
% 3.78/0.97 double_divide(double_divide(inverse(V), multiply(double_divide(X, U), U)), double_divide(multiply(double_divide(Z, W), W), multiply(double_divide(inverse(V), T), T)))
% 3.78/0.97 = { by lemma 6 }
% 3.78/0.97 double_divide(double_divide(inverse(V), multiply(double_divide(X, U), U)), double_divide(multiply(double_divide(Z, Y), Y), multiply(double_divide(inverse(V), T), T)))
% 3.78/0.97 = { by lemma 10 }
% 3.78/0.97 double_divide(X, double_divide(Y, double_divide(Z, Y)))
% 3.78/0.97
% 3.78/0.97 Lemma 18: multiply(double_divide(W, double_divide(Y, W)), Z) = multiply(double_divide(X, double_divide(Y, X)), Z).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(double_divide(W, double_divide(Y, W)), Z)
% 3.78/0.97 = { by axiom 1 (multiply) }
% 3.78/0.97 inverse(double_divide(Z, double_divide(W, double_divide(Y, W))))
% 3.78/0.97 = { by lemma 17 }
% 3.78/0.97 inverse(double_divide(Z, double_divide(X, double_divide(Y, X))))
% 3.78/0.97 = { by axiom 1 (multiply) R->L }
% 3.78/0.97 multiply(double_divide(X, double_divide(Y, X)), Z)
% 3.78/0.97
% 3.78/0.97 Lemma 19: double_divide(Z, double_divide(Y, Z)) = double_divide(X, double_divide(Y, X)).
% 3.78/0.97 Proof:
% 3.78/0.97 double_divide(Z, double_divide(Y, Z))
% 3.78/0.97 = { by lemma 3 R->L }
% 3.78/0.97 double_divide(double_divide(W, multiply(double_divide(Z, double_divide(Y, Z)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.97 = { by lemma 18 }
% 3.78/0.97 double_divide(double_divide(W, multiply(double_divide(X, double_divide(Y, X)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.97 = { by lemma 3 }
% 3.78/0.97 double_divide(X, double_divide(Y, X))
% 3.78/0.97
% 3.78/0.97 Lemma 20: double_divide(X, double_divide(inverse(multiply(double_divide(Y, Z), Z)), X)) = Y.
% 3.78/0.97 Proof:
% 3.78/0.97 double_divide(X, double_divide(inverse(multiply(double_divide(Y, Z), Z)), X))
% 3.78/0.97 = { by lemma 19 }
% 3.78/0.97 double_divide(multiply(double_divide(W, V), V), double_divide(inverse(multiply(double_divide(Y, Z), Z)), multiply(double_divide(W, V), V)))
% 3.78/0.97 = { by lemma 15 R->L }
% 3.78/0.97 double_divide(double_divide(W, multiply(Y, multiply(double_divide(Y, Z), Z))), double_divide(inverse(multiply(double_divide(Y, Z), Z)), multiply(double_divide(W, V), V)))
% 3.78/0.97 = { by lemma 3 }
% 3.78/0.97 Y
% 3.78/0.97
% 3.78/0.97 Lemma 21: multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, Z)) = multiply(double_divide(Z, W), W).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(double_divide(X, double_divide(Y, X)), double_divide(Y, Z))
% 3.78/0.97 = { by lemma 18 }
% 3.78/0.97 multiply(double_divide(Z, double_divide(Y, Z)), double_divide(Y, Z))
% 3.78/0.97 = { by lemma 6 R->L }
% 3.78/0.97 multiply(double_divide(Z, W), W)
% 3.78/0.97
% 3.78/0.97 Lemma 22: multiply(double_divide(multiply(X, multiply(double_divide(X, Y), Y)), Z), Z) = multiply(W, inverse(W)).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(double_divide(multiply(X, multiply(double_divide(X, Y), Y)), Z), Z)
% 3.78/0.97 = { by lemma 21 R->L }
% 3.78/0.97 multiply(double_divide(V, double_divide(inverse(multiply(double_divide(W, U), U)), V)), double_divide(inverse(multiply(double_divide(W, U), U)), multiply(X, multiply(double_divide(X, Y), Y))))
% 3.78/0.97 = { by lemma 20 }
% 3.78/0.97 multiply(W, double_divide(inverse(multiply(double_divide(W, U), U)), multiply(X, multiply(double_divide(X, Y), Y))))
% 3.78/0.97 = { by lemma 15 }
% 3.78/0.97 multiply(W, multiply(double_divide(inverse(multiply(double_divide(W, U), U)), T), T))
% 3.78/0.97 = { by lemma 16 }
% 3.78/0.97 multiply(W, inverse(W))
% 3.78/0.97
% 3.78/0.97 Lemma 23: double_divide(multiply(X, Y), multiply(Z, inverse(Z))) = double_divide(Y, double_divide(W, double_divide(X, W))).
% 3.78/0.97 Proof:
% 3.78/0.97 double_divide(multiply(X, Y), multiply(Z, inverse(Z)))
% 3.78/0.97 = { by lemma 22 R->L }
% 3.78/0.97 double_divide(multiply(X, Y), multiply(double_divide(multiply(V, multiply(double_divide(V, U), U)), T), T))
% 3.78/0.97 = { by lemma 12 }
% 3.78/0.97 double_divide(Y, double_divide(multiply(V, multiply(double_divide(V, U), U)), multiply(double_divide(X, S), S)))
% 3.78/0.97 = { by lemma 12 }
% 3.78/0.97 double_divide(Y, double_divide(multiply(double_divide(V, U), U), double_divide(X, multiply(double_divide(V, U), U))))
% 3.78/0.97 = { by lemma 17 R->L }
% 3.78/0.97 double_divide(Y, double_divide(W, double_divide(X, W)))
% 3.78/0.97
% 3.78/0.97 Lemma 24: multiply(double_divide(X, double_divide(Y, X)), Z) = multiply(multiply(W, inverse(W)), multiply(Y, Z)).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(double_divide(X, double_divide(Y, X)), Z)
% 3.78/0.97 = { by axiom 1 (multiply) }
% 3.78/0.97 inverse(double_divide(Z, double_divide(X, double_divide(Y, X))))
% 3.78/0.97 = { by lemma 23 R->L }
% 3.78/0.97 inverse(double_divide(multiply(Y, Z), multiply(W, inverse(W))))
% 3.78/0.97 = { by axiom 1 (multiply) R->L }
% 3.78/0.97 multiply(multiply(W, inverse(W)), multiply(Y, Z))
% 3.78/0.97
% 3.78/0.97 Lemma 25: multiply(multiply(X, inverse(X)), multiply(Y, double_divide(Y, Z))) = multiply(double_divide(Z, W), W).
% 3.78/0.97 Proof:
% 3.78/0.97 multiply(multiply(X, inverse(X)), multiply(Y, double_divide(Y, Z)))
% 3.78/0.97 = { by lemma 24 R->L }
% 3.78/0.97 multiply(double_divide(V, double_divide(Y, V)), double_divide(Y, Z))
% 3.78/0.97 = { by lemma 21 }
% 3.78/0.98 multiply(double_divide(Z, W), W)
% 3.78/0.98
% 3.78/0.98 Lemma 26: multiply(inverse(multiply(double_divide(X, Y), Y)), inverse(X)) = inverse(multiply(Z, inverse(Z))).
% 3.78/0.98 Proof:
% 3.78/0.98 multiply(inverse(multiply(double_divide(X, Y), Y)), inverse(X))
% 3.78/0.98 = { by lemma 4 R->L }
% 3.78/0.98 multiply(multiply(double_divide(inverse(multiply(W, multiply(double_divide(W, V), V))), multiply(double_divide(U, T), T)), double_divide(U, multiply(multiply(double_divide(X, Y), Y), multiply(W, multiply(double_divide(W, V), V))))), inverse(X))
% 3.78/0.98 = { by lemma 14 }
% 3.78/0.98 multiply(multiply(double_divide(inverse(multiply(W, multiply(double_divide(W, V), V))), multiply(double_divide(U, T), T)), double_divide(U, multiply(double_divide(X, S), S))), inverse(X))
% 3.78/0.98 = { by lemma 7 }
% 3.78/0.98 multiply(multiply(multiply(W, multiply(double_divide(W, V), V)), X), inverse(X))
% 3.78/0.98 = { by lemma 20 R->L }
% 3.78/0.98 multiply(multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(double_divide(W, V), V), double_divide(inverse(multiply(double_divide(X, X2), X2)), multiply(double_divide(W, V), V)))), inverse(X))
% 3.78/0.98 = { by lemma 12 R->L }
% 3.78/0.98 multiply(multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(W, multiply(double_divide(W, V), V)), multiply(double_divide(inverse(multiply(double_divide(X, X2), X2)), Y2), Y2))), inverse(X))
% 3.78/0.98 = { by lemma 16 }
% 3.78/0.98 multiply(multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(W, multiply(double_divide(W, V), V)), inverse(X))), inverse(X))
% 3.78/0.98 = { by lemma 7 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(W, multiply(double_divide(W, V), V)), inverse(X)))), multiply(double_divide(Z2, W2), W2)), double_divide(Z2, multiply(double_divide(inverse(X), V2), V2)))
% 3.78/0.98 = { by lemma 25 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(W, multiply(double_divide(W, V), V)), inverse(X)))), multiply(double_divide(Z2, W2), W2)), double_divide(Z2, multiply(multiply(Z, inverse(Z)), multiply(multiply(W, multiply(double_divide(W, V), V)), double_divide(multiply(W, multiply(double_divide(W, V), V)), inverse(X))))))
% 3.78/0.98 = { by lemma 4 }
% 3.78/0.98 inverse(multiply(Z, inverse(Z)))
% 3.78/0.98
% 3.78/0.98 Lemma 27: multiply(double_divide(X, Y), Y) = double_divide(X, inverse(multiply(Z, inverse(Z)))).
% 3.78/0.98 Proof:
% 3.78/0.98 multiply(double_divide(X, Y), Y)
% 3.78/0.98 = { by lemma 15 R->L }
% 3.78/0.98 double_divide(X, multiply(inverse(multiply(double_divide(W, V), V)), multiply(double_divide(inverse(multiply(double_divide(W, V), V)), U), U)))
% 3.78/0.98 = { by lemma 16 }
% 3.78/0.98 double_divide(X, multiply(inverse(multiply(double_divide(W, V), V)), inverse(W)))
% 3.78/0.98 = { by lemma 26 }
% 3.78/0.98 double_divide(X, inverse(multiply(Z, inverse(Z))))
% 3.78/0.98
% 3.78/0.98 Lemma 28: multiply(multiply(X, inverse(X)), multiply(Y, multiply(double_divide(Y, Z), Z))) = multiply(W, inverse(W)).
% 3.78/0.98 Proof:
% 3.78/0.98 multiply(multiply(X, inverse(X)), multiply(Y, multiply(double_divide(Y, Z), Z)))
% 3.78/0.98 = { by lemma 24 R->L }
% 3.78/0.98 multiply(double_divide(multiply(V, multiply(double_divide(V, U), U)), double_divide(Y, multiply(V, multiply(double_divide(V, U), U)))), multiply(double_divide(Y, Z), Z))
% 3.78/0.98 = { by lemma 15 R->L }
% 3.78/0.98 multiply(double_divide(multiply(V, multiply(double_divide(V, U), U)), double_divide(Y, multiply(V, multiply(double_divide(V, U), U)))), double_divide(Y, multiply(V, multiply(double_divide(V, U), U))))
% 3.78/0.98 = { by lemma 22 }
% 3.78/0.98 multiply(W, inverse(W))
% 3.78/0.98
% 3.78/0.98 Lemma 29: double_divide(inverse(multiply(X, inverse(X))), inverse(multiply(Y, inverse(Y)))) = multiply(Z, inverse(Z)).
% 3.78/0.98 Proof:
% 3.78/0.98 double_divide(inverse(multiply(X, inverse(X))), inverse(multiply(Y, inverse(Y))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(X, inverse(X))), W), W)
% 3.78/0.98 = { by lemma 25 R->L }
% 3.78/0.98 multiply(multiply(V, inverse(V)), multiply(U, double_divide(U, inverse(multiply(X, inverse(X))))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(multiply(V, inverse(V)), multiply(U, multiply(double_divide(U, T), T)))
% 3.78/0.98 = { by lemma 28 }
% 3.78/0.98 multiply(Z, inverse(Z))
% 3.78/0.98
% 3.78/0.98 Lemma 30: multiply(X, double_divide(X, inverse(multiply(Y, inverse(Y))))) = inverse(multiply(Z, inverse(Z))).
% 3.78/0.98 Proof:
% 3.78/0.98 multiply(X, double_divide(X, inverse(multiply(Y, inverse(Y)))))
% 3.78/0.98 = { by lemma 3 R->L }
% 3.78/0.98 double_divide(double_divide(W, multiply(multiply(X, double_divide(X, inverse(multiply(Y, inverse(Y))))), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 double_divide(double_divide(W, multiply(multiply(X, multiply(double_divide(X, T), T)), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 7 R->L }
% 3.78/0.98 double_divide(double_divide(W, multiply(double_divide(inverse(multiply(X, multiply(double_divide(X, T), T))), multiply(double_divide(S, X2), X2)), double_divide(S, multiply(double_divide(V, Y2), Y2)))), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 14 R->L }
% 3.78/0.98 double_divide(double_divide(W, multiply(double_divide(inverse(multiply(X, multiply(double_divide(X, T), T))), multiply(double_divide(S, X2), X2)), double_divide(S, multiply(multiply(double_divide(V, Z2), Z2), multiply(X, multiply(double_divide(X, T), T)))))), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 4 }
% 3.78/0.98 double_divide(double_divide(W, inverse(multiply(double_divide(V, Z2), Z2))), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 27 }
% 3.78/0.98 double_divide(double_divide(W, inverse(double_divide(V, inverse(multiply(Z, inverse(Z)))))), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by axiom 1 (multiply) R->L }
% 3.78/0.98 double_divide(double_divide(W, multiply(inverse(multiply(Z, inverse(Z))), V)), double_divide(inverse(V), multiply(double_divide(W, U), U)))
% 3.78/0.98 = { by lemma 3 }
% 3.78/0.98 inverse(multiply(Z, inverse(Z)))
% 3.78/0.98
% 3.78/0.98 Lemma 31: double_divide(multiply(inverse(multiply(X, inverse(X))), Y), inverse(multiply(Z, inverse(Z)))) = inverse(Y).
% 3.78/0.98 Proof:
% 3.78/0.98 double_divide(multiply(inverse(multiply(X, inverse(X))), Y), inverse(multiply(Z, inverse(Z))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(double_divide(multiply(inverse(multiply(X, inverse(X))), Y), inverse(multiply(W, inverse(W)))), inverse(multiply(W, inverse(W))))
% 3.78/0.98 = { by lemma 30 R->L }
% 3.78/0.98 multiply(double_divide(multiply(inverse(multiply(X, inverse(X))), Y), inverse(multiply(W, inverse(W)))), multiply(V, double_divide(V, inverse(multiply(U, inverse(U))))))
% 3.78/0.98 = { by axiom 1 (multiply) }
% 3.78/0.98 multiply(double_divide(inverse(double_divide(Y, inverse(multiply(X, inverse(X))))), inverse(multiply(W, inverse(W)))), multiply(V, double_divide(V, inverse(multiply(U, inverse(U))))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(multiply(double_divide(inverse(double_divide(Y, inverse(multiply(X, inverse(X))))), T), T), multiply(V, double_divide(V, inverse(multiply(U, inverse(U))))))
% 3.78/0.98 = { by lemma 13 R->L }
% 3.78/0.98 multiply(double_divide(inverse(double_divide(Y, inverse(multiply(X, inverse(X))))), multiply(double_divide(V, S), S)), double_divide(V, inverse(multiply(U, inverse(U)))))
% 3.78/0.98 = { by lemma 30 R->L }
% 3.78/0.98 multiply(double_divide(inverse(double_divide(Y, inverse(multiply(X, inverse(X))))), multiply(double_divide(V, S), S)), double_divide(V, multiply(Y, double_divide(Y, inverse(multiply(X, inverse(X)))))))
% 3.78/0.98 = { by lemma 4 }
% 3.78/0.98 inverse(Y)
% 3.78/0.98
% 3.78/0.98 Lemma 32: inverse(multiply(X, inverse(X))) = multiply(Y, inverse(Y)).
% 3.78/0.98 Proof:
% 3.78/0.98 inverse(multiply(X, inverse(X)))
% 3.78/0.98 = { by lemma 29 R->L }
% 3.78/0.98 inverse(double_divide(inverse(multiply(Z, inverse(Z))), inverse(multiply(W, inverse(W)))))
% 3.78/0.98 = { by lemma 31 R->L }
% 3.78/0.98 double_divide(multiply(inverse(multiply(Z, inverse(Z))), double_divide(inverse(multiply(Z, inverse(Z))), inverse(multiply(W, inverse(W))))), inverse(multiply(V, inverse(V))))
% 3.78/0.98 = { by lemma 30 }
% 3.78/0.98 double_divide(inverse(multiply(U, inverse(U))), inverse(multiply(V, inverse(V))))
% 3.78/0.98 = { by lemma 29 }
% 3.78/0.98 multiply(Y, inverse(Y))
% 3.78/0.98
% 3.78/0.98 Lemma 33: multiply(inverse(X), multiply(Y, inverse(Y))) = inverse(X).
% 3.78/0.98 Proof:
% 3.78/0.98 multiply(inverse(X), multiply(Y, inverse(Y)))
% 3.78/0.98 = { by lemma 32 R->L }
% 3.78/0.98 multiply(inverse(X), inverse(multiply(Z, inverse(Z))))
% 3.78/0.98 = { by lemma 31 R->L }
% 3.78/0.98 multiply(double_divide(multiply(inverse(multiply(W, inverse(W))), X), inverse(multiply(Z, inverse(Z)))), inverse(multiply(Z, inverse(Z))))
% 3.78/0.98 = { by lemma 27 }
% 3.78/0.98 double_divide(multiply(inverse(multiply(W, inverse(W))), X), inverse(multiply(V, inverse(V))))
% 3.78/0.98 = { by lemma 31 }
% 3.78/0.98 inverse(X)
% 3.78/0.98
% 3.78/0.98 Lemma 34: double_divide(inverse(X), multiply(Y, inverse(Y))) = inverse(inverse(X)).
% 3.78/0.98 Proof:
% 3.78/0.98 double_divide(inverse(X), multiply(Y, inverse(Y)))
% 3.78/0.98 = { by lemma 32 R->L }
% 3.78/0.98 double_divide(inverse(X), inverse(multiply(Z, inverse(Z))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(double_divide(inverse(X), W), W)
% 3.78/0.98 = { by lemma 25 R->L }
% 3.78/0.98 multiply(multiply(V, inverse(V)), multiply(U, double_divide(U, inverse(X))))
% 3.78/0.98 = { by lemma 29 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(T, inverse(T))), inverse(multiply(S, inverse(S)))), multiply(U, double_divide(U, inverse(X))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(multiply(double_divide(inverse(multiply(T, inverse(T))), X2), X2), multiply(U, double_divide(U, inverse(X))))
% 3.78/0.98 = { by lemma 13 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(T, inverse(T))), multiply(double_divide(U, Y2), Y2)), double_divide(U, inverse(X)))
% 3.78/0.98 = { by lemma 33 R->L }
% 3.78/0.98 multiply(double_divide(inverse(multiply(T, inverse(T))), multiply(double_divide(U, Y2), Y2)), double_divide(U, multiply(inverse(X), multiply(T, inverse(T)))))
% 3.78/0.98 = { by lemma 4 }
% 3.78/0.98 inverse(inverse(X))
% 3.78/0.98
% 3.78/0.98 Lemma 35: double_divide(X, double_divide(inverse(inverse(Y)), X)) = Y.
% 3.78/0.98 Proof:
% 3.78/0.98 double_divide(X, double_divide(inverse(inverse(Y)), X))
% 3.78/0.98 = { by lemma 19 }
% 3.78/0.98 double_divide(inverse(inverse(Z)), double_divide(inverse(inverse(Y)), inverse(inverse(Z))))
% 3.78/0.98 = { by lemma 34 R->L }
% 3.78/0.98 double_divide(inverse(inverse(Z)), double_divide(inverse(inverse(Y)), double_divide(inverse(Z), multiply(W, inverse(W)))))
% 3.78/0.98 = { by lemma 32 R->L }
% 3.78/0.98 double_divide(inverse(inverse(Z)), double_divide(inverse(inverse(Y)), double_divide(inverse(Z), inverse(multiply(V, inverse(V))))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 double_divide(inverse(inverse(Z)), double_divide(inverse(inverse(Y)), multiply(double_divide(inverse(Z), U), U)))
% 3.78/0.98 = { by lemma 34 R->L }
% 3.78/0.98 double_divide(double_divide(inverse(Z), multiply(Y, inverse(Y))), double_divide(inverse(inverse(Y)), multiply(double_divide(inverse(Z), U), U)))
% 3.78/0.98 = { by lemma 3 }
% 3.78/0.98 Y
% 3.78/0.98
% 3.78/0.98 Lemma 36: inverse(inverse(inverse(X))) = inverse(X).
% 3.78/0.98 Proof:
% 3.78/0.98 inverse(inverse(inverse(X)))
% 3.78/0.98 = { by lemma 34 R->L }
% 3.78/0.98 double_divide(inverse(inverse(X)), multiply(Y, inverse(Y)))
% 3.78/0.98 = { by lemma 32 R->L }
% 3.78/0.98 double_divide(inverse(inverse(X)), inverse(multiply(Z, inverse(Z))))
% 3.78/0.98 = { by lemma 27 R->L }
% 3.78/0.98 multiply(double_divide(inverse(inverse(X)), W), W)
% 3.78/0.98 = { by axiom 1 (multiply) }
% 3.78/0.98 inverse(double_divide(W, double_divide(inverse(inverse(X)), W)))
% 3.78/0.98 = { by lemma 35 }
% 3.78/0.98 inverse(X)
% 3.78/0.98
% 3.78/0.98 Lemma 37: inverse(inverse(X)) = X.
% 3.78/0.98 Proof:
% 3.78/0.98 inverse(inverse(X))
% 3.78/0.98 = { by lemma 3 R->L }
% 3.78/0.98 double_divide(double_divide(Y, multiply(inverse(inverse(X)), inverse(X))), double_divide(inverse(inverse(X)), multiply(double_divide(Y, Z), Z)))
% 3.78/0.98 = { by lemma 7 R->L }
% 3.78/0.98 double_divide(double_divide(Y, multiply(double_divide(inverse(inverse(inverse(X))), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(inverse(X), X), X)))), double_divide(inverse(inverse(X)), multiply(double_divide(Y, Z), Z)))
% 3.78/0.99 = { by lemma 36 }
% 3.78/0.99 double_divide(double_divide(Y, multiply(double_divide(inverse(X), multiply(double_divide(W, V), V)), double_divide(W, multiply(double_divide(inverse(X), X), X)))), double_divide(inverse(inverse(X)), multiply(double_divide(Y, Z), Z)))
% 3.78/0.99 = { by lemma 4 }
% 3.78/0.99 double_divide(double_divide(Y, inverse(double_divide(inverse(X), X))), double_divide(inverse(inverse(X)), multiply(double_divide(Y, Z), Z)))
% 3.78/0.99 = { by axiom 1 (multiply) R->L }
% 3.78/0.99 double_divide(double_divide(Y, multiply(X, inverse(X))), double_divide(inverse(inverse(X)), multiply(double_divide(Y, Z), Z)))
% 3.78/0.99 = { by lemma 27 }
% 3.78/0.99 double_divide(double_divide(Y, multiply(X, inverse(X))), double_divide(inverse(inverse(X)), double_divide(Y, inverse(multiply(U, inverse(U))))))
% 3.78/0.99 = { by lemma 32 }
% 3.78/0.99 double_divide(double_divide(Y, multiply(X, inverse(X))), double_divide(inverse(inverse(X)), double_divide(Y, multiply(X, inverse(X)))))
% 3.78/0.99 = { by lemma 35 }
% 3.78/0.99 X
% 3.78/0.99
% 3.78/0.99 Lemma 38: multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X)))) = inverse(V).
% 3.78/0.99 Proof:
% 3.78/0.99 multiply(double_divide(multiply(X, Y), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X))))
% 3.78/0.99 = { by axiom 1 (multiply) }
% 3.78/0.99 multiply(double_divide(inverse(double_divide(Y, X)), multiply(double_divide(Z, W), W)), double_divide(Z, multiply(V, double_divide(Y, X))))
% 3.78/0.99 = { by lemma 4 }
% 3.78/0.99 inverse(V)
% 3.78/0.99
% 3.78/0.99 Lemma 39: multiply(double_divide(multiply(X, Y), multiply(double_divide(inverse(Z), W), W)), double_divide(inverse(Z), multiply(double_divide(V, U), U))) = multiply(double_divide(Y, X), V).
% 3.78/0.99 Proof:
% 3.78/0.99 multiply(double_divide(multiply(X, Y), multiply(double_divide(inverse(Z), W), W)), double_divide(inverse(Z), multiply(double_divide(V, U), U)))
% 3.78/0.99 = { by lemma 5 }
% 3.78/0.99 multiply(double_divide(multiply(X, Y), multiply(double_divide(inverse(Z), W), W)), double_divide(inverse(Z), multiply(double_divide(V, double_divide(Y, X)), double_divide(Y, X))))
% 3.78/0.99 = { by lemma 38 }
% 3.78/0.99 inverse(double_divide(V, double_divide(Y, X)))
% 3.78/0.99 = { by axiom 1 (multiply) R->L }
% 3.78/0.99 multiply(double_divide(Y, X), V)
% 3.78/0.99
% 3.78/0.99 Lemma 40: double_divide(multiply(double_divide(X, Y), Y), double_divide(Z, double_divide(X, Z))) = double_divide(inverse(W), W).
% 3.78/0.99 Proof:
% 3.78/0.99 double_divide(multiply(double_divide(X, Y), Y), double_divide(Z, double_divide(X, Z)))
% 3.78/0.99 = { by lemma 3 R->L }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(double_divide(X, Y), Y), double_divide(Z, double_divide(X, Z))), U)), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 23 R->L }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(X, multiply(double_divide(X, Y), Y)), multiply(S, inverse(S))), U)), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 39 R->L }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(multiply(S, inverse(S)), multiply(X, multiply(double_divide(X, Y), Y))), multiply(double_divide(inverse(X2), Y2), Y2)), double_divide(inverse(X2), multiply(double_divide(U, Z2), Z2)))), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 28 }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(W2, inverse(W2)), multiply(double_divide(inverse(X2), Y2), Y2)), double_divide(inverse(X2), multiply(double_divide(U, Z2), Z2)))), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 28 R->L }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(multiply(V2, inverse(V2)), multiply(inverse(multiply(double_divide(W, U2), U2)), multiply(double_divide(inverse(multiply(double_divide(W, U2), U2)), T2), T2))), multiply(double_divide(inverse(X2), Y2), Y2)), double_divide(inverse(X2), multiply(double_divide(U, Z2), Z2)))), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 39 }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(inverse(multiply(double_divide(W, U2), U2)), multiply(double_divide(inverse(multiply(double_divide(W, U2), U2)), T2), T2)), multiply(V2, inverse(V2))), U)), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 23 }
% 3.78/0.99 double_divide(double_divide(V, multiply(double_divide(multiply(double_divide(inverse(multiply(double_divide(W, U2), U2)), T2), T2), double_divide(S2, double_divide(inverse(multiply(double_divide(W, U2), U2)), S2))), U)), double_divide(inverse(U), multiply(double_divide(V, T), T)))
% 3.78/0.99 = { by lemma 3 }
% 3.78/0.99 double_divide(multiply(double_divide(inverse(multiply(double_divide(W, U2), U2)), T2), T2), double_divide(S2, double_divide(inverse(multiply(double_divide(W, U2), U2)), S2)))
% 3.78/0.99 = { by lemma 20 }
% 3.78/0.99 double_divide(multiply(double_divide(inverse(multiply(double_divide(W, U2), U2)), T2), T2), W)
% 3.78/0.99 = { by lemma 16 }
% 3.78/0.99 double_divide(inverse(W), W)
% 3.78/0.99
% 3.78/0.99 Lemma 41: multiply(X, inverse(X)) = double_divide(inverse(Y), Y).
% 3.78/0.99 Proof:
% 3.78/0.99 multiply(X, inverse(X))
% 3.78/0.99 = { by lemma 32 R->L }
% 3.78/0.99 inverse(multiply(Z, inverse(Z)))
% 3.78/0.99 = { by lemma 32 R->L }
% 3.78/0.99 inverse(inverse(multiply(W, inverse(W))))
% 3.78/0.99 = { by lemma 34 R->L }
% 3.78/0.99 double_divide(inverse(multiply(W, inverse(W))), multiply(V, inverse(V)))
% 3.78/0.99 = { by lemma 26 R->L }
% 3.78/0.99 double_divide(multiply(inverse(multiply(double_divide(Y, U), U)), inverse(Y)), multiply(V, inverse(V)))
% 3.78/0.99 = { by lemma 23 }
% 3.78/0.99 double_divide(inverse(Y), double_divide(T, double_divide(inverse(multiply(double_divide(Y, U), U)), T)))
% 3.78/0.99 = { by lemma 20 }
% 3.78/0.99 double_divide(inverse(Y), Y)
% 3.78/0.99
% 3.78/0.99 Lemma 42: double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z))) = double_divide(W, double_divide(Y, W)).
% 3.78/0.99 Proof:
% 3.78/0.99 double_divide(double_divide(X, Y), double_divide(Z, double_divide(X, Z)))
% 3.78/0.99 = { by lemma 17 }
% 3.78/0.99 double_divide(double_divide(X, Y), double_divide(Y, double_divide(X, Y)))
% 3.78/0.99 = { by lemma 19 R->L }
% 3.78/0.99 double_divide(W, double_divide(Y, W))
% 3.78/0.99
% 3.78/0.99 Lemma 43: double_divide(double_divide(inverse(multiply(double_divide(X, Y), Y)), Z), X) = double_divide(W, double_divide(Z, W)).
% 3.78/0.99 Proof:
% 3.78/0.99 double_divide(double_divide(inverse(multiply(double_divide(X, Y), Y)), Z), X)
% 3.78/0.99 = { by lemma 20 R->L }
% 3.78/0.99 double_divide(double_divide(inverse(multiply(double_divide(X, Y), Y)), Z), double_divide(V, double_divide(inverse(multiply(double_divide(X, Y), Y)), V)))
% 3.78/0.99 = { by lemma 42 }
% 3.78/0.99 double_divide(W, double_divide(Z, W))
% 3.78/0.99
% 3.78/0.99 Lemma 44: double_divide(X, double_divide(inverse(Y), Y)) = inverse(X).
% 3.78/0.99 Proof:
% 3.78/0.99 double_divide(X, double_divide(inverse(Y), Y))
% 3.78/0.99 = { by lemma 41 R->L }
% 3.78/0.99 double_divide(X, multiply(Z, inverse(Z)))
% 3.78/0.99 = { by lemma 32 R->L }
% 3.78/0.99 double_divide(X, inverse(multiply(W, inverse(W))))
% 3.78/0.99 = { by lemma 27 R->L }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(inverse(U), U))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 40 R->L }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(multiply(double_divide(inverse(inverse(V)), T), T), double_divide(S, double_divide(inverse(inverse(V)), S))))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 35 }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(multiply(double_divide(inverse(inverse(V)), T), T), V))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 27 }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(double_divide(inverse(inverse(V)), inverse(multiply(X2, inverse(X2)))), V))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 37 }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(double_divide(V, inverse(multiply(X2, inverse(X2)))), V))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 32 }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(double_divide(V, multiply(Y2, inverse(Y2))), V))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by lemma 41 }
% 3.78/0.99 multiply(double_divide(X, double_divide(V, double_divide(double_divide(V, double_divide(inverse(U), U)), V))), double_divide(V, double_divide(inverse(U), U)))
% 3.78/0.99 = { by axiom 1 (multiply) }
% 3.78/0.99 inverse(double_divide(double_divide(V, double_divide(inverse(U), U)), double_divide(X, double_divide(V, double_divide(double_divide(V, double_divide(inverse(U), U)), V)))))
% 3.78/0.99 = { by lemma 43 R->L }
% 3.78/0.99 inverse(double_divide(double_divide(V, double_divide(inverse(U), U)), double_divide(X, double_divide(double_divide(inverse(multiply(double_divide(X, Z2), Z2)), double_divide(V, double_divide(inverse(U), U))), X))))
% 3.78/0.99 = { by lemma 23 R->L }
% 3.78/0.99 inverse(double_divide(multiply(double_divide(inverse(multiply(double_divide(X, Z2), Z2)), double_divide(V, double_divide(inverse(U), U))), double_divide(V, double_divide(inverse(U), U))), multiply(W2, inverse(W2))))
% 3.78/0.99 = { by lemma 16 }
% 3.78/0.99 inverse(double_divide(inverse(X), multiply(W2, inverse(W2))))
% 3.78/0.99 = { by lemma 34 }
% 3.78/0.99 inverse(inverse(inverse(X)))
% 3.78/0.99 = { by lemma 36 }
% 3.78/0.99 inverse(X)
% 3.78/0.99
% 3.78/0.99 Lemma 45: double_divide(inverse(X), X) = double_divide(Y, inverse(Y)).
% 3.78/0.99 Proof:
% 3.78/0.99 double_divide(inverse(X), X)
% 3.78/0.99 = { by lemma 40 R->L }
% 3.78/0.99 double_divide(multiply(double_divide(inverse(Y), Z), Z), double_divide(Y, double_divide(inverse(Y), Y)))
% 3.78/0.99 = { by lemma 44 }
% 3.78/0.99 double_divide(multiply(double_divide(inverse(Y), Z), Z), inverse(Y))
% 3.78/0.99 = { by lemma 27 }
% 3.78/0.99 double_divide(double_divide(inverse(Y), inverse(multiply(W, inverse(W)))), inverse(Y))
% 3.78/0.99 = { by lemma 32 }
% 3.78/0.99 double_divide(double_divide(inverse(Y), multiply(V, inverse(V))), inverse(Y))
% 3.78/0.99 = { by lemma 34 }
% 3.78/0.99 double_divide(inverse(inverse(Y)), inverse(Y))
% 3.78/0.99 = { by lemma 37 }
% 3.78/0.99 double_divide(Y, inverse(Y))
% 3.78/0.99
% 3.78/0.99 Lemma 46: multiply(inverse(multiply(X, inverse(X))), multiply(Y, inverse(Y))) = inverse(multiply(Z, inverse(Z))).
% 3.78/0.99 Proof:
% 3.78/0.99 multiply(inverse(multiply(X, inverse(X))), multiply(Y, inverse(Y)))
% 3.78/0.99 = { by lemma 29 R->L }
% 3.78/0.99 multiply(inverse(multiply(X, inverse(X))), double_divide(inverse(multiply(X, inverse(X))), inverse(multiply(W, inverse(W)))))
% 3.78/0.99 = { by lemma 30 }
% 3.78/0.99 inverse(multiply(Z, inverse(Z)))
% 3.78/0.99
% 3.78/0.99 Lemma 47: inverse(inverse(multiply(X, inverse(X)))) = multiply(Y, inverse(Y)).
% 3.78/0.99 Proof:
% 3.78/0.99 inverse(inverse(multiply(X, inverse(X))))
% 3.78/0.99 = { by lemma 4 R->L }
% 3.78/0.99 multiply(double_divide(inverse(multiply(Z, inverse(Z))), multiply(double_divide(W, V), V)), double_divide(W, multiply(inverse(multiply(X, inverse(X))), multiply(Z, inverse(Z)))))
% 3.78/0.99 = { by lemma 46 }
% 3.78/0.99 multiply(double_divide(inverse(multiply(Z, inverse(Z))), multiply(double_divide(W, V), V)), double_divide(W, inverse(multiply(U, inverse(U)))))
% 3.78/0.99 = { by lemma 13 }
% 3.78/0.99 multiply(multiply(double_divide(inverse(multiply(Z, inverse(Z))), T), T), multiply(W, double_divide(W, inverse(multiply(U, inverse(U))))))
% 3.78/0.99 = { by lemma 27 }
% 3.78/0.99 multiply(double_divide(inverse(multiply(Z, inverse(Z))), inverse(multiply(S, inverse(S)))), multiply(W, double_divide(W, inverse(multiply(U, inverse(U))))))
% 3.78/0.99 = { by lemma 29 }
% 3.78/0.99 multiply(multiply(X2, inverse(X2)), multiply(W, double_divide(W, inverse(multiply(U, inverse(U))))))
% 3.78/0.99 = { by lemma 25 }
% 3.78/0.99 multiply(double_divide(inverse(multiply(U, inverse(U))), Y2), Y2)
% 3.78/0.99 = { by lemma 27 }
% 3.78/0.99 double_divide(inverse(multiply(U, inverse(U))), inverse(multiply(Z2, inverse(Z2))))
% 3.78/0.99 = { by lemma 29 }
% 3.78/0.99 multiply(Y, inverse(Y))
% 3.78/0.99
% 3.78/0.99 Lemma 48: multiply(double_divide(X, inverse(X)), double_divide(Y, inverse(Y))) = double_divide(Z, inverse(Z)).
% 3.78/0.99 Proof:
% 3.78/0.99 multiply(double_divide(X, inverse(X)), double_divide(Y, inverse(Y)))
% 3.78/0.99 = { by lemma 45 R->L }
% 3.78/0.99 multiply(double_divide(X, inverse(X)), double_divide(inverse(W), W))
% 3.78/0.99 = { by lemma 41 R->L }
% 3.78/0.99 multiply(double_divide(X, inverse(X)), multiply(V, inverse(V)))
% 3.78/1.00 = { by lemma 47 R->L }
% 3.78/1.00 multiply(double_divide(X, inverse(X)), inverse(inverse(multiply(U, inverse(U)))))
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 multiply(double_divide(inverse(T), T), inverse(inverse(multiply(U, inverse(U)))))
% 3.78/1.00 = { by lemma 41 R->L }
% 3.78/1.00 multiply(multiply(S, inverse(S)), inverse(inverse(multiply(U, inverse(U)))))
% 3.78/1.00 = { by axiom 1 (multiply) }
% 3.78/1.00 inverse(double_divide(inverse(inverse(multiply(U, inverse(U)))), multiply(S, inverse(S))))
% 3.78/1.00 = { by lemma 34 }
% 3.78/1.00 inverse(inverse(inverse(inverse(multiply(U, inverse(U))))))
% 3.78/1.00 = { by lemma 47 }
% 3.78/1.00 inverse(inverse(multiply(X2, inverse(X2))))
% 3.78/1.00 = { by lemma 47 }
% 3.78/1.00 multiply(Y2, inverse(Y2))
% 3.78/1.00 = { by lemma 41 }
% 3.78/1.00 double_divide(inverse(Z2), Z2)
% 3.78/1.00 = { by lemma 45 }
% 3.78/1.00 double_divide(Z, inverse(Z))
% 3.78/1.00
% 3.78/1.00 Lemma 49: inverse(double_divide(X, inverse(X))) = double_divide(Y, inverse(Y)).
% 3.78/1.00 Proof:
% 3.78/1.00 inverse(double_divide(X, inverse(X)))
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 inverse(double_divide(inverse(Z), Z))
% 3.78/1.00 = { by axiom 1 (multiply) R->L }
% 3.78/1.00 multiply(Z, inverse(Z))
% 3.78/1.00 = { by lemma 41 }
% 3.78/1.00 double_divide(inverse(W), W)
% 3.78/1.00 = { by lemma 45 }
% 3.78/1.00 double_divide(Y, inverse(Y))
% 3.78/1.00
% 3.78/1.00 Lemma 50: multiply(multiply(X, Y), double_divide(Z, inverse(Z))) = multiply(X, Y).
% 3.78/1.00 Proof:
% 3.78/1.00 multiply(multiply(X, Y), double_divide(Z, inverse(Z)))
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 multiply(multiply(X, Y), double_divide(inverse(W), W))
% 3.78/1.00 = { by lemma 41 R->L }
% 3.78/1.00 multiply(multiply(X, Y), multiply(V, inverse(V)))
% 3.78/1.00 = { by axiom 1 (multiply) }
% 3.78/1.00 multiply(inverse(double_divide(Y, X)), multiply(V, inverse(V)))
% 3.78/1.00 = { by lemma 33 }
% 3.78/1.00 inverse(double_divide(Y, X))
% 3.78/1.00 = { by axiom 1 (multiply) R->L }
% 3.78/1.00 multiply(X, Y)
% 3.78/1.00
% 3.78/1.00 Lemma 51: double_divide(X, double_divide(multiply(Y, Z), X)) = multiply(Y, Z).
% 3.78/1.00 Proof:
% 3.78/1.00 double_divide(X, double_divide(multiply(Y, Z), X))
% 3.78/1.00 = { by lemma 42 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, double_divide(V, inverse(V)))))
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, double_divide(inverse(U), U))))
% 3.78/1.00 = { by lemma 41 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, multiply(T, inverse(T)))))
% 3.78/1.00 = { by lemma 32 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, inverse(multiply(S, inverse(S))))))
% 3.78/1.00 = { by lemma 27 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(double_divide(V, inverse(V)), multiply(double_divide(W, X2), X2)))
% 3.78/1.00 = { by lemma 49 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(Y, Z)), double_divide(inverse(double_divide(Y2, inverse(Y2))), multiply(double_divide(W, X2), X2)))
% 3.78/1.00 = { by lemma 50 R->L }
% 3.78/1.00 double_divide(double_divide(W, multiply(multiply(Y, Z), double_divide(Y2, inverse(Y2)))), double_divide(inverse(double_divide(Y2, inverse(Y2))), multiply(double_divide(W, X2), X2)))
% 3.78/1.00 = { by lemma 3 }
% 3.78/1.00 multiply(Y, Z)
% 3.78/1.00
% 3.78/1.00 Lemma 52: multiply(inverse(X), double_divide(Y, inverse(Y))) = inverse(X).
% 3.78/1.00 Proof:
% 3.78/1.00 multiply(inverse(X), double_divide(Y, inverse(Y)))
% 3.78/1.00 = { by lemma 16 R->L }
% 3.78/1.00 multiply(multiply(double_divide(inverse(multiply(double_divide(X, Z), Z)), W), W), double_divide(Y, inverse(Y)))
% 3.78/1.00 = { by lemma 50 }
% 3.78/1.00 multiply(double_divide(inverse(multiply(double_divide(X, Z), Z)), W), W)
% 3.78/1.00 = { by lemma 16 }
% 3.78/1.00 inverse(X)
% 3.78/1.00
% 3.78/1.00 Lemma 53: multiply(double_divide(X, Y), Y) = multiply(inverse(Z), double_divide(inverse(Z), X)).
% 3.78/1.00 Proof:
% 3.78/1.00 multiply(double_divide(X, Y), Y)
% 3.78/1.00 = { by lemma 6 }
% 3.78/1.00 multiply(double_divide(X, double_divide(inverse(Z), X)), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 42 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, double_divide(V, inverse(V))))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, double_divide(inverse(U), U)))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 41 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, multiply(T, inverse(T))))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 32 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(double_divide(V, inverse(V)), double_divide(W, inverse(multiply(S, inverse(S)))))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 27 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(double_divide(V, inverse(V)), multiply(double_divide(W, X2), X2))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 49 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, inverse(Z)), double_divide(inverse(double_divide(Y2, inverse(Y2))), multiply(double_divide(W, X2), X2))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 52 R->L }
% 3.78/1.00 multiply(double_divide(double_divide(W, multiply(inverse(Z), double_divide(Y2, inverse(Y2)))), double_divide(inverse(double_divide(Y2, inverse(Y2))), multiply(double_divide(W, X2), X2))), double_divide(inverse(Z), X))
% 3.78/1.00 = { by lemma 3 }
% 3.78/1.00 multiply(inverse(Z), double_divide(inverse(Z), X))
% 3.78/1.00
% 3.78/1.00 Lemma 54: double_divide(double_divide(X, Y), X) = double_divide(Z, double_divide(Y, Z)).
% 3.78/1.00 Proof:
% 3.78/1.00 double_divide(double_divide(X, Y), X)
% 3.78/1.00 = { by lemma 37 R->L }
% 3.78/1.00 double_divide(double_divide(inverse(inverse(X)), Y), X)
% 3.78/1.00 = { by lemma 52 R->L }
% 3.78/1.00 double_divide(double_divide(inverse(multiply(inverse(X), double_divide(W, inverse(W)))), Y), X)
% 3.78/1.00 = { by lemma 45 R->L }
% 3.78/1.00 double_divide(double_divide(inverse(multiply(inverse(X), double_divide(inverse(X), X))), Y), X)
% 3.78/1.00 = { by lemma 53 R->L }
% 3.78/1.00 double_divide(double_divide(inverse(multiply(double_divide(X, V), V)), Y), X)
% 3.78/1.00 = { by lemma 43 }
% 3.78/1.00 double_divide(Z, double_divide(Y, Z))
% 3.78/1.00
% 3.78/1.00 Lemma 55: multiply(X, double_divide(X, Y)) = inverse(Y).
% 3.78/1.00 Proof:
% 3.78/1.00 multiply(X, double_divide(X, Y))
% 3.78/1.00 = { by lemma 51 R->L }
% 3.78/1.00 double_divide(Z, double_divide(multiply(X, double_divide(X, Y)), Z))
% 3.78/1.00 = { by lemma 54 R->L }
% 3.78/1.00 double_divide(double_divide(Y, multiply(X, double_divide(X, Y))), Y)
% 3.78/1.00 = { by lemma 8 R->L }
% 3.78/1.00 double_divide(double_divide(double_divide(W, multiply(double_divide(Y, V), V)), double_divide(inverse(multiply(X, double_divide(X, Y))), multiply(double_divide(W, U), U))), Y)
% 3.78/1.00 = { by lemma 25 R->L }
% 3.78/1.00 double_divide(double_divide(double_divide(W, multiply(multiply(T, inverse(T)), multiply(X, double_divide(X, Y)))), double_divide(inverse(multiply(X, double_divide(X, Y))), multiply(double_divide(W, U), U))), Y)
% 3.78/1.00 = { by lemma 3 }
% 3.78/1.00 double_divide(multiply(T, inverse(T)), Y)
% 3.78/1.00 = { by lemma 41 }
% 3.78/1.00 double_divide(double_divide(inverse(S), S), Y)
% 3.78/1.00 = { by lemma 45 }
% 3.78/1.00 double_divide(double_divide(Y, inverse(Y)), Y)
% 3.78/1.00 = { by lemma 54 }
% 3.78/1.00 double_divide(Y, double_divide(inverse(Y), Y))
% 3.78/1.00 = { by lemma 44 }
% 3.78/1.00 inverse(Y)
% 3.78/1.00
% 3.78/1.00 Lemma 56: double_divide(X, double_divide(Y, X)) = Y.
% 3.78/1.00 Proof:
% 3.78/1.00 double_divide(X, double_divide(Y, X))
% 3.78/1.00 = { by lemma 19 }
% 3.78/1.00 double_divide(double_divide(Z, inverse(W)), double_divide(Y, double_divide(Z, inverse(W))))
% 3.78/1.01 = { by lemma 55 R->L }
% 3.78/1.01 double_divide(double_divide(Z, inverse(W)), double_divide(Y, double_divide(Z, multiply(inverse(V), double_divide(inverse(V), W)))))
% 3.78/1.01 = { by lemma 53 R->L }
% 3.78/1.01 double_divide(double_divide(Z, inverse(W)), double_divide(Y, double_divide(Z, multiply(double_divide(W, U), U))))
% 3.78/1.01 = { by lemma 12 R->L }
% 3.78/1.01 double_divide(double_divide(Z, inverse(W)), double_divide(multiply(W, Y), multiply(double_divide(Z, T), T)))
% 3.78/1.01 = { by lemma 55 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(Y, double_divide(Y, W))), double_divide(multiply(W, Y), multiply(double_divide(Z, T), T)))
% 3.78/1.01 = { by lemma 9 }
% 3.78/1.01 Y
% 3.78/1.01
% 3.78/1.01 Lemma 57: double_divide(double_divide(X, inverse(X)), double_divide(Y, inverse(Y))) = double_divide(Z, inverse(Z)).
% 3.78/1.01 Proof:
% 3.78/1.01 double_divide(double_divide(X, inverse(X)), double_divide(Y, inverse(Y)))
% 3.78/1.01 = { by lemma 49 R->L }
% 3.78/1.01 double_divide(double_divide(X, inverse(X)), inverse(double_divide(W, inverse(W))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 double_divide(double_divide(inverse(V), V), inverse(double_divide(W, inverse(W))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 double_divide(double_divide(inverse(V), V), inverse(double_divide(inverse(U), U)))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 double_divide(multiply(T, inverse(T)), inverse(double_divide(inverse(U), U)))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 double_divide(multiply(T, inverse(T)), inverse(multiply(S, inverse(S))))
% 3.78/1.01 = { by lemma 27 R->L }
% 3.78/1.01 multiply(double_divide(multiply(T, inverse(T)), X2), X2)
% 3.78/1.01 = { by lemma 25 R->L }
% 3.78/1.01 multiply(multiply(Y2, inverse(Y2)), multiply(Z2, double_divide(Z2, multiply(T, inverse(T)))))
% 3.78/1.01 = { by lemma 29 R->L }
% 3.78/1.01 multiply(double_divide(inverse(multiply(W2, inverse(W2))), inverse(multiply(V2, inverse(V2)))), multiply(Z2, double_divide(Z2, multiply(T, inverse(T)))))
% 3.78/1.01 = { by lemma 27 R->L }
% 3.78/1.01 multiply(multiply(double_divide(inverse(multiply(W2, inverse(W2))), U2), U2), multiply(Z2, double_divide(Z2, multiply(T, inverse(T)))))
% 3.78/1.01 = { by lemma 13 R->L }
% 3.78/1.01 multiply(double_divide(inverse(multiply(W2, inverse(W2))), multiply(double_divide(Z2, T2), T2)), double_divide(Z2, multiply(T, inverse(T))))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 multiply(double_divide(inverse(multiply(W2, inverse(W2))), multiply(double_divide(Z2, T2), T2)), double_divide(Z2, inverse(multiply(S2, inverse(S2)))))
% 3.78/1.01 = { by lemma 46 R->L }
% 3.78/1.01 multiply(double_divide(inverse(multiply(W2, inverse(W2))), multiply(double_divide(Z2, T2), T2)), double_divide(Z2, multiply(inverse(multiply(X3, inverse(X3))), multiply(W2, inverse(W2)))))
% 3.78/1.01 = { by lemma 32 }
% 3.78/1.01 multiply(double_divide(inverse(multiply(W2, inverse(W2))), multiply(double_divide(Z2, T2), T2)), double_divide(Z2, multiply(multiply(Y3, inverse(Y3)), multiply(W2, inverse(W2)))))
% 3.78/1.01 = { by lemma 4 }
% 3.78/1.01 inverse(multiply(Y3, inverse(Y3)))
% 3.78/1.01 = { by lemma 32 }
% 3.78/1.01 multiply(Z3, inverse(Z3))
% 3.78/1.01 = { by lemma 41 }
% 3.78/1.01 double_divide(inverse(W3), W3)
% 3.78/1.01 = { by lemma 45 }
% 3.78/1.01 double_divide(Z, inverse(Z))
% 3.78/1.01
% 3.78/1.01 Lemma 58: double_divide(X, double_divide(Y, inverse(Y))) = inverse(X).
% 3.78/1.01 Proof:
% 3.78/1.01 double_divide(X, double_divide(Y, inverse(Y)))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 double_divide(X, double_divide(inverse(Z), Z))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 double_divide(X, multiply(W, inverse(W)))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 double_divide(X, inverse(multiply(V, inverse(V))))
% 3.78/1.01 = { by lemma 27 R->L }
% 3.78/1.01 multiply(double_divide(X, double_divide(U, inverse(U))), double_divide(U, inverse(U)))
% 3.78/1.01 = { by lemma 48 R->L }
% 3.78/1.01 multiply(double_divide(X, double_divide(U, inverse(U))), multiply(double_divide(T, inverse(T)), double_divide(S, inverse(S))))
% 3.78/1.01 = { by lemma 56 R->L }
% 3.78/1.01 multiply(double_divide(X, double_divide(X2, double_divide(double_divide(U, inverse(U)), X2))), multiply(double_divide(T, inverse(T)), double_divide(S, inverse(S))))
% 3.78/1.01 = { by lemma 57 R->L }
% 3.78/1.01 multiply(double_divide(X, double_divide(X2, double_divide(double_divide(U, inverse(U)), X2))), multiply(double_divide(T, inverse(T)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2)))))
% 3.78/1.01 = { by lemma 23 R->L }
% 3.78/1.01 multiply(double_divide(multiply(double_divide(U, inverse(U)), X), multiply(Z2, inverse(Z2))), multiply(double_divide(T, inverse(T)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2)))))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 multiply(double_divide(multiply(double_divide(U, inverse(U)), X), inverse(multiply(W2, inverse(W2)))), multiply(double_divide(T, inverse(T)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2)))))
% 3.78/1.01 = { by lemma 27 R->L }
% 3.78/1.01 multiply(multiply(double_divide(multiply(double_divide(U, inverse(U)), X), V2), V2), multiply(double_divide(T, inverse(T)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2)))))
% 3.78/1.01 = { by lemma 13 R->L }
% 3.78/1.01 multiply(double_divide(multiply(double_divide(U, inverse(U)), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 multiply(double_divide(multiply(double_divide(inverse(T2), T2), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), double_divide(Y2, inverse(Y2))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 multiply(double_divide(multiply(double_divide(inverse(T2), T2), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), double_divide(inverse(S2), S2)))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 multiply(double_divide(multiply(multiply(X3, inverse(X3)), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), double_divide(inverse(S2), S2)))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 multiply(double_divide(multiply(multiply(X3, inverse(X3)), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), multiply(Y3, inverse(Y3))))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 multiply(double_divide(multiply(inverse(multiply(Z3, inverse(Z3))), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), multiply(Y3, inverse(Y3))))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 multiply(double_divide(multiply(inverse(multiply(Z3, inverse(Z3))), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), inverse(multiply(W3, inverse(W3)))))
% 3.78/1.01 = { by lemma 30 R->L }
% 3.78/1.01 multiply(double_divide(multiply(inverse(multiply(Z3, inverse(Z3))), X), multiply(double_divide(double_divide(T, inverse(T)), U2), U2)), double_divide(double_divide(T, inverse(T)), multiply(X, double_divide(X, inverse(multiply(Z3, inverse(Z3)))))))
% 3.78/1.01 = { by lemma 38 }
% 3.78/1.01 inverse(X)
% 3.78/1.01
% 3.78/1.01 Lemma 59: multiply(X, double_divide(Y, inverse(Y))) = X.
% 3.78/1.01 Proof:
% 3.78/1.01 multiply(X, double_divide(Y, inverse(Y)))
% 3.78/1.01 = { by lemma 51 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(multiply(X, double_divide(Y, inverse(Y))), double_divide(Z, multiply(X, double_divide(Y, inverse(Y))))))
% 3.78/1.01 = { by lemma 17 }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(double_divide(W, inverse(W)), double_divide(Z, double_divide(W, inverse(W)))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(double_divide(W, inverse(W)), double_divide(Z, double_divide(inverse(V), V))))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(double_divide(W, inverse(W)), double_divide(Z, multiply(U, inverse(U)))))
% 3.78/1.01 = { by lemma 32 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(double_divide(W, inverse(W)), double_divide(Z, inverse(multiply(T, inverse(T))))))
% 3.78/1.01 = { by lemma 27 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(double_divide(W, inverse(W)), multiply(double_divide(Z, S), S)))
% 3.78/1.01 = { by lemma 48 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(Y, inverse(Y)))), double_divide(multiply(double_divide(X2, inverse(X2)), double_divide(Y2, inverse(Y2))), multiply(double_divide(Z, S), S)))
% 3.78/1.01 = { by lemma 57 R->L }
% 3.78/1.01 double_divide(double_divide(Z, multiply(X, double_divide(double_divide(Y2, inverse(Y2)), double_divide(X2, inverse(X2))))), double_divide(multiply(double_divide(X2, inverse(X2)), double_divide(Y2, inverse(Y2))), multiply(double_divide(Z, S), S)))
% 3.78/1.01 = { by lemma 9 }
% 3.78/1.01 X
% 3.78/1.01
% 3.78/1.01 Lemma 60: multiply(double_divide(X, inverse(X)), Y) = Y.
% 3.78/1.01 Proof:
% 3.78/1.01 multiply(double_divide(X, inverse(X)), Y)
% 3.78/1.01 = { by lemma 59 R->L }
% 3.78/1.01 multiply(double_divide(X, inverse(X)), multiply(Y, double_divide(Z, inverse(Z))))
% 3.78/1.01 = { by lemma 45 R->L }
% 3.78/1.01 multiply(double_divide(inverse(W), W), multiply(Y, double_divide(Z, inverse(Z))))
% 3.78/1.01 = { by lemma 41 R->L }
% 3.78/1.01 multiply(multiply(V, inverse(V)), multiply(Y, double_divide(Z, inverse(Z))))
% 3.78/1.01 = { by lemma 24 R->L }
% 3.78/1.01 multiply(double_divide(U, double_divide(Y, U)), double_divide(Z, inverse(Z)))
% 3.78/1.01 = { by lemma 59 }
% 3.78/1.01 double_divide(U, double_divide(Y, U))
% 3.78/1.01 = { by lemma 56 }
% 3.78/1.01 Y
% 3.78/1.01
% 3.78/1.01 Lemma 61: multiply(double_divide(double_divide(X, multiply(Y, Z)), W), W) = multiply(Y, double_divide(inverse(Z), multiply(double_divide(X, V), V))).
% 3.78/1.01 Proof:
% 3.78/1.01 multiply(double_divide(double_divide(X, multiply(Y, Z)), W), W)
% 3.78/1.01 = { by lemma 6 }
% 3.78/1.01 multiply(double_divide(double_divide(X, multiply(Y, Z)), double_divide(inverse(Z), multiply(double_divide(X, V), V))), double_divide(inverse(Z), multiply(double_divide(X, V), V)))
% 3.78/1.01 = { by lemma 3 }
% 3.78/1.02 multiply(Y, double_divide(inverse(Z), multiply(double_divide(X, V), V)))
% 3.78/1.02
% 3.78/1.02 Lemma 62: multiply(X, Y) = double_divide(inverse(X), inverse(Y)).
% 3.78/1.02 Proof:
% 3.78/1.02 multiply(X, Y)
% 3.78/1.02 = { by axiom 1 (multiply) }
% 3.78/1.02 inverse(double_divide(Y, X))
% 3.78/1.02 = { by lemma 58 R->L }
% 3.78/1.02 double_divide(double_divide(Y, X), double_divide(Z, inverse(Z)))
% 3.78/1.02 = { by lemma 45 R->L }
% 3.78/1.02 double_divide(double_divide(Y, X), double_divide(inverse(W), W))
% 3.78/1.02 = { by lemma 41 R->L }
% 3.78/1.02 double_divide(double_divide(Y, X), multiply(V, inverse(V)))
% 3.78/1.02 = { by lemma 32 R->L }
% 3.78/1.02 double_divide(double_divide(Y, X), inverse(multiply(U, inverse(U))))
% 3.78/1.02 = { by lemma 27 R->L }
% 3.78/1.02 multiply(double_divide(double_divide(Y, X), T), T)
% 3.78/1.02 = { by lemma 60 R->L }
% 3.78/1.02 multiply(double_divide(double_divide(Y, multiply(double_divide(S, inverse(S)), X)), T), T)
% 3.78/1.02 = { by lemma 61 }
% 3.78/1.02 multiply(double_divide(S, inverse(S)), double_divide(inverse(X), multiply(double_divide(Y, X2), X2)))
% 3.78/1.02 = { by lemma 60 }
% 3.78/1.02 double_divide(inverse(X), multiply(double_divide(Y, X2), X2))
% 3.78/1.02 = { by lemma 27 }
% 3.78/1.02 double_divide(inverse(X), double_divide(Y, inverse(multiply(Y2, inverse(Y2)))))
% 3.78/1.02 = { by lemma 32 }
% 3.78/1.02 double_divide(inverse(X), double_divide(Y, multiply(Z2, inverse(Z2))))
% 3.78/1.02 = { by lemma 41 }
% 3.78/1.02 double_divide(inverse(X), double_divide(Y, double_divide(inverse(W2), W2)))
% 3.78/1.02 = { by lemma 45 }
% 3.78/1.02 double_divide(inverse(X), double_divide(Y, double_divide(V2, inverse(V2))))
% 3.78/1.02 = { by lemma 58 }
% 3.78/1.02 double_divide(inverse(X), inverse(Y))
% 3.78/1.02
% 3.78/1.02 Lemma 63: inverse(multiply(X, Y)) = double_divide(Y, X).
% 3.78/1.02 Proof:
% 3.78/1.02 inverse(multiply(X, Y))
% 3.78/1.02 = { by axiom 1 (multiply) }
% 3.78/1.02 inverse(inverse(double_divide(Y, X)))
% 3.78/1.02 = { by lemma 37 }
% 3.78/1.02 double_divide(Y, X)
% 3.78/1.02
% 3.78/1.02 Lemma 64: multiply(inverse(X), X) = double_divide(Y, inverse(Y)).
% 3.78/1.02 Proof:
% 3.78/1.02 multiply(inverse(X), X)
% 3.78/1.02 = { by lemma 56 R->L }
% 3.78/1.02 multiply(inverse(X), double_divide(Z, double_divide(X, Z)))
% 3.78/1.02 = { by lemma 63 R->L }
% 3.78/1.02 multiply(inverse(X), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 58 R->L }
% 3.78/1.02 multiply(double_divide(X, double_divide(W, inverse(W))), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 45 R->L }
% 3.78/1.02 multiply(double_divide(X, double_divide(inverse(V), V)), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 41 R->L }
% 3.78/1.02 multiply(double_divide(X, multiply(U, inverse(U))), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 32 R->L }
% 3.78/1.02 multiply(double_divide(X, inverse(multiply(T, inverse(T)))), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 27 R->L }
% 3.78/1.02 multiply(multiply(double_divide(X, Z), Z), inverse(multiply(double_divide(X, Z), Z)))
% 3.78/1.02 = { by lemma 41 }
% 3.78/1.02 double_divide(inverse(S), S)
% 3.78/1.02 = { by lemma 45 }
% 3.78/1.02 double_divide(Y, inverse(Y))
% 3.78/1.02
% 3.78/1.02 Lemma 65: multiply(double_divide(X, double_divide(multiply(Y, Z), X)), W) = multiply(Y, double_divide(inverse(Z), inverse(W))).
% 3.78/1.02 Proof:
% 3.78/1.02 multiply(double_divide(X, double_divide(multiply(Y, Z), X)), W)
% 3.78/1.02 = { by lemma 43 R->L }
% 3.78/1.02 multiply(double_divide(double_divide(inverse(multiply(double_divide(W, V), V)), multiply(Y, Z)), W), W)
% 3.78/1.02 = { by lemma 61 }
% 3.78/1.02 multiply(Y, double_divide(inverse(Z), multiply(double_divide(inverse(multiply(double_divide(W, V), V)), U), U)))
% 3.78/1.02 = { by lemma 16 }
% 3.78/1.02 multiply(Y, double_divide(inverse(Z), inverse(W)))
% 3.78/1.02
% 3.78/1.02 Goal 1 (prove_these_axioms): tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3)) = tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3))).
% 3.78/1.02 Proof:
% 3.78/1.02 tuple(multiply(inverse(a1), a1), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 3.78/1.02 = { by lemma 64 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), multiply(multiply(inverse(b2), b2), a2), multiply(multiply(a3, b3), c3))
% 3.78/1.02 = { by lemma 64 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), multiply(double_divide(Y, inverse(Y)), a2), multiply(multiply(a3, b3), c3))
% 3.78/1.02 = { by lemma 60 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(multiply(a3, b3), c3))
% 3.78/1.02 = { by lemma 56 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(multiply(a3, b3), double_divide(double_divide(inverse(Z), Z), double_divide(c3, double_divide(inverse(Z), Z)))))
% 3.78/1.02 = { by lemma 44 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(multiply(a3, b3), double_divide(double_divide(inverse(Z), Z), inverse(c3))))
% 3.78/1.02 = { by lemma 45 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(multiply(a3, b3), double_divide(double_divide(W, inverse(W)), inverse(c3))))
% 3.78/1.02 = { by lemma 49 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(multiply(a3, b3), double_divide(inverse(double_divide(V, inverse(V))), inverse(c3))))
% 3.78/1.02 = { by lemma 65 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(double_divide(U, double_divide(multiply(multiply(a3, b3), double_divide(V, inverse(V))), U)), c3))
% 3.78/1.02 = { by lemma 50 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(double_divide(U, double_divide(multiply(a3, b3), U)), c3))
% 3.78/1.02 = { by lemma 65 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(a3, double_divide(inverse(b3), inverse(c3))))
% 3.78/1.02 = { by lemma 62 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), inverse(double_divide(inverse(b3), inverse(c3)))))
% 3.78/1.02 = { by axiom 1 (multiply) R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), multiply(inverse(c3), inverse(b3))))
% 3.78/1.02 = { by lemma 62 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), double_divide(inverse(inverse(c3)), inverse(inverse(b3)))))
% 3.78/1.02 = { by lemma 8 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), double_divide(double_divide(T, multiply(double_divide(inverse(inverse(c3)), S), S)), double_divide(inverse(inverse(inverse(b3))), multiply(double_divide(T, X2), X2)))))
% 3.78/1.02 = { by lemma 36 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), double_divide(double_divide(T, multiply(double_divide(inverse(inverse(c3)), S), S)), double_divide(inverse(b3), multiply(double_divide(T, X2), X2)))))
% 3.78/1.02 = { by lemma 8 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), double_divide(inverse(inverse(c3)), b3)))
% 3.78/1.02 = { by lemma 37 }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), double_divide(c3, b3)))
% 3.78/1.02 = { by lemma 63 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, double_divide(inverse(a3), inverse(multiply(b3, c3))))
% 3.78/1.02 = { by lemma 62 R->L }
% 3.78/1.02 tuple(double_divide(X, inverse(X)), a2, multiply(a3, multiply(b3, c3)))
% 3.78/1.02 = { by lemma 64 R->L }
% 3.78/1.02 tuple(multiply(inverse(b1), b1), a2, multiply(a3, multiply(b3, c3)))
% 3.78/1.02 % SZS output end Proof
% 3.78/1.02
% 3.78/1.02 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------