%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : GRP654+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n017.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 10:11:34 AM UTC 2026
% Result : Theorem 10.23s 1.77s
% Output : Proof 11.72s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : GRP654+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n017.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sun Sep 27 10:26:34 UTC 2026
% 0.10/0.37 % CPUTime :
% 0.10/0.37 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 10.23/1.77 Command-line arguments: --no-flatten-goal
% 10.23/1.77
% 10.23/1.77 % SZS status Theorem
% 10.23/1.77
% 11.72/1.93 % SZS output start Proof
% 11.72/1.93 Axiom 1 (f02): ld(X, mult(X, Y)) = Y.
% 11.72/1.93 Axiom 2 (f04): rd(mult(X, Y), Y) = X.
% 11.72/1.93 Axiom 3 (f01): mult(X, ld(X, Y)) = Y.
% 11.72/1.93 Axiom 4 (f03): mult(rd(X, Y), Y) = X.
% 11.72/1.93 Axiom 5 (f05): mult(X, mult(Y, mult(X, Z))) = mult(mult(mult(X, Y), X), Z).
% 11.72/1.93
% 11.72/1.93 Lemma 6: rd(mult(X, mult(Y, mult(X, Z))), Z) = mult(mult(X, Y), X).
% 11.72/1.93 Proof:
% 11.72/1.93 rd(mult(X, mult(Y, mult(X, Z))), Z)
% 11.72/1.93 = { by axiom 5 (f05) }
% 11.72/1.93 rd(mult(mult(mult(X, Y), X), Z), Z)
% 11.72/1.93 = { by axiom 2 (f04) }
% 11.72/1.93 mult(mult(X, Y), X)
% 11.72/1.93
% 11.72/1.93 Lemma 7: rd(mult(X, mult(Y, Z)), ld(X, Z)) = mult(mult(X, Y), X).
% 11.72/1.93 Proof:
% 11.72/1.93 rd(mult(X, mult(Y, Z)), ld(X, Z))
% 11.72/1.93 = { by axiom 3 (f01) R->L }
% 11.72/1.93 rd(mult(X, mult(Y, mult(X, ld(X, Z)))), ld(X, Z))
% 11.72/1.93 = { by lemma 6 }
% 11.72/1.93 mult(mult(X, Y), X)
% 11.72/1.93
% 11.72/1.94 Lemma 8: mult(mult(X, rd(Y, Z)), X) = rd(mult(X, Y), ld(X, Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(mult(X, rd(Y, Z)), X)
% 11.72/1.94 = { by lemma 7 R->L }
% 11.72/1.94 rd(mult(X, mult(rd(Y, Z), Z)), ld(X, Z))
% 11.72/1.94 = { by axiom 4 (f03) }
% 11.72/1.94 rd(mult(X, Y), ld(X, Z))
% 11.72/1.94
% 11.72/1.94 Lemma 9: rd(rd(mult(X, Y), ld(X, Z)), X) = mult(X, rd(Y, Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(rd(mult(X, Y), ld(X, Z)), X)
% 11.72/1.94 = { by lemma 8 R->L }
% 11.72/1.94 rd(mult(mult(X, rd(Y, Z)), X), X)
% 11.72/1.94 = { by axiom 2 (f04) }
% 11.72/1.94 mult(X, rd(Y, Z))
% 11.72/1.94
% 11.72/1.94 Lemma 10: mult(X, rd(ld(X, Y), Y)) = rd(X, X).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(X, rd(ld(X, Y), Y))
% 11.72/1.94 = { by lemma 9 R->L }
% 11.72/1.94 rd(rd(mult(X, ld(X, Y)), ld(X, Y)), X)
% 11.72/1.94 = { by axiom 2 (f04) }
% 11.72/1.94 rd(X, X)
% 11.72/1.94
% 11.72/1.94 Lemma 11: rd(ld(X, Y), Y) = ld(X, rd(X, X)).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(ld(X, Y), Y)
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(X, mult(X, rd(ld(X, Y), Y)))
% 11.72/1.94 = { by lemma 10 }
% 11.72/1.94 ld(X, rd(X, X))
% 11.72/1.94
% 11.72/1.94 Lemma 12: rd(Y, mult(X, Y)) = ld(X, rd(X, X)).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(Y, mult(X, Y))
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 rd(ld(X, mult(X, Y)), mult(X, Y))
% 11.72/1.94 = { by lemma 11 }
% 11.72/1.94 ld(X, rd(X, X))
% 11.72/1.94
% 11.72/1.94 Lemma 13: rd(ld(X, Y), Y) = rd(Z, mult(X, Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(ld(X, Y), Y)
% 11.72/1.94 = { by lemma 11 }
% 11.72/1.94 ld(X, rd(X, X))
% 11.72/1.94 = { by lemma 12 R->L }
% 11.72/1.94 rd(Z, mult(X, Z))
% 11.72/1.94
% 11.72/1.94 Lemma 14: ld(rd(X, Y), X) = Y.
% 11.72/1.94 Proof:
% 11.72/1.94 ld(rd(X, Y), X)
% 11.72/1.94 = { by axiom 4 (f03) R->L }
% 11.72/1.94 ld(rd(X, Y), mult(rd(X, Y), Y))
% 11.72/1.94 = { by axiom 1 (f02) }
% 11.72/1.94 Y
% 11.72/1.94
% 11.72/1.94 Lemma 15: rd(X, mult(rd(Y, Z), X)) = rd(Z, Y).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(X, mult(rd(Y, Z), X))
% 11.72/1.94 = { by lemma 13 R->L }
% 11.72/1.94 rd(ld(rd(Y, Z), Y), Y)
% 11.72/1.94 = { by lemma 14 }
% 11.72/1.94 rd(Z, Y)
% 11.72/1.94
% 11.72/1.94 Lemma 16: mult(X, mult(ld(X, Y), mult(X, Z))) = mult(mult(Y, X), Z).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(X, mult(ld(X, Y), mult(X, Z)))
% 11.72/1.94 = { by axiom 5 (f05) }
% 11.72/1.94 mult(mult(mult(X, ld(X, Y)), X), Z)
% 11.72/1.94 = { by axiom 3 (f01) }
% 11.72/1.94 mult(mult(Y, X), Z)
% 11.72/1.94
% 11.72/1.94 Lemma 17: mult(mult(X, Y), ld(Y, Z)) = mult(Y, mult(ld(Y, X), Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(mult(X, Y), ld(Y, Z))
% 11.72/1.94 = { by lemma 16 R->L }
% 11.72/1.94 mult(Y, mult(ld(Y, X), mult(Y, ld(Y, Z))))
% 11.72/1.94 = { by axiom 3 (f01) }
% 11.72/1.94 mult(Y, mult(ld(Y, X), Z))
% 11.72/1.94
% 11.72/1.94 Lemma 18: rd(ld(X, Z), Z) = rd(ld(X, Y), Y).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(ld(X, Z), Z)
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(X, mult(X, rd(ld(X, Z), Z)))
% 11.72/1.94 = { by lemma 10 }
% 11.72/1.94 ld(X, rd(X, X))
% 11.72/1.94 = { by lemma 10 R->L }
% 11.72/1.94 ld(X, mult(X, rd(ld(X, Y), Y)))
% 11.72/1.94 = { by axiom 1 (f02) }
% 11.72/1.94 rd(ld(X, Y), Y)
% 11.72/1.94
% 11.72/1.94 Lemma 19: ld(rd(X, mult(Y, X)), Z) = mult(Y, Z).
% 11.72/1.94 Proof:
% 11.72/1.94 ld(rd(X, mult(Y, X)), Z)
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(rd(ld(Y, mult(Y, X)), mult(Y, X)), Z)
% 11.72/1.94 = { by lemma 18 R->L }
% 11.72/1.94 ld(rd(ld(Y, mult(Y, Z)), mult(Y, Z)), Z)
% 11.72/1.94 = { by axiom 1 (f02) }
% 11.72/1.94 ld(rd(Z, mult(Y, Z)), Z)
% 11.72/1.94 = { by lemma 14 }
% 11.72/1.94 mult(Y, Z)
% 11.72/1.94
% 11.72/1.94 Lemma 20: mult(ld(X, Y), mult(X, Z)) = ld(X, mult(mult(Y, X), Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(ld(X, Y), mult(X, Z))
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(X, mult(X, mult(ld(X, Y), mult(X, Z))))
% 11.72/1.94 = { by lemma 16 }
% 11.72/1.94 ld(X, mult(mult(Y, X), Z))
% 11.72/1.94
% 11.72/1.94 Lemma 21: mult(X, mult(ld(X, rd(Y, X)), Z)) = mult(Y, ld(X, Z)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(X, mult(ld(X, rd(Y, X)), Z))
% 11.72/1.94 = { by lemma 17 R->L }
% 11.72/1.94 mult(mult(rd(Y, X), X), ld(X, Z))
% 11.72/1.94 = { by axiom 4 (f03) }
% 11.72/1.94 mult(Y, ld(X, Z))
% 11.72/1.94
% 11.72/1.94 Lemma 22: mult(ld(X, rd(Y, X)), Z) = ld(X, mult(Y, ld(X, Z))).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(ld(X, rd(Y, X)), Z)
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(X, mult(X, mult(ld(X, rd(Y, X)), Z)))
% 11.72/1.94 = { by lemma 21 }
% 11.72/1.94 ld(X, mult(Y, ld(X, Z)))
% 11.72/1.94
% 11.72/1.94 Lemma 23: mult(rd(X, mult(Y, X)), Z) = ld(Y, Z).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(rd(X, mult(Y, X)), Z)
% 11.72/1.94 = { by lemma 13 R->L }
% 11.72/1.94 mult(rd(ld(Y, W), W), Z)
% 11.72/1.94 = { by lemma 11 }
% 11.72/1.94 mult(ld(Y, rd(Y, Y)), Z)
% 11.72/1.94 = { by lemma 22 }
% 11.72/1.94 ld(Y, mult(Y, ld(Y, Z)))
% 11.72/1.94 = { by axiom 1 (f02) }
% 11.72/1.94 ld(Y, Z)
% 11.72/1.94
% 11.72/1.94 Lemma 24: mult(rd(X, Y), Z) = ld(rd(Y, X), Z).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(rd(X, Y), Z)
% 11.72/1.94 = { by lemma 15 R->L }
% 11.72/1.94 mult(rd(W, mult(rd(Y, X), W)), Z)
% 11.72/1.94 = { by lemma 23 }
% 11.72/1.94 ld(rd(Y, X), Z)
% 11.72/1.94
% 11.72/1.94 Lemma 25: mult(X, mult(ld(X, X), Y)) = mult(X, ld(rd(X, X), Y)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(X, mult(ld(X, X), Y))
% 11.72/1.94 = { by lemma 17 R->L }
% 11.72/1.94 mult(mult(X, X), ld(X, Y))
% 11.72/1.94 = { by lemma 19 R->L }
% 11.72/1.94 mult(ld(rd(Z, mult(X, Z)), X), ld(X, Y))
% 11.72/1.94 = { by lemma 13 R->L }
% 11.72/1.94 mult(ld(rd(ld(X, Y), Y), X), ld(X, Y))
% 11.72/1.94 = { by axiom 4 (f03) R->L }
% 11.72/1.94 mult(ld(rd(ld(X, Y), Y), X), mult(rd(ld(X, Y), Y), Y))
% 11.72/1.94 = { by lemma 20 }
% 11.72/1.94 ld(rd(ld(X, Y), Y), mult(mult(X, rd(ld(X, Y), Y)), Y))
% 11.72/1.94 = { by lemma 10 }
% 11.72/1.94 ld(rd(ld(X, Y), Y), mult(rd(X, X), Y))
% 11.72/1.94 = { by lemma 18 R->L }
% 11.72/1.94 ld(rd(ld(X, W), W), mult(rd(X, X), Y))
% 11.72/1.94 = { by lemma 13 }
% 11.72/1.94 ld(rd(V, mult(X, V)), mult(rd(X, X), Y))
% 11.72/1.94 = { by lemma 19 }
% 11.72/1.94 mult(X, mult(rd(X, X), Y))
% 11.72/1.94 = { by lemma 24 }
% 11.72/1.94 mult(X, ld(rd(X, X), Y))
% 11.72/1.94
% 11.72/1.94 Lemma 26: mult(ld(X, X), Y) = ld(rd(X, X), Y).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(ld(X, X), Y)
% 11.72/1.94 = { by axiom 1 (f02) R->L }
% 11.72/1.94 ld(X, mult(X, mult(ld(X, X), Y)))
% 11.72/1.94 = { by lemma 25 }
% 11.72/1.94 ld(X, mult(X, ld(rd(X, X), Y)))
% 11.72/1.94 = { by axiom 1 (f02) }
% 11.72/1.94 ld(rd(X, X), Y)
% 11.72/1.94
% 11.72/1.94 Lemma 27: rd(X, X) = ld(X, X).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(X, X)
% 11.72/1.94 = { by lemma 15 R->L }
% 11.72/1.94 rd(Y, mult(rd(X, X), Y))
% 11.72/1.94 = { by lemma 13 R->L }
% 11.72/1.94 rd(ld(rd(X, X), Z), Z)
% 11.72/1.94 = { by lemma 26 R->L }
% 11.72/1.94 rd(mult(ld(X, X), Z), Z)
% 11.72/1.94 = { by axiom 2 (f04) }
% 11.72/1.94 ld(X, X)
% 11.72/1.94
% 11.72/1.94 Lemma 28: ld(ld(X, X), X) = X.
% 11.72/1.94 Proof:
% 11.72/1.94 ld(ld(X, X), X)
% 11.72/1.94 = { by lemma 27 R->L }
% 11.72/1.94 ld(rd(X, X), X)
% 11.72/1.94 = { by lemma 14 }
% 11.72/1.94 X
% 11.72/1.94
% 11.72/1.94 Lemma 29: rd(X, ld(Y, X)) = Y.
% 11.72/1.94 Proof:
% 11.72/1.94 rd(X, ld(Y, X))
% 11.72/1.94 = { by axiom 3 (f01) R->L }
% 11.72/1.94 rd(mult(Y, ld(Y, X)), ld(Y, X))
% 11.72/1.94 = { by axiom 2 (f04) }
% 11.72/1.94 Y
% 11.72/1.94
% 11.72/1.94 Lemma 30: rd(X, mult(Y, X)) = ld(Y, ld(Y, Y)).
% 11.72/1.94 Proof:
% 11.72/1.94 rd(X, mult(Y, X))
% 11.72/1.94 = { by lemma 12 }
% 11.72/1.94 ld(Y, rd(Y, Y))
% 11.72/1.94 = { by lemma 27 }
% 11.72/1.94 ld(Y, ld(Y, Y))
% 11.72/1.94
% 11.72/1.94 Lemma 31: mult(rd(X, Y), mult(Y, mult(rd(X, Y), Z))) = mult(mult(X, rd(X, Y)), Z).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(rd(X, Y), mult(Y, mult(rd(X, Y), Z)))
% 11.72/1.94 = { by axiom 5 (f05) }
% 11.72/1.94 mult(mult(mult(rd(X, Y), Y), rd(X, Y)), Z)
% 11.72/1.94 = { by axiom 4 (f03) }
% 11.72/1.94 mult(mult(X, rd(X, Y)), Z)
% 11.72/1.94
% 11.72/1.94 Lemma 32: mult(mult(X, rd(X, Y)), Y) = mult(rd(X, Y), mult(Y, X)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(mult(X, rd(X, Y)), Y)
% 11.72/1.94 = { by lemma 31 R->L }
% 11.72/1.94 mult(rd(X, Y), mult(Y, mult(rd(X, Y), Y)))
% 11.72/1.94 = { by axiom 4 (f03) }
% 11.72/1.94 mult(rd(X, Y), mult(Y, X))
% 11.72/1.94
% 11.72/1.94 Lemma 33: mult(rd(X, X), mult(X, X)) = rd(mult(X, X), ld(X, X)).
% 11.72/1.94 Proof:
% 11.72/1.94 mult(rd(X, X), mult(X, X))
% 11.72/1.94 = { by lemma 32 R->L }
% 11.72/1.94 mult(mult(X, rd(X, X)), X)
% 11.72/1.94 = { by lemma 8 }
% 11.72/1.94 rd(mult(X, X), ld(X, X))
% 11.72/1.94
% 11.72/1.94 Lemma 34: ld(ld(X, X), mult(X, Y)) = ld(X, mult(mult(X, X), Y)).
% 11.72/1.94 Proof:
% 11.72/1.94 ld(ld(X, X), mult(X, Y))
% 11.72/1.94 = { by lemma 19 R->L }
% 11.72/1.94 ld(ld(X, X), ld(rd(Z, mult(X, Z)), Y))
% 11.72/1.94 = { by lemma 27 R->L }
% 11.72/1.94 ld(rd(X, X), ld(rd(Z, mult(X, Z)), Y))
% 11.72/1.95 = { by lemma 24 R->L }
% 11.72/1.95 mult(rd(X, X), ld(rd(Z, mult(X, Z)), Y))
% 11.72/1.95 = { by lemma 21 R->L }
% 11.72/1.95 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(rd(X, X), rd(Z, mult(X, Z)))), Y))
% 11.72/1.95 = { by lemma 13 R->L }
% 11.72/1.95 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(rd(X, X), rd(ld(X, W), W))), Y))
% 11.72/1.95 = { by lemma 10 R->L }
% 11.72/1.95 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), rd(mult(X, rd(ld(X, W), W)), rd(ld(X, W), W))), Y))
% 11.72/1.95 = { by axiom 2 (f04) }
% 11.72/1.95 mult(rd(Z, mult(X, Z)), mult(ld(rd(Z, mult(X, Z)), X), Y))
% 11.72/1.95 = { by lemma 23 }
% 11.72/1.95 ld(X, mult(ld(rd(Z, mult(X, Z)), X), Y))
% 11.72/1.95 = { by lemma 19 }
% 11.72/1.95 ld(X, mult(mult(X, X), Y))
% 11.72/1.95
% 11.72/1.95 Lemma 35: rd(mult(X, X), ld(X, X)) = ld(X, mult(mult(X, X), X)).
% 11.72/1.95 Proof:
% 11.72/1.95 rd(mult(X, X), ld(X, X))
% 11.72/1.95 = { by axiom 4 (f03) R->L }
% 11.72/1.95 mult(rd(rd(mult(X, X), ld(X, X)), mult(X, X)), mult(X, X))
% 11.72/1.95 = { by lemma 33 R->L }
% 11.72/1.95 mult(rd(mult(rd(X, X), mult(X, X)), mult(X, X)), mult(X, X))
% 11.72/1.95 = { by axiom 2 (f04) }
% 11.72/1.95 mult(rd(X, X), mult(X, X))
% 11.72/1.95 = { by lemma 27 }
% 11.72/1.95 mult(ld(X, X), mult(X, X))
% 11.72/1.95 = { by lemma 26 }
% 11.72/1.95 ld(rd(X, X), mult(X, X))
% 11.72/1.95 = { by lemma 27 }
% 11.72/1.95 ld(ld(X, X), mult(X, X))
% 11.72/1.95 = { by lemma 34 }
% 11.72/1.95 ld(X, mult(mult(X, X), X))
% 11.72/1.95
% 11.72/1.95 Lemma 36: ld(X, mult(mult(X, X), X)) = mult(X, X).
% 11.72/1.95 Proof:
% 11.72/1.95 ld(X, mult(mult(X, X), X))
% 11.72/1.95 = { by lemma 35 R->L }
% 11.72/1.95 rd(mult(X, X), ld(X, X))
% 11.72/1.95 = { by lemma 33 R->L }
% 11.72/1.95 mult(rd(X, X), mult(X, X))
% 11.72/1.95 = { by lemma 32 R->L }
% 11.72/1.95 mult(mult(X, rd(X, X)), X)
% 11.72/1.95 = { by lemma 27 }
% 11.72/1.95 mult(mult(X, ld(X, X)), X)
% 11.72/1.95 = { by axiom 3 (f01) }
% 11.72/1.95 mult(X, X)
% 11.72/1.95
% 11.72/1.95 Lemma 37: mult(ld(X, Y), mult(ld(ld(X, Y), X), Z)) = mult(Y, ld(ld(X, Y), Z)).
% 11.72/1.95 Proof:
% 11.72/1.95 mult(ld(X, Y), mult(ld(ld(X, Y), X), Z))
% 11.72/1.95 = { by lemma 17 R->L }
% 11.72/1.95 mult(mult(X, ld(X, Y)), ld(ld(X, Y), Z))
% 11.72/1.95 = { by axiom 3 (f01) }
% 11.72/1.95 mult(Y, ld(ld(X, Y), Z))
% 11.72/1.95
% 11.72/1.95 Lemma 38: mult(mult(rd(X, Y), Z), rd(X, Y)) = rd(mult(rd(X, Y), mult(Z, X)), Y).
% 11.72/1.95 Proof:
% 11.72/1.95 mult(mult(rd(X, Y), Z), rd(X, Y))
% 11.72/1.95 = { by lemma 6 R->L }
% 11.72/1.95 rd(mult(rd(X, Y), mult(Z, mult(rd(X, Y), Y))), Y)
% 11.72/1.95 = { by axiom 4 (f03) }
% 11.72/1.95 rd(mult(rd(X, Y), mult(Z, X)), Y)
% 11.72/1.95
% 11.72/1.95 Lemma 39: rd(ld(X, Y), mult(X, Z)) = ld(X, rd(rd(Y, Z), X)).
% 11.72/1.95 Proof:
% 11.72/1.95 rd(ld(X, Y), mult(X, Z))
% 11.72/1.95 = { by axiom 4 (f03) R->L }
% 11.72/1.95 rd(ld(X, mult(rd(Y, Z), Z)), mult(X, Z))
% 11.72/1.95 = { by lemma 19 R->L }
% 11.72/1.95 rd(ld(X, mult(rd(Y, Z), Z)), ld(rd(W, mult(X, W)), Z))
% 11.72/1.95 = { by lemma 13 R->L }
% 11.72/1.95 rd(ld(X, mult(rd(Y, Z), Z)), ld(rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), Z))
% 11.72/1.95 = { by axiom 4 (f03) R->L }
% 11.72/1.95 rd(mult(rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), ld(rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), Z))
% 11.72/1.95 = { by lemma 7 }
% 11.72/1.95 mult(mult(rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), rd(Y, Z)), rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)))
% 11.72/1.95 = { by lemma 38 }
% 11.72/1.95 rd(mult(rd(ld(X, mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z)), mult(rd(Y, Z), ld(X, mult(rd(Y, Z), Z)))), mult(rd(Y, Z), Z))
% 11.72/1.95 = { by lemma 24 }
% 11.72/1.95 rd(ld(rd(mult(rd(Y, Z), Z), ld(X, mult(rd(Y, Z), Z))), mult(rd(Y, Z), ld(X, mult(rd(Y, Z), Z)))), mult(rd(Y, Z), Z))
% 11.72/1.95 = { by lemma 29 }
% 11.72/1.95 rd(ld(X, mult(rd(Y, Z), ld(X, mult(rd(Y, Z), Z)))), mult(rd(Y, Z), Z))
% 11.72/1.95 = { by lemma 22 R->L }
% 11.72/1.95 rd(mult(ld(X, rd(rd(Y, Z), X)), mult(rd(Y, Z), Z)), mult(rd(Y, Z), Z))
% 11.72/1.95 = { by axiom 2 (f04) }
% 11.72/1.95 ld(X, rd(rd(Y, Z), X))
% 11.72/1.95
% 11.72/1.95 Lemma 40: rd(mult(rd(X, Y), mult(Y, X)), Y) = mult(X, rd(X, Y)).
% 11.72/1.95 Proof:
% 11.72/1.95 rd(mult(rd(X, Y), mult(Y, X)), Y)
% 11.72/1.95 = { by lemma 32 R->L }
% 11.72/1.95 rd(mult(mult(X, rd(X, Y)), Y), Y)
% 11.72/1.95 = { by axiom 2 (f04) }
% 11.72/1.95 mult(X, rd(X, Y))
% 11.72/1.95
% 11.72/1.95 Lemma 41: ld(ld(X, X), ld(X, X)) = ld(X, X).
% 11.72/1.95 Proof:
% 11.72/1.95 ld(ld(X, X), ld(X, X))
% 11.72/1.95 = { by lemma 27 R->L }
% 11.72/1.95 ld(rd(X, X), ld(X, X))
% 11.72/1.95 = { by lemma 26 R->L }
% 11.72/1.95 mult(ld(X, X), ld(X, X))
% 11.72/1.95 = { by axiom 1 (f02) R->L }
% 11.72/1.95 ld(X, mult(X, mult(ld(X, X), ld(X, X))))
% 11.72/1.95 = { by lemma 17 R->L }
% 11.72/1.95 ld(X, mult(mult(X, X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 34 R->L }
% 11.72/1.95 ld(ld(X, X), mult(X, ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 28 R->L }
% 11.72/1.95 ld(ld(X, X), mult(ld(ld(X, X), X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 27 R->L }
% 11.72/1.95 ld(rd(X, X), mult(ld(ld(X, X), X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 26 R->L }
% 11.72/1.95 mult(ld(X, X), mult(ld(ld(X, X), X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 37 }
% 11.72/1.95 mult(X, ld(ld(X, X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 27 R->L }
% 11.72/1.95 mult(X, ld(rd(X, X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 25 R->L }
% 11.72/1.95 mult(X, mult(ld(X, X), ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 17 R->L }
% 11.72/1.95 mult(mult(X, X), ld(X, ld(X, ld(X, X))))
% 11.72/1.95 = { by lemma 30 R->L }
% 11.72/1.95 mult(mult(X, X), ld(X, rd(Y, mult(X, Y))))
% 11.72/1.95 = { by lemma 13 R->L }
% 11.72/1.95 mult(mult(X, X), ld(X, rd(ld(X, X), X)))
% 11.72/1.95 = { by lemma 27 R->L }
% 11.72/1.95 mult(mult(X, X), ld(X, rd(rd(X, X), X)))
% 11.72/1.95 = { by lemma 39 R->L }
% 11.72/1.95 mult(mult(X, X), rd(ld(X, X), mult(X, X)))
% 11.72/1.95 = { by lemma 29 R->L }
% 11.72/1.95 mult(mult(X, X), rd(Z, ld(rd(ld(X, X), mult(X, X)), Z)))
% 11.72/1.95 = { by lemma 24 R->L }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(rd(mult(X, X), ld(X, X)), Z)))
% 11.72/1.95 = { by lemma 33 R->L }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(mult(rd(X, X), mult(X, X)), Z)))
% 11.72/1.95 = { by lemma 32 R->L }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(mult(mult(X, rd(X, X)), X), Z)))
% 11.72/1.95 = { by axiom 5 (f05) R->L }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(X, mult(rd(X, X), mult(X, Z)))))
% 11.72/1.95 = { by lemma 24 }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(X, ld(rd(X, X), mult(X, Z)))))
% 11.72/1.95 = { by lemma 27 }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(X, ld(ld(X, X), mult(X, Z)))))
% 11.72/1.95 = { by lemma 34 }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(X, ld(X, mult(mult(X, X), Z)))))
% 11.72/1.95 = { by axiom 3 (f01) }
% 11.72/1.95 mult(mult(X, X), rd(Z, mult(mult(X, X), Z)))
% 11.72/1.95 = { by lemma 12 }
% 11.72/1.95 mult(mult(X, X), ld(mult(X, X), rd(mult(X, X), mult(X, X))))
% 11.72/1.95 = { by axiom 1 (f02) R->L }
% 11.72/1.95 mult(mult(X, X), ld(mult(X, X), rd(ld(X, mult(X, mult(X, X))), mult(X, X))))
% 11.72/1.95 = { by axiom 2 (f04) R->L }
% 11.72/1.95 mult(mult(X, X), ld(mult(X, X), rd(ld(rd(mult(X, X), X), mult(X, mult(X, X))), mult(X, X))))
% 11.72/1.95 = { by lemma 24 R->L }
% 11.72/1.95 mult(mult(X, X), ld(mult(X, X), rd(mult(rd(X, mult(X, X)), mult(X, mult(X, X))), mult(X, X))))
% 11.72/1.95 = { by lemma 36 R->L }
% 11.72/1.95 mult(mult(X, X), ld(mult(X, X), rd(mult(rd(X, mult(X, X)), mult(X, ld(X, mult(mult(X, X), X)))), mult(X, X))))
% 11.72/1.95 = { by axiom 3 (f01) }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), rd(mult(rd(X, mult(X, X)), mult(mult(X, X), X)), mult(X, X))))
% 11.72/1.96 = { by lemma 40 }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), mult(X, rd(X, mult(X, X)))))
% 11.72/1.96 = { by lemma 9 R->L }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), rd(rd(mult(X, X), ld(X, mult(X, X))), X)))
% 11.72/1.96 = { by axiom 1 (f02) }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), rd(rd(mult(X, X), X), X)))
% 11.72/1.96 = { by axiom 2 (f04) }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), rd(X, X)))
% 11.72/1.96 = { by lemma 27 }
% 11.72/1.96 mult(mult(X, X), ld(mult(X, X), ld(X, X)))
% 11.72/1.96 = { by axiom 3 (f01) }
% 11.72/1.96 ld(X, X)
% 11.72/1.96
% 11.72/1.96 Lemma 42: ld(ld(X, X), ld(ld(X, X), Y)) = Y.
% 11.72/1.96 Proof:
% 11.72/1.96 ld(ld(X, X), ld(ld(X, X), Y))
% 11.72/1.96 = { by lemma 41 R->L }
% 11.72/1.96 ld(ld(X, X), ld(ld(ld(X, X), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 27 R->L }
% 11.72/1.96 ld(rd(X, X), ld(ld(ld(X, X), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 26 R->L }
% 11.72/1.96 mult(ld(X, X), ld(ld(ld(X, X), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 37 R->L }
% 11.72/1.96 mult(ld(ld(X, X), ld(X, X)), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 41 }
% 11.72/1.96 mult(ld(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 26 }
% 11.72/1.96 ld(rd(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 27 }
% 11.72/1.96 ld(ld(X, X), mult(ld(ld(ld(X, X), ld(X, X)), ld(X, X)), Y))
% 11.72/1.96 = { by lemma 28 }
% 11.72/1.96 ld(ld(X, X), mult(ld(X, X), Y))
% 11.72/1.96 = { by axiom 1 (f02) }
% 11.72/1.96 Y
% 11.72/1.96
% 11.72/1.96 Lemma 43: rd(mult(X, Y), ld(X, ld(Z, Y))) = mult(mult(X, Z), X).
% 11.72/1.96 Proof:
% 11.72/1.96 rd(mult(X, Y), ld(X, ld(Z, Y)))
% 11.72/1.96 = { by axiom 3 (f01) R->L }
% 11.72/1.96 rd(mult(X, mult(Z, ld(Z, Y))), ld(X, ld(Z, Y)))
% 11.72/1.96 = { by lemma 7 }
% 11.72/1.96 mult(mult(X, Z), X)
% 11.72/1.96
% 11.72/1.96 Lemma 44: ld(ld(X, rd(Y, X)), ld(X, Z)) = mult(X, ld(Y, Z)).
% 11.72/1.96 Proof:
% 11.72/1.96 ld(ld(X, rd(Y, X)), ld(X, Z))
% 11.72/1.96 = { by axiom 3 (f01) R->L }
% 11.72/1.96 ld(ld(X, rd(Y, X)), ld(X, mult(mult(rd(Y, X), X), ld(mult(rd(Y, X), X), Z))))
% 11.72/1.96 = { by lemma 20 R->L }
% 11.72/1.96 ld(ld(X, rd(Y, X)), mult(ld(X, rd(Y, X)), mult(X, ld(mult(rd(Y, X), X), Z))))
% 11.72/1.96 = { by axiom 1 (f02) }
% 11.72/1.96 mult(X, ld(mult(rd(Y, X), X), Z))
% 11.72/1.96 = { by axiom 4 (f03) }
% 11.72/1.96 mult(X, ld(Y, Z))
% 11.72/1.96
% 11.72/1.96 Lemma 45: ld(rd(X, Y), mult(mult(X, rd(X, Y)), Z)) = mult(Y, mult(rd(X, Y), Z)).
% 11.72/1.96 Proof:
% 11.72/1.96 ld(rd(X, Y), mult(mult(X, rd(X, Y)), Z))
% 11.72/1.96 = { by lemma 31 R->L }
% 11.72/1.96 ld(rd(X, Y), mult(rd(X, Y), mult(Y, mult(rd(X, Y), Z))))
% 11.72/1.96 = { by axiom 1 (f02) }
% 11.72/1.96 mult(Y, mult(rd(X, Y), Z))
% 11.72/1.96
% 11.72/1.96 Lemma 46: ld(rd(X, Y), mult(X, ld(rd(X, Y), Y))) = rd(mult(Y, Y), ld(Y, X)).
% 11.72/1.96 Proof:
% 11.72/1.96 ld(rd(X, Y), mult(X, ld(rd(X, Y), Y)))
% 11.72/1.96 = { by lemma 24 R->L }
% 11.72/1.96 mult(rd(Y, X), mult(X, ld(rd(X, Y), Y)))
% 11.72/1.96 = { by lemma 24 R->L }
% 11.72/1.96 mult(rd(Y, X), mult(X, mult(rd(Y, X), Y)))
% 11.72/1.96 = { by lemma 45 R->L }
% 11.72/1.96 mult(rd(Y, X), ld(rd(Y, X), mult(mult(Y, rd(Y, X)), Y)))
% 11.72/1.96 = { by lemma 8 }
% 11.72/1.96 mult(rd(Y, X), ld(rd(Y, X), rd(mult(Y, Y), ld(Y, X))))
% 11.72/1.96 = { by axiom 3 (f01) }
% 11.72/1.96 rd(mult(Y, Y), ld(Y, X))
% 11.72/1.96
% 11.72/1.96 Lemma 47: ld(rd(X, ld(Y, mult(Z, ld(Y, X)))), W) = ld(Y, mult(Z, ld(Y, W))).
% 11.72/1.96 Proof:
% 11.72/1.96 ld(rd(X, ld(Y, mult(Z, ld(Y, X)))), W)
% 11.72/1.96 = { by lemma 29 R->L }
% 11.72/1.96 ld(rd(X, ld(rd(X, ld(Y, X)), mult(Z, ld(Y, X)))), W)
% 11.72/1.96 = { by lemma 24 R->L }
% 11.72/1.96 ld(rd(X, mult(rd(ld(Y, X), X), mult(Z, ld(Y, X)))), W)
% 11.72/1.96 = { by lemma 24 R->L }
% 11.72/1.96 mult(rd(mult(rd(ld(Y, X), X), mult(Z, ld(Y, X))), X), W)
% 11.72/1.96 = { by lemma 38 R->L }
% 11.72/1.96 mult(mult(mult(rd(ld(Y, X), X), Z), rd(ld(Y, X), X)), W)
% 11.72/1.96 = { by axiom 5 (f05) R->L }
% 11.72/1.96 mult(rd(ld(Y, X), X), mult(Z, mult(rd(ld(Y, X), X), W)))
% 11.72/1.96 = { by lemma 24 }
% 11.72/1.96 ld(rd(X, ld(Y, X)), mult(Z, mult(rd(ld(Y, X), X), W)))
% 11.72/1.96 = { by lemma 24 }
% 11.72/1.96 ld(rd(X, ld(Y, X)), mult(Z, ld(rd(X, ld(Y, X)), W)))
% 11.72/1.96 = { by lemma 29 }
% 11.72/1.96 ld(Y, mult(Z, ld(rd(X, ld(Y, X)), W)))
% 11.72/1.96 = { by lemma 29 }
% 11.72/1.96 ld(Y, mult(Z, ld(Y, W)))
% 11.72/1.96
% 11.72/1.96 Lemma 48: rd(mult(X, Y), ld(X, mult(Z, Y))) = rd(W, ld(X, mult(Z, ld(X, W)))).
% 11.72/1.96 Proof:
% 11.72/1.96 rd(mult(X, Y), ld(X, mult(Z, Y)))
% 11.72/1.96 = { by axiom 1 (f02) R->L }
% 11.72/1.96 rd(mult(X, Y), ld(X, mult(Z, ld(X, mult(X, Y)))))
% 11.72/1.96 = { by lemma 29 R->L }
% 11.72/1.96 rd(W, ld(rd(mult(X, Y), ld(X, mult(Z, ld(X, mult(X, Y))))), W))
% 11.72/1.96 = { by lemma 47 }
% 11.72/1.96 rd(W, ld(X, mult(Z, ld(X, W))))
% 11.72/1.96
% 11.72/1.96 Lemma 49: rd(X, ld(Y, ld(Z, ld(Y, X)))) = mult(mult(Y, Z), Y).
% 11.72/1.96 Proof:
% 11.72/1.96 rd(X, ld(Y, ld(Z, ld(Y, X))))
% 11.72/1.96 = { by axiom 3 (f01) R->L }
% 11.72/1.96 rd(mult(Y, ld(Y, X)), ld(Y, ld(Z, ld(Y, X))))
% 11.72/1.96 = { by lemma 43 }
% 11.72/1.96 mult(mult(Y, Z), Y)
% 11.72/1.96
% 11.72/1.96 Lemma 50: ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))) = rd(mult(Y, X), ld(Y, X)).
% 11.72/1.96 Proof:
% 11.72/1.96 ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y)))
% 11.72/1.96 = { by lemma 27 R->L }
% 11.72/1.96 ld(ld(X, X), mult(ld(rd(X, X), Y), ld(ld(X, X), Y)))
% 11.72/1.96 = { by axiom 1 (f02) R->L }
% 11.72/1.96 ld(ld(X, X), mult(ld(X, mult(X, ld(rd(X, X), Y))), ld(ld(X, X), Y)))
% 11.72/1.96 = { by lemma 25 R->L }
% 11.72/1.96 ld(ld(X, X), mult(ld(X, mult(X, mult(ld(X, X), Y))), ld(ld(X, X), Y)))
% 11.72/1.96 = { by axiom 1 (f02) }
% 11.72/1.96 ld(ld(X, X), mult(mult(ld(X, X), Y), ld(ld(X, X), Y)))
% 11.72/1.96 = { by axiom 2 (f04) R->L }
% 11.72/1.96 ld(ld(X, X), mult(mult(ld(X, X), Y), ld(rd(mult(ld(X, X), Y), Y), Y)))
% 11.72/1.96 = { by axiom 2 (f04) R->L }
% 11.72/1.96 ld(rd(mult(ld(X, X), Y), Y), mult(mult(ld(X, X), Y), ld(rd(mult(ld(X, X), Y), Y), Y)))
% 11.72/1.96 = { by lemma 46 }
% 11.72/1.96 rd(mult(Y, Y), ld(Y, mult(ld(X, X), Y)))
% 11.72/1.96 = { by lemma 48 }
% 11.72/1.96 rd(Z, ld(Y, mult(ld(X, X), ld(Y, Z))))
% 11.72/1.96 = { by lemma 26 }
% 11.72/1.96 rd(Z, ld(Y, ld(rd(X, X), ld(Y, Z))))
% 11.72/1.96 = { by lemma 49 }
% 11.72/1.96 mult(mult(Y, rd(X, X)), Y)
% 11.72/1.96 = { by lemma 8 }
% 11.72/1.96 rd(mult(Y, X), ld(Y, X))
% 11.72/1.96
% 11.72/1.97 Lemma 51: ld(ld(X, X), rd(ld(X, X), Y)) = rd(Z, mult(Y, Z)).
% 11.72/1.97 Proof:
% 11.72/1.97 ld(ld(X, X), rd(ld(X, X), Y))
% 11.72/1.97 = { by axiom 1 (f02) R->L }
% 11.72/1.97 ld(Y, mult(Y, ld(ld(X, X), rd(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 27 R->L }
% 11.72/1.97 ld(Y, mult(Y, ld(rd(X, X), rd(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 27 R->L }
% 11.72/1.97 ld(Y, mult(Y, ld(rd(X, X), rd(rd(X, X), Y))))
% 11.72/1.97 = { by lemma 44 R->L }
% 11.72/1.97 ld(Y, ld(ld(Y, rd(rd(X, X), Y)), ld(Y, rd(rd(X, X), Y))))
% 11.72/1.97 = { by lemma 39 R->L }
% 11.72/1.97 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), ld(Y, rd(rd(X, X), Y))))
% 11.72/1.97 = { by lemma 39 R->L }
% 11.72/1.97 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), rd(ld(Y, X), mult(Y, X))))
% 11.72/1.97 = { by lemma 15 R->L }
% 11.72/1.97 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), rd(W, mult(rd(mult(Y, X), ld(Y, X)), W))))
% 11.72/1.97 = { by lemma 13 R->L }
% 11.72/1.97 ld(Y, ld(rd(ld(Y, X), mult(Y, X)), rd(ld(rd(mult(Y, X), ld(Y, X)), V), V)))
% 11.72/1.97 = { by lemma 24 R->L }
% 11.72/1.97 ld(Y, mult(rd(mult(Y, X), ld(Y, X)), rd(ld(rd(mult(Y, X), ld(Y, X)), V), V)))
% 11.72/1.97 = { by lemma 10 }
% 11.72/1.97 ld(Y, rd(rd(mult(Y, X), ld(Y, X)), rd(mult(Y, X), ld(Y, X))))
% 11.72/1.97 = { by lemma 50 R->L }
% 11.72/1.97 ld(Y, rd(ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 11.72/1.97 = { by lemma 27 R->L }
% 11.72/1.97 ld(Y, rd(ld(rd(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 11.72/1.97 = { by lemma 26 R->L }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), rd(mult(Y, X), ld(Y, X))))
% 11.72/1.97 = { by lemma 50 R->L }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y)))))
% 11.72/1.97 = { by lemma 36 R->L }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), ld(ld(ld(X, X), Y), mult(mult(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(X, X), Y))))))
% 11.72/1.97 = { by lemma 35 R->L }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), rd(mult(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(ld(X, X), Y), ld(ld(X, X), Y))))))
% 11.72/1.97 = { by lemma 33 R->L }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(ld(X, X), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), mult(ld(ld(X, X), Y), ld(ld(X, X), Y))))))
% 11.72/1.97 = { by lemma 48 }
% 11.72/1.97 ld(Y, rd(U, ld(ld(X, X), mult(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(X, X), U)))))
% 11.72/1.97 = { by lemma 24 }
% 11.72/1.97 ld(Y, rd(U, ld(ld(X, X), ld(rd(ld(ld(X, X), Y), ld(ld(X, X), Y)), ld(ld(X, X), U)))))
% 11.72/1.97 = { by lemma 49 }
% 11.72/1.97 ld(Y, mult(mult(ld(X, X), rd(ld(ld(X, X), Y), ld(ld(X, X), Y))), ld(X, X)))
% 11.72/1.97 = { by lemma 8 }
% 11.72/1.97 ld(Y, rd(mult(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 26 }
% 11.72/1.97 ld(Y, rd(ld(rd(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 27 }
% 11.72/1.97 ld(Y, rd(ld(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 27 }
% 11.72/1.97 ld(Y, ld(ld(ld(X, X), ld(ld(X, X), Y)), ld(ld(X, X), ld(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 42 }
% 11.72/1.97 ld(Y, ld(Y, ld(ld(X, X), ld(ld(X, X), Y))))
% 11.72/1.97 = { by lemma 42 }
% 11.72/1.97 ld(Y, ld(Y, Y))
% 11.72/1.97 = { by lemma 27 R->L }
% 11.72/1.97 ld(Y, rd(Y, Y))
% 11.72/1.97 = { by lemma 11 R->L }
% 11.72/1.97 rd(ld(Y, T), T)
% 11.72/1.97 = { by lemma 13 }
% 11.72/1.97 rd(Z, mult(Y, Z))
% 11.72/1.97
% 11.72/1.97 Lemma 52: ld(ld(X, X), mult(Y, ld(X, X))) = ld(ld(X, X), Y).
% 11.72/1.97 Proof:
% 11.72/1.97 ld(ld(X, X), mult(Y, ld(X, X)))
% 11.72/1.97 = { by lemma 41 R->L }
% 11.72/1.97 ld(ld(X, X), mult(Y, ld(ld(X, X), ld(X, X))))
% 11.72/1.97 = { by lemma 21 R->L }
% 11.72/1.97 ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), rd(Y, ld(X, X))), ld(X, X))))
% 11.72/1.97 = { by lemma 27 R->L }
% 11.72/1.97 ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), rd(Y, ld(X, X))), rd(X, X))))
% 11.72/1.97 = { by lemma 29 R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), rd(Y, ld(X, X))), rd(Z, ld(rd(X, X), Z)))))
% 11.72/1.98 = { by lemma 26 R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), mult(ld(ld(X, X), rd(Y, ld(X, X))), rd(Z, mult(ld(X, X), Z)))))
% 11.72/1.98 = { by axiom 2 (f04) R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), mult(ld(rd(mult(ld(X, X), Z), Z), rd(Y, ld(X, X))), rd(Z, mult(ld(X, X), Z)))))
% 11.72/1.98 = { by lemma 24 R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), mult(mult(rd(Z, mult(ld(X, X), Z)), rd(Y, ld(X, X))), rd(Z, mult(ld(X, X), Z)))))
% 11.72/1.98 = { by lemma 43 R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), rd(mult(rd(Z, mult(ld(X, X), Z)), mult(ld(X, X), Z)), ld(rd(Z, mult(ld(X, X), Z)), ld(rd(Y, ld(X, X)), mult(ld(X, X), Z))))))
% 11.72/1.98 = { by axiom 4 (f03) }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), rd(Z, ld(rd(Z, mult(ld(X, X), Z)), ld(rd(Y, ld(X, X)), mult(ld(X, X), Z))))))
% 11.72/1.98 = { by lemma 19 }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), rd(Z, mult(ld(X, X), ld(rd(Y, ld(X, X)), mult(ld(X, X), Z))))))
% 11.72/1.98 = { by lemma 44 R->L }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), rd(Z, ld(ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X))), ld(ld(X, X), mult(ld(X, X), Z))))))
% 11.72/1.98 = { by axiom 1 (f02) }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), rd(Z, ld(ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X))), Z))))
% 11.72/1.98 = { by lemma 29 }
% 11.72/1.98 ld(ld(X, X), mult(ld(X, X), ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X)))))
% 11.72/1.98 = { by lemma 26 }
% 11.72/1.98 ld(ld(X, X), ld(rd(X, X), ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X)))))
% 11.72/1.98 = { by lemma 27 }
% 11.72/1.98 ld(ld(X, X), ld(ld(X, X), ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X)))))
% 11.72/1.98 = { by lemma 42 }
% 11.72/1.98 ld(ld(X, X), rd(rd(Y, ld(X, X)), ld(X, X)))
% 11.72/1.98 = { by lemma 39 R->L }
% 11.72/1.98 rd(ld(ld(X, X), Y), mult(ld(X, X), ld(X, X)))
% 11.72/1.98 = { by lemma 15 R->L }
% 11.72/1.98 rd(W, mult(rd(mult(ld(X, X), ld(X, X)), ld(ld(X, X), Y)), W))
% 11.72/1.98 = { by lemma 51 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), rd(mult(ld(X, X), ld(X, X)), ld(ld(X, X), Y))))
% 11.72/1.98 = { by lemma 46 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(Y, ld(X, X)), mult(Y, ld(rd(Y, ld(X, X)), ld(X, X))))))
% 11.72/1.98 = { by lemma 47 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), ld(rd(Y, ld(X, X)), mult(Y, ld(rd(Y, ld(X, X)), ld(mult(ld(X, X), rd(ld(X, X), Y)), V))))), ld(X, X))))
% 11.72/1.98 = { by lemma 24 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), ld(rd(Y, ld(X, X)), mult(Y, mult(rd(ld(X, X), Y), ld(mult(ld(X, X), rd(ld(X, X), Y)), V))))), ld(X, X))))
% 11.72/1.98 = { by lemma 45 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), ld(rd(Y, ld(X, X)), ld(rd(ld(X, X), Y), mult(mult(ld(X, X), rd(ld(X, X), Y)), ld(mult(ld(X, X), rd(ld(X, X), Y)), V))))), ld(X, X))))
% 11.72/1.98 = { by axiom 3 (f01) }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), ld(rd(Y, ld(X, X)), ld(rd(ld(X, X), Y), V))), ld(X, X))))
% 11.72/1.98 = { by lemma 24 R->L }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), ld(rd(Y, ld(X, X)), mult(rd(Y, ld(X, X)), V))), ld(X, X))))
% 11.72/1.98 = { by axiom 1 (f02) }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(ld(mult(ld(X, X), rd(ld(X, X), Y)), V), V), ld(X, X))))
% 11.72/1.98 = { by lemma 13 }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), ld(rd(U, mult(mult(ld(X, X), rd(ld(X, X), Y)), U)), ld(X, X))))
% 11.72/1.98 = { by lemma 19 }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), mult(mult(ld(X, X), rd(ld(X, X), Y)), ld(X, X))))
% 11.72/1.98 = { by lemma 26 }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), mult(ld(rd(X, X), rd(ld(X, X), Y)), ld(X, X))))
% 11.72/1.98 = { by lemma 27 }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), mult(ld(ld(X, X), rd(ld(X, X), Y)), ld(X, X))))
% 11.72/1.98 = { by lemma 51 }
% 11.72/1.98 ld(ld(X, X), rd(ld(X, X), mult(rd(T, mult(Y, T)), ld(X, X))))
% 11.72/1.98 = { by lemma 15 }
% 11.72/1.98 ld(ld(X, X), rd(mult(Y, T), T))
% 11.72/1.98 = { by axiom 2 (f04) }
% 11.72/1.98 ld(ld(X, X), Y)
% 11.72/1.98
% 11.72/1.98 Lemma 53: ld(ld(X, mult(mult(X, Y), X)), Y) = ld(X, ld(X, X)).
% 11.72/1.98 Proof:
% 11.72/1.98 ld(ld(X, mult(mult(X, Y), X)), Y)
% 11.72/1.98 = { by axiom 2 (f04) R->L }
% 11.72/1.98 ld(ld(X, mult(mult(X, Y), rd(mult(X, Y), Y))), Y)
% 11.72/1.98 = { by axiom 2 (f04) R->L }
% 11.72/1.98 ld(ld(rd(mult(X, Y), Y), mult(mult(X, Y), rd(mult(X, Y), Y))), Y)
% 11.72/1.98 = { by lemma 24 R->L }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), mult(mult(X, Y), rd(mult(X, Y), Y))), Y)
% 11.72/1.98 = { by axiom 1 (f02) R->L }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), mult(mult(X, Y), rd(ld(Y, mult(Y, mult(X, Y))), Y))), Y)
% 11.72/1.98 = { by axiom 1 (f02) R->L }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), mult(ld(Y, mult(Y, mult(X, Y))), rd(ld(Y, mult(Y, mult(X, Y))), Y))), Y)
% 11.72/1.98 = { by lemma 40 R->L }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), rd(mult(rd(ld(Y, mult(Y, mult(X, Y))), Y), mult(Y, ld(Y, mult(Y, mult(X, Y))))), Y)), Y)
% 11.72/1.98 = { by axiom 3 (f01) }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), rd(mult(rd(ld(Y, mult(Y, mult(X, Y))), Y), mult(Y, mult(X, Y))), Y)), Y)
% 11.72/1.98 = { by lemma 24 }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), rd(ld(rd(Y, ld(Y, mult(Y, mult(X, Y)))), mult(Y, mult(X, Y))), Y)), Y)
% 11.72/1.98 = { by axiom 1 (f02) }
% 11.72/1.98 ld(mult(rd(Y, mult(X, Y)), rd(ld(rd(Y, mult(X, Y)), mult(Y, mult(X, Y))), Y)), Y)
% 11.72/1.99 = { by lemma 9 R->L }
% 11.72/1.99 ld(rd(rd(mult(rd(Y, mult(X, Y)), ld(rd(Y, mult(X, Y)), mult(Y, mult(X, Y)))), ld(rd(Y, mult(X, Y)), Y)), rd(Y, mult(X, Y))), Y)
% 11.72/1.99 = { by axiom 3 (f01) }
% 11.72/1.99 ld(rd(rd(mult(Y, mult(X, Y)), ld(rd(Y, mult(X, Y)), Y)), rd(Y, mult(X, Y))), Y)
% 11.72/1.99 = { by lemma 14 }
% 11.72/1.99 ld(rd(rd(mult(Y, mult(X, Y)), mult(X, Y)), rd(Y, mult(X, Y))), Y)
% 11.72/1.99 = { by axiom 2 (f04) }
% 11.72/1.99 ld(rd(Y, rd(Y, mult(X, Y))), Y)
% 11.72/1.99 = { by lemma 30 }
% 11.72/1.99 ld(rd(Y, ld(X, ld(X, X))), Y)
% 11.72/1.99 = { by lemma 14 }
% 11.72/1.99 ld(X, ld(X, X))
% 11.72/1.99
% 11.72/1.99 Goal 1 (goals): tuple(mult(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2))) = tuple(x0, x0_2).
% 11.72/1.99 Proof:
% 11.72/1.99 tuple(mult(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2)))
% 11.72/1.99 = { by lemma 24 }
% 11.72/1.99 tuple(ld(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2)))
% 11.72/1.99 = { by lemma 27 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2)))
% 11.72/1.99 = { by lemma 27 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), mult(x0_2, ld(x1_2, x1_2)))
% 11.72/1.99 = { by axiom 3 (f01) R->L }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), mult(ld(x1_2, x1_2), ld(ld(x1_2, x1_2), mult(x0_2, ld(x1_2, x1_2)))))
% 11.72/1.99 = { by lemma 52 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), mult(ld(x1_2, x1_2), ld(ld(x1_2, x1_2), x0_2)))
% 11.72/1.99 = { by lemma 26 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), ld(rd(x1_2, x1_2), ld(ld(x1_2, x1_2), x0_2)))
% 11.72/1.99 = { by lemma 27 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), ld(ld(x1_2, x1_2), ld(ld(x1_2, x1_2), x0_2)))
% 11.72/1.99 = { by lemma 42 }
% 11.72/1.99 tuple(ld(ld(x1, x1), x0), x0_2)
% 11.72/1.99 = { by axiom 1 (f02) R->L }
% 11.72/1.99 tuple(ld(ld(ld(ld(X, X), mult(ld(X, X), x1)), x1), x0), x0_2)
% 11.72/1.99 = { by lemma 52 R->L }
% 11.72/1.99 tuple(ld(ld(ld(ld(X, X), mult(mult(ld(X, X), x1), ld(X, X))), x1), x0), x0_2)
% 11.72/1.99 = { by lemma 53 }
% 11.72/1.99 tuple(ld(ld(ld(X, X), ld(ld(X, X), ld(X, X))), x0), x0_2)
% 11.72/1.99 = { by lemma 53 R->L }
% 11.72/1.99 tuple(ld(ld(ld(ld(X, X), mult(mult(ld(X, X), x0), ld(X, X))), x0), x0), x0_2)
% 11.72/1.99 = { by lemma 52 }
% 11.72/1.99 tuple(ld(ld(ld(ld(X, X), mult(ld(X, X), x0)), x0), x0), x0_2)
% 11.72/1.99 = { by axiom 1 (f02) }
% 11.72/1.99 tuple(ld(ld(x0, x0), x0), x0_2)
% 11.72/1.99 = { by lemma 28 }
% 11.72/1.99 tuple(x0, x0_2)
% 11.72/1.99 % SZS output end Proof
% 11.72/1.99
% 11.72/1.99 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------