TSTP Solution File: GRP667-10 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP667-10 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n028.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:19:35 EDT 2023
% Result : Unsatisfiable 26.36s 3.68s
% Output : Proof 27.02s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.06/0.11 % Problem : GRP667-10 : TPTP v8.1.2. Released v7.5.0.
% 0.06/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.31 % Computer : n028.cluster.edu
% 0.12/0.31 % Model : x86_64 x86_64
% 0.12/0.31 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.31 % Memory : 8042.1875MB
% 0.12/0.31 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.31 % CPULimit : 300
% 0.12/0.31 % WCLimit : 300
% 0.12/0.31 % DateTime : Mon Aug 28 23:42:25 EDT 2023
% 0.12/0.31 % CPUTime :
% 26.36/3.68 Command-line arguments: --no-flatten-goal
% 26.36/3.68
% 26.36/3.68 % SZS status Unsatisfiable
% 26.59/3.68
% 27.02/3.75 % SZS output start Proof
% 27.02/3.75 Axiom 1 (f05): mult(X, unit) = X.
% 27.02/3.75 Axiom 2 (f06): mult(unit, X) = X.
% 27.02/3.75 Axiom 3 (f02): ld(X, mult(X, Y)) = Y.
% 27.02/3.75 Axiom 4 (f04): rd(mult(X, Y), Y) = X.
% 27.02/3.75 Axiom 5 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 27.02/3.75 Axiom 6 (f01): mult(X, ld(X, Y)) = Y.
% 27.02/3.75 Axiom 7 (f09): mult(f(X), f(X)) = X.
% 27.02/3.75 Axiom 8 (f03): mult(rd(X, Y), Y) = X.
% 27.02/3.75 Axiom 9 (f08): mult(mult(X, Y), X) = mult(X, mult(Y, X)).
% 27.02/3.75 Axiom 10 (f07): mult(mult(X, Y), mult(mult(Z, Y), Z)) = mult(mult(X, mult(mult(Y, Z), Y)), Z).
% 27.02/3.75 Axiom 11 (f10): ifeq(mult(X, mult(Y, mult(Z, Y))), mult(mult(mult(X, Y), Z), Y), mult(Y, mult(X, mult(Y, Z))), mult(mult(mult(Y, X), Y), Z)) = mult(mult(mult(Y, X), Y), Z).
% 27.02/3.75
% 27.02/3.75 Lemma 12: mult(f(X), X) = mult(X, f(X)).
% 27.02/3.75 Proof:
% 27.02/3.75 mult(f(X), X)
% 27.02/3.75 = { by axiom 7 (f09) R->L }
% 27.02/3.75 mult(f(X), mult(f(X), f(X)))
% 27.02/3.75 = { by axiom 9 (f08) R->L }
% 27.02/3.75 mult(mult(f(X), f(X)), f(X))
% 27.02/3.75 = { by axiom 7 (f09) }
% 27.02/3.75 mult(X, f(X))
% 27.02/3.75
% 27.02/3.75 Lemma 13: rd(mult(X, mult(Y, X)), X) = mult(X, Y).
% 27.02/3.75 Proof:
% 27.02/3.75 rd(mult(X, mult(Y, X)), X)
% 27.02/3.75 = { by axiom 9 (f08) R->L }
% 27.02/3.75 rd(mult(mult(X, Y), X), X)
% 27.02/3.75 = { by axiom 4 (f04) }
% 27.02/3.75 mult(X, Y)
% 27.02/3.75
% 27.02/3.75 Lemma 14: mult(mult(X, mult(Y, mult(Z, Y))), Z) = mult(mult(X, Y), mult(Z, mult(Y, Z))).
% 27.02/3.75 Proof:
% 27.02/3.75 mult(mult(X, mult(Y, mult(Z, Y))), Z)
% 27.02/3.75 = { by axiom 9 (f08) R->L }
% 27.02/3.75 mult(mult(X, mult(mult(Y, Z), Y)), Z)
% 27.02/3.75 = { by axiom 10 (f07) R->L }
% 27.02/3.75 mult(mult(X, Y), mult(mult(Z, Y), Z))
% 27.02/3.75 = { by axiom 9 (f08) }
% 27.02/3.75 mult(mult(X, Y), mult(Z, mult(Y, Z)))
% 27.02/3.75
% 27.02/3.75 Lemma 15: mult(mult(X, mult(Y, X)), Y) = mult(X, mult(Y, mult(X, Y))).
% 27.02/3.75 Proof:
% 27.02/3.75 mult(mult(X, mult(Y, X)), Y)
% 27.02/3.75 = { by axiom 2 (f06) R->L }
% 27.02/3.75 mult(mult(unit, mult(X, mult(Y, X))), Y)
% 27.02/3.75 = { by lemma 14 }
% 27.02/3.75 mult(mult(unit, X), mult(Y, mult(X, Y)))
% 27.02/3.75 = { by axiom 2 (f06) }
% 27.02/3.75 mult(X, mult(Y, mult(X, Y)))
% 27.02/3.75
% 27.02/3.75 Lemma 16: mult(mult(X, Y), mult(X, mult(Y, X))) = mult(X, mult(Y, mult(X, mult(Y, X)))).
% 27.02/3.75 Proof:
% 27.02/3.75 mult(mult(X, Y), mult(X, mult(Y, X)))
% 27.02/3.75 = { by lemma 14 R->L }
% 27.02/3.75 mult(mult(X, mult(Y, mult(X, Y))), X)
% 27.02/3.75 = { by axiom 9 (f08) }
% 27.02/3.75 mult(X, mult(mult(Y, mult(X, Y)), X))
% 27.02/3.75 = { by lemma 15 }
% 27.02/3.76 mult(X, mult(Y, mult(X, mult(Y, X))))
% 27.02/3.76
% 27.02/3.76 Lemma 17: ifeq(mult(X, mult(Y, mult(Z, Y))), mult(mult(mult(X, Y), Z), Y), mult(Y, mult(X, mult(Y, Z))), mult(mult(Y, mult(X, Y)), Z)) = mult(mult(Y, mult(X, Y)), Z).
% 27.02/3.76 Proof:
% 27.02/3.76 ifeq(mult(X, mult(Y, mult(Z, Y))), mult(mult(mult(X, Y), Z), Y), mult(Y, mult(X, mult(Y, Z))), mult(mult(Y, mult(X, Y)), Z))
% 27.02/3.76 = { by axiom 9 (f08) R->L }
% 27.02/3.76 ifeq(mult(X, mult(Y, mult(Z, Y))), mult(mult(mult(X, Y), Z), Y), mult(Y, mult(X, mult(Y, Z))), mult(mult(mult(Y, X), Y), Z))
% 27.02/3.76 = { by axiom 11 (f10) }
% 27.02/3.76 mult(mult(mult(Y, X), Y), Z)
% 27.02/3.76 = { by axiom 9 (f08) }
% 27.02/3.76 mult(mult(Y, mult(X, Y)), Z)
% 27.02/3.76
% 27.02/3.76 Lemma 18: mult(mult(X, X), Y) = mult(X, mult(X, Y)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(mult(X, X), Y)
% 27.02/3.76 = { by axiom 2 (f06) R->L }
% 27.02/3.76 mult(mult(X, mult(unit, X)), Y)
% 27.02/3.76 = { by lemma 17 R->L }
% 27.02/3.76 ifeq(mult(unit, mult(X, mult(Y, X))), mult(mult(mult(unit, X), Y), X), mult(X, mult(unit, mult(X, Y))), mult(mult(X, mult(unit, X)), Y))
% 27.02/3.76 = { by axiom 2 (f06) }
% 27.02/3.76 ifeq(mult(X, mult(Y, X)), mult(mult(mult(unit, X), Y), X), mult(X, mult(unit, mult(X, Y))), mult(mult(X, mult(unit, X)), Y))
% 27.02/3.76 = { by axiom 2 (f06) }
% 27.02/3.76 ifeq(mult(X, mult(Y, X)), mult(mult(X, Y), X), mult(X, mult(unit, mult(X, Y))), mult(mult(X, mult(unit, X)), Y))
% 27.02/3.76 = { by axiom 2 (f06) }
% 27.02/3.76 ifeq(mult(X, mult(Y, X)), mult(mult(X, Y), X), mult(X, mult(unit, mult(X, Y))), mult(mult(X, X), Y))
% 27.02/3.76 = { by axiom 9 (f08) }
% 27.02/3.76 ifeq(mult(X, mult(Y, X)), mult(X, mult(Y, X)), mult(X, mult(unit, mult(X, Y))), mult(mult(X, X), Y))
% 27.02/3.76 = { by axiom 5 (ifeq_axiom) }
% 27.02/3.76 mult(X, mult(unit, mult(X, Y)))
% 27.02/3.76 = { by axiom 2 (f06) }
% 27.02/3.76 mult(X, mult(X, Y))
% 27.02/3.76
% 27.02/3.76 Lemma 19: mult(mult(X, Y), mult(X, Y)) = mult(X, mult(Y, mult(X, Y))).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(mult(X, Y), mult(X, Y))
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 ld(X, mult(X, mult(mult(X, Y), mult(X, Y))))
% 27.02/3.76 = { by axiom 5 (ifeq_axiom) R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(X, mult(Y, mult(X, mult(Y, X)))), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(X, mult(Y, X))), Y)))
% 27.02/3.76 = { by lemma 16 R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(mult(X, Y), mult(X, mult(Y, X))), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(X, mult(Y, X))), Y)))
% 27.02/3.76 = { by lemma 14 R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(mult(X, mult(Y, mult(X, Y))), X), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(X, mult(Y, X))), Y)))
% 27.02/3.76 = { by lemma 15 R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(mult(mult(X, mult(Y, X)), Y), X), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(X, mult(Y, X))), Y)))
% 27.02/3.76 = { by axiom 9 (f08) R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(mult(mult(mult(X, Y), X), Y), X), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(X, mult(Y, X))), Y)))
% 27.02/3.76 = { by axiom 9 (f08) R->L }
% 27.02/3.76 ld(X, ifeq(mult(X, mult(Y, mult(X, mult(Y, X)))), mult(mult(mult(mult(X, Y), X), Y), X), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(mult(X, Y), X)), Y)))
% 27.02/3.76 = { by lemma 16 R->L }
% 27.02/3.76 ld(X, ifeq(mult(mult(X, Y), mult(X, mult(Y, X))), mult(mult(mult(mult(X, Y), X), Y), X), mult(X, mult(mult(X, Y), mult(X, Y))), mult(mult(X, mult(mult(X, Y), X)), Y)))
% 27.02/3.76 = { by lemma 17 }
% 27.02/3.76 ld(X, mult(mult(X, mult(mult(X, Y), X)), Y))
% 27.02/3.76 = { by axiom 9 (f08) }
% 27.02/3.76 ld(X, mult(mult(X, mult(X, mult(Y, X))), Y))
% 27.02/3.76 = { by lemma 14 }
% 27.02/3.76 ld(X, mult(mult(X, X), mult(Y, mult(X, Y))))
% 27.02/3.76 = { by lemma 18 }
% 27.02/3.76 ld(X, mult(X, mult(X, mult(Y, mult(X, Y)))))
% 27.02/3.76 = { by axiom 3 (f02) }
% 27.02/3.76 mult(X, mult(Y, mult(X, Y)))
% 27.02/3.76
% 27.02/3.76 Lemma 20: mult(X, mult(ld(X, Y), Y)) = mult(Y, Y).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(X, mult(ld(X, Y), Y))
% 27.02/3.76 = { by axiom 6 (f01) R->L }
% 27.02/3.76 mult(X, mult(ld(X, Y), mult(X, ld(X, Y))))
% 27.02/3.76 = { by lemma 19 R->L }
% 27.02/3.76 mult(mult(X, ld(X, Y)), mult(X, ld(X, Y)))
% 27.02/3.76 = { by axiom 6 (f01) }
% 27.02/3.76 mult(Y, mult(X, ld(X, Y)))
% 27.02/3.76 = { by axiom 6 (f01) }
% 27.02/3.76 mult(Y, Y)
% 27.02/3.76
% 27.02/3.76 Lemma 21: mult(ld(X, Y), Y) = ld(X, mult(Y, Y)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(ld(X, Y), Y)
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 ld(X, mult(X, mult(ld(X, Y), Y)))
% 27.02/3.76 = { by lemma 20 }
% 27.02/3.76 ld(X, mult(Y, Y))
% 27.02/3.76
% 27.02/3.76 Lemma 22: mult(X, mult(ld(X, Y), X)) = mult(Y, X).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(X, mult(ld(X, Y), X))
% 27.02/3.76 = { by axiom 9 (f08) R->L }
% 27.02/3.76 mult(mult(X, ld(X, Y)), X)
% 27.02/3.76 = { by axiom 6 (f01) }
% 27.02/3.76 mult(Y, X)
% 27.02/3.76
% 27.02/3.76 Lemma 23: rd(mult(X, X), ld(Y, X)) = mult(X, Y).
% 27.02/3.76 Proof:
% 27.02/3.76 rd(mult(X, X), ld(Y, X))
% 27.02/3.76 = { by lemma 20 R->L }
% 27.02/3.76 rd(mult(Y, mult(ld(Y, X), X)), ld(Y, X))
% 27.02/3.76 = { by axiom 6 (f01) R->L }
% 27.02/3.76 rd(mult(Y, mult(ld(Y, X), mult(Y, ld(Y, X)))), ld(Y, X))
% 27.02/3.76 = { by lemma 15 R->L }
% 27.02/3.76 rd(mult(mult(Y, mult(ld(Y, X), Y)), ld(Y, X)), ld(Y, X))
% 27.02/3.76 = { by lemma 22 }
% 27.02/3.76 rd(mult(mult(X, Y), ld(Y, X)), ld(Y, X))
% 27.02/3.76 = { by axiom 4 (f04) }
% 27.02/3.76 mult(X, Y)
% 27.02/3.76
% 27.02/3.76 Lemma 24: ld(rd(X, Y), X) = Y.
% 27.02/3.76 Proof:
% 27.02/3.76 ld(rd(X, Y), X)
% 27.02/3.76 = { by axiom 8 (f03) R->L }
% 27.02/3.76 ld(rd(X, Y), mult(rd(X, Y), Y))
% 27.02/3.76 = { by axiom 3 (f02) }
% 27.02/3.76 Y
% 27.02/3.76
% 27.02/3.76 Lemma 25: ld(mult(X, Y), mult(X, X)) = ld(Y, X).
% 27.02/3.76 Proof:
% 27.02/3.76 ld(mult(X, Y), mult(X, X))
% 27.02/3.76 = { by lemma 23 R->L }
% 27.02/3.76 ld(rd(mult(X, X), ld(Y, X)), mult(X, X))
% 27.02/3.76 = { by lemma 24 }
% 27.02/3.76 ld(Y, X)
% 27.02/3.76
% 27.02/3.76 Lemma 26: mult(ld(X, Y), X) = ld(X, mult(Y, X)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(ld(X, Y), X)
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 ld(X, mult(X, mult(ld(X, Y), X)))
% 27.02/3.76 = { by lemma 22 }
% 27.02/3.76 ld(X, mult(Y, X))
% 27.02/3.76
% 27.02/3.76 Lemma 27: mult(mult(X, Y), Y) = mult(X, mult(Y, Y)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(mult(X, Y), Y)
% 27.02/3.76 = { by axiom 1 (f05) R->L }
% 27.02/3.76 mult(mult(X, Y), mult(Y, unit))
% 27.02/3.76 = { by axiom 2 (f06) R->L }
% 27.02/3.76 mult(mult(X, Y), mult(unit, mult(Y, unit)))
% 27.02/3.76 = { by lemma 14 R->L }
% 27.02/3.76 mult(mult(X, mult(Y, mult(unit, Y))), unit)
% 27.02/3.76 = { by axiom 1 (f05) }
% 27.02/3.76 mult(X, mult(Y, mult(unit, Y)))
% 27.02/3.76 = { by axiom 2 (f06) }
% 27.02/3.76 mult(X, mult(Y, Y))
% 27.02/3.76
% 27.02/3.76 Lemma 28: mult(X, mult(ld(X, Y), ld(X, Y))) = mult(Y, ld(X, Y)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(X, mult(ld(X, Y), ld(X, Y)))
% 27.02/3.76 = { by lemma 27 R->L }
% 27.02/3.76 mult(mult(X, ld(X, Y)), ld(X, Y))
% 27.02/3.76 = { by axiom 6 (f01) }
% 27.02/3.76 mult(Y, ld(X, Y))
% 27.02/3.76
% 27.02/3.76 Lemma 29: mult(rd(X, Y), X) = mult(X, ld(Y, X)).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(rd(X, Y), X)
% 27.02/3.76 = { by axiom 6 (f01) R->L }
% 27.02/3.76 mult(Y, ld(Y, mult(rd(X, Y), X)))
% 27.02/3.76 = { by axiom 8 (f03) R->L }
% 27.02/3.76 mult(Y, ld(Y, mult(rd(X, Y), mult(rd(X, Y), Y))))
% 27.02/3.76 = { by lemma 22 R->L }
% 27.02/3.76 mult(Y, ld(Y, mult(rd(X, Y), mult(Y, mult(ld(Y, rd(X, Y)), Y)))))
% 27.02/3.76 = { by axiom 6 (f01) R->L }
% 27.02/3.76 mult(Y, ld(Y, mult(mult(Y, ld(Y, rd(X, Y))), mult(Y, mult(ld(Y, rd(X, Y)), Y)))))
% 27.02/3.76 = { by lemma 16 }
% 27.02/3.76 mult(Y, ld(Y, mult(Y, mult(ld(Y, rd(X, Y)), mult(Y, mult(ld(Y, rd(X, Y)), Y))))))
% 27.02/3.76 = { by lemma 22 }
% 27.02/3.76 mult(Y, ld(Y, mult(Y, mult(ld(Y, rd(X, Y)), mult(rd(X, Y), Y)))))
% 27.02/3.76 = { by axiom 3 (f02) }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), mult(rd(X, Y), Y)))
% 27.02/3.76 = { by axiom 8 (f03) }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), X))
% 27.02/3.76 = { by axiom 6 (f01) R->L }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), mult(Y, ld(Y, X))))
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), mult(ld(ld(Y, rd(X, Y)), mult(ld(Y, rd(X, Y)), Y)), ld(Y, X))))
% 27.02/3.76 = { by lemma 26 }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), mult(ld(ld(Y, rd(X, Y)), ld(Y, mult(rd(X, Y), Y))), ld(Y, X))))
% 27.02/3.76 = { by axiom 8 (f03) }
% 27.02/3.76 mult(Y, mult(ld(Y, rd(X, Y)), mult(ld(ld(Y, rd(X, Y)), ld(Y, X)), ld(Y, X))))
% 27.02/3.76 = { by lemma 20 }
% 27.02/3.76 mult(Y, mult(ld(Y, X), ld(Y, X)))
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 mult(Y, ld(Y, mult(Y, mult(ld(Y, X), ld(Y, X)))))
% 27.02/3.76 = { by lemma 28 }
% 27.02/3.76 mult(Y, ld(Y, mult(X, ld(Y, X))))
% 27.02/3.76 = { by axiom 6 (f01) }
% 27.02/3.76 mult(X, ld(Y, X))
% 27.02/3.76
% 27.02/3.76 Lemma 30: mult(X, rd(X, Y)) = rd(mult(X, X), Y).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(X, rd(X, Y))
% 27.02/3.76 = { by lemma 23 R->L }
% 27.02/3.76 rd(mult(X, X), ld(rd(X, Y), X))
% 27.02/3.76 = { by lemma 24 }
% 27.02/3.76 rd(mult(X, X), Y)
% 27.02/3.76
% 27.02/3.76 Lemma 31: mult(rd(X, Y), mult(Y, X)) = mult(X, X).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(rd(X, Y), mult(Y, X))
% 27.02/3.76 = { by axiom 8 (f03) R->L }
% 27.02/3.76 mult(rd(X, Y), mult(Y, mult(rd(X, Y), Y)))
% 27.02/3.76 = { by lemma 19 R->L }
% 27.02/3.76 mult(mult(rd(X, Y), Y), mult(rd(X, Y), Y))
% 27.02/3.76 = { by axiom 8 (f03) }
% 27.02/3.76 mult(X, mult(rd(X, Y), Y))
% 27.02/3.76 = { by axiom 8 (f03) }
% 27.02/3.76 mult(X, X)
% 27.02/3.76
% 27.02/3.76 Lemma 32: rd(mult(X, X), mult(Y, X)) = rd(X, Y).
% 27.02/3.76 Proof:
% 27.02/3.76 rd(mult(X, X), mult(Y, X))
% 27.02/3.76 = { by lemma 31 R->L }
% 27.02/3.76 rd(mult(rd(X, Y), mult(Y, X)), mult(Y, X))
% 27.02/3.76 = { by axiom 4 (f04) }
% 27.02/3.76 rd(X, Y)
% 27.02/3.76
% 27.02/3.76 Lemma 33: rd(mult(X, X), Y) = rd(X, rd(Y, X)).
% 27.02/3.76 Proof:
% 27.02/3.76 rd(mult(X, X), Y)
% 27.02/3.76 = { by axiom 8 (f03) R->L }
% 27.02/3.76 rd(mult(X, X), mult(rd(Y, X), X))
% 27.02/3.76 = { by lemma 32 }
% 27.02/3.76 rd(X, rd(Y, X))
% 27.02/3.76
% 27.02/3.76 Lemma 34: mult(X, ld(Y, X)) = ld(rd(Y, X), X).
% 27.02/3.76 Proof:
% 27.02/3.76 mult(X, ld(Y, X))
% 27.02/3.76 = { by lemma 29 R->L }
% 27.02/3.76 mult(rd(X, Y), X)
% 27.02/3.76 = { by axiom 3 (f02) R->L }
% 27.02/3.76 mult(ld(X, mult(X, rd(X, Y))), X)
% 27.02/3.76 = { by lemma 30 }
% 27.02/3.76 mult(ld(X, rd(mult(X, X), Y)), X)
% 27.02/3.76 = { by lemma 33 }
% 27.02/3.76 mult(ld(X, rd(X, rd(Y, X))), X)
% 27.02/3.76 = { by lemma 26 }
% 27.02/3.76 ld(X, mult(rd(X, rd(Y, X)), X))
% 27.02/3.76 = { by lemma 29 }
% 27.02/3.76 ld(X, mult(X, ld(rd(Y, X), X)))
% 27.02/3.76 = { by axiom 3 (f02) }
% 27.02/3.76 ld(rd(Y, X), X)
% 27.02/3.76
% 27.02/3.76 Lemma 35: rd(X, mult(Y, f(X))) = rd(f(X), Y).
% 27.02/3.76 Proof:
% 27.02/3.76 rd(X, mult(Y, f(X)))
% 27.02/3.77 = { by axiom 7 (f09) R->L }
% 27.02/3.77 rd(mult(f(X), f(X)), mult(Y, f(X)))
% 27.02/3.77 = { by lemma 32 }
% 27.02/3.77 rd(f(X), Y)
% 27.02/3.77
% 27.02/3.77 Lemma 36: mult(f(X), mult(f(X), Y)) = mult(X, Y).
% 27.02/3.77 Proof:
% 27.02/3.77 mult(f(X), mult(f(X), Y))
% 27.02/3.77 = { by lemma 18 R->L }
% 27.02/3.77 mult(mult(f(X), f(X)), Y)
% 27.02/3.77 = { by axiom 7 (f09) }
% 27.02/3.77 mult(X, Y)
% 27.02/3.77
% 27.02/3.77 Lemma 37: ld(X, mult(f(X), Y)) = ld(f(X), Y).
% 27.02/3.77 Proof:
% 27.02/3.77 ld(X, mult(f(X), Y))
% 27.02/3.77 = { by axiom 6 (f01) R->L }
% 27.02/3.77 ld(X, mult(f(X), mult(f(X), ld(f(X), Y))))
% 27.02/3.77 = { by lemma 36 }
% 27.02/3.77 ld(X, mult(X, ld(f(X), Y)))
% 27.02/3.77 = { by axiom 3 (f02) }
% 27.02/3.77 ld(f(X), Y)
% 27.02/3.77
% 27.02/3.77 Lemma 38: rd(f(X), rd(Y, f(X))) = rd(X, Y).
% 27.02/3.77 Proof:
% 27.02/3.77 rd(f(X), rd(Y, f(X)))
% 27.02/3.77 = { by lemma 33 R->L }
% 27.02/3.77 rd(mult(f(X), f(X)), Y)
% 27.02/3.77 = { by axiom 7 (f09) }
% 27.02/3.77 rd(X, Y)
% 27.02/3.77
% 27.02/3.77 Lemma 39: ld(f(X), rd(f(X), Y)) = ld(X, rd(X, Y)).
% 27.02/3.77 Proof:
% 27.02/3.77 ld(f(X), rd(f(X), Y))
% 27.02/3.77 = { by lemma 37 R->L }
% 27.02/3.77 ld(X, mult(f(X), rd(f(X), Y)))
% 27.02/3.77 = { by lemma 30 }
% 27.02/3.77 ld(X, rd(mult(f(X), f(X)), Y))
% 27.02/3.77 = { by lemma 33 }
% 27.02/3.77 ld(X, rd(f(X), rd(Y, f(X))))
% 27.02/3.77 = { by lemma 38 }
% 27.02/3.77 ld(X, rd(X, Y))
% 27.02/3.77
% 27.02/3.77 Lemma 40: mult(f(X), ld(X, Y)) = ld(f(X), Y).
% 27.02/3.77 Proof:
% 27.02/3.77 mult(f(X), ld(X, Y))
% 27.02/3.77 = { by axiom 3 (f02) R->L }
% 27.02/3.77 ld(f(X), mult(f(X), mult(f(X), ld(X, Y))))
% 27.02/3.77 = { by lemma 36 }
% 27.02/3.77 ld(f(X), mult(X, ld(X, Y)))
% 27.02/3.77 = { by axiom 6 (f01) }
% 27.02/3.77 ld(f(X), Y)
% 27.02/3.77
% 27.02/3.77 Lemma 41: ld(mult(X, Y), X) = ld(X, rd(X, Y)).
% 27.02/3.77 Proof:
% 27.02/3.77 ld(mult(X, Y), X)
% 27.02/3.77 = { by lemma 13 R->L }
% 27.02/3.77 ld(rd(mult(X, mult(Y, X)), X), X)
% 27.02/3.77 = { by axiom 8 (f03) R->L }
% 27.02/3.77 ld(rd(mult(X, mult(rd(mult(Y, X), X), X)), X), X)
% 27.02/3.77 = { by lemma 13 }
% 27.02/3.77 ld(mult(X, rd(mult(Y, X), X)), X)
% 27.02/3.77 = { by axiom 4 (f04) R->L }
% 27.02/3.77 rd(mult(ld(mult(X, rd(mult(Y, X), X)), X), X), X)
% 27.02/3.77 = { by lemma 21 }
% 27.02/3.77 rd(ld(mult(X, rd(mult(Y, X), X)), mult(X, X)), X)
% 27.02/3.77 = { by lemma 25 }
% 27.02/3.77 rd(ld(rd(mult(Y, X), X), X), X)
% 27.02/3.77 = { by lemma 34 R->L }
% 27.02/3.77 rd(mult(X, ld(mult(Y, X), X)), X)
% 27.02/3.77 = { by lemma 29 R->L }
% 27.02/3.77 rd(mult(rd(X, mult(Y, X)), X), X)
% 27.02/3.77 = { by axiom 4 (f04) }
% 27.02/3.77 rd(X, mult(Y, X))
% 27.02/3.77 = { by axiom 7 (f09) R->L }
% 27.02/3.77 rd(X, mult(Y, mult(f(X), f(X))))
% 27.02/3.77 = { by lemma 27 R->L }
% 27.02/3.77 rd(X, mult(mult(Y, f(X)), f(X)))
% 27.02/3.77 = { by lemma 35 }
% 27.02/3.77 rd(f(X), mult(Y, f(X)))
% 27.02/3.77 = { by axiom 6 (f01) R->L }
% 27.02/3.77 mult(f(X), ld(f(X), rd(f(X), mult(Y, f(X)))))
% 27.02/3.77 = { by lemma 39 }
% 27.02/3.77 mult(f(X), ld(X, rd(X, mult(Y, f(X)))))
% 27.02/3.77 = { by lemma 40 }
% 27.02/3.77 ld(f(X), rd(X, mult(Y, f(X))))
% 27.02/3.77 = { by lemma 35 }
% 27.02/3.77 ld(f(X), rd(f(X), Y))
% 27.02/3.77 = { by lemma 39 }
% 27.02/3.77 ld(X, rd(X, Y))
% 27.02/3.77
% 27.02/3.77 Lemma 42: mult(mult(X, f(Y)), Y) = mult(mult(X, Y), f(Y)).
% 27.02/3.77 Proof:
% 27.02/3.77 mult(mult(X, f(Y)), Y)
% 27.02/3.77 = { by axiom 7 (f09) R->L }
% 27.02/3.77 mult(mult(X, f(Y)), mult(f(Y), f(Y)))
% 27.02/3.77 = { by lemma 27 R->L }
% 27.02/3.77 mult(mult(mult(X, f(Y)), f(Y)), f(Y))
% 27.02/3.77 = { by lemma 27 }
% 27.02/3.77 mult(mult(X, mult(f(Y), f(Y))), f(Y))
% 27.02/3.77 = { by axiom 7 (f09) }
% 27.02/3.77 mult(mult(X, Y), f(Y))
% 27.02/3.77
% 27.02/3.77 Lemma 43: rd(X, ld(Y, f(Y))) = mult(X, f(Y)).
% 27.02/3.77 Proof:
% 27.02/3.77 rd(X, ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 8 (f03) R->L }
% 27.02/3.77 rd(mult(rd(X, Y), Y), ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 1 (f05) R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), unit), ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 3 (f02) R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), ld(f(Y), mult(f(Y), unit))), ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 1 (f05) }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), ld(f(Y), f(Y))), ld(Y, f(Y)))
% 27.02/3.77 = { by lemma 37 R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), ld(Y, mult(f(Y), f(Y)))), ld(Y, f(Y)))
% 27.02/3.77 = { by lemma 21 R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), mult(ld(Y, f(Y)), f(Y))), ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 6 (f01) R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), Y), mult(ld(Y, f(Y)), mult(Y, ld(Y, f(Y))))), ld(Y, f(Y)))
% 27.02/3.77 = { by lemma 14 R->L }
% 27.02/3.77 rd(mult(mult(rd(X, Y), mult(Y, mult(ld(Y, f(Y)), Y))), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.77 = { by lemma 22 }
% 27.02/3.77 rd(mult(mult(rd(X, Y), mult(f(Y), Y)), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.77 = { by lemma 12 }
% 27.02/3.77 rd(mult(mult(rd(X, Y), mult(Y, f(Y))), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.77 = { by axiom 4 (f04) }
% 27.02/3.77 mult(rd(X, Y), mult(Y, f(Y)))
% 27.02/3.77 = { by axiom 3 (f02) R->L }
% 27.02/3.77 ld(f(Y), mult(f(Y), mult(rd(X, Y), mult(Y, f(Y)))))
% 27.02/3.77 = { by lemma 12 R->L }
% 27.02/3.77 ld(f(Y), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))))
% 27.02/3.77 = { by axiom 5 (ifeq_axiom) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(Y, Y)), mult(rd(X, Y), mult(Y, Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 27 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(Y, Y)), mult(mult(rd(X, Y), Y), Y), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 7 (f09) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(Y, Y)), mult(mult(rd(X, Y), Y), mult(f(Y), f(Y))), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 27 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(Y, Y)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 7 (f09) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(Y, mult(f(Y), f(Y)))), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), mult(f(Y), mult(f(Y), mult(f(Y), f(Y))))), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 5 (ifeq_axiom) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(Z, mult(W, f(Z))), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(f(Z), mult(f(Z), mult(W, f(Z)))), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 9 (f08) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(f(Z), mult(mult(f(Z), W), f(Z))), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 9 (f08) R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(mult(f(Z), mult(f(Z), W)), f(Z)), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 18 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(mult(mult(f(Z), f(Z)), W), f(Z)), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 R->L }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(f(Z), mult(f(Z), mult(W, f(Z)))), mult(mult(mult(f(Z), f(Z)), W), f(Z)), mult(f(Y), mult(f(Y), mult(f(Y), f(Y)))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(f(Z), mult(f(Z), mult(W, f(Z)))), mult(mult(mult(f(Z), f(Z)), W), f(Z)), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(mult(mult(f(Z), f(Z)), W), f(Z)), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 18 }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(mult(f(Z), mult(f(Z), W)), f(Z)), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 9 (f08) }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(f(Z), mult(mult(f(Z), W), f(Z))), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by axiom 9 (f08) }
% 27.02/3.77 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(f(Z), mult(f(Z), mult(W, f(Z)))), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.77 = { by lemma 36 }
% 27.02/3.78 ld(f(Y), ifeq(mult(rd(X, Y), ifeq(mult(Z, mult(W, f(Z))), mult(Z, mult(W, f(Z))), mult(f(Y), mult(Y, f(Y))), V)), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.78 = { by axiom 5 (ifeq_axiom) }
% 27.02/3.78 ld(f(Y), ifeq(mult(rd(X, Y), mult(f(Y), mult(Y, f(Y)))), mult(mult(mult(rd(X, Y), Y), f(Y)), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.78 = { by lemma 42 R->L }
% 27.02/3.78 ld(f(Y), ifeq(mult(rd(X, Y), mult(f(Y), mult(Y, f(Y)))), mult(mult(mult(rd(X, Y), f(Y)), Y), f(Y)), mult(f(Y), mult(rd(X, Y), mult(f(Y), Y))), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y)))
% 27.02/3.78 = { by lemma 17 }
% 27.02/3.78 ld(f(Y), mult(mult(f(Y), mult(rd(X, Y), f(Y))), Y))
% 27.02/3.78 = { by axiom 7 (f09) R->L }
% 27.02/3.78 ld(f(Y), mult(mult(f(Y), mult(rd(X, Y), f(Y))), mult(f(Y), f(Y))))
% 27.02/3.78 = { by lemma 27 R->L }
% 27.02/3.78 ld(f(Y), mult(mult(mult(f(Y), mult(rd(X, Y), f(Y))), f(Y)), f(Y)))
% 27.02/3.78 = { by axiom 9 (f08) }
% 27.02/3.78 ld(f(Y), mult(mult(f(Y), mult(mult(rd(X, Y), f(Y)), f(Y))), f(Y)))
% 27.02/3.78 = { by axiom 9 (f08) }
% 27.02/3.78 ld(f(Y), mult(f(Y), mult(mult(mult(rd(X, Y), f(Y)), f(Y)), f(Y))))
% 27.02/3.78 = { by lemma 27 }
% 27.02/3.78 ld(f(Y), mult(f(Y), mult(mult(rd(X, Y), f(Y)), mult(f(Y), f(Y)))))
% 27.02/3.78 = { by axiom 7 (f09) }
% 27.02/3.78 ld(f(Y), mult(f(Y), mult(mult(rd(X, Y), f(Y)), Y)))
% 27.02/3.78 = { by lemma 42 }
% 27.02/3.78 ld(f(Y), mult(f(Y), mult(mult(rd(X, Y), Y), f(Y))))
% 27.02/3.78 = { by axiom 3 (f02) }
% 27.02/3.78 mult(mult(rd(X, Y), Y), f(Y))
% 27.02/3.78 = { by axiom 8 (f03) }
% 27.02/3.78 mult(X, f(Y))
% 27.02/3.78
% 27.02/3.78 Lemma 44: rd(X, ld(Y, unit)) = mult(X, Y).
% 27.02/3.78 Proof:
% 27.02/3.78 rd(X, ld(Y, unit))
% 27.02/3.78 = { by axiom 4 (f04) R->L }
% 27.02/3.78 rd(X, ld(Y, rd(mult(unit, f(Y)), f(Y))))
% 27.02/3.78 = { by axiom 2 (f06) }
% 27.02/3.78 rd(X, ld(Y, rd(f(Y), f(Y))))
% 27.02/3.78 = { by lemma 24 R->L }
% 27.02/3.78 rd(X, ld(Y, ld(rd(f(Y), rd(f(Y), f(Y))), f(Y))))
% 27.02/3.78 = { by lemma 38 }
% 27.02/3.78 rd(X, ld(Y, ld(rd(Y, f(Y)), f(Y))))
% 27.02/3.78 = { by lemma 34 R->L }
% 27.02/3.78 rd(X, ld(Y, mult(f(Y), ld(Y, f(Y)))))
% 27.02/3.78 = { by lemma 28 R->L }
% 27.02/3.78 rd(X, ld(Y, mult(Y, mult(ld(Y, f(Y)), ld(Y, f(Y))))))
% 27.02/3.78 = { by axiom 3 (f02) }
% 27.02/3.78 rd(X, mult(ld(Y, f(Y)), ld(Y, f(Y))))
% 27.02/3.78 = { by axiom 4 (f04) R->L }
% 27.02/3.78 rd(mult(rd(X, mult(ld(Y, f(Y)), ld(Y, f(Y)))), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.78 = { by axiom 4 (f04) R->L }
% 27.02/3.78 rd(rd(mult(mult(rd(X, mult(ld(Y, f(Y)), ld(Y, f(Y)))), ld(Y, f(Y))), ld(Y, f(Y))), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.78 = { by lemma 27 }
% 27.02/3.78 rd(rd(mult(rd(X, mult(ld(Y, f(Y)), ld(Y, f(Y)))), mult(ld(Y, f(Y)), ld(Y, f(Y)))), ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.78 = { by axiom 8 (f03) }
% 27.02/3.78 rd(rd(X, ld(Y, f(Y))), ld(Y, f(Y)))
% 27.02/3.78 = { by lemma 43 }
% 27.02/3.78 mult(rd(X, ld(Y, f(Y))), f(Y))
% 27.02/3.78 = { by lemma 43 }
% 27.02/3.78 mult(mult(X, f(Y)), f(Y))
% 27.02/3.78 = { by lemma 27 }
% 27.02/3.78 mult(X, mult(f(Y), f(Y)))
% 27.02/3.78 = { by axiom 7 (f09) }
% 27.02/3.78 mult(X, Y)
% 27.02/3.78
% 27.02/3.78 Lemma 45: ld(X, rd(X, Y)) = ld(Y, unit).
% 27.02/3.78 Proof:
% 27.02/3.78 ld(X, rd(X, Y))
% 27.02/3.78 = { by lemma 41 R->L }
% 27.02/3.78 ld(mult(X, Y), X)
% 27.02/3.78 = { by lemma 44 R->L }
% 27.02/3.78 ld(rd(X, ld(Y, unit)), X)
% 27.02/3.78 = { by lemma 24 }
% 27.02/3.78 ld(Y, unit)
% 27.02/3.78
% 27.02/3.78 Lemma 46: ld(ld(X, Y), unit) = ld(Y, X).
% 27.02/3.78 Proof:
% 27.02/3.78 ld(ld(X, Y), unit)
% 27.02/3.78 = { by lemma 45 R->L }
% 27.02/3.78 ld(Y, rd(Y, ld(X, Y)))
% 27.02/3.78 = { by axiom 6 (f01) R->L }
% 27.02/3.78 ld(Y, rd(mult(X, ld(X, Y)), ld(X, Y)))
% 27.02/3.78 = { by axiom 4 (f04) }
% 27.02/3.78 ld(Y, X)
% 27.02/3.78
% 27.02/3.78 Lemma 47: mult(X, ld(Y, Z)) = rd(X, ld(Z, Y)).
% 27.02/3.78 Proof:
% 27.02/3.78 mult(X, ld(Y, Z))
% 27.02/3.78 = { by lemma 44 R->L }
% 27.02/3.78 rd(X, ld(ld(Y, Z), unit))
% 27.02/3.78 = { by lemma 46 }
% 27.02/3.78 rd(X, ld(Z, Y))
% 27.02/3.78
% 27.02/3.78 Lemma 48: ld(f(X), ld(f(X), Y)) = ld(X, Y).
% 27.02/3.78 Proof:
% 27.02/3.78 ld(f(X), ld(f(X), Y))
% 27.02/3.78 = { by lemma 40 R->L }
% 27.02/3.78 ld(f(X), mult(f(X), ld(X, Y)))
% 27.02/3.78 = { by axiom 3 (f02) }
% 27.02/3.78 ld(X, Y)
% 27.02/3.78
% 27.02/3.78 Lemma 49: rd(mult(X, mult(Y, Z)), ld(Y, Z)) = rd(mult(X, Z), ld(Y, ld(Y, Z))).
% 27.02/3.78 Proof:
% 27.02/3.78 rd(mult(X, mult(Y, Z)), ld(Y, Z))
% 27.02/3.78 = { by axiom 6 (f01) R->L }
% 27.02/3.78 rd(mult(X, mult(Z, ld(Z, mult(Y, Z)))), ld(Y, Z))
% 27.02/3.78 = { by lemma 47 R->L }
% 27.02/3.78 mult(mult(X, mult(Z, ld(Z, mult(Y, Z)))), ld(Z, Y))
% 27.02/3.78 = { by lemma 26 R->L }
% 27.02/3.78 mult(mult(X, mult(Z, mult(ld(Z, Y), Z))), ld(Z, Y))
% 27.02/3.78 = { by lemma 14 }
% 27.02/3.78 mult(mult(X, Z), mult(ld(Z, Y), mult(Z, ld(Z, Y))))
% 27.02/3.78 = { by lemma 46 R->L }
% 27.02/3.78 mult(mult(X, Z), mult(ld(ld(Y, Z), unit), mult(Z, ld(Z, Y))))
% 27.02/3.78 = { by axiom 3 (f02) R->L }
% 27.02/3.78 mult(mult(X, Z), ld(rd(mult(Z, ld(Z, Y)), ld(ld(Y, Z), unit)), mult(rd(mult(Z, ld(Z, Y)), ld(ld(Y, Z), unit)), mult(ld(ld(Y, Z), unit), mult(Z, ld(Z, Y))))))
% 27.02/3.78 = { by lemma 31 }
% 27.02/3.78 mult(mult(X, Z), ld(rd(mult(Z, ld(Z, Y)), ld(ld(Y, Z), unit)), mult(mult(Z, ld(Z, Y)), mult(Z, ld(Z, Y)))))
% 27.02/3.78 = { by lemma 44 }
% 27.02/3.78 mult(mult(X, Z), ld(mult(mult(Z, ld(Z, Y)), ld(Y, Z)), mult(mult(Z, ld(Z, Y)), mult(Z, ld(Z, Y)))))
% 27.02/3.78 = { by lemma 25 }
% 27.02/3.78 mult(mult(X, Z), ld(ld(Y, Z), mult(Z, ld(Z, Y))))
% 27.02/3.78 = { by lemma 47 }
% 27.02/3.78 rd(mult(X, Z), ld(mult(Z, ld(Z, Y)), ld(Y, Z)))
% 27.02/3.78 = { by axiom 6 (f01) }
% 27.02/3.78 rd(mult(X, Z), ld(Y, ld(Y, Z)))
% 27.02/3.78
% 27.02/3.78 Goal 1 (goals): mult(a, mult(b, mult(a, c))) = mult(mult(mult(a, b), a), c).
% 27.02/3.78 Proof:
% 27.02/3.78 mult(a, mult(b, mult(a, c)))
% 27.02/3.78 = { by axiom 5 (ifeq_axiom) R->L }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(b, mult(a, mult(c, a))), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by axiom 8 (f03) R->L }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(a, mult(c, a))), a), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by axiom 9 (f08) R->L }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(mult(a, c), a)), a), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by axiom 3 (f02) R->L }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(mult(a, c), a)), ld(f(mult(a, c)), mult(f(mult(a, c)), a))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 36 R->L }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(f(mult(a, c)), mult(f(mult(a, c)), a))), ld(f(mult(a, c)), mult(f(mult(a, c)), a))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 49 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(f(mult(a, c)), a)), ld(f(mult(a, c)), ld(f(mult(a, c)), mult(f(mult(a, c)), a)))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 48 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(f(mult(a, c)), a)), ld(mult(a, c), mult(f(mult(a, c)), a))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 37 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, mult(f(mult(a, c)), a)), ld(f(mult(a, c)), a)), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 49 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, a), ld(f(mult(a, c)), ld(f(mult(a, c)), a))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 48 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, a), ld(mult(a, c), a)), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 41 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, a), ld(a, rd(a, c))), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 45 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(rd(mult(b, a), ld(c, unit)), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 44 }
% 27.02/3.78 ifeq(mult(b, mult(a, mult(c, a))), mult(mult(mult(b, a), c), a), mult(a, mult(b, mult(a, c))), mult(mult(a, mult(b, a)), c))
% 27.02/3.78 = { by lemma 17 }
% 27.02/3.78 mult(mult(a, mult(b, a)), c)
% 27.02/3.78 = { by axiom 9 (f08) R->L }
% 27.02/3.78 mult(mult(mult(a, b), a), c)
% 27.02/3.78 % SZS output end Proof
% 27.02/3.78
% 27.02/3.78 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------