0.05/0.09	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.05/0.09	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.09/0.29	% Computer   : n025.cluster.edu
0.09/0.29	% Model      : x86_64 x86_64
0.09/0.29	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.09/0.29	% Memory     : 8042.1875MB
0.09/0.29	% OS         : Linux 3.10.0-693.el7.x86_64
0.09/0.29	% CPULimit   : 1200
0.09/0.29	% WCLimit    : 120
0.09/0.29	% DateTime   : Tue Jul 13 14:52:27 EDT 2021
0.09/0.29	% CPUTime    : 
11.61/1.79	% SZS status Theorem
11.61/1.79	
12.09/1.89	% SZS output start Proof
12.09/1.89	Take the following subset of the input axioms:
12.09/1.89	  fof(f01, axiom, ![B, A]: mult(A, ld(A, B))=B).
12.09/1.89	  fof(f02, axiom, ![B, A]: ld(A, mult(A, B))=B).
12.09/1.89	  fof(f03, axiom, ![B, A]: mult(rd(A, B), B)=A).
12.09/1.89	  fof(f04, axiom, ![B, A]: rd(mult(A, B), B)=A).
12.09/1.89	  fof(f05, axiom, ![B, A, C]: mult(A, mult(B, mult(A, C)))=mult(mult(mult(A, B), A), C)).
12.09/1.89	  fof(goals, conjecture, ?[X0]: ![X1]: (mult(X1, X0)=X1 & mult(X0, X1)=X1)).
12.09/1.89	
12.09/1.89	Now clausify the problem and encode Horn clauses using encoding 3 of
12.09/1.89	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
12.09/1.89	We repeatedly replace C & s=t => u=v by the two clauses:
12.09/1.89	  fresh(y, y, x1...xn) = u
12.09/1.89	  C => fresh(s, t, x1...xn) = v
12.09/1.89	where fresh is a fresh function symbol and x1..xn are the free
12.09/1.89	variables of u and v.
12.09/1.89	A predicate p(X) is encoded as p(X)=true (this is sound, because the
12.09/1.89	input problem has no model of domain size 1).
12.09/1.89	
12.09/1.89	The encoding turns the above axioms into the following unit equations and goals:
12.09/1.89	
12.09/1.89	Axiom 1 (f01): mult(X, ld(X, Y)) = Y.
12.09/1.89	Axiom 2 (f03): mult(rd(X, Y), Y) = X.
12.09/1.89	Axiom 3 (f04): rd(mult(X, Y), Y) = X.
12.09/1.89	Axiom 4 (f02): ld(X, mult(X, Y)) = Y.
12.09/1.89	Axiom 5 (f05): mult(X, mult(Y, mult(X, Z))) = mult(mult(mult(X, Y), X), Z).
12.09/1.89	
12.09/1.89	Lemma 6: rd(mult(X, mult(Y, mult(X, Z))), Z) = mult(mult(X, Y), X).
12.09/1.89	Proof:
12.09/1.89	  rd(mult(X, mult(Y, mult(X, Z))), Z)
12.09/1.89	= { by axiom 5 (f05) }
12.09/1.89	  rd(mult(mult(mult(X, Y), X), Z), Z)
12.09/1.89	= { by axiom 3 (f04) }
12.09/1.89	  mult(mult(X, Y), X)
12.09/1.89	
12.09/1.89	Lemma 7: rd(mult(X, mult(Y, Z)), ld(X, Z)) = mult(mult(X, Y), X).
12.09/1.89	Proof:
12.09/1.89	  rd(mult(X, mult(Y, Z)), ld(X, Z))
12.09/1.89	= { by axiom 1 (f01) R->L }
12.09/1.89	  rd(mult(X, mult(Y, mult(X, ld(X, Z)))), ld(X, Z))
12.09/1.89	= { by lemma 6 }
12.09/1.89	  mult(mult(X, Y), X)
12.09/1.89	
12.09/1.89	Lemma 8: mult(rd(X, Y), mult(Y, mult(rd(X, Y), Z))) = mult(mult(X, rd(X, Y)), Z).
12.09/1.89	Proof:
12.09/1.89	  mult(rd(X, Y), mult(Y, mult(rd(X, Y), Z)))
12.09/1.89	= { by axiom 5 (f05) }
12.09/1.89	  mult(mult(mult(rd(X, Y), Y), rd(X, Y)), Z)
12.09/1.89	= { by axiom 2 (f03) }
12.09/1.89	  mult(mult(X, rd(X, Y)), Z)
12.09/1.89	
12.09/1.89	Lemma 9: mult(X, rd(Y, mult(X, Y))) = rd(X, X).
12.09/1.89	Proof:
12.09/1.89	  mult(X, rd(Y, mult(X, Y)))
12.09/1.89	= { by axiom 3 (f04) R->L }
12.09/1.89	  rd(mult(mult(X, rd(Y, mult(X, Y))), X), X)
12.09/1.89	= { by lemma 6 R->L }
12.09/1.89	  rd(rd(mult(X, mult(rd(Y, mult(X, Y)), mult(X, Y))), Y), X)
12.09/1.89	= { by axiom 2 (f03) }
12.09/1.89	  rd(rd(mult(X, Y), Y), X)
12.09/1.89	= { by axiom 3 (f04) }
12.09/1.89	  rd(X, X)
12.09/1.89	
12.09/1.89	Lemma 10: ld(mult(mult(X, Y), X), mult(X, mult(Y, mult(X, Z)))) = Z.
12.09/1.89	Proof:
12.09/1.89	  ld(mult(mult(X, Y), X), mult(X, mult(Y, mult(X, Z))))
12.09/1.89	= { by axiom 5 (f05) }
12.09/1.89	  ld(mult(mult(X, Y), X), mult(mult(mult(X, Y), X), Z))
12.09/1.89	= { by axiom 4 (f02) }
12.09/1.89	  Z
12.09/1.89	
12.09/1.89	Lemma 11: ld(mult(mult(X, Y), X), mult(X, mult(Y, Z))) = ld(X, Z).
12.09/1.89	Proof:
12.09/1.89	  ld(mult(mult(X, Y), X), mult(X, mult(Y, Z)))
12.09/1.89	= { by axiom 1 (f01) R->L }
12.09/1.89	  ld(mult(mult(X, Y), X), mult(X, mult(Y, mult(X, ld(X, Z)))))
12.09/1.89	= { by lemma 10 }
12.09/1.89	  ld(X, Z)
12.09/1.89	
12.09/1.89	Lemma 12: mult(rd(X, mult(Y, X)), Z) = ld(Y, Z).
12.09/1.89	Proof:
12.09/1.89	  mult(rd(X, mult(Y, X)), Z)
12.09/1.89	= { by axiom 4 (f02) R->L }
12.09/1.89	  ld(Y, mult(Y, mult(rd(X, mult(Y, X)), Z)))
12.09/1.89	= { by axiom 2 (f03) R->L }
12.09/1.89	  ld(mult(rd(Y, Y), Y), mult(Y, mult(rd(X, mult(Y, X)), Z)))
12.09/1.89	= { by lemma 9 R->L }
12.09/1.89	  ld(mult(mult(Y, rd(X, mult(Y, X))), Y), mult(Y, mult(rd(X, mult(Y, X)), Z)))
12.09/1.89	= { by lemma 11 }
12.09/1.89	  ld(Y, Z)
12.09/1.89	
12.09/1.89	Lemma 13: rd(mult(rd(X, Y), mult(Z, X)), Y) = mult(mult(rd(X, Y), Z), rd(X, Y)).
12.09/1.89	Proof:
12.09/1.89	  rd(mult(rd(X, Y), mult(Z, X)), Y)
12.09/1.89	= { by axiom 2 (f03) R->L }
12.09/1.89	  rd(mult(rd(X, Y), mult(Z, mult(rd(X, Y), Y))), Y)
12.09/1.89	= { by lemma 6 }
12.09/1.89	  mult(mult(rd(X, Y), Z), rd(X, Y))
12.09/1.89	
12.09/1.89	Lemma 14: mult(mult(rd(X, mult(Y, X)), Y), rd(X, mult(Y, X))) = rd(X, mult(Y, X)).
12.09/1.89	Proof:
12.09/1.89	  mult(mult(rd(X, mult(Y, X)), Y), rd(X, mult(Y, X)))
12.09/1.89	= { by lemma 13 R->L }
12.09/1.89	  rd(mult(rd(X, mult(Y, X)), mult(Y, X)), mult(Y, X))
12.09/1.89	= { by axiom 2 (f03) }
12.09/1.89	  rd(X, mult(Y, X))
12.09/1.89	
12.09/1.89	Lemma 15: ld(rd(X, mult(Y, X)), Z) = mult(Y, Z).
12.09/1.89	Proof:
12.09/1.89	  ld(rd(X, mult(Y, X)), Z)
12.09/1.89	= { by lemma 11 R->L }
12.09/1.89	  ld(mult(mult(rd(X, mult(Y, X)), Y), rd(X, mult(Y, X))), mult(rd(X, mult(Y, X)), mult(Y, Z)))
12.09/1.89	= { by lemma 14 }
12.09/1.89	  ld(rd(X, mult(Y, X)), mult(rd(X, mult(Y, X)), mult(Y, Z)))
12.09/1.89	= { by axiom 4 (f02) }
12.09/1.89	  mult(Y, Z)
12.09/1.89	
12.09/1.89	Lemma 16: mult(X, mult(Y, mult(X, ld(mult(mult(X, Y), X), Z)))) = Z.
12.09/1.89	Proof:
12.09/1.89	  mult(X, mult(Y, mult(X, ld(mult(mult(X, Y), X), Z))))
12.09/1.89	= { by axiom 5 (f05) }
12.09/1.89	  mult(mult(mult(X, Y), X), ld(mult(mult(X, Y), X), Z))
12.09/1.89	= { by axiom 1 (f01) }
12.09/1.89	  Z
12.09/1.89	
12.09/1.89	Lemma 17: mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z))) = ld(Y, Z).
12.09/1.89	Proof:
12.09/1.89	  mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z)))
12.09/1.89	= { by axiom 4 (f02) R->L }
12.09/1.89	  ld(Y, mult(Y, mult(X, mult(Y, ld(mult(mult(Y, X), Y), Z)))))
12.09/1.89	= { by lemma 16 }
12.09/1.89	  ld(Y, Z)
12.09/1.89	
12.09/1.89	Lemma 18: ld(rd(X, Y), X) = Y.
12.09/1.89	Proof:
12.09/1.90	  ld(rd(X, Y), X)
12.09/1.90	= { by axiom 2 (f03) R->L }
12.09/1.90	  ld(rd(X, Y), mult(rd(X, Y), Y))
12.09/1.90	= { by axiom 4 (f02) }
12.09/1.90	  Y
12.09/1.90	
12.09/1.90	Lemma 19: mult(X, rd(X, X)) = X.
12.09/1.90	Proof:
12.09/1.90	  mult(X, rd(X, X))
12.09/1.90	= { by axiom 2 (f03) R->L }
12.09/1.90	  mult(mult(rd(X, X), X), rd(X, X))
12.09/1.90	= { by lemma 7 R->L }
12.09/1.90	  rd(mult(rd(X, X), mult(X, X)), ld(rd(X, X), X))
12.09/1.90	= { by axiom 2 (f03) R->L }
12.09/1.90	  rd(mult(rd(X, X), mult(X, mult(rd(X, X), X))), ld(rd(X, X), X))
12.09/1.90	= { by lemma 8 }
12.09/1.90	  rd(mult(mult(X, rd(X, X)), X), ld(rd(X, X), X))
12.09/1.90	= { by lemma 7 R->L }
12.09/1.90	  rd(rd(mult(X, mult(rd(X, X), ld(rd(X, X), Y))), ld(X, ld(rd(X, X), Y))), ld(rd(X, X), X))
12.09/1.90	= { by axiom 1 (f01) }
12.09/1.90	  rd(rd(mult(X, Y), ld(X, ld(rd(X, X), Y))), ld(rd(X, X), X))
12.09/1.90	= { by lemma 12 R->L }
12.09/1.90	  rd(rd(mult(X, Y), mult(rd(Z, mult(X, Z)), ld(rd(X, X), Y))), ld(rd(X, X), X))
12.09/1.90	= { by lemma 15 R->L }
12.09/1.90	  rd(rd(ld(rd(Z, mult(X, Z)), Y), mult(rd(Z, mult(X, Z)), ld(rd(X, X), Y))), ld(rd(X, X), X))
12.09/1.90	= { by lemma 9 R->L }
12.09/1.90	  rd(rd(ld(rd(Z, mult(X, Z)), Y), mult(rd(Z, mult(X, Z)), ld(mult(X, rd(Z, mult(X, Z))), Y))), ld(rd(X, X), X))
12.09/1.90	= { by lemma 17 R->L }
12.09/1.90	  rd(rd(mult(ld(rd(Z, mult(X, Z)), X), mult(rd(Z, mult(X, Z)), ld(mult(mult(rd(Z, mult(X, Z)), ld(rd(Z, mult(X, Z)), X)), rd(Z, mult(X, Z))), Y))), mult(rd(Z, mult(X, Z)), ld(mult(X, rd(Z, mult(X, Z))), Y))), ld(rd(X, X), X))
12.09/1.90	= { by axiom 1 (f01) }
12.09/1.90	  rd(rd(mult(ld(rd(Z, mult(X, Z)), X), mult(rd(Z, mult(X, Z)), ld(mult(X, rd(Z, mult(X, Z))), Y))), mult(rd(Z, mult(X, Z)), ld(mult(X, rd(Z, mult(X, Z))), Y))), ld(rd(X, X), X))
12.09/1.90	= { by axiom 3 (f04) }
12.09/1.90	  rd(ld(rd(Z, mult(X, Z)), X), ld(rd(X, X), X))
12.09/1.90	= { by lemma 15 }
12.09/1.90	  rd(mult(X, X), ld(rd(X, X), X))
12.09/1.90	= { by lemma 18 }
12.09/1.90	  rd(mult(X, X), X)
12.09/1.90	= { by axiom 3 (f04) }
12.09/1.90	  X
12.09/1.90	
12.09/1.90	Lemma 20: ld(X, X) = rd(X, X).
12.09/1.90	Proof:
12.09/1.90	  ld(X, X)
12.09/1.90	= { by lemma 19 R->L }
12.09/1.90	  ld(X, mult(X, rd(X, X)))
12.09/1.90	= { by axiom 4 (f02) }
12.09/1.90	  rd(X, X)
12.09/1.90	
12.09/1.90	Lemma 21: rd(Z, mult(Y, Z)) = rd(X, mult(Y, X)).
12.09/1.90	Proof:
12.09/1.90	  rd(Z, mult(Y, Z))
12.09/1.90	= { by axiom 4 (f02) R->L }
12.09/1.90	  ld(Y, mult(Y, rd(Z, mult(Y, Z))))
12.09/1.90	= { by lemma 9 }
12.09/1.90	  ld(Y, rd(Y, Y))
12.09/1.90	= { by lemma 9 R->L }
12.09/1.90	  ld(Y, mult(Y, rd(X, mult(Y, X))))
12.09/1.90	= { by axiom 4 (f02) }
12.09/1.90	  rd(X, mult(Y, X))
12.09/1.90	
12.09/1.90	Lemma 22: mult(mult(X, rd(ld(X, Y), Y)), X) = X.
12.09/1.90	Proof:
12.09/1.90	  mult(mult(X, rd(ld(X, Y), Y)), X)
12.09/1.90	= { by lemma 7 R->L }
12.09/1.90	  rd(mult(X, mult(rd(ld(X, Y), Y), Y)), ld(X, Y))
12.09/1.90	= { by axiom 2 (f03) }
12.09/1.90	  rd(mult(X, ld(X, Y)), ld(X, Y))
12.09/1.90	= { by axiom 3 (f04) }
12.09/1.90	  X
12.09/1.90	
12.09/1.90	Lemma 23: mult(rd(X, X), rd(Y, mult(X, Y))) = rd(Z, mult(X, Z)).
12.09/1.90	Proof:
12.09/1.90	  mult(rd(X, X), rd(Y, mult(X, Y)))
12.09/1.90	= { by lemma 21 }
12.09/1.90	  mult(rd(X, X), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by lemma 20 R->L }
12.09/1.90	  mult(ld(X, X), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by lemma 11 R->L }
12.09/1.90	  mult(ld(mult(mult(X, rd(ld(X, W), W)), X), mult(X, mult(rd(ld(X, W), W), X))), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by lemma 22 }
12.09/1.90	  mult(ld(X, mult(X, mult(rd(ld(X, W), W), X))), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by axiom 4 (f02) }
12.09/1.90	  mult(mult(rd(ld(X, W), W), X), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by axiom 1 (f01) R->L }
12.09/1.90	  mult(mult(rd(ld(X, W), mult(X, ld(X, W))), X), rd(ld(X, W), mult(X, ld(X, W))))
12.09/1.90	= { by lemma 14 }
12.09/1.90	  rd(ld(X, W), mult(X, ld(X, W)))
12.09/1.90	= { by lemma 21 R->L }
12.09/1.90	  rd(Z, mult(X, Z))
12.09/1.90	
12.09/1.90	Lemma 24: mult(rd(X, X), rd(X, X)) = rd(X, X).
12.09/1.90	Proof:
12.09/1.90	  mult(rd(X, X), rd(X, X))
12.09/1.90	= { by lemma 9 R->L }
12.09/1.90	  mult(rd(X, X), mult(X, rd(Y, mult(X, Y))))
12.09/1.90	= { by lemma 23 R->L }
12.09/1.90	  mult(rd(X, X), mult(X, mult(rd(X, X), rd(Z, mult(X, Z)))))
12.09/1.90	= { by lemma 8 }
12.09/1.90	  mult(mult(X, rd(X, X)), rd(Z, mult(X, Z)))
12.09/1.90	= { by lemma 19 }
12.09/1.90	  mult(X, rd(Z, mult(X, Z)))
12.09/1.90	= { by lemma 9 }
12.09/1.90	  rd(X, X)
12.09/1.90	
12.09/1.90	Lemma 25: rd(rd(X, X), rd(X, X)) = rd(X, X).
12.09/1.90	Proof:
12.09/1.90	  rd(rd(X, X), rd(X, X))
12.09/1.90	= { by lemma 9 R->L }
12.09/1.90	  rd(mult(X, rd(Y, mult(X, Y))), rd(X, X))
12.09/1.90	= { by lemma 9 R->L }
12.09/1.90	  rd(mult(X, rd(Y, mult(X, Y))), mult(X, rd(Z, mult(X, Z))))
12.09/1.90	= { by lemma 19 R->L }
12.09/1.90	  rd(mult(mult(X, rd(X, X)), rd(Y, mult(X, Y))), mult(X, rd(Z, mult(X, Z))))
12.09/1.90	= { by lemma 23 R->L }
12.09/1.90	  rd(mult(mult(X, rd(X, X)), rd(Y, mult(X, Y))), mult(X, mult(rd(X, X), rd(Y, mult(X, Y)))))
12.09/1.90	= { by lemma 8 R->L }
12.09/1.90	  rd(mult(rd(X, X), mult(X, mult(rd(X, X), rd(Y, mult(X, Y))))), mult(X, mult(rd(X, X), rd(Y, mult(X, Y)))))
12.09/1.90	= { by axiom 3 (f04) }
12.09/1.90	  rd(X, X)
12.09/1.90	
12.09/1.90	Lemma 26: ld(rd(X, Y), Z) = mult(rd(Y, X), Z).
12.09/1.90	Proof:
12.09/1.90	  ld(rd(X, Y), Z)
12.09/1.90	= { by axiom 2 (f03) R->L }
12.09/1.90	  ld(rd(X, mult(rd(Y, X), X)), Z)
12.09/1.90	= { by lemma 15 }
12.09/1.90	  mult(rd(Y, X), Z)
12.09/1.90	
12.09/1.90	Lemma 27: mult(X, mult(ld(X, Y), mult(X, Z))) = mult(mult(Y, X), Z).
12.09/1.90	Proof:
12.09/1.90	  mult(X, mult(ld(X, Y), mult(X, Z)))
12.09/1.90	= { by axiom 5 (f05) }
12.09/1.90	  mult(mult(mult(X, ld(X, Y)), X), Z)
12.09/1.90	= { by axiom 1 (f01) }
12.09/1.90	  mult(mult(Y, X), Z)
12.09/1.90	
12.09/1.90	Lemma 28: mult(rd(X, Y), mult(mult(rd(Y, X), Z), X)) = mult(mult(Z, rd(X, Y)), Y).
12.09/1.90	Proof:
12.09/1.90	  mult(rd(X, Y), mult(mult(rd(Y, X), Z), X))
12.09/1.90	= { by lemma 26 R->L }
12.09/1.90	  mult(rd(X, Y), mult(ld(rd(X, Y), Z), X))
12.09/1.90	= { by axiom 1 (f01) R->L }
12.09/1.90	  mult(rd(X, Y), mult(ld(rd(X, Y), Z), mult(rd(X, Y), ld(rd(X, Y), X))))
12.09/1.90	= { by lemma 27 }
12.09/1.90	  mult(mult(Z, rd(X, Y)), ld(rd(X, Y), X))
12.09/1.90	= { by lemma 18 }
12.09/1.90	  mult(mult(Z, rd(X, Y)), Y)
12.09/1.90	
12.09/1.90	Lemma 29: mult(X, rd(ld(X, Y), Y)) = rd(X, X).
12.09/1.90	Proof:
12.09/1.90	  mult(X, rd(ld(X, Y), Y))
12.09/1.90	= { by axiom 3 (f04) R->L }
12.09/1.90	  rd(mult(mult(X, rd(ld(X, Y), Y)), X), X)
12.09/1.90	= { by lemma 22 }
12.09/1.90	  rd(X, X)
12.09/1.90	
12.09/1.90	Lemma 30: mult(rd(X, Y), mult(rd(Y, X), Z)) = Z.
12.09/1.90	Proof:
12.09/1.90	  mult(rd(X, Y), mult(rd(Y, X), Z))
12.09/1.90	= { by axiom 2 (f03) R->L }
12.09/1.90	  mult(rd(X, mult(rd(Y, X), X)), mult(rd(Y, X), Z))
12.09/1.90	= { by axiom 1 (f01) R->L }
12.09/1.90	  mult(rd(X, mult(rd(Y, X), X)), mult(rd(Y, X), mult(rd(X, mult(rd(Y, X), X)), ld(rd(X, mult(rd(Y, X), X)), Z))))
12.09/1.90	= { by lemma 14 R->L }
12.09/1.90	  mult(rd(X, mult(rd(Y, X), X)), mult(rd(Y, X), mult(rd(X, mult(rd(Y, X), X)), ld(mult(mult(rd(X, mult(rd(Y, X), X)), rd(Y, X)), rd(X, mult(rd(Y, X), X))), Z))))
12.09/1.90	= { by lemma 16 }
12.70/1.91	  Z
12.70/1.91	
12.70/1.91	Lemma 31: mult(mult(X, rd(Y, Y)), rd(Y, Y)) = rd(X, rd(Y, Y)).
12.70/1.91	Proof:
12.70/1.91	  mult(mult(X, rd(Y, Y)), rd(Y, Y))
12.70/1.91	= { by axiom 3 (f04) R->L }
12.70/1.91	  mult(mult(X, rd(mult(rd(Y, Y), mult(rd(Y, Y), ld(mult(mult(rd(Y, Y), rd(Y, Y)), rd(Y, Y)), Z))), mult(rd(Y, Y), ld(mult(mult(rd(Y, Y), rd(Y, Y)), rd(Y, Y)), Z)))), rd(Y, Y))
12.70/1.91	= { by lemma 17 }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(mult(mult(rd(Y, Y), rd(Y, Y)), rd(Y, Y)), Z)))), rd(Y, Y))
12.70/1.91	= { by axiom 2 (f03) R->L }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(mult(mult(rd(Y, Y), rd(Y, Y)), rd(Y, mult(rd(Y, Y), Y))), Z)))), rd(Y, Y))
12.70/1.91	= { by axiom 2 (f03) R->L }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(mult(mult(rd(Y, mult(rd(Y, Y), Y)), rd(Y, Y)), rd(Y, mult(rd(Y, Y), Y))), Z)))), rd(Y, Y))
12.70/1.91	= { by lemma 14 }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(rd(Y, mult(rd(Y, Y), Y)), Z)))), rd(Y, Y))
12.70/1.91	= { by axiom 2 (f03) }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(rd(Y, Y), Z)))), rd(Y, Y))
12.70/1.91	= { by lemma 24 R->L }
12.70/1.91	  mult(mult(X, rd(ld(rd(Y, Y), Z), mult(rd(Y, Y), ld(rd(Y, Y), Z)))), mult(rd(Y, Y), rd(Y, Y)))
12.70/1.91	= { by lemma 21 }
12.70/1.91	  mult(mult(X, rd(rd(Y, Y), mult(rd(Y, Y), rd(Y, Y)))), mult(rd(Y, Y), rd(Y, Y)))
12.70/1.91	= { by lemma 28 R->L }
12.70/1.91	  mult(rd(rd(Y, Y), mult(rd(Y, Y), rd(Y, Y))), mult(mult(rd(mult(rd(Y, Y), rd(Y, Y)), rd(Y, Y)), X), rd(Y, Y)))
12.70/1.91	= { by axiom 3 (f04) }
12.70/1.91	  mult(rd(rd(Y, Y), mult(rd(Y, Y), rd(Y, Y))), mult(mult(rd(Y, Y), X), rd(Y, Y)))
12.70/1.91	= { by lemma 12 }
12.70/1.91	  ld(rd(Y, Y), mult(mult(rd(Y, Y), X), rd(Y, Y)))
12.70/1.91	= { by lemma 26 }
12.70/1.91	  mult(rd(Y, Y), mult(mult(rd(Y, Y), X), rd(Y, Y)))
12.70/1.91	= { by lemma 13 R->L }
12.70/1.91	  mult(rd(Y, Y), rd(mult(rd(Y, Y), mult(X, Y)), Y))
12.70/1.91	= { by lemma 26 R->L }
12.70/1.91	  mult(rd(Y, Y), rd(ld(rd(Y, Y), mult(X, Y)), Y))
12.70/1.91	= { by lemma 29 R->L }
12.70/1.91	  mult(rd(Y, Y), rd(ld(mult(Y, rd(ld(Y, W), W)), mult(X, Y)), Y))
12.70/1.91	= { by lemma 22 R->L }
12.70/1.91	  mult(rd(Y, Y), rd(ld(mult(Y, rd(ld(Y, W), W)), mult(X, Y)), mult(mult(Y, rd(ld(Y, W), W)), Y)))
12.70/1.91	= { by axiom 2 (f03) R->L }
12.70/1.91	  mult(rd(Y, Y), rd(ld(mult(Y, rd(ld(Y, W), W)), mult(mult(rd(X, mult(Y, rd(ld(Y, W), W))), mult(Y, rd(ld(Y, W), W))), Y)), mult(mult(Y, rd(ld(Y, W), W)), Y)))
12.70/1.91	= { by lemma 27 R->L }
12.70/1.91	  mult(rd(Y, Y), rd(ld(mult(Y, rd(ld(Y, W), W)), mult(mult(Y, rd(ld(Y, W), W)), mult(ld(mult(Y, rd(ld(Y, W), W)), rd(X, mult(Y, rd(ld(Y, W), W)))), mult(mult(Y, rd(ld(Y, W), W)), Y)))), mult(mult(Y, rd(ld(Y, W), W)), Y)))
12.70/1.91	= { by axiom 4 (f02) }
12.70/1.91	  mult(rd(Y, Y), rd(mult(ld(mult(Y, rd(ld(Y, W), W)), rd(X, mult(Y, rd(ld(Y, W), W)))), mult(mult(Y, rd(ld(Y, W), W)), Y)), mult(mult(Y, rd(ld(Y, W), W)), Y)))
12.70/1.91	= { by axiom 3 (f04) }
12.70/1.91	  mult(rd(Y, Y), ld(mult(Y, rd(ld(Y, W), W)), rd(X, mult(Y, rd(ld(Y, W), W)))))
12.70/1.91	= { by lemma 29 }
12.70/1.91	  mult(rd(Y, Y), ld(rd(Y, Y), rd(X, mult(Y, rd(ld(Y, W), W)))))
12.70/1.91	= { by lemma 29 }
12.70/1.91	  mult(rd(Y, Y), ld(rd(Y, Y), rd(X, rd(Y, Y))))
12.70/1.91	= { by lemma 26 }
12.70/1.91	  mult(rd(Y, Y), mult(rd(Y, Y), rd(X, rd(Y, Y))))
12.70/1.91	= { by lemma 30 }
12.70/1.91	  rd(X, rd(Y, Y))
12.70/1.91	
12.70/1.91	Lemma 32: ld(mult(rd(X, X), mult(Y, Y)), mult(mult(rd(X, X), Y), mult(mult(Y, rd(X, X)), X))) = X.
12.70/1.91	Proof:
12.70/1.91	  ld(mult(rd(X, X), mult(Y, Y)), mult(mult(rd(X, X), Y), mult(mult(Y, rd(X, X)), X)))
12.70/1.91	= { by lemma 30 R->L }
12.70/1.91	  ld(mult(rd(X, X), mult(Y, mult(rd(X, X), mult(rd(X, X), Y)))), mult(mult(rd(X, X), Y), mult(mult(Y, rd(X, X)), X)))
12.70/1.91	= { by lemma 28 R->L }
12.70/1.91	  ld(mult(rd(X, X), mult(Y, mult(rd(X, X), mult(rd(X, X), Y)))), mult(mult(rd(X, X), Y), mult(rd(X, X), mult(mult(rd(X, X), Y), X))))
12.70/1.91	= { by axiom 5 (f05) }
12.70/1.91	  ld(mult(mult(mult(rd(X, X), Y), rd(X, X)), mult(rd(X, X), Y)), mult(mult(rd(X, X), Y), mult(rd(X, X), mult(mult(rd(X, X), Y), X))))
12.70/1.91	= { by lemma 10 }
12.70/1.91	  X
12.70/1.91	
12.70/1.91	Lemma 33: mult(mult(rd(X, X), rd(Y, Y)), Y) = Y.
12.70/1.91	Proof:
12.70/1.91	  mult(mult(rd(X, X), rd(Y, Y)), Y)
12.70/1.91	= { by axiom 4 (f02) R->L }
12.70/1.91	  ld(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.91	= { by lemma 24 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), mult(rd(X, X), rd(X, X))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.91	= { by lemma 32 }
12.70/1.91	  Y
12.70/1.91	
12.70/1.91	Lemma 34: mult(rd(X, X), Y) = Y.
12.70/1.91	Proof:
12.70/1.91	  mult(rd(X, X), Y)
12.70/1.91	= { by lemma 24 R->L }
12.70/1.91	  mult(mult(rd(X, X), rd(X, X)), Y)
12.70/1.91	= { by axiom 4 (f02) R->L }
12.70/1.91	  ld(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(X, X)), Y)))
12.70/1.91	= { by lemma 20 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(X, X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(X, X)), Y)))
12.70/1.91	= { by lemma 19 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(X, X)), Y)))
12.70/1.91	= { by axiom 3 (f04) R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(mult(rd(X, X), rd(Z, Z)), rd(Z, Z))), Y)))
12.70/1.91	= { by lemma 25 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(mult(rd(X, X), rd(rd(Z, Z), rd(Z, Z))), rd(Z, Z))), Y)))
12.70/1.91	= { by lemma 31 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), mult(mult(mult(rd(X, X), rd(rd(Z, Z), rd(Z, Z))), rd(Z, Z)), rd(Z, Z))), Y)))
12.70/1.91	= { by lemma 33 }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), mult(rd(Z, Z), rd(Z, Z))), Y)))
12.70/1.91	= { by lemma 33 R->L }
12.70/1.91	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), mult(mult(mult(rd(Y, Y), rd(rd(Z, Z), rd(Z, Z))), rd(Z, Z)), rd(Z, Z))), Y)))
12.70/1.91	= { by lemma 31 }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(mult(rd(Y, Y), rd(rd(Z, Z), rd(Z, Z))), rd(Z, Z))), Y)))
12.70/1.92	= { by lemma 25 }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(mult(rd(Y, Y), rd(Z, Z)), rd(Z, Z))), Y)))
12.70/1.92	= { by axiom 3 (f04) }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(X, rd(X, X)), X)), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 10 R->L }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(mult(rd(X, X), X), rd(X, X)), mult(rd(X, X), mult(X, mult(rd(X, X), ld(mult(X, rd(X, X)), X)))))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by axiom 2 (f03) R->L }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(mult(rd(X, X), X), rd(X, X)), mult(rd(X, X), mult(X, mult(rd(X, X), ld(mult(mult(rd(X, X), X), rd(X, X)), X)))))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 17 }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(mult(rd(X, X), X), rd(X, X)), mult(rd(X, X), ld(rd(X, X), X)))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 26 }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(mult(rd(X, X), X), rd(X, X)), mult(rd(X, X), mult(rd(X, X), X)))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by axiom 1 (f01) R->L }
12.70/1.92	  ld(mult(rd(Y, Y), ld(mult(mult(rd(X, X), X), rd(X, X)), mult(rd(X, X), mult(X, ld(X, mult(rd(X, X), X)))))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 11 }
12.70/1.92	  ld(mult(rd(Y, Y), ld(rd(X, X), ld(X, mult(rd(X, X), X)))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 26 }
12.70/1.92	  ld(mult(rd(Y, Y), mult(rd(X, X), ld(X, mult(rd(X, X), X)))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by axiom 2 (f03) }
12.70/1.92	  ld(mult(rd(Y, Y), mult(rd(X, X), ld(X, X))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 20 }
12.70/1.92	  ld(mult(rd(Y, Y), mult(rd(X, X), rd(X, X))), mult(mult(rd(Y, Y), rd(X, X)), mult(mult(rd(X, X), rd(Y, Y)), Y)))
12.70/1.92	= { by lemma 32 }
12.70/1.92	  Y
12.70/1.92	
12.70/1.92	Goal 1 (goals): tuple(mult(X, x1(X)), mult(x1_2(X), X)) = tuple(x1(X), x1_2(X)).
12.70/1.92	The goal is true when:
12.70/1.92	  X = rd(X, X)
12.70/1.92	
12.70/1.92	Proof:
12.70/1.92	  tuple(mult(rd(X, X), x1(rd(X, X))), mult(x1_2(rd(X, X)), rd(X, X)))
12.70/1.92	= { by lemma 34 R->L }
12.70/1.92	  tuple(mult(rd(X, X), x1(rd(X, X))), mult(mult(rd(X, X), x1_2(rd(X, X))), rd(X, X)))
12.70/1.92	= { by lemma 13 R->L }
12.70/1.92	  tuple(mult(rd(X, X), x1(rd(X, X))), rd(mult(rd(X, X), mult(x1_2(rd(X, X)), X)), X))
12.70/1.92	= { by lemma 34 }
12.70/1.92	  tuple(mult(rd(X, X), x1(rd(X, X))), rd(mult(x1_2(rd(X, X)), X), X))
12.70/1.92	= { by axiom 3 (f04) }
12.70/1.92	  tuple(mult(rd(X, X), x1(rd(X, X))), x1_2(rd(X, X)))
12.70/1.92	= { by lemma 34 }
12.70/1.92	  tuple(x1(rd(X, X)), x1_2(rd(X, X)))
12.70/1.92	% SZS output end Proof
12.70/1.92	
12.70/1.92	RESULT: Theorem (the conjecture is true).
12.70/1.92	EOF