0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.23	% Computer   : n132.star.cs.uiowa.edu
0.03/0.23	% Model      : x86_64 x86_64
0.03/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.23	% Memory     : 32218.625MB
0.03/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.23	% CPULimit   : 300
0.03/0.23	% DateTime   : Sat Jul 14 05:27:55 CDT 2018
0.03/0.23	% CPUTime    : 
3.47/3.69	% SZS status Theorem
3.47/3.69	
4.20/4.42	% SZS output start Proof
4.20/4.42	Take the following subset of the input axioms:
4.20/4.43	  fof(f01, axiom, ![B, A]: B=mult(A, ld(A, B))).
4.20/4.43	  fof(f02, axiom, ![B, A]: B=ld(A, mult(A, B))).
4.20/4.43	  fof(f03, axiom, ![B, A]: A=mult(rd(A, B), B)).
4.20/4.43	  fof(f04, axiom, ![B, A]: rd(mult(A, B), B)=A).
4.20/4.43	  fof(f05, axiom,
4.20/4.43	      ![B, A, C]:
4.20/4.43	        mult(mult(mult(A, B), C), B)=mult(A, mult(B, mult(C, B)))).
4.20/4.43	  fof(goals, conjecture,
4.20/4.43	      ?[X0]: ![X1]: (mult(X0, X1)=X1 & X1=mult(X1, X0))).
4.20/4.43	
4.20/4.43	Now clausify the problem and encode Horn clauses using encoding 3 of
4.20/4.43	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
4.20/4.43	We repeatedly replace C & s=t => u=v by the two clauses:
4.20/4.43	  $$fresh(y, y, x1...xn) = u
4.20/4.43	  C => $$fresh(s, t, x1...xn) = v
4.20/4.43	where $$fresh is a fresh function symbol and x1..xn are the free
4.20/4.43	variables of u and v.
4.20/4.43	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
4.20/4.43	input problem has no model of domain size 1).
4.20/4.43	
4.20/4.43	The encoding turns the above axioms into the following unit equations and goals:
4.20/4.43	
4.20/4.43	Axiom 1 (f05): mult(mult(mult(X, Y), Z), Y) = mult(X, mult(Y, mult(Z, Y))).
4.20/4.43	Axiom 2 (f02): X = ld(Y, mult(Y, X)).
4.20/4.43	Axiom 3 (f04): rd(mult(X, Y), Y) = X.
4.20/4.43	Axiom 4 (f03): X = mult(rd(X, Y), Y).
4.20/4.43	Axiom 5 (f01): X = mult(Y, ld(Y, X)).
4.20/4.43	
4.20/4.43	Lemma 6: ld(rd(Y, X), Y) = X.
4.20/4.43	Proof:
4.20/4.43	  ld(rd(Y, X), Y)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  ld(rd(Y, X), mult(rd(Y, X), X))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  X
4.20/4.43	
4.20/4.43	Lemma 7: rd(Y, ld(X, Y)) = X.
4.20/4.43	Proof:
4.20/4.43	  rd(Y, ld(X, Y))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  rd(mult(X, ld(X, Y)), ld(X, Y))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  X
4.20/4.43	
4.20/4.43	Lemma 8: rd(mult(X, mult(Y, mult(Z, Y))), Y) = mult(mult(X, Y), Z).
4.20/4.43	Proof:
4.20/4.43	  rd(mult(X, mult(Y, mult(Z, Y))), Y)
4.20/4.43	= { by axiom 1 (f05) }
4.20/4.43	  rd(mult(mult(mult(X, Y), Z), Y), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  mult(mult(X, Y), Z)
4.20/4.43	
4.20/4.43	Lemma 9: rd(mult(X, mult(Y, Z)), Y) = mult(mult(X, Y), rd(Z, Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(mult(X, mult(Y, Z)), Y)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  rd(mult(X, mult(Y, mult(rd(Z, Y), Y))), Y)
4.20/4.43	= { by lemma 8 }
4.20/4.43	  mult(mult(X, Y), rd(Z, Y))
4.20/4.43	
4.20/4.43	Lemma 10: mult(rd(X, Y), mult(Y, mult(Z, Y))) = mult(mult(X, Z), Y).
4.20/4.43	Proof:
4.20/4.43	  mult(rd(X, Y), mult(Y, mult(Z, Y)))
4.20/4.43	= { by axiom 1 (f05) }
4.20/4.43	  mult(mult(mult(rd(X, Y), Y), Z), Y)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  mult(mult(X, Z), Y)
4.20/4.43	
4.20/4.43	Lemma 11: mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))) = mult(mult(Y, Z), ld(X, Y)).
4.20/4.43	Proof:
4.20/4.43	  mult(X, mult(ld(X, Y), mult(Z, ld(X, Y))))
4.20/4.43	= { by axiom 1 (f05) }
4.20/4.43	  mult(mult(mult(X, ld(X, Y)), Z), ld(X, Y))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  mult(mult(Y, Z), ld(X, Y))
4.20/4.43	
4.20/4.43	Lemma 12: ld(rd(Z, X), mult(mult(Z, Y), X)) = mult(X, mult(Y, X)).
4.20/4.43	Proof:
4.20/4.43	  ld(rd(Z, X), mult(mult(Z, Y), X))
4.20/4.43	= { by lemma 10 }
4.20/4.43	  ld(rd(Z, X), mult(rd(Z, X), mult(X, mult(Y, X))))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  mult(X, mult(Y, X))
4.20/4.43	
4.20/4.43	Lemma 13: ld(rd(Y, X), mult(Z, X)) = mult(X, mult(ld(Y, Z), X)).
4.20/4.43	Proof:
4.20/4.43	  ld(rd(Y, X), mult(Z, X))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  ld(rd(Y, X), mult(mult(Y, ld(Y, Z)), X))
4.20/4.43	= { by lemma 12 }
4.20/4.43	  mult(X, mult(ld(Y, Z), X))
4.20/4.43	
4.20/4.43	Lemma 14: mult(X, mult(ld(Y, rd(Y, X)), X)) = X.
4.20/4.43	Proof:
4.20/4.43	  mult(X, mult(ld(Y, rd(Y, X)), X))
4.20/4.43	= { by lemma 13 }
4.20/4.43	  ld(rd(Y, X), mult(rd(Y, X), X))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  X
4.20/4.43	
4.20/4.43	Lemma 15: mult(ld(X, rd(X, Y)), Y) = ld(Y, Y).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(X, rd(X, Y)), Y)
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  ld(Y, mult(Y, mult(ld(X, rd(X, Y)), Y)))
4.20/4.43	= { by lemma 14 }
4.20/4.43	  ld(Y, Y)
4.20/4.43	
4.20/4.43	Lemma 16: rd(ld(Y, Y), Y) = ld(?, rd(?, Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(ld(Y, Y), Y)
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(mult(ld(?, rd(?, Y)), Y), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  ld(?, rd(?, Y))
4.20/4.43	
4.20/4.43	Lemma 17: ld(X, rd(X, Y)) = ld(?, rd(?, Y)).
4.20/4.43	Proof:
4.20/4.43	  ld(X, rd(X, Y))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(mult(ld(X, rd(X, Y)), Y), Y)
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(ld(Y, Y), Y)
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(mult(ld(?, rd(?, Y)), Y), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  ld(?, rd(?, Y))
4.20/4.43	
4.20/4.43	Lemma 18: mult(X, mult(ld(mult(Y, X), Y), X)) = X.
4.20/4.43	Proof:
4.20/4.43	  mult(X, mult(ld(mult(Y, X), Y), X))
4.20/4.43	= { by lemma 13 }
4.20/4.43	  ld(rd(mult(Y, X), X), mult(Y, X))
4.20/4.43	= { by lemma 6 }
4.20/4.43	  X
4.20/4.43	
4.20/4.43	Lemma 19: mult(ld(mult(X, Y), X), Y) = ld(Y, Y).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(mult(X, Y), X), Y)
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  ld(Y, mult(Y, mult(ld(mult(X, Y), X), Y)))
4.20/4.43	= { by lemma 18 }
4.20/4.43	  ld(Y, Y)
4.20/4.43	
4.20/4.43	Lemma 20: ld(mult(X, Y), X) = ld(?, rd(?, Y)).
4.20/4.43	Proof:
4.20/4.43	  ld(mult(X, Y), X)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(mult(ld(mult(X, Y), X), Y), Y)
4.20/4.43	= { by lemma 19 }
4.20/4.43	  rd(ld(Y, Y), Y)
4.20/4.43	= { by lemma 16 }
4.20/4.43	  ld(?, rd(?, Y))
4.20/4.43	
4.20/4.43	Lemma 21: rd(X, ld(Y, Y)) = mult(X, ld(Y, Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(X, ld(Y, Y))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(rd(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(rd(mult(X, mult(ld(?, rd(?, Y)), Y)), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 18 }
4.20/4.43	  rd(rd(mult(X, mult(mult(ld(?, rd(?, Y)), Y), mult(ld(mult(Y, mult(ld(?, rd(?, Y)), Y)), Y), mult(ld(?, rd(?, Y)), Y)))), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(rd(mult(X, mult(ld(Y, Y), mult(ld(mult(Y, mult(ld(?, rd(?, Y)), Y)), Y), mult(ld(?, rd(?, Y)), Y)))), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 14 }
4.20/4.43	  rd(rd(mult(X, mult(ld(Y, Y), mult(ld(Y, Y), mult(ld(?, rd(?, Y)), Y)))), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(rd(mult(X, mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by lemma 8 }
4.20/4.43	  rd(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  mult(X, ld(Y, Y))
4.20/4.43	
4.20/4.43	Lemma 22: mult(X, mult(ld(mult(Y, X), Z), X)) = ld(Y, mult(Z, X)).
4.20/4.43	Proof:
4.20/4.43	  mult(X, mult(ld(mult(Y, X), Z), X))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  ld(Y, mult(Y, mult(X, mult(ld(mult(Y, X), Z), X))))
4.20/4.43	= { by axiom 1 (f05) }
4.20/4.43	  ld(Y, mult(mult(mult(Y, X), ld(mult(Y, X), Z)), X))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  ld(Y, mult(Z, X))
4.20/4.43	
4.20/4.43	Lemma 23: ld(Y, ld(X, mult(Z, Y))) = mult(ld(mult(X, Y), Z), Y).
4.20/4.43	Proof:
4.20/4.43	  ld(Y, ld(X, mult(Z, Y)))
4.20/4.43	= { by lemma 22 }
4.20/4.43	  ld(Y, mult(Y, mult(ld(mult(X, Y), Z), Y)))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  mult(ld(mult(X, Y), Z), Y)
4.20/4.43	
4.20/4.43	Lemma 24: ld(X, mult(mult(Y, Z), ld(X, Y))) = mult(ld(X, Y), mult(Z, ld(X, Y))).
4.20/4.43	Proof:
4.20/4.43	  ld(X, mult(mult(Y, Z), ld(X, Y)))
4.20/4.43	= { by lemma 11 }
4.20/4.43	  ld(X, mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  mult(ld(X, Y), mult(Z, ld(X, Y)))
4.20/4.43	
4.20/4.43	Lemma 25: mult(mult(X, Y), rd(ld(Y, Z), Y)) = rd(mult(X, Z), Y).
4.20/4.43	Proof:
4.20/4.43	  mult(mult(X, Y), rd(ld(Y, Z), Y))
4.20/4.43	= { by lemma 9 }
4.20/4.43	  rd(mult(X, mult(Y, ld(Y, Z))), Y)
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  rd(mult(X, Z), Y)
4.20/4.43	
4.20/4.43	Lemma 26: rd(mult(rd(X, Y), Z), Y) = mult(X, rd(ld(Y, Z), Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(mult(rd(X, Y), Z), Y)
4.20/4.43	= { by lemma 25 }
4.20/4.43	  mult(mult(rd(X, Y), Y), rd(ld(Y, Z), Y))
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  mult(X, rd(ld(Y, Z), Y))
4.20/4.43	
4.20/4.43	Lemma 27: mult(X, ld(?, rd(?, Y))) = rd(X, Y).
4.20/4.43	Proof:
4.20/4.43	  mult(X, ld(?, rd(?, Y)))
4.20/4.43	= { by lemma 16 }
4.20/4.43	  mult(X, rd(ld(Y, Y), Y))
4.20/4.43	= { by lemma 26 }
4.20/4.43	  rd(mult(rd(X, Y), Y), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(X, Y)
4.20/4.43	
4.20/4.43	Lemma 28: ld(ld(?, rd(?, X)), ld(?, rd(?, X))) = rd(X, X).
4.20/4.43	Proof:
4.20/4.43	  ld(ld(?, rd(?, X)), ld(?, rd(?, X)))
4.20/4.43	= { by lemma 19 }
4.20/4.43	  mult(ld(mult(?, ld(?, rd(?, X))), ?), ld(?, rd(?, X)))
4.20/4.43	= { by lemma 27 }
4.20/4.43	  mult(ld(rd(?, X), ?), ld(?, rd(?, X)))
4.20/4.43	= { by lemma 27 }
4.20/4.43	  rd(ld(rd(?, X), ?), X)
4.20/4.43	= { by lemma 6 }
4.20/4.43	  rd(X, X)
4.20/4.43	
4.20/4.43	Lemma 29: mult(rd(X, mult(Y, mult(Z, Y))), Y) = rd(rd(X, Y), Z).
4.20/4.43	Proof:
4.20/4.43	  mult(rd(X, mult(Y, mult(Z, Y))), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), Z), Z)
4.20/4.43	= { by lemma 8 }
4.20/4.43	  rd(rd(mult(rd(X, mult(Y, mult(Z, Y))), mult(Y, mult(Z, Y))), Y), Z)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  rd(rd(X, Y), Z)
4.20/4.43	
4.20/4.43	Lemma 30: rd(rd(X, Y), rd(Z, Y)) = mult(rd(X, mult(Y, Z)), Y).
4.20/4.43	Proof:
4.20/4.43	  rd(rd(X, Y), rd(Z, Y))
4.20/4.43	= { by lemma 29 }
4.20/4.43	  mult(rd(X, mult(Y, mult(rd(Z, Y), Y))), Y)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  mult(rd(X, mult(Y, Z)), Y)
4.20/4.43	
4.20/4.43	Lemma 31: rd(mult(X, Z), mult(Z, mult(Y, Z))) = rd(rd(X, Y), Z).
4.20/4.43	Proof:
4.20/4.43	  rd(mult(X, Z), mult(Z, mult(Y, Z)))
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  rd(mult(mult(rd(X, Y), Y), Z), mult(Z, mult(Y, Z)))
4.20/4.43	= { by lemma 10 }
4.20/4.43	  rd(mult(rd(rd(X, Y), Z), mult(Z, mult(Y, Z))), mult(Z, mult(Y, Z)))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(rd(X, Y), Z)
4.20/4.43	
4.20/4.43	Lemma 32: rd(X, ld(?, rd(?, Y))) = mult(X, Y).
4.20/4.43	Proof:
4.20/4.43	  rd(X, ld(?, rd(?, Y)))
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  rd(rd(mult(X, Y), Y), ld(?, rd(?, Y)))
4.20/4.43	= { by lemma 29 }
4.20/4.43	  mult(rd(mult(X, Y), mult(Y, mult(ld(?, rd(?, Y)), Y))), Y)
4.20/4.43	= { by lemma 14 }
4.20/4.43	  mult(rd(mult(X, Y), Y), Y)
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  mult(X, Y)
4.20/4.43	
4.20/4.43	Lemma 33: mult(ld(mult(X, Y), rd(Z, Y)), Y) = ld(Y, ld(X, Z)).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(mult(X, Y), rd(Z, Y)), Y)
4.20/4.43	= { by lemma 23 }
4.20/4.43	  ld(Y, ld(X, mult(rd(Z, Y), Y)))
4.20/4.43	= { by axiom 4 (f03) }
4.20/4.43	  ld(Y, ld(X, Z))
4.20/4.43	
4.20/4.43	Lemma 34: rd(ld(Y, ld(X, Z)), Y) = ld(mult(X, Y), rd(Z, Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(ld(Y, ld(X, Z)), Y)
4.20/4.43	= { by lemma 33 }
4.20/4.43	  rd(mult(ld(mult(X, Y), rd(Z, Y)), Y), Y)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  ld(mult(X, Y), rd(Z, Y))
4.20/4.43	
4.20/4.43	Lemma 35: ld(mult(Z, X), rd(mult(Z, Y), X)) = rd(ld(X, Y), X).
4.20/4.43	Proof:
4.20/4.43	  ld(mult(Z, X), rd(mult(Z, Y), X))
4.20/4.43	= { by lemma 25 }
4.20/4.43	  ld(mult(Z, X), mult(mult(Z, X), rd(ld(X, Y), X)))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  rd(ld(X, Y), X)
4.20/4.43	
4.20/4.43	Lemma 36: ld(?, rd(?, ld(X, Y))) = ld(Y, X).
4.20/4.43	Proof:
4.20/4.43	  ld(?, rd(?, ld(X, Y)))
4.20/4.43	= { by lemma 20 }
4.20/4.43	  ld(mult(X, ld(X, Y)), X)
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  ld(Y, X)
4.20/4.43	
4.20/4.43	Lemma 37: rd(X, ld(Y, Z)) = mult(X, ld(Z, Y)).
4.20/4.43	Proof:
4.20/4.43	  rd(X, ld(Y, Z))
4.20/4.43	= { by lemma 27 }
4.20/4.43	  mult(X, ld(?, rd(?, ld(Y, Z))))
4.20/4.43	= { by lemma 36 }
4.20/4.43	  mult(X, ld(Z, Y))
4.20/4.43	
4.20/4.43	Lemma 38: mult(X, mult(ld(X, X), X)) = mult(X, X).
4.20/4.43	Proof:
4.20/4.43	  mult(X, mult(ld(X, X), X))
4.20/4.43	= { by lemma 13 }
4.20/4.43	  ld(rd(X, X), mult(X, X))
4.20/4.43	= { by lemma 27 }
4.20/4.43	  ld(mult(X, ld(?, rd(?, X))), mult(X, X))
4.20/4.43	= { by lemma 32 }
4.20/4.43	  ld(mult(X, ld(?, rd(?, X))), rd(X, ld(?, rd(?, X))))
4.20/4.43	= { by lemma 34 }
4.20/4.43	  rd(ld(ld(?, rd(?, X)), ld(X, X)), ld(?, rd(?, X)))
4.20/4.43	= { by lemma 15 }
4.20/4.43	  rd(ld(ld(?, rd(?, X)), mult(ld(?, rd(?, X)), X)), ld(?, rd(?, X)))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  rd(X, ld(?, rd(?, X)))
4.20/4.43	= { by lemma 32 }
4.20/4.43	  mult(X, X)
4.20/4.43	
4.20/4.43	Lemma 39: mult(ld(X, X), X) = X.
4.20/4.43	Proof:
4.20/4.43	  mult(ld(X, X), X)
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  ld(X, mult(X, mult(ld(X, X), X)))
4.20/4.43	= { by lemma 38 }
4.20/4.43	  ld(X, mult(X, X))
4.20/4.43	= { by axiom 2 (f02) }
4.20/4.43	  X
4.20/4.43	
4.20/4.43	Lemma 40: rd(X, X) = ld(X, X).
4.20/4.43	Proof:
4.20/4.43	  rd(X, X)
4.20/4.43	= { by lemma 39 }
4.20/4.43	  rd(mult(ld(X, X), X), X)
4.20/4.43	= { by axiom 3 (f04) }
4.20/4.43	  ld(X, X)
4.20/4.43	
4.20/4.43	Lemma 41: mult(ld(mult(X, ld(Y, Z)), Y), ld(Y, Z)) = ld(ld(Y, Z), ld(X, Z)).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(mult(X, ld(Y, Z)), Y), ld(Y, Z))
4.20/4.43	= { by lemma 23 }
4.20/4.43	  ld(ld(Y, Z), ld(X, mult(Y, ld(Y, Z))))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  ld(ld(Y, Z), ld(X, Z))
4.20/4.43	
4.20/4.43	Lemma 42: rd(Z, mult(ld(X, Z), mult(Y, ld(X, Z)))) = mult(rd(X, Y), ld(Z, X)).
4.20/4.43	Proof:
4.20/4.43	  rd(Z, mult(ld(X, Z), mult(Y, ld(X, Z))))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  rd(mult(X, ld(X, Z)), mult(ld(X, Z), mult(Y, ld(X, Z))))
4.20/4.43	= { by lemma 31 }
4.20/4.43	  rd(rd(X, Y), ld(X, Z))
4.20/4.43	= { by lemma 37 }
4.20/4.43	  mult(rd(X, Y), ld(Z, X))
4.20/4.43	
4.20/4.43	Lemma 44: mult(ld(X, X), ld(Y, X)) = rd(Y, mult(ld(X, Y), Y)).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(X, X), ld(Y, X))
4.20/4.43	= { by lemma 40 }
4.20/4.43	  mult(rd(X, X), ld(Y, X))
4.20/4.43	= { by lemma 42 }
4.20/4.43	  rd(Y, mult(ld(X, Y), mult(X, ld(X, Y))))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  rd(Y, mult(ld(X, Y), Y))
4.20/4.43	
4.20/4.43	Lemma 44: rd(Y, mult(ld(X, Y), Y)) = mult(ld(X, X), ld(Y, X)).
4.20/4.43	Proof:
4.20/4.43	  rd(Y, mult(ld(X, Y), Y))
4.20/4.43	= { by axiom 5 (f01) }
4.20/4.43	  rd(Y, mult(ld(X, Y), mult(X, ld(X, Y))))
4.20/4.43	= { by lemma 42 }
4.20/4.43	  mult(rd(X, X), ld(Y, X))
4.20/4.43	= { by lemma 40 }
4.20/4.43	  mult(ld(X, X), ld(Y, X))
4.20/4.43	
4.20/4.43	Lemma 45: mult(ld(X, X), ld(X, X)) = ld(X, X).
4.20/4.43	Proof:
4.20/4.43	  mult(ld(X, X), ld(X, X))
4.20/4.43	= { by lemma 40 }
4.20/4.43	  mult(rd(X, X), ld(X, X))
4.20/4.43	= { by lemma 28 }
4.20/4.43	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(X, X))
4.20/4.43	= { by lemma 40 }
4.20/4.44	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), rd(X, X))
4.20/4.44	= { by lemma 28 }
4.20/4.44	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(ld(?, rd(?, X)), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 44 }
4.20/4.44	  rd(ld(?, rd(?, X)), mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(?, rd(?, X))))
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  rd(rd(mult(ld(?, rd(?, X)), X), X), mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 28 }
4.20/4.44	  rd(rd(mult(ld(?, rd(?, X)), X), X), mult(rd(X, X), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 27 }
4.20/4.44	  rd(rd(mult(ld(?, rd(?, X)), X), X), rd(rd(X, X), X))
4.20/4.44	= { by lemma 40 }
4.20/4.44	  rd(rd(mult(ld(?, rd(?, X)), X), X), rd(ld(X, X), X))
4.20/4.44	= { by lemma 30 }
4.20/4.44	  mult(rd(mult(ld(?, rd(?, X)), X), mult(X, ld(X, X))), X)
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  mult(rd(mult(ld(?, rd(?, X)), X), X), X)
4.20/4.44	= { by axiom 4 (f03) }
4.20/4.44	  mult(ld(?, rd(?, X)), X)
4.20/4.44	= { by lemma 15 }
4.20/4.44	  ld(X, X)
4.20/4.44	
4.20/4.44	Lemma 46: mult(mult(Y, ld(X, Z)), ld(Z, X)) = Y.
4.20/4.44	Proof:
4.20/4.44	  mult(mult(Y, ld(X, Z)), ld(Z, X))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  mult(mult(X, ld(X, mult(Y, ld(X, Z)))), ld(Z, X))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(rd(X, ld(mult(Y, ld(X, Z)), X)), ld(Z, X))
4.20/4.44	= { by lemma 42 }
4.20/4.44	  rd(Z, mult(ld(X, Z), mult(ld(mult(Y, ld(X, Z)), X), ld(X, Z))))
4.20/4.44	= { by lemma 41 }
4.20/4.44	  rd(Z, mult(ld(X, Z), ld(ld(X, Z), ld(Y, Z))))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  rd(Z, ld(Y, Z))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(Z, ld(Z, Y))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  Y
4.20/4.44	
4.20/4.44	Lemma 47: mult(ld(ld(X, Y), Z), ld(Y, X)) = mult(ld(Y, X), mult(Z, ld(Y, X))).
4.20/4.44	Proof:
4.20/4.44	  mult(ld(ld(X, Y), Z), ld(Y, X))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  rd(ld(ld(X, Y), Z), ld(X, Y))
4.20/4.44	= { by lemma 35 }
4.20/4.44	  ld(mult(X, ld(X, Y)), rd(mult(X, Z), ld(X, Y)))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  ld(Y, rd(mult(X, Z), ld(X, Y)))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  ld(Y, mult(mult(X, Z), ld(Y, X)))
4.20/4.44	= { by lemma 24 }
4.20/4.44	  mult(ld(Y, X), mult(Z, ld(Y, X)))
4.20/4.44	
4.20/4.44	Lemma 48: mult(ld(X, Y), ld(Y, X)) = mult(ld(Y, X), ld(X, Y)).
4.20/4.44	Proof:
4.20/4.44	  mult(ld(X, Y), ld(Y, X))
4.20/4.44	= { by axiom 4 (f03) }
4.20/4.44	  mult(ld(X, Y), mult(rd(ld(Y, X), ld(X, Y)), ld(X, Y)))
4.20/4.44	= { by lemma 47 }
4.20/4.44	  mult(ld(ld(Y, X), rd(ld(Y, X), ld(X, Y))), ld(X, Y))
4.20/4.44	= { by lemma 15 }
4.20/4.44	  ld(ld(X, Y), ld(X, Y))
4.20/4.44	= { by lemma 15 }
4.20/4.44	  mult(ld(Y, rd(Y, ld(X, Y))), ld(X, Y))
4.20/4.44	= { by lemma 7 }
4.20/4.44	  mult(ld(Y, X), ld(X, Y))
4.20/4.44	
4.20/4.44	Lemma 49: mult(ld(?, rd(?, X)), rd(Y, X)) = rd(ld(X, Y), X).
4.20/4.44	Proof:
4.20/4.44	  mult(ld(?, rd(?, X)), rd(Y, X))
4.20/4.44	= { by lemma 27 }
4.20/4.44	  mult(ld(?, rd(?, X)), mult(Y, ld(?, rd(?, X))))
4.20/4.44	= { by lemma 24 }
4.20/4.44	  ld(?, mult(mult(rd(?, X), Y), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 27 }
4.20/4.44	  ld(?, rd(mult(rd(?, X), Y), X))
4.20/4.44	= { by lemma 26 }
4.20/4.44	  ld(?, mult(?, rd(ld(X, Y), X)))
4.20/4.44	= { by axiom 2 (f02) }
4.20/4.44	  rd(ld(X, Y), X)
4.20/4.44	
4.20/4.44	Lemma 50: rd(ld(X, Y), mult(X, Y)) = ld(?, rd(?, mult(X, X))).
4.20/4.44	Proof:
4.20/4.44	  rd(ld(X, Y), mult(X, Y))
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  rd(mult(rd(ld(X, Y), mult(X, Y)), X), X)
4.20/4.44	= { by lemma 30 }
4.20/4.44	  rd(rd(rd(ld(X, Y), X), rd(Y, X)), X)
4.20/4.44	= { by lemma 49 }
4.20/4.44	  rd(rd(mult(ld(?, rd(?, X)), rd(Y, X)), rd(Y, X)), X)
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  rd(ld(?, rd(?, X)), X)
4.20/4.44	= { by lemma 27 }
4.20/4.44	  mult(ld(?, rd(?, X)), ld(?, rd(?, X)))
4.20/4.44	= { by lemma 16 }
4.20/4.44	  mult(ld(?, rd(?, X)), rd(ld(X, X), X))
4.20/4.44	= { by lemma 49 }
4.20/4.44	  rd(ld(X, ld(X, X)), X)
4.20/4.44	= { by lemma 35 }
4.20/4.44	  ld(mult(?, X), rd(mult(?, ld(X, X)), X))
4.20/4.44	= { by lemma 27 }
4.20/4.44	  ld(mult(?, X), mult(mult(?, ld(X, X)), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  ld(mult(?, X), mult(rd(?, ld(X, X)), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 19 }
4.20/4.44	  ld(mult(?, X), mult(rd(?, mult(ld(mult(X, X), X), X)), ld(?, rd(?, X))))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  ld(mult(?, X), mult(rd(?, mult(ld(mult(X, X), X), mult(mult(X, X), ld(mult(X, X), X)))), ld(?, rd(?, X))))
4.20/4.44	= { by lemma 20 }
4.20/4.44	  ld(mult(?, X), mult(rd(?, mult(ld(mult(X, X), X), mult(mult(X, X), ld(mult(X, X), X)))), ld(mult(X, X), X)))
4.20/4.44	= { by lemma 29 }
4.20/4.44	  ld(mult(?, X), rd(rd(?, ld(mult(X, X), X)), mult(X, X)))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  ld(mult(?, X), rd(mult(?, ld(X, mult(X, X))), mult(X, X)))
4.20/4.44	= { by axiom 2 (f02) }
4.20/4.44	  ld(mult(?, X), rd(mult(?, X), mult(X, X)))
4.20/4.44	= { by lemma 17 }
4.20/4.44	  ld(?, rd(?, mult(X, X)))
4.20/4.44	
4.20/4.44	Lemma 51: mult(mult(X, ld(mult(Y, X), Z)), mult(X, ld(Z, mult(Y, X)))) = mult(ld(Y, mult(Z, X)), ld(Z, mult(Y, X))).
4.20/4.44	Proof:
4.20/4.44	  mult(mult(X, ld(mult(Y, X), Z)), mult(X, ld(Z, mult(Y, X))))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(mult(X, ld(mult(Y, X), Z)), rd(X, ld(mult(Y, X), Z)))
4.20/4.44	= { by lemma 9 }
4.20/4.44	  rd(mult(X, mult(ld(mult(Y, X), Z), X)), ld(mult(Y, X), Z))
4.20/4.44	= { by lemma 22 }
4.20/4.44	  rd(ld(Y, mult(Z, X)), ld(mult(Y, X), Z))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(ld(Y, mult(Z, X)), ld(Z, mult(Y, X)))
4.20/4.44	
4.20/4.44	Lemma 52: ld(X, mult(rd(mult(X, Y), Z), Y)) = mult(Y, mult(ld(?, rd(?, Z)), Y)).
4.20/4.44	Proof:
4.20/4.44	  ld(X, mult(rd(mult(X, Y), Z), Y))
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  rd(mult(ld(X, mult(rd(mult(X, Y), Z), Y)), Z), Z)
4.20/4.44	= { by lemma 6 }
4.20/4.44	  rd(mult(ld(X, mult(rd(mult(X, Y), Z), Y)), ld(rd(mult(X, Y), Z), mult(X, Y))), Z)
4.20/4.44	= { by lemma 51 }
4.20/4.44	  rd(mult(mult(Y, ld(mult(X, Y), rd(mult(X, Y), Z))), mult(Y, ld(rd(mult(X, Y), Z), mult(X, Y)))), Z)
4.20/4.44	= { by lemma 17 }
4.20/4.44	  rd(mult(mult(Y, ld(?, rd(?, Z))), mult(Y, ld(rd(mult(X, Y), Z), mult(X, Y)))), Z)
4.20/4.44	= { by lemma 27 }
4.20/4.44	  rd(mult(rd(Y, Z), mult(Y, ld(rd(mult(X, Y), Z), mult(X, Y)))), Z)
4.20/4.44	= { by lemma 6 }
4.20/4.44	  rd(mult(rd(Y, Z), mult(Y, Z)), Z)
4.20/4.44	= { by lemma 26 }
4.20/4.44	  mult(Y, rd(ld(Z, mult(Y, Z)), Z))
4.20/4.44	= { by lemma 49 }
4.20/4.44	  mult(Y, mult(ld(?, rd(?, Z)), rd(mult(Y, Z), Z)))
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  mult(Y, mult(ld(?, rd(?, Z)), Y))
4.20/4.44	
4.20/4.44	Lemma 53: ld(ld(X, Y), ld(X, Y)) = mult(ld(X, Y), ld(Y, X)).
4.20/4.44	Proof:
4.20/4.44	  ld(ld(X, Y), ld(X, Y))
4.20/4.44	= { by lemma 15 }
4.20/4.44	  mult(ld(Y, rd(Y, ld(X, Y))), ld(X, Y))
4.20/4.44	= { by lemma 7 }
4.20/4.44	  mult(ld(Y, X), ld(X, Y))
4.20/4.44	= { by lemma 48 }
4.20/4.44	  mult(ld(X, Y), ld(Y, X))
4.20/4.44	
4.20/4.44	Lemma 54: rd(X, mult(ld(Y, Z), ld(Z, Y))) = mult(X, mult(ld(Y, Z), ld(Z, Y))).
4.20/4.44	Proof:
4.20/4.44	  rd(X, mult(ld(Y, Z), ld(Z, Y)))
4.20/4.44	= { by lemma 48 }
4.20/4.44	  rd(X, mult(ld(Z, Y), ld(Y, Z)))
4.20/4.44	= { by lemma 46 }
4.20/4.44	  mult(mult(rd(X, mult(ld(Z, Y), ld(Y, Z))), ld(Z, Y)), ld(Y, Z))
4.20/4.44	= { by lemma 46 }
4.20/4.44	  mult(mult(rd(X, mult(ld(Z, Y), mult(mult(ld(Y, Z), ld(Y, Z)), ld(Z, Y)))), ld(Z, Y)), ld(Y, Z))
4.20/4.44	= { by lemma 29 }
4.20/4.44	  mult(rd(rd(X, ld(Z, Y)), mult(ld(Y, Z), ld(Y, Z))), ld(Y, Z))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(rd(mult(X, ld(Y, Z)), mult(ld(Y, Z), ld(Y, Z))), ld(Y, Z))
4.20/4.44	= { by axiom 2 (f02) }
4.20/4.44	  mult(rd(mult(X, ld(Y, Z)), mult(ld(Y, Z), ld(Y, Z))), ld(ld(Y, Z), mult(ld(Y, Z), ld(Y, Z))))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  rd(rd(mult(X, ld(Y, Z)), mult(ld(Y, Z), ld(Y, Z))), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z)))
4.20/4.44	= { by lemma 31 }
4.20/4.44	  rd(mult(mult(X, ld(Y, Z)), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z))), mult(ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z)), mult(mult(ld(Y, Z), ld(Y, Z)), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z)))))
4.20/4.44	= { by axiom 5 (f01) }
4.20/4.44	  rd(mult(mult(X, ld(Y, Z)), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z))), mult(ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z)), ld(Y, Z)))
4.20/4.44	= { by lemma 19 }
4.20/4.44	  rd(mult(mult(X, ld(Y, Z)), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z))), ld(ld(Y, Z), ld(Y, Z)))
4.20/4.44	= { by lemma 37 }
4.20/4.44	  mult(mult(mult(X, ld(Y, Z)), ld(mult(ld(Y, Z), ld(Y, Z)), ld(Y, Z))), ld(ld(Y, Z), ld(Y, Z)))
4.20/4.44	= { by lemma 20 }
4.20/4.44	  mult(mult(mult(X, ld(Y, Z)), ld(?, rd(?, ld(Y, Z)))), ld(ld(Y, Z), ld(Y, Z)))
4.20/4.44	= { by lemma 27 }
4.20/4.44	  mult(rd(mult(X, ld(Y, Z)), ld(Y, Z)), ld(ld(Y, Z), ld(Y, Z)))
4.20/4.44	= { by axiom 3 (f04) }
4.20/4.44	  mult(X, ld(ld(Y, Z), ld(Y, Z)))
4.20/4.44	= { by lemma 53 }
4.95/5.18	  mult(X, mult(ld(Y, Z), ld(Z, Y)))
4.95/5.18	
4.95/5.18	Lemma 55: mult(ld(Y, Y), mult(ld(X, X), ld(Y, Y))) = ld(ld(X, X), ld(Y, Y)).
4.95/5.18	Proof:
4.95/5.18	  mult(ld(Y, Y), mult(ld(X, X), ld(Y, Y)))
4.95/5.18	= { by lemma 45 }
4.95/5.18	  mult(ld(Y, Y), mult(mult(ld(X, X), ld(X, X)), ld(Y, Y)))
4.95/5.18	= { by lemma 53 }
4.95/5.18	  mult(ld(Y, Y), mult(ld(ld(X, X), ld(X, X)), ld(Y, Y)))
4.95/5.18	= { by lemma 47 }
4.95/5.18	  mult(ld(ld(Y, Y), ld(ld(X, X), ld(X, X))), ld(Y, Y))
4.95/5.18	= { by lemma 33 }
4.95/5.18	  mult(mult(ld(mult(ld(X, X), ld(Y, Y)), rd(ld(X, X), ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
4.95/5.18	= { by lemma 37 }
4.95/5.18	  mult(rd(ld(mult(ld(X, X), ld(Y, Y)), rd(ld(X, X), ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
4.95/5.18	= { by lemma 21 }
4.95/5.18	  mult(rd(ld(rd(ld(X, X), ld(Y, Y)), rd(ld(X, X), ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
4.95/5.18	= { by lemma 27 }
4.95/5.18	  mult(mult(ld(rd(ld(X, X), ld(Y, Y)), rd(ld(X, X), ld(Y, Y))), ld(?, rd(?, ld(Y, Y)))), ld(Y, Y))
4.95/5.18	= { by lemma 17 }
4.95/5.18	  mult(mult(ld(rd(ld(X, X), ld(Y, Y)), rd(ld(X, X), ld(Y, Y))), ld(ld(X, X), rd(ld(X, X), ld(Y, Y)))), ld(Y, Y))
4.95/5.18	= { by lemma 44 }
4.95/5.18	  mult(rd(ld(X, X), mult(ld(rd(ld(X, X), ld(Y, Y)), ld(X, X)), ld(X, X))), ld(Y, Y))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  mult(rd(ld(X, X), mult(ld(Y, Y), ld(X, X))), ld(Y, Y))
4.95/5.18	= { by lemma 36 }
4.95/5.18	  mult(rd(ld(X, X), mult(ld(?, rd(?, ld(Y, Y))), ld(X, X))), ld(Y, Y))
4.95/5.18	= { by lemma 20 }
4.95/5.18	  mult(rd(ld(X, X), mult(ld(mult(ld(X, X), ld(Y, Y)), ld(X, X)), ld(X, X))), ld(Y, Y))
4.95/5.18	= { by lemma 44 }
4.95/5.18	  mult(mult(ld(mult(ld(X, X), ld(Y, Y)), mult(ld(X, X), ld(Y, Y))), ld(ld(X, X), mult(ld(X, X), ld(Y, Y)))), ld(Y, Y))
4.95/5.18	= { by axiom 2 (f02) }
4.95/5.18	  mult(mult(ld(mult(ld(X, X), ld(Y, Y)), mult(ld(X, X), ld(Y, Y))), ld(Y, Y)), ld(Y, Y))
4.95/5.18	= { by lemma 46 }
4.95/5.18	  ld(mult(ld(X, X), ld(Y, Y)), mult(ld(X, X), ld(Y, Y)))
4.95/5.18	= { by lemma 45 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(Y, Y)))
4.95/5.18	= { by lemma 45 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))
4.95/5.18	= { by lemma 38 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(Y, Y), mult(ld(ld(Y, Y), ld(Y, Y)), ld(Y, Y)))))
4.95/5.18	= { by axiom 5 (f01) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(Y, Y), mult(ld(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), ld(ld(Y, Y), ld(Y, Y)))))))
4.95/5.18	= { by lemma 11 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), ld(Y, Y)), ld(ld(Y, Y), ld(Y, Y)))))
4.95/5.18	= { by lemma 40 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), ld(Y, Y)), rd(ld(Y, Y), ld(Y, Y)))))
4.95/5.18	= { by lemma 9 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y))))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(rd(?, mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ?), ld(Y, Y))))
4.95/5.18	= { by axiom 3 (f04) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(rd(mult(rd(?, mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)), ld(Y, Y)), ?), ld(Y, Y))))
4.95/5.18	= { by lemma 29 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(rd(rd(rd(?, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), ?), ld(Y, Y))))
4.95/5.18	= { by lemma 27 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(rd(rd(mult(?, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y)), ?), ld(Y, Y))))
4.95/5.18	= { by lemma 27 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(mult(rd(mult(?, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), ld(?, rd(?, ld(Y, Y)))), ?), ld(Y, Y))))
4.95/5.18	= { by lemma 27 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(mult(rd(mult(?, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), ld(?, rd(?, ld(Y, Y)))), ?), ld(?, rd(?, ld(Y, Y))))))
4.95/5.18	= { by lemma 23 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(?, rd(?, ld(Y, Y))), ld(rd(mult(?, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), mult(?, ld(?, rd(?, ld(Y, Y))))))))
4.95/5.18	= { by lemma 36 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(Y, Y), ld(rd(mult(?, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), mult(?, ld(?, rd(?, ld(Y, Y))))))))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(Y, Y), ld(Y, Y))))
4.95/5.18	= { by axiom 2 (f02) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(Y, Y), ld(?, mult(?, ld(Y, Y))))))
4.95/5.18	= { by lemma 37 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(Y, Y), ld(?, rd(?, ld(Y, Y))))))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(ld(Y, Y), ld(rd(Y, ld(?, rd(?, ld(Y, Y)))), Y))))
4.95/5.18	= { by lemma 41 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(Y, Y)), Y), ld(Y, Y))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(Y, mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))))), Y), ld(Y, Y))))
4.95/5.18	= { by lemma 37 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(rd(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y)), Y), ld(Y, Y))))
4.95/5.18	= { by axiom 5 (f01) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(rd(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y)), mult(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y))), ld(Y, Y))))
4.95/5.18	= { by lemma 12 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y), mult(ld(?, rd(?, ld(Y, Y))), ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y))), ld(Y, Y))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), mult(ld(?, rd(?, ld(Y, Y))), ld(mult(rd(Y, ld(?, rd(?, ld(Y, Y)))), ld(?, rd(?, ld(Y, Y)))), Y))), ld(Y, Y))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), mult(ld(?, rd(?, ld(Y, Y))), ld(Y, Y))), ld(Y, Y))))
4.95/5.18	= { by lemma 52 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(?, mult(rd(mult(?, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), ld(Y, Y))))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(?, mult(rd(mult(?, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y))))))
4.95/5.18	= { by lemma 51 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), ld(mult(?, ld(Y, Y)), rd(mult(?, ld(Y, Y)), ld(Y, Y)))), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by lemma 17 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), ld(?, rd(?, ld(Y, Y)))), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by lemma 27 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(rd(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by lemma 37 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X)), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), X)), mult(ld(Y, Y), ld(rd(mult(?, ld(Y, Y)), ld(Y, Y)), mult(?, ld(Y, Y)))))))
4.95/5.18	= { by lemma 6 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), X)), mult(ld(Y, Y), ld(Y, Y)))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), X)), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X)))))
4.95/5.18	= { by axiom 4 (f03) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), X)), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))))))))
4.95/5.18	= { by lemma 51 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.18	= { by lemma 13 }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), mult(ld(X, X), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.18	= { by axiom 2 (f02) }
4.95/5.18	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), mult(ld(mult(mult(?, ld(X, X)), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(mult(mult(?, ld(X, X)), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.19	= { by axiom 1 (f05) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), mult(ld(mult(mult(?, ld(X, X)), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(?, mult(ld(X, X), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X))))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.19	= { by axiom 4 (f03) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), mult(ld(mult(mult(?, ld(X, X)), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.19	= { by lemma 22 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(mult(?, ld(X, X)), mult(mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, mult(rd(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))))))
4.95/5.19	= { by axiom 4 (f03) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(mult(?, ld(X, X)), mult(mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))), ld(X, X))))
4.95/5.19	= { by lemma 22 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(?, mult(mult(mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X))))
4.95/5.19	= { by lemma 10 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(?, mult(rd(mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))), ld(X, X)), mult(ld(X, X), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X))))))
4.95/5.19	= { by axiom 4 (f03) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(?, mult(rd(mult(?, mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))), ld(X, X)), mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))))
4.95/5.19	= { by lemma 52 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(?, rd(?, ld(X, X))), mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))))
4.95/5.19	= { by lemma 37 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(?, mult(?, ld(X, X))), mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))))
4.95/5.19	= { by axiom 2 (f02) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))))))
4.95/5.19	= { by axiom 4 (f03) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(X, X), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X))))))
4.95/5.19	= { by lemma 24 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(X, mult(mult(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X))))))
4.95/5.19	= { by lemma 14 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), mult(mult(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X))))))
4.95/5.19	= { by lemma 21 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), rd(mult(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X))))))
4.95/5.19	= { by lemma 15 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), rd(mult(X, rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(ld(?, rd(?, X)), X))))))
4.95/5.19	= { by lemma 35 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(mult(ld(?, rd(?, X)), X), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(ld(?, rd(?, X)), X)))))
4.95/5.19	= { by lemma 15 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(ld(X, X), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), mult(ld(?, rd(?, X)), X)))))
4.95/5.19	= { by lemma 15 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), rd(ld(ld(X, X), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X)))))
4.95/5.19	= { by lemma 21 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(ld(X, X), mult(ld(ld(X, X), rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))), ld(X, X)))))
4.95/5.19	= { by lemma 13 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(rd(ld(X, X), ld(X, X)), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X)))))
4.95/5.19	= { by lemma 37 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(mult(ld(X, X), ld(X, X)), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X)))))
4.95/5.19	= { by lemma 45 }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(ld(X, X), mult(rd(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)), ld(X, X)))))
4.95/5.19	= { by axiom 4 (f03) }
4.95/5.19	  ld(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), mult(mult(ld(X, X), mult(ld(Y, Y), ld(Y, Y))), ld(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))))
4.95/5.19	= { by axiom 2 (f02) }
4.95/5.19	  ld(ld(X, X), mult(ld(Y, Y), ld(Y, Y)))
4.95/5.19	= { by lemma 45 }
4.95/5.19	  ld(ld(X, X), ld(Y, Y))
4.95/5.19	
4.95/5.19	Lemma 56: ld(X, X) = ld(?, ?).
4.95/5.19	Proof:
4.95/5.19	  ld(X, X)
4.95/5.19	= { by axiom 3 (f04) }
4.95/5.19	  rd(mult(ld(X, X), mult(ld(Y, Y), ld(X, X))), mult(ld(Y, Y), ld(X, X)))
4.95/5.19	= { by lemma 55 }
4.95/5.19	  rd(ld(ld(Y, Y), ld(X, X)), mult(ld(Y, Y), ld(X, X)))
4.95/5.19	= { by lemma 50 }
4.95/5.19	  ld(?, rd(?, mult(ld(Y, Y), ld(Y, Y))))
4.95/5.19	= { by lemma 50 }
4.95/5.19	  rd(ld(ld(Y, Y), ld(?, ?)), mult(ld(Y, Y), ld(?, ?)))
4.95/5.19	= { by lemma 55 }
4.95/5.19	  rd(mult(ld(?, ?), mult(ld(Y, Y), ld(?, ?))), mult(ld(Y, Y), ld(?, ?)))
4.95/5.19	= { by axiom 3 (f04) }
4.95/5.19	  ld(?, ?)
4.95/5.19	
4.95/5.19	Goal 1 (goals): tuple(mult(X, sK2_goals_X1(X)), sK1_goals_X1(X)) = tuple(sK2_goals_X1(X), mult(sK1_goals_X1(X), X)).
4.95/5.19	The goal is true when:
4.95/5.19	  X = ld(?, ?)
4.95/5.19	where "?" stands for an arbitrary term of your choice.
4.95/5.19	
4.95/5.19	Proof:
4.95/5.19	  tuple(mult(ld(?, ?), sK2_goals_X1(ld(?, ?))), sK1_goals_X1(ld(?, ?)))
4.95/5.19	= { by lemma 56 }
4.95/5.19	  tuple(mult(ld(sK2_goals_X1(ld(?, ?)), sK2_goals_X1(ld(?, ?))), sK2_goals_X1(ld(?, ?))), sK1_goals_X1(ld(?, ?)))
4.95/5.19	= { by lemma 39 }
4.95/5.19	  tuple(sK2_goals_X1(ld(?, ?)), sK1_goals_X1(ld(?, ?)))
4.95/5.19	= { by axiom 5 (f01) }
4.95/5.19	  tuple(sK2_goals_X1(ld(?, ?)), mult(sK1_goals_X1(ld(?, ?)), ld(sK1_goals_X1(ld(?, ?)), sK1_goals_X1(ld(?, ?)))))
4.95/5.19	= { by lemma 56 }
4.95/5.19	  tuple(sK2_goals_X1(ld(?, ?)), mult(sK1_goals_X1(ld(?, ?)), ld(?, ?)))
4.95/5.19	% SZS output end Proof
4.95/5.19	
4.95/5.19	RESULT: Theorem (the conjecture is true).
4.95/5.19	EOF