0.03/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n023.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:22:51 EDT 2019
0.12/0.33	% CPUTime    : 
5.27/5.51	% SZS status Theorem
5.27/5.51	
5.27/5.51	% SZS output start Proof
5.27/5.51	Take the following subset of the input axioms:
5.27/5.51	  fof(f01, axiom, ![B, A]: B=mult(A, ld(A, B))).
5.27/5.51	  fof(f02, axiom, ![B, A]: ld(A, mult(A, B))=B).
5.27/5.51	  fof(f03, axiom, ![B, A]: A=mult(rd(A, B), B)).
5.27/5.51	  fof(f04, axiom, ![B, A]: rd(mult(A, B), B)=A).
5.27/5.51	  fof(f05, axiom, ![B, A, C]: mult(mult(mult(A, B), C), B)=mult(A, mult(B, mult(C, B)))).
5.27/5.51	  fof(goals, conjecture, ![X0, X1]: (X0=mult(X0, ld(X1, X1)) & mult(ld(X1, X1), X0)=X0)).
5.27/5.51	
5.27/5.51	Now clausify the problem and encode Horn clauses using encoding 3 of
5.27/5.51	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
5.27/5.51	We repeatedly replace C & s=t => u=v by the two clauses:
5.27/5.51	  fresh(y, y, x1...xn) = u
5.27/5.51	  C => fresh(s, t, x1...xn) = v
5.27/5.51	where fresh is a fresh function symbol and x1..xn are the free
5.27/5.51	variables of u and v.
5.27/5.51	A predicate p(X) is encoded as p(X)=true (this is sound, because the
5.27/5.51	input problem has no model of domain size 1).
5.27/5.51	
5.27/5.51	The encoding turns the above axioms into the following unit equations and goals:
5.27/5.51	
5.27/5.51	Axiom 1 (f05): mult(mult(mult(X, Y), Z), Y) = mult(X, mult(Y, mult(Z, Y))).
5.27/5.51	Axiom 2 (f04): rd(mult(X, Y), Y) = X.
5.27/5.51	Axiom 3 (f03): X = mult(rd(X, Y), Y).
5.27/5.51	Axiom 4 (f01): X = mult(Y, ld(Y, X)).
5.67/5.85	Axiom 5 (f02): ld(X, mult(X, Y)) = Y.
5.67/5.85	
5.67/5.85	Lemma 6: mult(rd(X, Y), mult(Y, mult(Z, Y))) = mult(mult(X, Z), Y).
5.67/5.85	Proof:
5.67/5.85	  mult(rd(X, Y), mult(Y, mult(Z, Y)))
5.67/5.85	= { by axiom 1 (f05) }
5.67/5.85	  mult(mult(mult(rd(X, Y), Y), Z), Y)
5.67/5.85	= { by axiom 3 (f03) }
5.67/5.85	  mult(mult(X, Z), Y)
5.67/5.85	
5.67/5.85	Lemma 7: ld(rd(Z, X), mult(mult(Z, Y), X)) = mult(X, mult(Y, X)).
5.67/5.85	Proof:
5.67/5.85	  ld(rd(Z, X), mult(mult(Z, Y), X))
5.67/5.85	= { by lemma 6 }
5.67/5.85	  ld(rd(Z, X), mult(rd(Z, X), mult(X, mult(Y, X))))
5.67/5.85	= { by axiom 5 (f02) }
5.67/5.85	  mult(X, mult(Y, X))
5.67/5.85	
5.67/5.85	Lemma 8: ld(rd(Y, X), mult(Z, X)) = mult(X, mult(ld(Y, Z), X)).
5.67/5.85	Proof:
5.67/5.85	  ld(rd(Y, X), mult(Z, X))
5.67/5.85	= { by axiom 4 (f01) }
5.67/5.85	  ld(rd(Y, X), mult(mult(Y, ld(Y, Z)), X))
5.67/5.85	= { by lemma 7 }
5.67/5.85	  mult(X, mult(ld(Y, Z), X))
5.67/5.85	
5.67/5.85	Lemma 9: ld(rd(Y, X), Y) = X.
5.67/5.85	Proof:
5.67/5.85	  ld(rd(Y, X), Y)
5.67/5.85	= { by axiom 3 (f03) }
5.67/5.85	  ld(rd(Y, X), mult(rd(Y, X), X))
5.67/5.85	= { by axiom 5 (f02) }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 10: mult(X, mult(ld(mult(Y, X), Y), X)) = X.
5.67/5.86	Proof:
5.67/5.86	  mult(X, mult(ld(mult(Y, X), Y), X))
5.67/5.86	= { by lemma 8 }
5.67/5.86	  ld(rd(mult(Y, X), X), mult(Y, X))
5.67/5.86	= { by lemma 9 }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 11: mult(ld(mult(X, Y), X), Y) = ld(Y, Y).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(mult(X, Y), X), Y)
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  ld(Y, mult(Y, mult(ld(mult(X, Y), X), Y)))
5.67/5.86	= { by lemma 10 }
5.67/5.86	  ld(Y, Y)
5.67/5.86	
5.67/5.86	Lemma 12: mult(X, mult(ld(Y, rd(Y, X)), X)) = X.
5.67/5.86	Proof:
5.67/5.86	  mult(X, mult(ld(Y, rd(Y, X)), X))
5.67/5.86	= { by lemma 8 }
5.67/5.86	  ld(rd(Y, X), mult(rd(Y, X), X))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 13: mult(ld(X, rd(X, Y)), Y) = ld(Y, Y).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(X, rd(X, Y)), Y)
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  ld(Y, mult(Y, mult(ld(X, rd(X, Y)), Y)))
5.67/5.86	= { by lemma 12 }
5.67/5.86	  ld(Y, Y)
5.67/5.86	
5.67/5.86	Lemma 14: rd(ld(Y, Y), Y) = ld(?, rd(?, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(ld(Y, Y), Y)
5.67/5.86	= { by lemma 13 }
5.67/5.86	  rd(mult(ld(?, rd(?, Y)), Y), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  ld(?, rd(?, Y))
5.67/5.86	
5.67/5.86	Lemma 15: ld(mult(X, Y), X) = ld(?, rd(?, Y)).
5.67/5.86	Proof:
5.67/5.86	  ld(mult(X, Y), X)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  rd(mult(ld(mult(X, Y), X), Y), Y)
5.67/5.86	= { by lemma 11 }
5.67/5.86	  rd(ld(Y, Y), Y)
5.67/5.86	= { by lemma 14 }
5.67/5.86	  ld(?, rd(?, Y))
5.67/5.86	
5.67/5.86	Lemma 16: ld(?, rd(?, ld(X, Y))) = ld(Y, X).
5.67/5.86	Proof:
5.67/5.86	  ld(?, rd(?, ld(X, Y)))
5.67/5.86	= { by lemma 15 }
5.67/5.86	  ld(mult(X, ld(X, Y)), X)
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  ld(Y, X)
5.67/5.86	
5.67/5.86	Lemma 17: mult(mult(X, rd(Y, Z)), Z) = mult(rd(X, Z), mult(Z, Y)).
5.67/5.86	Proof:
5.67/5.86	  mult(mult(X, rd(Y, Z)), Z)
5.67/5.86	= { by lemma 6 }
5.67/5.86	  mult(rd(X, Z), mult(Z, mult(rd(Y, Z), Z)))
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  mult(rd(X, Z), mult(Z, Y))
5.67/5.86	
5.67/5.86	Lemma 18: rd(mult(X, mult(Y, mult(Z, Y))), Y) = mult(mult(X, Y), Z).
5.67/5.86	Proof:
5.67/5.86	  rd(mult(X, mult(Y, mult(Z, Y))), Y)
5.67/5.86	= { by axiom 1 (f05) }
5.67/5.86	  rd(mult(mult(mult(X, Y), Z), Y), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  mult(mult(X, Y), Z)
5.67/5.86	
5.67/5.86	Lemma 19: rd(mult(X, mult(Y, Z)), Y) = mult(mult(X, Y), rd(Z, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(mult(X, mult(Y, Z)), Y)
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  rd(mult(X, mult(Y, mult(rd(Z, Y), Y))), Y)
5.67/5.86	= { by lemma 18 }
5.67/5.86	  mult(mult(X, Y), rd(Z, Y))
5.67/5.86	
5.67/5.86	Lemma 20: mult(mult(X, Y), rd(ld(Y, Z), Y)) = rd(mult(X, Z), Y).
5.67/5.86	Proof:
5.67/5.86	  mult(mult(X, Y), rd(ld(Y, Z), Y))
5.67/5.86	= { by lemma 19 }
5.67/5.86	  rd(mult(X, mult(Y, ld(Y, Z))), Y)
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  rd(mult(X, Z), Y)
5.67/5.86	
5.67/5.86	Lemma 21: rd(mult(rd(X, Y), Z), Y) = mult(X, rd(ld(Y, Z), Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(mult(rd(X, Y), Z), Y)
5.67/5.86	= { by lemma 20 }
5.67/5.86	  mult(mult(rd(X, Y), Y), rd(ld(Y, Z), Y))
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  mult(X, rd(ld(Y, Z), Y))
5.67/5.86	
5.67/5.86	Lemma 22: mult(X, ld(?, rd(?, Y))) = rd(X, Y).
5.67/5.86	Proof:
5.67/5.86	  mult(X, ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 14 }
5.67/5.86	  mult(X, rd(ld(Y, Y), Y))
5.67/5.86	= { by lemma 21 }
5.67/5.86	  rd(mult(rd(X, Y), Y), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  rd(X, Y)
5.67/5.86	
5.67/5.86	Lemma 23: ld(ld(?, rd(?, X)), ld(?, rd(?, X))) = rd(X, X).
5.67/5.86	Proof:
5.67/5.86	  ld(ld(?, rd(?, X)), ld(?, rd(?, X)))
5.67/5.86	= { by lemma 11 }
5.67/5.86	  mult(ld(mult(?, ld(?, rd(?, X))), ?), ld(?, rd(?, X)))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(ld(rd(?, X), ?), ld(?, rd(?, X)))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  rd(ld(rd(?, X), ?), X)
5.67/5.86	= { by lemma 9 }
5.67/5.86	  rd(X, X)
5.67/5.86	
5.67/5.86	Lemma 24: mult(ld(X, X), mult(ld(X, X), ld(X, X))) = ld(X, X).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(X, X), mult(ld(X, X), ld(X, X)))
5.67/5.86	= { by lemma 13 }
5.67/5.86	  mult(mult(ld(?, rd(?, X)), X), mult(ld(X, X), ld(X, X)))
5.67/5.86	= { by lemma 12 }
5.67/5.86	  mult(mult(ld(?, rd(?, X)), X), mult(ld(mult(X, mult(ld(?, rd(?, X)), X)), X), ld(X, X)))
5.67/5.86	= { by lemma 13 }
5.67/5.86	  mult(mult(ld(?, rd(?, X)), X), mult(ld(mult(X, mult(ld(?, rd(?, X)), X)), X), mult(ld(?, rd(?, X)), X)))
5.67/5.86	= { by lemma 10 }
5.67/5.86	  mult(ld(?, rd(?, X)), X)
5.67/5.86	= { by lemma 13 }
5.67/5.86	  ld(X, X)
5.67/5.86	
5.67/5.86	Lemma 25: mult(mult(X, ld(Y, Y)), ld(Y, Y)) = X.
5.67/5.86	Proof:
5.67/5.86	  mult(mult(X, ld(Y, Y)), ld(Y, Y))
5.67/5.86	= { by lemma 18 }
5.67/5.86	  rd(mult(X, mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y))
5.67/5.86	= { by lemma 24 }
5.67/5.86	  rd(mult(X, ld(Y, Y)), ld(Y, Y))
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 26: rd(X, ld(Y, Y)) = mult(X, ld(Y, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(X, ld(Y, Y))
5.67/5.86	= { by lemma 25 }
5.67/5.86	  rd(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  mult(X, ld(Y, Y))
5.67/5.86	
5.67/5.86	Lemma 27: rd(X, rd(Y, Y)) = mult(X, rd(Y, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(X, rd(Y, Y))
5.67/5.86	= { by lemma 23 }
5.67/5.86	  rd(X, ld(ld(?, rd(?, Y)), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 26 }
5.67/5.86	  mult(X, ld(ld(?, rd(?, Y)), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 23 }
5.67/5.86	  mult(X, rd(Y, Y))
5.67/5.86	
5.67/5.86	Lemma 28: mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), Z) = rd(X, Y).
5.67/5.86	Proof:
5.67/5.86	  mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), Z)
5.67/5.86	= { by lemma 18 }
5.67/5.86	  rd(mult(rd(X, mult(Y, mult(Z, Y))), mult(Y, mult(Z, Y))), Y)
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  rd(X, Y)
5.67/5.86	
5.67/5.86	Lemma 29: mult(rd(X, mult(Y, mult(Z, Y))), Y) = rd(rd(X, Y), Z).
5.67/5.86	Proof:
5.67/5.86	  mult(rd(X, mult(Y, mult(Z, Y))), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  rd(mult(mult(rd(X, mult(Y, mult(Z, Y))), Y), Z), Z)
5.67/5.86	= { by lemma 28 }
5.67/5.86	  rd(rd(X, Y), Z)
5.67/5.86	
5.67/5.86	Lemma 30: rd(rd(X, Y), rd(Z, Y)) = mult(rd(X, mult(Y, Z)), Y).
5.67/5.86	Proof:
5.67/5.86	  rd(rd(X, Y), rd(Z, Y))
5.67/5.86	= { by lemma 29 }
5.67/5.86	  mult(rd(X, mult(Y, mult(rd(Z, Y), Y))), Y)
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  mult(rd(X, mult(Y, Z)), Y)
5.67/5.86	
5.67/5.86	Lemma 31: rd(X, ld(Y, Z)) = mult(X, ld(Z, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(X, ld(Y, Z))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(X, ld(?, rd(?, ld(Y, Z))))
5.67/5.86	= { by lemma 16 }
5.67/5.86	  mult(X, ld(Z, Y))
5.67/5.86	
5.67/5.86	Lemma 32: rd(mult(X, ld(Y, Y)), Y) = rd(mult(X, Y), mult(Y, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(mult(X, ld(Y, Y)), Y)
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(mult(X, ld(Y, Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 31 }
5.67/5.86	  mult(rd(X, ld(Y, Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 11 }
5.67/5.86	  mult(rd(X, mult(ld(mult(Y, Y), Y), Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  mult(rd(X, mult(ld(mult(Y, Y), Y), mult(mult(Y, Y), ld(mult(Y, Y), Y)))), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 15 }
5.67/5.86	  mult(rd(X, mult(ld(mult(Y, Y), Y), mult(mult(Y, Y), ld(mult(Y, Y), Y)))), ld(mult(Y, Y), Y))
5.67/5.86	= { by lemma 29 }
5.67/5.86	  rd(rd(X, ld(mult(Y, Y), Y)), mult(Y, Y))
5.67/5.86	= { by lemma 31 }
5.67/5.86	  rd(mult(X, ld(Y, mult(Y, Y))), mult(Y, Y))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  rd(mult(X, Y), mult(Y, Y))
5.67/5.86	
5.67/5.86	Lemma 33: ld(X, rd(X, Y)) = ld(?, rd(?, Y)).
5.67/5.86	Proof:
5.67/5.86	  ld(X, rd(X, Y))
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  rd(mult(ld(X, rd(X, Y)), Y), Y)
5.67/5.86	= { by lemma 13 }
5.67/5.86	  rd(ld(Y, Y), Y)
5.67/5.86	= { by lemma 13 }
5.67/5.86	  rd(mult(ld(?, rd(?, Y)), Y), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  ld(?, rd(?, Y))
5.67/5.86	
5.67/5.86	Lemma 34: rd(X, ld(?, rd(?, Y))) = mult(X, Y).
5.67/5.86	Proof:
5.67/5.86	  rd(X, ld(?, rd(?, Y)))
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  rd(rd(mult(X, Y), Y), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 29 }
5.67/5.86	  mult(rd(mult(X, Y), mult(Y, mult(ld(?, rd(?, Y)), Y))), Y)
5.67/5.86	= { by lemma 12 }
5.67/5.86	  mult(rd(mult(X, Y), Y), Y)
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  mult(X, Y)
5.67/5.86	
5.67/5.86	Lemma 35: mult(X, mult(ld(mult(Y, X), Z), X)) = ld(Y, mult(Z, X)).
5.67/5.86	Proof:
5.67/5.86	  mult(X, mult(ld(mult(Y, X), Z), X))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  ld(Y, mult(Y, mult(X, mult(ld(mult(Y, X), Z), X))))
5.67/5.86	= { by axiom 1 (f05) }
5.67/5.86	  ld(Y, mult(mult(mult(Y, X), ld(mult(Y, X), Z)), X))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  ld(Y, mult(Z, X))
5.67/5.86	
5.67/5.86	Lemma 36: ld(Y, ld(X, mult(Z, Y))) = mult(ld(mult(X, Y), Z), Y).
5.67/5.86	Proof:
5.67/5.86	  ld(Y, ld(X, mult(Z, Y)))
5.67/5.86	= { by lemma 35 }
5.67/5.86	  ld(Y, mult(Y, mult(ld(mult(X, Y), Z), Y)))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  mult(ld(mult(X, Y), Z), Y)
5.67/5.86	
5.67/5.86	Lemma 37: mult(ld(mult(X, Y), rd(Z, Y)), Y) = ld(Y, ld(X, Z)).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(mult(X, Y), rd(Z, Y)), Y)
5.67/5.86	= { by lemma 36 }
5.67/5.86	  ld(Y, ld(X, mult(rd(Z, Y), Y)))
5.67/5.86	= { by axiom 3 (f03) }
5.67/5.86	  ld(Y, ld(X, Z))
5.67/5.86	
5.67/5.86	Lemma 38: rd(ld(Y, ld(X, Z)), Y) = ld(mult(X, Y), rd(Z, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(ld(Y, ld(X, Z)), Y)
5.67/5.86	= { by lemma 37 }
5.67/5.86	  rd(mult(ld(mult(X, Y), rd(Z, Y)), Y), Y)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  ld(mult(X, Y), rd(Z, Y))
5.67/5.86	
5.67/5.86	Lemma 39: mult(X, mult(ld(X, X), X)) = mult(X, X).
5.67/5.86	Proof:
5.67/5.86	  mult(X, mult(ld(X, X), X))
5.67/5.86	= { by lemma 8 }
5.67/5.86	  ld(rd(X, X), mult(X, X))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  ld(mult(X, ld(?, rd(?, X))), mult(X, X))
5.67/5.86	= { by lemma 34 }
5.67/5.86	  ld(mult(X, ld(?, rd(?, X))), rd(X, ld(?, rd(?, X))))
5.67/5.86	= { by lemma 38 }
5.67/5.86	  rd(ld(ld(?, rd(?, X)), ld(X, X)), ld(?, rd(?, X)))
5.67/5.86	= { by lemma 13 }
5.67/5.86	  rd(ld(ld(?, rd(?, X)), mult(ld(?, rd(?, X)), X)), ld(?, rd(?, X)))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  rd(X, ld(?, rd(?, X)))
5.67/5.86	= { by lemma 34 }
5.67/5.86	  mult(X, X)
5.67/5.86	
5.67/5.86	Lemma 40: mult(ld(X, X), X) = X.
5.67/5.86	Proof:
5.67/5.86	  mult(ld(X, X), X)
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  ld(X, mult(X, mult(ld(X, X), X)))
5.67/5.86	= { by lemma 39 }
5.67/5.86	  ld(X, mult(X, X))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 41: rd(X, X) = ld(X, X).
5.67/5.86	Proof:
5.67/5.86	  rd(X, X)
5.67/5.86	= { by lemma 40 }
5.67/5.86	  rd(mult(ld(X, X), X), X)
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  ld(X, X)
5.67/5.86	
5.67/5.86	Lemma 42: mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))) = mult(mult(Y, Z), ld(X, Y)).
5.67/5.86	Proof:
5.67/5.86	  mult(X, mult(ld(X, Y), mult(Z, ld(X, Y))))
5.67/5.86	= { by axiom 1 (f05) }
5.67/5.86	  mult(mult(mult(X, ld(X, Y)), Z), ld(X, Y))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  mult(mult(Y, Z), ld(X, Y))
5.67/5.86	
5.67/5.86	Lemma 43: ld(X, mult(mult(Y, Z), ld(X, Y))) = mult(ld(X, Y), mult(Z, ld(X, Y))).
5.67/5.86	Proof:
5.67/5.86	  ld(X, mult(mult(Y, Z), ld(X, Y)))
5.67/5.86	= { by lemma 42 }
5.67/5.86	  ld(X, mult(X, mult(ld(X, Y), mult(Z, ld(X, Y)))))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  mult(ld(X, Y), mult(Z, ld(X, Y)))
5.67/5.86	
5.67/5.86	Lemma 44: rd(mult(X, Y), mult(Y, Y)) = mult(rd(X, Y), ld(Y, Y)).
5.67/5.86	Proof:
5.67/5.86	  rd(mult(X, Y), mult(Y, Y))
5.67/5.86	= { by lemma 32 }
5.67/5.86	  rd(mult(X, ld(Y, Y)), Y)
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(mult(X, ld(Y, Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 41 }
5.67/5.86	  mult(mult(X, rd(Y, Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 27 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), ld(?, rd(?, Y)))
5.67/5.86	= { by lemma 14 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), rd(ld(Y, Y), Y))
5.67/5.86	= { by lemma 41 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), rd(rd(Y, Y), Y))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 15 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), ld(mult(?, Y), ?)))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), ld(mult(?, Y), mult(mult(?, Y), ld(mult(?, Y), ?)))))
5.67/5.86	= { by lemma 43 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), mult(ld(mult(?, Y), ?), mult(Y, ld(mult(?, Y), ?)))))
5.67/5.86	= { by lemma 15 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), mult(ld(?, rd(?, Y)), mult(Y, ld(mult(?, Y), ?)))))
5.67/5.86	= { by lemma 15 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), mult(ld(?, rd(?, Y)), mult(Y, ld(?, rd(?, Y))))))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), mult(ld(?, rd(?, Y)), rd(Y, Y))))
5.67/5.86	= { by lemma 6 }
5.67/5.86	  mult(mult(X, ld(?, rd(?, Y))), rd(Y, Y))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(rd(X, Y), rd(Y, Y))
5.67/5.86	= { by lemma 41 }
5.67/5.86	  mult(rd(X, Y), ld(Y, Y))
5.67/5.86	
5.67/5.86	Lemma 45: mult(rd(X, Y), mult(Y, Y)) = mult(mult(X, Y), ld(Y, Y)).
5.67/5.86	Proof:
5.67/5.86	  mult(rd(X, Y), mult(Y, Y))
5.67/5.86	= { by lemma 17 }
5.67/5.86	  mult(mult(X, rd(Y, Y)), Y)
5.67/5.86	= { by lemma 27 }
5.67/5.86	  mult(rd(X, rd(Y, Y)), Y)
5.67/5.86	= { by lemma 22 }
5.67/5.86	  mult(rd(X, mult(Y, ld(?, rd(?, Y)))), Y)
5.67/5.86	= { by lemma 30 }
5.67/5.86	  rd(rd(X, Y), rd(ld(?, rd(?, Y)), Y))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  rd(mult(X, ld(?, rd(?, Y))), rd(ld(?, rd(?, Y)), Y))
5.67/5.86	= { by lemma 22 }
5.67/5.86	  rd(mult(X, ld(?, rd(?, Y))), mult(ld(?, rd(?, Y)), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 44 }
5.67/5.86	  mult(rd(X, ld(?, rd(?, Y))), ld(ld(?, rd(?, Y)), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 34 }
5.67/5.86	  mult(mult(X, Y), ld(ld(?, rd(?, Y)), ld(?, rd(?, Y))))
5.67/5.86	= { by lemma 23 }
5.67/5.86	  mult(mult(X, Y), rd(Y, Y))
5.67/5.86	= { by lemma 41 }
5.67/5.86	  mult(mult(X, Y), ld(Y, Y))
5.67/5.86	
5.67/5.86	Lemma 46: mult(ld(Y, X), mult(ld(X, Y), ld(X, Y))) = ld(X, Y).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(Y, X), mult(ld(X, Y), ld(X, Y)))
5.67/5.86	= { by lemma 16 }
5.67/5.86	  mult(ld(?, rd(?, ld(X, Y))), mult(ld(X, Y), ld(X, Y)))
5.67/5.86	= { by lemma 14 }
5.67/5.86	  mult(rd(ld(ld(X, Y), ld(X, Y)), ld(X, Y)), mult(ld(X, Y), ld(X, Y)))
5.67/5.86	= { by lemma 45 }
5.67/5.86	  mult(mult(ld(ld(X, Y), ld(X, Y)), ld(X, Y)), ld(ld(X, Y), ld(X, Y)))
5.67/5.86	= { by lemma 40 }
5.67/5.86	  mult(ld(X, Y), ld(ld(X, Y), ld(X, Y)))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  ld(X, Y)
5.67/5.86	
5.67/5.86	Lemma 47: ld(mult(Z, X), rd(mult(Z, Y), X)) = rd(ld(X, Y), X).
5.67/5.86	Proof:
5.67/5.86	  ld(mult(Z, X), rd(mult(Z, Y), X))
5.67/5.86	= { by lemma 20 }
5.67/5.86	  ld(mult(Z, X), mult(mult(Z, X), rd(ld(X, Y), X)))
5.67/5.86	= { by axiom 5 (f02) }
5.67/5.86	  rd(ld(X, Y), X)
5.67/5.86	
5.67/5.86	Lemma 48: mult(ld(ld(X, Y), Z), ld(Y, X)) = mult(ld(Y, X), mult(Z, ld(Y, X))).
5.67/5.86	Proof:
5.67/5.86	  mult(ld(ld(X, Y), Z), ld(Y, X))
5.67/5.86	= { by lemma 31 }
5.67/5.86	  rd(ld(ld(X, Y), Z), ld(X, Y))
5.67/5.86	= { by lemma 47 }
5.67/5.86	  ld(mult(X, ld(X, Y)), rd(mult(X, Z), ld(X, Y)))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  ld(Y, rd(mult(X, Z), ld(X, Y)))
5.67/5.86	= { by lemma 31 }
5.67/5.86	  ld(Y, mult(mult(X, Z), ld(Y, X)))
5.67/5.86	= { by lemma 43 }
5.67/5.86	  mult(ld(Y, X), mult(Z, ld(Y, X)))
5.67/5.86	
5.67/5.86	Lemma 49: rd(Y, ld(X, Y)) = X.
5.67/5.86	Proof:
5.67/5.86	  rd(Y, ld(X, Y))
5.67/5.86	= { by axiom 4 (f01) }
5.67/5.86	  rd(mult(X, ld(X, Y)), ld(X, Y))
5.67/5.86	= { by axiom 2 (f04) }
5.67/5.86	  X
5.67/5.86	
5.67/5.86	Lemma 50: ld(ld(X, Y), ld(X, Y)) = mult(ld(Y, X), ld(X, Y)).
5.67/5.86	Proof:
5.67/5.86	  ld(ld(X, Y), ld(X, Y))
5.67/5.86	= { by lemma 13 }
5.67/5.86	  mult(ld(Y, rd(Y, ld(X, Y))), ld(X, Y))
5.67/5.86	= { by lemma 49 }
5.67/5.87	  mult(ld(Y, X), ld(X, Y))
5.67/5.87	
5.67/5.87	Lemma 51: mult(ld(X, Y), ld(Y, X)) = mult(ld(Y, X), ld(X, Y)).
5.67/5.87	Proof:
5.67/5.87	  mult(ld(X, Y), ld(Y, X))
5.67/5.87	= { by axiom 3 (f03) }
5.67/5.87	  mult(ld(X, Y), mult(rd(ld(Y, X), ld(X, Y)), ld(X, Y)))
5.67/5.87	= { by lemma 48 }
5.67/5.87	  mult(ld(ld(Y, X), rd(ld(Y, X), ld(X, Y))), ld(X, Y))
5.67/5.87	= { by lemma 13 }
5.67/5.87	  ld(ld(X, Y), ld(X, Y))
5.67/5.87	= { by lemma 50 }
5.67/5.87	  mult(ld(Y, X), ld(X, Y))
5.67/5.87	
5.67/5.87	Lemma 52: rd(mult(X, Z), mult(Z, mult(Y, Z))) = rd(rd(X, Y), Z).
5.67/5.87	Proof:
5.67/5.87	  rd(mult(X, Z), mult(Z, mult(Y, Z)))
5.67/5.87	= { by axiom 3 (f03) }
5.67/5.87	  rd(mult(mult(rd(X, Y), Y), Z), mult(Z, mult(Y, Z)))
5.67/5.87	= { by lemma 6 }
5.67/5.87	  rd(mult(rd(rd(X, Y), Z), mult(Z, mult(Y, Z))), mult(Z, mult(Y, Z)))
5.67/5.87	= { by axiom 2 (f04) }
5.67/5.87	  rd(rd(X, Y), Z)
5.67/5.87	
5.67/5.87	Lemma 53: rd(Z, mult(ld(X, Z), mult(Y, ld(X, Z)))) = mult(rd(X, Y), ld(Z, X)).
5.67/5.87	Proof:
5.67/5.87	  rd(Z, mult(ld(X, Z), mult(Y, ld(X, Z))))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  rd(mult(X, ld(X, Z)), mult(ld(X, Z), mult(Y, ld(X, Z))))
5.67/5.87	= { by lemma 52 }
5.67/5.87	  rd(rd(X, Y), ld(X, Z))
5.67/5.87	= { by lemma 31 }
5.67/5.87	  mult(rd(X, Y), ld(Z, X))
5.67/5.87	
5.67/5.87	Lemma 54: mult(ld(mult(X, ld(Y, Z)), Y), ld(Y, Z)) = ld(ld(Y, Z), ld(X, Z)).
5.67/5.87	Proof:
5.67/5.87	  mult(ld(mult(X, ld(Y, Z)), Y), ld(Y, Z))
5.67/5.87	= { by lemma 36 }
5.67/5.87	  ld(ld(Y, Z), ld(X, mult(Y, ld(Y, Z))))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  ld(ld(Y, Z), ld(X, Z))
5.67/5.87	
5.67/5.87	Lemma 55: mult(mult(Y, ld(X, Z)), ld(Z, X)) = Y.
5.67/5.87	Proof:
5.67/5.87	  mult(mult(Y, ld(X, Z)), ld(Z, X))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  mult(mult(X, ld(X, mult(Y, ld(X, Z)))), ld(Z, X))
5.67/5.87	= { by lemma 31 }
5.67/5.87	  mult(rd(X, ld(mult(Y, ld(X, Z)), X)), ld(Z, X))
5.67/5.87	= { by lemma 53 }
5.67/5.87	  rd(Z, mult(ld(X, Z), mult(ld(mult(Y, ld(X, Z)), X), ld(X, Z))))
5.67/5.87	= { by lemma 54 }
5.67/5.87	  rd(Z, mult(ld(X, Z), ld(ld(X, Z), ld(Y, Z))))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  rd(Z, ld(Y, Z))
5.67/5.87	= { by lemma 31 }
5.67/5.87	  mult(Z, ld(Z, Y))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  Y
5.67/5.87	
5.67/5.87	Lemma 56: mult(rd(mult(X, Y), mult(Y, Z)), Y) = rd(X, rd(Z, Y)).
5.67/5.87	Proof:
5.67/5.87	  mult(rd(mult(X, Y), mult(Y, Z)), Y)
5.67/5.87	= { by lemma 30 }
5.67/5.87	  rd(rd(mult(X, Y), Y), rd(Z, Y))
5.67/5.87	= { by axiom 2 (f04) }
5.67/5.87	  rd(X, rd(Z, Y))
5.67/5.87	
5.67/5.87	Lemma 57: rd(X, mult(ld(Y, Z), ld(Z, Y))) = mult(X, mult(ld(Y, Z), ld(Z, Y))).
5.67/5.87	Proof:
5.67/5.87	  rd(X, mult(ld(Y, Z), ld(Z, Y)))
5.67/5.87	= { by lemma 51 }
5.67/5.87	  rd(X, mult(ld(Z, Y), ld(Y, Z)))
5.67/5.87	= { by lemma 55 }
5.67/5.87	  mult(mult(rd(X, mult(ld(Z, Y), ld(Y, Z))), ld(Z, Y)), ld(Y, Z))
5.67/5.87	= { by lemma 55 }
5.67/5.87	  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))
5.67/5.87	= { by lemma 29 }
5.67/5.87	  mult(rd(rd(X, ld(Z, Y)), mult(ld(Y, Z), ld(Y, Z))), ld(Y, Z))
5.67/5.87	= { by lemma 31 }
5.67/5.87	  mult(rd(mult(X, ld(Y, Z)), mult(ld(Y, Z), ld(Y, Z))), ld(Y, Z))
5.67/5.87	= { by lemma 56 }
5.67/5.87	  rd(X, rd(ld(Y, Z), ld(Y, Z)))
5.67/5.87	= { by lemma 27 }
5.67/5.87	  mult(X, rd(ld(Y, Z), ld(Y, Z)))
5.67/5.87	= { by lemma 31 }
5.67/5.87	  mult(X, mult(ld(Y, Z), ld(Z, Y)))
5.67/5.87	
5.67/5.87	Lemma 58: mult(X, mult(ld(Y, rd(Z, X)), X)) = ld(rd(Y, X), Z).
5.67/5.87	Proof:
5.67/5.87	  mult(X, mult(ld(Y, rd(Z, X)), X))
5.67/5.87	= { by lemma 8 }
5.67/5.87	  ld(rd(Y, X), mult(rd(Z, X), X))
5.67/5.87	= { by axiom 3 (f03) }
5.67/5.87	  ld(rd(Y, X), Z)
5.67/5.87	
5.67/5.87	Lemma 59: rd(Y, mult(ld(X, Y), Y)) = mult(ld(X, X), ld(Y, X)).
5.67/5.87	Proof:
5.67/5.87	  rd(Y, mult(ld(X, Y), Y))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  rd(Y, mult(ld(X, Y), mult(X, ld(X, Y))))
5.67/5.87	= { by lemma 53 }
5.67/5.87	  mult(rd(X, X), ld(Y, X))
5.67/5.87	= { by lemma 41 }
5.67/5.87	  mult(ld(X, X), ld(Y, X))
5.67/5.87	
5.67/5.87	Lemma 60: rd(X, rd(ld(Y, Z), Y)) = mult(rd(mult(X, Y), Z), Y).
5.67/5.87	Proof:
5.67/5.87	  rd(X, rd(ld(Y, Z), Y))
5.67/5.87	= { by lemma 56 }
5.67/5.87	  mult(rd(mult(X, Y), mult(Y, ld(Y, Z))), Y)
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  mult(rd(mult(X, Y), Z), Y)
5.67/5.87	
5.67/5.87	Lemma 61: mult(ld(X, X), ld(X, X)) = ld(X, X).
5.67/5.87	Proof:
5.67/5.87	  mult(ld(X, X), ld(X, X))
5.67/5.87	= { by lemma 41 }
5.67/5.87	  mult(rd(X, X), ld(X, X))
5.67/5.87	= { by lemma 23 }
5.67/5.87	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(X, X))
5.67/5.87	= { by lemma 41 }
5.67/5.87	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), rd(X, X))
5.67/5.87	= { by lemma 23 }
5.67/5.87	  mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(ld(?, rd(?, X)), ld(?, rd(?, X))))
5.67/5.87	= { by lemma 59 }
5.67/5.87	  rd(ld(?, rd(?, X)), mult(ld(ld(?, rd(?, X)), ld(?, rd(?, X))), ld(?, rd(?, X))))
5.67/5.87	= { by lemma 23 }
5.67/5.87	  rd(ld(?, rd(?, X)), mult(rd(X, X), ld(?, rd(?, X))))
5.67/5.87	= { by lemma 22 }
5.67/5.87	  rd(ld(?, rd(?, X)), rd(rd(X, X), X))
5.67/5.87	= { by lemma 41 }
5.67/5.87	  rd(ld(?, rd(?, X)), rd(ld(X, X), X))
5.67/5.87	= { by lemma 60 }
5.67/5.87	  mult(rd(mult(ld(?, rd(?, X)), X), X), X)
5.67/5.87	= { by axiom 3 (f03) }
5.67/5.87	  mult(ld(?, rd(?, X)), X)
5.67/5.87	= { by lemma 13 }
5.67/5.87	  ld(X, X)
5.67/5.87	
5.67/5.87	Lemma 62: mult(ld(X, X), mult(ld(X, X), Y)) = ld(ld(X, X), Y).
5.67/5.87	Proof:
5.67/5.87	  mult(ld(X, X), mult(ld(X, X), Y))
5.67/5.87	= { by axiom 3 (f03) }
5.67/5.87	  mult(ld(X, X), mult(ld(X, X), mult(rd(Y, ld(X, X)), ld(X, X))))
5.67/5.87	= { by lemma 43 }
5.67/5.87	  mult(ld(X, X), ld(X, mult(mult(X, rd(Y, ld(X, X))), ld(X, X))))
5.67/5.87	= { by lemma 12 }
5.67/5.87	  mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), mult(mult(X, rd(Y, ld(X, X))), ld(X, X))))
5.67/5.87	= { by lemma 26 }
5.67/5.87	  mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), rd(mult(X, rd(Y, ld(X, X))), ld(X, X))))
5.67/5.87	= { by lemma 13 }
5.67/5.87	  mult(ld(X, X), ld(mult(X, mult(ld(?, rd(?, X)), X)), rd(mult(X, rd(Y, ld(X, X))), mult(ld(?, rd(?, X)), X))))
5.67/5.87	= { by lemma 47 }
5.67/5.87	  mult(ld(X, X), rd(ld(mult(ld(?, rd(?, X)), X), rd(Y, ld(X, X))), mult(ld(?, rd(?, X)), X)))
5.67/5.87	= { by lemma 13 }
5.67/5.87	  mult(ld(X, X), rd(ld(ld(X, X), rd(Y, ld(X, X))), mult(ld(?, rd(?, X)), X)))
5.67/5.87	= { by lemma 13 }
5.67/5.87	  mult(ld(X, X), rd(ld(ld(X, X), rd(Y, ld(X, X))), ld(X, X)))
5.67/5.87	= { by lemma 26 }
5.67/5.87	  mult(ld(X, X), mult(ld(ld(X, X), rd(Y, ld(X, X))), ld(X, X)))
5.67/5.87	= { by lemma 58 }
5.67/5.87	  ld(rd(ld(X, X), ld(X, X)), Y)
5.67/5.87	= { by lemma 31 }
5.67/5.87	  ld(mult(ld(X, X), ld(X, X)), Y)
5.67/5.87	= { by lemma 61 }
5.67/5.87	  ld(ld(X, X), Y)
5.67/5.87	
5.67/5.87	Lemma 63: ld(mult(X, X), mult(X, X)) = ld(X, X).
5.67/5.87	Proof:
5.67/5.87	  ld(mult(X, X), mult(X, X))
5.67/5.87	= { by lemma 39 }
5.67/5.87	  ld(mult(X, X), mult(X, mult(ld(X, X), X)))
5.67/5.87	= { by axiom 4 (f01) }
5.67/5.87	  ld(mult(X, X), mult(X, mult(ld(X, X), mult(X, ld(X, X)))))
5.67/5.87	= { by lemma 42 }
5.67/5.87	  ld(mult(X, X), mult(mult(X, X), ld(X, X)))
5.67/5.87	= { by axiom 5 (f02) }
5.81/6.02	  ld(X, X)
5.81/6.02	
5.81/6.02	Lemma 64: mult(mult(X, X), ld(Y, Y)) = mult(X, mult(X, ld(Y, Y))).
5.81/6.02	Proof:
5.81/6.02	  mult(mult(X, X), ld(Y, Y))
5.81/6.02	= { by lemma 55 }
5.81/6.02	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), X), ld(Y, Y))
5.81/6.02	= { by lemma 46 }
5.81/6.02	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), X), mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y))))
5.81/6.02	= { by axiom 1 (f05) }
5.81/6.02	  mult(mult(mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), X), ld(Y, Y)), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by axiom 1 (f05) }
5.81/6.02	  mult(mult(mult(mult(X, ld(Y, Y)), mult(ld(Y, Y), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 61 }
5.81/6.02	  mult(mult(mult(mult(X, ld(Y, Y)), mult(mult(ld(Y, Y), ld(Y, Y)), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by axiom 5 (f02) }
5.81/6.02	  mult(mult(ld(?, mult(?, mult(mult(X, ld(Y, Y)), mult(mult(ld(Y, Y), ld(Y, Y)), mult(X, ld(Y, Y)))))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by axiom 1 (f05) }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 25 }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), mult(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y)))), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by axiom 1 (f05) }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(mult(mult(?, mult(X, ld(Y, Y))), ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 31 }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(rd(mult(?, mult(X, ld(Y, Y))), ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 25 }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(rd(mult(?, mult(X, ld(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 24 }
5.81/6.02	  mult(mult(ld(?, mult(mult(mult(rd(mult(?, mult(X, ld(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y))))), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 28 }
5.81/6.02	  mult(mult(ld(?, mult(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(X, ld(Y, Y)))), ld(Y, Y)), ld(Y, Y))
5.81/6.02	= { by lemma 61 }
5.81/6.02	  mult(mult(ld(?, mult(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(X, ld(Y, Y)))), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y))
5.81/6.02	= { by lemma 9 }
5.81/6.02	  mult(mult(ld(?, mult(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(X, ld(Y, Y)))), ld(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(?, mult(X, ld(Y, Y))))), ld(Y, Y))
5.81/6.02	= { by lemma 31 }
5.81/6.02	  mult(rd(ld(?, mult(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(X, ld(Y, Y)))), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))))), ld(Y, Y))
5.81/6.02	= { by lemma 35 }
5.81/6.02	  mult(rd(mult(mult(X, ld(Y, Y)), mult(ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))), mult(X, ld(Y, Y)))), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))))), ld(Y, Y))
5.81/6.02	= { by lemma 19 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))))), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 33 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(?, rd(?, mult(ld(Y, Y), ld(Y, Y))))), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 22 }
5.81/6.03	  mult(mult(rd(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 55 }
5.81/6.03	  mult(mult(mult(mult(rd(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by axiom 5 (f02) }
5.81/6.03	  mult(mult(mult(mult(rd(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(mult(mult(rd(rd(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 52 }
5.81/6.03	  mult(mult(mult(rd(mult(mult(X, ld(Y, Y)), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), mult(ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y)), mult(mult(ld(Y, Y), ld(Y, Y)), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by axiom 4 (f01) }
5.81/6.03	  mult(mult(mult(rd(mult(mult(X, ld(Y, Y)), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), mult(ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 11 }
5.81/6.03	  mult(mult(mult(rd(mult(mult(X, ld(Y, Y)), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(mult(mult(mult(mult(mult(X, ld(Y, Y)), ld(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 15 }
5.81/6.03	  mult(mult(mult(mult(mult(mult(X, ld(Y, Y)), ld(?, rd(?, ld(Y, Y)))), ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 22 }
5.81/6.03	  mult(mult(mult(mult(rd(mult(X, ld(Y, Y)), ld(Y, Y)), ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(mult(mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 50 }
5.81/6.03	  mult(mult(mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by axiom 1 (f05) }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), mult(ld(Y, Y), mult(mult(ld(Y, Y), ld(Y, Y)), ld(Y, Y)))), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 55 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 61 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), rd(mult(X, ld(Y, Y)), ld(mult(?, mult(X, ld(Y, Y))), rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(X, ld(Y, Y)), ld(rd(mult(?, mult(X, ld(Y, Y))), mult(ld(Y, Y), ld(Y, Y))), mult(?, mult(X, ld(Y, Y)))))), ld(Y, Y))
5.81/6.03	= { by lemma 9 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y))
5.81/6.03	= { by lemma 61 }
5.81/6.03	  mult(mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), mult(mult(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y))
5.81/6.03	= { by axiom 1 (f05) }
5.81/6.03	  mult(mult(X, ld(Y, Y)), mult(ld(Y, Y), mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(mult(mult(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))))
5.81/6.03	= { by axiom 1 (f05) }
5.81/6.03	  mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(X, mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y))))))
5.81/6.03	= { by lemma 46 }
5.81/6.03	  mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(X, ld(Y, Y))))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))
5.81/6.03	= { by lemma 40 }
5.81/6.03	  mult(ld(mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 55 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(ld(mult(rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by axiom 3 (f03) }
5.81/6.03	  mult(ld(mult(rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), mult(rd(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(ld(mult(rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), rd(rd(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 29 }
5.81/6.03	  mult(ld(mult(rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), mult(ld(Y, Y), mult(rd(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 6 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), rd(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(ld(Y, Y), ld(Y, Y))))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 61 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), rd(rd(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 57 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y)))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 61 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 55 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by axiom 1 (f05) }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(ld(mult(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y)), rd(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 38 }
5.81/6.03	  mult(rd(ld(ld(Y, Y), ld(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))), mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y))))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 63 }
5.81/6.03	  mult(rd(ld(ld(Y, Y), ld(mult(rd(X, ld(Y, Y)), ld(Y, Y)), mult(rd(X, ld(Y, Y)), ld(Y, Y)))), ld(Y, Y)), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 38 }
5.81/6.03	  mult(ld(mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y)), rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 55 }
5.81/6.03	  mult(ld(rd(X, ld(Y, Y)), rd(mult(rd(X, ld(Y, Y)), ld(Y, Y)), ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by axiom 2 (f04) }
5.81/6.03	  mult(ld(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 41 }
5.81/6.03	  mult(rd(rd(X, ld(Y, Y)), rd(X, ld(Y, Y))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 23 }
5.81/6.03	  mult(ld(ld(?, rd(?, rd(X, ld(Y, Y)))), ld(?, rd(?, rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 13 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(?, rd(?, rd(X, ld(Y, Y))))), mult(rd(X, ld(Y, Y)), mult(ld(Y, Y), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 7 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(?, rd(?, rd(X, ld(Y, Y))))), ld(rd(?, rd(X, ld(Y, Y))), mult(mult(?, ld(Y, Y)), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 22 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(?, rd(?, rd(X, ld(Y, Y))))), ld(mult(?, ld(?, rd(?, rd(X, ld(Y, Y))))), mult(mult(?, ld(Y, Y)), rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 34 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(?, rd(?, rd(X, ld(Y, Y))))), ld(mult(?, ld(?, rd(?, rd(X, ld(Y, Y))))), rd(mult(?, ld(Y, Y)), ld(?, rd(?, rd(X, ld(Y, Y)))))))
5.81/6.03	= { by lemma 47 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(?, rd(?, rd(X, ld(Y, Y))))), rd(ld(ld(?, rd(?, rd(X, ld(Y, Y)))), ld(Y, Y)), ld(?, rd(?, rd(X, ld(Y, Y))))))
5.81/6.03	= { by lemma 20 }
5.81/6.03	  rd(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(Y, Y)), ld(?, rd(?, rd(X, ld(Y, Y)))))
5.81/6.03	= { by lemma 34 }
5.81/6.03	  mult(mult(ld(?, rd(?, ld(?, rd(?, rd(X, ld(Y, Y)))))), ld(Y, Y)), rd(X, ld(Y, Y)))
5.81/6.03	= { by lemma 16 }
5.81/6.03	  mult(mult(ld(rd(?, rd(X, ld(Y, Y))), ?), ld(Y, Y)), rd(X, ld(Y, Y)))
5.81/6.03	= { by lemma 9 }
5.81/6.03	  mult(mult(rd(X, ld(Y, Y)), ld(Y, Y)), rd(X, ld(Y, Y)))
5.81/6.03	= { by axiom 3 (f03) }
5.81/6.03	  mult(X, rd(X, ld(Y, Y)))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(X, mult(X, ld(Y, Y)))
5.81/6.03	
5.81/6.03	Lemma 65: ld(ld(X, Y), mult(Z, ld(X, Y))) = mult(mult(ld(Y, X), Z), ld(X, Y)).
5.81/6.03	Proof:
5.81/6.03	  ld(ld(X, Y), mult(Z, ld(X, Y)))
5.81/6.03	= { by lemma 9 }
5.81/6.03	  ld(ld(X, Y), ld(rd(Y, mult(Z, ld(X, Y))), Y))
5.81/6.03	= { by lemma 54 }
5.81/6.03	  mult(ld(mult(rd(Y, mult(Z, ld(X, Y))), ld(X, Y)), X), ld(X, Y))
5.81/6.03	= { by lemma 31 }
5.81/6.03	  mult(ld(mult(rd(Y, rd(Z, ld(Y, X))), ld(X, Y)), X), ld(X, Y))
5.81/6.03	= { by lemma 53 }
5.81/6.03	  mult(ld(rd(X, mult(ld(Y, X), mult(rd(Z, ld(Y, X)), ld(Y, X)))), X), ld(X, Y))
5.81/6.03	= { by axiom 3 (f03) }
5.81/6.03	  mult(ld(rd(X, mult(ld(Y, X), Z)), X), ld(X, Y))
5.81/6.03	= { by lemma 9 }
5.81/6.03	  mult(mult(ld(Y, X), Z), ld(X, Y))
5.81/6.03	
5.81/6.03	Lemma 66: ld(ld(X, X), X) = X.
5.81/6.03	Proof:
5.81/6.03	  ld(ld(X, X), X)
5.81/6.03	= { by lemma 40 }
5.81/6.03	  ld(ld(X, X), mult(ld(X, X), X))
5.81/6.03	= { by axiom 5 (f02) }
5.98/6.19	  X
5.98/6.19	
5.98/6.19	Lemma 67: ld(ld(Y, Y), ld(X, X)) = ld(X, X).
5.98/6.19	Proof:
5.98/6.19	  ld(ld(Y, Y), ld(X, X))
5.98/6.19	= { by lemma 62 }
5.98/6.19	  mult(ld(Y, Y), mult(ld(Y, Y), ld(X, X)))
5.98/6.19	= { by lemma 64 }
5.98/6.19	  mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X))
5.98/6.19	= { by lemma 49 }
5.98/6.19	  mult(mult(ld(Y, rd(Y, ld(Y, Y))), ld(Y, Y)), ld(X, X))
5.98/6.19	= { by lemma 13 }
5.98/6.19	  mult(ld(ld(Y, Y), ld(Y, Y)), ld(X, X))
5.98/6.19	= { by axiom 3 (f03) }
5.98/6.19	  mult(ld(ld(Y, Y), mult(rd(ld(Y, Y), ld(Y, Y)), ld(Y, Y))), ld(X, X))
5.98/6.19	= { by lemma 65 }
5.98/6.19	  mult(mult(mult(ld(Y, Y), rd(ld(Y, Y), ld(Y, Y))), ld(Y, Y)), ld(X, X))
5.98/6.19	= { by lemma 17 }
5.98/6.19	  mult(mult(rd(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(X, X))
5.98/6.19	= { by lemma 31 }
5.98/6.19	  mult(mult(mult(ld(Y, Y), ld(Y, Y)), mult(ld(Y, Y), ld(Y, Y))), ld(X, X))
5.98/6.19	= { by lemma 64 }
5.98/6.19	  mult(mult(ld(Y, Y), ld(Y, Y)), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 4 (f01) }
5.98/6.19	  mult(mult(Y, ld(Y, mult(ld(Y, Y), ld(Y, Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 5 (f02) }
5.98/6.19	  mult(mult(Y, ld(ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), mult(ld(Y, Y), ld(Y, Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 41 }
5.98/6.19	  mult(mult(Y, ld(ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), mult(rd(Y, Y), ld(Y, Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 4 (f01) }
5.98/6.19	  mult(mult(Y, ld(ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), mult(rd(Y, Y), mult(Y, ld(Y, ld(Y, Y)))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 17 }
5.98/6.19	  mult(mult(Y, ld(ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 66 }
5.98/6.19	  mult(mult(Y, ld(ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), ld(ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 54 }
5.98/6.19	  mult(mult(Y, mult(ld(mult(ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y))), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 41 }
5.98/6.19	  mult(mult(Y, mult(ld(mult(rd(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y))), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 53 }
5.98/6.19	  mult(mult(Y, mult(ld(rd(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), mult(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(Y, rd(ld(Y, ld(Y, Y)), Y)))))), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 4 (f01) }
5.98/6.19	  mult(mult(Y, mult(ld(rd(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), mult(Y, rd(ld(Y, ld(Y, Y)), Y)))), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 9 }
5.98/6.19	  mult(mult(Y, mult(mult(ld(mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 15 }
5.98/6.19	  mult(mult(Y, mult(mult(ld(?, rd(?, Y)), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 14 }
5.98/6.19	  mult(mult(Y, mult(mult(rd(ld(Y, Y), Y), mult(Y, rd(ld(Y, ld(Y, Y)), Y))), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 17 }
5.98/6.19	  mult(mult(Y, mult(mult(mult(ld(Y, Y), rd(rd(ld(Y, ld(Y, Y)), Y), Y)), Y), ld(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), mult(mult(Y, rd(ld(Y, ld(Y, Y)), Y)), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 5 (f02) }
5.98/6.19	  mult(mult(Y, mult(mult(mult(ld(Y, Y), rd(rd(ld(Y, ld(Y, Y)), Y), Y)), Y), Y)), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 3 (f03) }
5.98/6.19	  mult(mult(Y, mult(rd(mult(mult(mult(ld(Y, Y), rd(rd(ld(Y, ld(Y, Y)), Y), Y)), Y), Y), mult(Y, Y)), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 44 }
5.98/6.19	  mult(mult(Y, mult(mult(rd(mult(mult(ld(Y, Y), rd(rd(ld(Y, ld(Y, Y)), Y), Y)), Y), Y), ld(Y, Y)), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 2 (f04) }
5.98/6.19	  mult(mult(Y, mult(mult(mult(ld(Y, Y), rd(rd(ld(Y, ld(Y, Y)), Y), Y)), ld(Y, Y)), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 65 }
5.98/6.19	  mult(mult(Y, mult(ld(ld(Y, Y), mult(rd(rd(ld(Y, ld(Y, Y)), Y), Y), ld(Y, Y))), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 63 }
5.98/6.19	  mult(mult(Y, mult(ld(ld(mult(Y, Y), mult(Y, Y)), mult(rd(rd(ld(Y, ld(Y, Y)), Y), Y), ld(Y, Y))), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 13 }
5.98/6.19	  mult(mult(Y, mult(ld(mult(ld(?, rd(?, mult(Y, Y))), mult(Y, Y)), mult(rd(rd(ld(Y, ld(Y, Y)), Y), Y), ld(Y, Y))), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 44 }
5.98/6.19	  mult(mult(Y, mult(ld(mult(ld(?, rd(?, mult(Y, Y))), mult(Y, Y)), rd(mult(rd(ld(Y, ld(Y, Y)), Y), Y), mult(Y, Y))), mult(Y, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 37 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(?, rd(?, mult(Y, Y))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 33 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(mult(?, Y), rd(mult(?, Y), mult(Y, Y))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 32 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(mult(?, Y), rd(mult(?, ld(Y, Y)), Y)), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 47 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(rd(ld(Y, ld(Y, Y)), Y), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by axiom 5 (f02) }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(?, mult(?, rd(ld(Y, ld(Y, Y)), Y))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 21 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(?, rd(mult(rd(?, Y), ld(Y, Y)), Y)), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 22 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(ld(?, mult(mult(rd(?, Y), ld(Y, Y)), ld(?, rd(?, Y)))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 43 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(mult(ld(?, rd(?, Y)), mult(ld(Y, Y), ld(?, rd(?, Y)))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 22 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(mult(ld(?, rd(?, Y)), rd(ld(Y, Y), Y)), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 14 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(mult(ld(?, rd(?, Y)), ld(?, rd(?, Y))), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 31 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), ld(rd(ld(?, rd(?, Y)), ld(rd(?, Y), ?)), mult(rd(ld(Y, ld(Y, Y)), Y), Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 58 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), mult(ld(rd(?, Y), ?), mult(ld(ld(?, rd(?, Y)), rd(mult(rd(ld(Y, ld(Y, Y)), Y), Y), ld(rd(?, Y), ?))), ld(rd(?, Y), ?))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.19	= { by lemma 9 }
5.98/6.19	  mult(mult(Y, ld(mult(Y, Y), mult(Y, mult(ld(ld(?, rd(?, Y)), rd(mult(rd(ld(Y, ld(Y, Y)), Y), Y), ld(rd(?, Y), ?))), ld(rd(?, Y), ?))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 48 }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), mult(Y, mult(ld(rd(?, Y), ?), mult(rd(mult(rd(ld(Y, ld(Y, Y)), Y), Y), ld(rd(?, Y), ?)), ld(rd(?, Y), ?)))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 9 }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), mult(Y, mult(Y, mult(rd(mult(rd(ld(Y, ld(Y, Y)), Y), Y), ld(rd(?, Y), ?)), ld(rd(?, Y), ?)))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by axiom 3 (f03) }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), mult(Y, mult(Y, mult(rd(ld(Y, ld(Y, Y)), Y), Y))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by axiom 3 (f03) }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), mult(Y, mult(Y, ld(Y, ld(Y, Y)))))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by axiom 4 (f01) }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), mult(Y, ld(Y, Y)))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by axiom 4 (f01) }
5.98/6.20	  mult(mult(Y, ld(mult(Y, Y), Y)), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 15 }
5.98/6.20	  mult(mult(Y, ld(?, rd(?, Y))), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 22 }
5.98/6.20	  mult(rd(Y, Y), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 41 }
5.98/6.20	  mult(ld(Y, Y), mult(mult(ld(Y, Y), ld(Y, Y)), ld(X, X)))
5.98/6.20	= { by lemma 64 }
5.98/6.20	  mult(ld(Y, Y), mult(ld(Y, Y), mult(ld(Y, Y), ld(X, X))))
5.98/6.20	= { by lemma 62 }
5.98/6.20	  mult(ld(Y, Y), ld(ld(Y, Y), ld(X, X)))
5.98/6.20	= { by axiom 4 (f01) }
5.98/6.20	  ld(X, X)
5.98/6.20	
5.98/6.20	Lemma 68: ld(X, X) = ld(?, ?).
5.98/6.20	Proof:
5.98/6.20	  ld(X, X)
5.98/6.20	= { by lemma 46 }
5.98/6.20	  mult(ld(X, X), mult(ld(X, X), ld(X, X)))
5.98/6.20	= { by lemma 57 }
5.98/6.20	  rd(ld(X, X), mult(ld(X, X), ld(X, X)))
5.98/6.20	= { by lemma 67 }
5.98/6.20	  rd(ld(X, X), mult(ld(ld(Y, Y), ld(X, X)), ld(X, X)))
5.98/6.20	= { by axiom 4 (f01) }
5.98/6.20	  rd(ld(X, X), mult(ld(ld(Y, Y), ld(X, X)), mult(ld(Y, Y), ld(ld(Y, Y), ld(X, X)))))
5.98/6.20	= { by lemma 53 }
5.98/6.20	  mult(rd(ld(Y, Y), ld(Y, Y)), ld(ld(X, X), ld(Y, Y)))
5.98/6.20	= { by lemma 67 }
5.98/6.20	  mult(rd(ld(Y, Y), ld(Y, Y)), ld(Y, Y))
5.98/6.20	= { by lemma 67 }
5.98/6.20	  mult(rd(ld(Y, Y), ld(Y, Y)), ld(ld(?, ?), ld(Y, Y)))
5.98/6.20	= { by lemma 53 }
5.98/6.20	  rd(ld(?, ?), mult(ld(ld(Y, Y), ld(?, ?)), mult(ld(Y, Y), ld(ld(Y, Y), ld(?, ?)))))
5.98/6.20	= { by lemma 67 }
5.98/6.20	  rd(ld(?, ?), mult(ld(?, ?), mult(ld(Y, Y), ld(ld(Y, Y), ld(?, ?)))))
5.98/6.20	= { by axiom 4 (f01) }
5.98/6.20	  rd(ld(?, ?), mult(ld(?, ?), ld(?, ?)))
5.98/6.20	= { by lemma 57 }
5.98/6.20	  mult(ld(?, ?), mult(ld(?, ?), ld(?, ?)))
5.98/6.20	= { by lemma 46 }
6.04/6.23	  ld(?, ?)
6.04/6.23	
6.04/6.23	Lemma 69: mult(ld(?, ?), X) = X.
6.04/6.23	Proof:
6.04/6.23	  mult(ld(?, ?), X)
6.04/6.23	= { by lemma 16 }
6.04/6.23	  mult(ld(?, rd(?, ld(?, ?))), X)
6.04/6.23	= { by lemma 15 }
6.04/6.23	  mult(ld(mult(X, ld(?, ?)), X), X)
6.04/6.23	= { by lemma 9 }
6.04/6.23	  ld(rd(X, mult(ld(mult(X, ld(?, ?)), X), X)), X)
6.04/6.23	= { by lemma 59 }
6.04/6.23	  ld(mult(ld(mult(X, ld(?, ?)), mult(X, ld(?, ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 25 }
6.04/6.23	  ld(mult(mult(mult(ld(mult(X, ld(?, ?)), mult(X, ld(?, ?))), ld(?, ?)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 41 }
6.04/6.23	  ld(mult(mult(mult(rd(mult(X, ld(?, ?)), mult(X, ld(?, ?))), ld(?, ?)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 5 (f02) }
6.04/6.23	  ld(mult(mult(mult(rd(mult(X, ld(?, ?)), mult(X, ld(?, ?))), ld(X, mult(X, ld(?, ?)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 53 }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(mult(X, ld(?, ?)), X), mult(mult(X, ld(?, ?)), ld(mult(X, ld(?, ?)), X)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 15 }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(?, rd(?, ld(?, ?))), mult(mult(X, ld(?, ?)), ld(mult(X, ld(?, ?)), X)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 16 }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(?, ?), mult(mult(X, ld(?, ?)), ld(mult(X, ld(?, ?)), X)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 9 }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(rd(X, ld(?, ?)), X), mult(mult(X, ld(?, ?)), ld(mult(X, ld(?, ?)), X)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 4 (f01) }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(rd(X, ld(?, ?)), X), X)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 4 (f01) }
6.04/6.23	  ld(mult(mult(rd(X, mult(ld(rd(X, ld(?, ?)), X), mult(rd(X, ld(?, ?)), ld(rd(X, ld(?, ?)), X)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 53 }
6.04/6.23	  ld(mult(mult(mult(rd(rd(X, ld(?, ?)), rd(X, ld(?, ?))), ld(X, rd(X, ld(?, ?)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 41 }
6.04/6.23	  ld(mult(mult(mult(ld(rd(X, ld(?, ?)), rd(X, ld(?, ?))), ld(X, rd(X, ld(?, ?)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 33 }
6.04/6.23	  ld(mult(mult(mult(ld(rd(X, ld(?, ?)), rd(X, ld(?, ?))), ld(?, rd(?, ld(?, ?)))), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 22 }
6.04/6.23	  ld(mult(mult(rd(ld(rd(X, ld(?, ?)), rd(X, ld(?, ?))), ld(?, ?)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 26 }
6.04/6.23	  ld(mult(mult(rd(ld(mult(X, ld(?, ?)), rd(X, ld(?, ?))), ld(?, ?)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 31 }
6.04/6.23	  ld(mult(mult(mult(ld(mult(X, ld(?, ?)), rd(X, ld(?, ?))), ld(?, ?)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 37 }
6.04/6.23	  ld(mult(mult(ld(ld(?, ?), ld(X, X)), ld(?, ?)), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 48 }
6.04/6.23	  ld(mult(mult(ld(?, ?), mult(ld(X, X), ld(?, ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  ld(mult(mult(ld(mult(rd(?, ld(X, X)), ld(X, X)), ?), mult(ld(X, X), ld(?, ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  ld(mult(mult(ld(mult(rd(?, ld(X, X)), ld(X, X)), ?), mult(ld(X, X), ld(mult(rd(?, ld(X, X)), ld(X, X)), ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 7 }
6.04/6.23	  ld(mult(ld(rd(rd(?, ld(X, X)), ld(mult(rd(?, ld(X, X)), ld(X, X)), ?)), mult(mult(rd(?, ld(X, X)), ld(X, X)), ld(mult(rd(?, ld(X, X)), ld(X, X)), ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by lemma 31 }
6.04/6.23	  ld(mult(ld(mult(rd(?, ld(X, X)), ld(?, mult(rd(?, ld(X, X)), ld(X, X)))), mult(mult(rd(?, ld(X, X)), ld(X, X)), ld(mult(rd(?, ld(X, X)), ld(X, X)), ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  ld(mult(ld(mult(rd(?, ld(X, X)), ld(?, ?)), mult(mult(rd(?, ld(X, X)), ld(X, X)), ld(mult(rd(?, ld(X, X)), ld(X, X)), ?))), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 4 (f01) }
6.04/6.23	  ld(mult(ld(mult(rd(?, ld(X, X)), ld(?, ?)), ?), ld(X, mult(X, ld(?, ?)))), X)
6.04/6.23	= { by axiom 5 (f02) }
6.04/6.23	  ld(mult(ld(mult(rd(?, ld(X, X)), ld(?, ?)), ?), ld(?, ?)), X)
6.04/6.23	= { by lemma 54 }
6.04/6.23	  ld(ld(ld(?, ?), ld(rd(?, ld(X, X)), ?)), X)
6.04/6.23	= { by lemma 9 }
6.04/6.23	  ld(ld(ld(?, ?), ld(X, X)), X)
6.04/6.23	= { by lemma 67 }
6.04/6.23	  ld(ld(X, X), X)
6.04/6.23	= { by lemma 66 }
6.04/6.23	  X
6.04/6.23	
6.04/6.23	Goal 1 (goals): tuple(mult(ld(sK1_goals_X1, sK1_goals_X1), sK2_goals_X0), sK4_goals_X0) = tuple(sK2_goals_X0, mult(sK4_goals_X0, ld(sK3_goals_X1, sK3_goals_X1))).
6.04/6.23	Proof:
6.04/6.23	  tuple(mult(ld(sK1_goals_X1, sK1_goals_X1), sK2_goals_X0), sK4_goals_X0)
6.04/6.23	= { by lemma 68 }
6.04/6.23	  tuple(mult(ld(?, ?), sK2_goals_X0), sK4_goals_X0)
6.04/6.23	= { by lemma 69 }
6.04/6.23	  tuple(sK2_goals_X0, sK4_goals_X0)
6.04/6.23	= { by lemma 40 }
6.04/6.23	  tuple(sK2_goals_X0, mult(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0))
6.04/6.23	= { by lemma 69 }
6.04/6.23	  tuple(sK2_goals_X0, mult(mult(ld(?, ?), ld(sK4_goals_X0, sK4_goals_X0)), sK4_goals_X0))
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  tuple(sK2_goals_X0, mult(rd(mult(mult(ld(?, ?), ld(sK4_goals_X0, sK4_goals_X0)), sK4_goals_X0), sK4_goals_X0), sK4_goals_X0))
6.04/6.23	= { by lemma 60 }
6.04/6.23	  tuple(sK2_goals_X0, rd(mult(ld(?, ?), ld(sK4_goals_X0, sK4_goals_X0)), rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0)))
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  tuple(sK2_goals_X0, rd(mult(ld(?, ?), mult(rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0), sK4_goals_X0)), rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0)))
6.04/6.23	= { by lemma 19 }
6.04/6.23	  tuple(sK2_goals_X0, mult(mult(ld(?, ?), rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0)), rd(sK4_goals_X0, rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0))))
6.04/6.23	= { by lemma 14 }
6.04/6.23	  tuple(sK2_goals_X0, mult(mult(ld(?, ?), ld(?, rd(?, sK4_goals_X0))), rd(sK4_goals_X0, rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0))))
6.04/6.23	= { by lemma 22 }
6.04/6.23	  tuple(sK2_goals_X0, mult(rd(ld(?, ?), sK4_goals_X0), rd(sK4_goals_X0, rd(ld(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0))))
6.04/6.23	= { by lemma 60 }
6.04/6.23	  tuple(sK2_goals_X0, mult(rd(ld(?, ?), sK4_goals_X0), mult(rd(mult(sK4_goals_X0, sK4_goals_X0), sK4_goals_X0), sK4_goals_X0)))
6.04/6.23	= { by axiom 3 (f03) }
6.04/6.23	  tuple(sK2_goals_X0, mult(rd(ld(?, ?), sK4_goals_X0), mult(sK4_goals_X0, sK4_goals_X0)))
6.04/6.23	= { by lemma 45 }
6.04/6.23	  tuple(sK2_goals_X0, mult(mult(ld(?, ?), sK4_goals_X0), ld(sK4_goals_X0, sK4_goals_X0)))
6.04/6.23	= { by lemma 69 }
6.04/6.23	  tuple(sK2_goals_X0, mult(sK4_goals_X0, ld(sK4_goals_X0, sK4_goals_X0)))
6.04/6.23	= { by lemma 68 }
6.04/6.23	  tuple(sK2_goals_X0, mult(sK4_goals_X0, ld(?, ?)))
6.04/6.23	= { by lemma 68 }
6.04/6.23	  tuple(sK2_goals_X0, mult(sK4_goals_X0, ld(sK3_goals_X1, sK3_goals_X1)))
6.04/6.23	% SZS output end Proof
6.04/6.23	
6.04/6.23	RESULT: Theorem (the conjecture is true).
6.04/6.23	EOF