0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.13/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.13/0.34	% Computer   : n027.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 1200
0.13/0.34	% WCLimit    : 120
0.13/0.34	% DateTime   : Tue Jul 13 14:38:25 EDT 2021
0.13/0.34	% CPUTime    : 
15.66/2.33	% SZS status Theorem
15.66/2.33	
16.30/2.41	% SZS output start Proof
16.30/2.41	Take the following subset of the input axioms:
16.30/2.41	  fof(f01, axiom, ![B, A]: B=mult(A, ld(A, B))).
16.30/2.41	  fof(f02, axiom, ![B, A]: B=ld(A, mult(A, B))).
16.30/2.41	  fof(f03, axiom, ![B, A]: mult(rd(A, B), B)=A).
16.30/2.41	  fof(f04, axiom, ![B, A]: rd(mult(A, B), B)=A).
16.30/2.41	  fof(f05, axiom, ![B, A, C]: mult(mult(mult(A, B), C), B)=mult(A, mult(B, mult(C, B)))).
16.30/2.41	  fof(goals, conjecture, ![X0, X1]: (X0=mult(X0, rd(X1, X1)) & X0=mult(rd(X1, X1), X0))).
16.30/2.41	
16.30/2.41	Now clausify the problem and encode Horn clauses using encoding 3 of
16.30/2.41	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
16.30/2.41	We repeatedly replace C & s=t => u=v by the two clauses:
16.30/2.41	  fresh(y, y, x1...xn) = u
16.30/2.41	  C => fresh(s, t, x1...xn) = v
16.30/2.41	where fresh is a fresh function symbol and x1..xn are the free
16.30/2.41	variables of u and v.
16.30/2.41	A predicate p(X) is encoded as p(X)=true (this is sound, because the
16.30/2.41	input problem has no model of domain size 1).
16.30/2.41	
16.30/2.41	The encoding turns the above axioms into the following unit equations and goals:
16.30/2.41	
16.30/2.41	Axiom 1 (f01): X = mult(Y, ld(Y, X)).
16.30/2.41	Axiom 2 (f03): mult(rd(X, Y), Y) = X.
16.30/2.41	Axiom 3 (f04): rd(mult(X, Y), Y) = X.
16.30/2.41	Axiom 4 (f02): X = ld(Y, mult(Y, X)).
16.30/2.41	Axiom 5 (f05): mult(mult(mult(X, Y), Z), Y) = mult(X, mult(Y, mult(Z, Y))).
16.30/2.41	
16.30/2.41	Lemma 6: mult(rd(X, Y), mult(Y, mult(Z, Y))) = mult(mult(X, Z), Y).
16.30/2.41	Proof:
16.30/2.41	  mult(rd(X, Y), mult(Y, mult(Z, Y)))
16.30/2.41	= { by axiom 5 (f05) R->L }
16.30/2.41	  mult(mult(mult(rd(X, Y), Y), Z), Y)
16.30/2.41	= { by axiom 2 (f03) }
16.30/2.41	  mult(mult(X, Z), Y)
16.30/2.41	
16.30/2.41	Lemma 7: ld(X, mult(mult(mult(X, Y), Z), Y)) = mult(Y, mult(Z, Y)).
16.30/2.41	Proof:
16.30/2.41	  ld(X, mult(mult(mult(X, Y), Z), Y))
16.30/2.41	= { by axiom 5 (f05) }
16.30/2.41	  ld(X, mult(X, mult(Y, mult(Z, Y))))
16.30/2.41	= { by axiom 4 (f02) R->L }
16.30/2.41	  mult(Y, mult(Z, Y))
16.30/2.41	
16.30/2.41	Lemma 8: mult(X, mult(ld(mult(Y, X), Z), X)) = ld(Y, mult(Z, X)).
16.30/2.41	Proof:
16.30/2.41	  mult(X, mult(ld(mult(Y, X), Z), X))
16.30/2.41	= { by lemma 7 R->L }
16.30/2.41	  ld(Y, mult(mult(mult(Y, X), ld(mult(Y, X), Z)), X))
16.30/2.41	= { by axiom 1 (f01) R->L }
16.30/2.41	  ld(Y, mult(Z, X))
16.30/2.41	
16.30/2.41	Lemma 9: ld(X, ld(Y, mult(Z, X))) = mult(ld(mult(Y, X), Z), X).
16.30/2.41	Proof:
16.30/2.41	  ld(X, ld(Y, mult(Z, X)))
16.30/2.41	= { by lemma 8 R->L }
16.30/2.41	  ld(X, mult(X, mult(ld(mult(Y, X), Z), X)))
16.30/2.41	= { by axiom 4 (f02) R->L }
16.30/2.41	  mult(ld(mult(Y, X), Z), X)
16.30/2.41	
16.30/2.41	Lemma 10: mult(mult(X, ld(mult(Y, Z), Y)), Z) = X.
16.30/2.41	Proof:
16.30/2.41	  mult(mult(X, ld(mult(Y, Z), Y)), Z)
16.30/2.41	= { by lemma 6 R->L }
16.30/2.41	  mult(rd(X, Z), mult(Z, mult(ld(mult(Y, Z), Y), Z)))
16.30/2.41	= { by lemma 9 R->L }
16.30/2.41	  mult(rd(X, Z), mult(Z, ld(Z, ld(Y, mult(Y, Z)))))
16.30/2.41	= { by axiom 4 (f02) R->L }
16.30/2.41	  mult(rd(X, Z), mult(Z, ld(Z, Z)))
16.30/2.41	= { by axiom 1 (f01) R->L }
16.30/2.41	  mult(rd(X, Z), Z)
16.30/2.41	= { by axiom 2 (f03) }
16.30/2.41	  X
16.30/2.41	
16.30/2.41	Lemma 11: ld(rd(X, Y), mult(mult(X, Z), Y)) = mult(Y, mult(Z, Y)).
16.30/2.41	Proof:
16.30/2.41	  ld(rd(X, Y), mult(mult(X, Z), Y))
16.30/2.41	= { by lemma 6 R->L }
16.30/2.41	  ld(rd(X, Y), mult(rd(X, Y), mult(Y, mult(Z, Y))))
16.30/2.41	= { by axiom 4 (f02) R->L }
16.30/2.41	  mult(Y, mult(Z, Y))
16.30/2.41	
16.30/2.41	Lemma 12: ld(rd(rd(X, Y), Z), mult(X, Z)) = mult(Z, mult(Y, Z)).
16.30/2.41	Proof:
16.30/2.41	  ld(rd(rd(X, Y), Z), mult(X, Z))
16.30/2.41	= { by axiom 2 (f03) R->L }
16.30/2.41	  ld(rd(rd(X, Y), Z), mult(mult(rd(X, Y), Y), Z))
16.30/2.41	= { by lemma 11 }
16.30/2.41	  mult(Z, mult(Y, Z))
16.30/2.41	
16.30/2.41	Lemma 13: rd(X, ld(Y, X)) = Y.
16.30/2.41	Proof:
16.30/2.41	  rd(X, ld(Y, X))
16.30/2.41	= { by axiom 1 (f01) }
16.30/2.41	  rd(mult(Y, ld(Y, X)), ld(Y, X))
16.30/2.41	= { by axiom 3 (f04) }
16.30/2.41	  Y
16.30/2.41	
16.30/2.41	Lemma 14: mult(ld(X, rd(X, X)), rd(X, X)) = ld(X, rd(X, X)).
16.30/2.41	Proof:
16.30/2.41	  mult(ld(X, rd(X, X)), rd(X, X))
16.30/2.41	= { by axiom 1 (f01) }
16.30/2.41	  mult(ld(X, rd(X, X)), mult(X, ld(X, rd(X, X))))
16.30/2.41	= { by lemma 12 R->L }
16.30/2.41	  ld(rd(rd(X, X), ld(X, rd(X, X))), mult(X, ld(X, rd(X, X))))
16.30/2.41	= { by axiom 1 (f01) R->L }
16.30/2.41	  ld(rd(rd(X, X), ld(X, rd(X, X))), rd(X, X))
16.30/2.41	= { by lemma 13 }
16.30/2.41	  ld(X, rd(X, X))
16.30/2.41	
16.30/2.41	Lemma 15: ld(mult(X, rd(Y, Y)), X) = rd(Y, Y).
16.30/2.41	Proof:
16.30/2.41	  ld(mult(X, rd(Y, Y)), X)
16.30/2.41	= { by axiom 4 (f02) }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), mult(ld(Y, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)))
16.30/2.41	= { by axiom 3 (f04) R->L }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), rd(mult(mult(ld(Y, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X)), rd(Y, Y)), rd(Y, Y)))
16.30/2.41	= { by lemma 10 }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), rd(ld(Y, rd(Y, Y)), rd(Y, Y)))
16.30/2.41	= { by lemma 14 R->L }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), rd(mult(ld(Y, rd(Y, Y)), rd(Y, Y)), rd(Y, Y)))
16.30/2.41	= { by axiom 3 (f04) }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), ld(Y, rd(Y, Y)))
16.30/2.41	= { by lemma 14 R->L }
16.30/2.41	  ld(ld(Y, rd(Y, Y)), mult(ld(Y, rd(Y, Y)), rd(Y, Y)))
16.30/2.41	= { by axiom 4 (f02) R->L }
16.30/2.41	  rd(Y, Y)
16.30/2.41	
16.30/2.41	Lemma 16: mult(mult(X, rd(Y, Y)), rd(Y, Y)) = X.
16.30/2.41	Proof:
16.30/2.41	  mult(mult(X, rd(Y, Y)), rd(Y, Y))
16.30/2.41	= { by lemma 15 R->L }
16.30/2.41	  mult(mult(X, rd(Y, Y)), ld(mult(X, rd(Y, Y)), X))
16.30/2.41	= { by axiom 1 (f01) R->L }
16.30/2.41	  X
16.30/2.41	
16.30/2.41	Lemma 17: rd(X, rd(Y, Y)) = mult(X, rd(Y, Y)).
16.30/2.41	Proof:
16.30/2.41	  rd(X, rd(Y, Y))
16.30/2.41	= { by lemma 15 R->L }
16.30/2.41	  rd(X, ld(mult(Z, rd(Y, Y)), Z))
16.30/2.41	= { by lemma 10 R->L }
16.30/2.41	  mult(mult(rd(X, ld(mult(Z, rd(Y, Y)), Z)), ld(mult(Z, rd(Y, Y)), Z)), rd(Y, Y))
16.30/2.41	= { by axiom 2 (f03) }
16.30/2.41	  mult(X, rd(Y, Y))
16.30/2.41	
16.30/2.41	Lemma 18: rd(mult(X, mult(Y, mult(Z, Y))), Y) = mult(mult(X, Y), Z).
16.30/2.41	Proof:
16.30/2.41	  rd(mult(X, mult(Y, mult(Z, Y))), Y)
16.30/2.41	= { by axiom 5 (f05) R->L }
16.30/2.41	  rd(mult(mult(mult(X, Y), Z), Y), Y)
16.30/2.41	= { by axiom 3 (f04) }
16.30/2.41	  mult(mult(X, Y), Z)
16.30/2.41	
16.30/2.41	Lemma 19: rd(mult(X, mult(Y, Z)), Y) = mult(mult(X, Y), rd(Z, Y)).
16.30/2.41	Proof:
16.30/2.41	  rd(mult(X, mult(Y, Z)), Y)
16.30/2.41	= { by axiom 2 (f03) R->L }
16.30/2.41	  rd(mult(X, mult(Y, mult(rd(Z, Y), Y))), Y)
16.30/2.41	= { by lemma 18 }
16.30/2.41	  mult(mult(X, Y), rd(Z, Y))
16.30/2.41	
16.30/2.41	Lemma 20: mult(mult(X, Y), rd(ld(Y, Z), Y)) = rd(mult(X, Z), Y).
16.30/2.41	Proof:
16.30/2.41	  mult(mult(X, Y), rd(ld(Y, Z), Y))
16.30/2.41	= { by lemma 19 R->L }
16.30/2.41	  rd(mult(X, mult(Y, ld(Y, Z))), Y)
16.30/2.41	= { by axiom 1 (f01) R->L }
16.30/2.41	  rd(mult(X, Z), Y)
16.30/2.41	
16.30/2.41	Lemma 21: mult(mult(X, rd(Y, Z)), Z) = mult(rd(X, Z), mult(Z, Y)).
16.30/2.41	Proof:
16.30/2.41	  mult(mult(X, rd(Y, Z)), Z)
16.30/2.41	= { by lemma 6 R->L }
16.30/2.41	  mult(rd(X, Z), mult(Z, mult(rd(Y, Z), Z)))
16.30/2.41	= { by axiom 2 (f03) }
16.30/2.41	  mult(rd(X, Z), mult(Z, Y))
16.30/2.41	
16.30/2.41	Lemma 22: mult(ld(rd(X, X), mult(Y, rd(X, X))), rd(X, X)) = mult(rd(X, X), Y).
16.30/2.41	Proof:
16.30/2.41	  mult(ld(rd(X, X), mult(Y, rd(X, X))), rd(X, X))
16.30/2.41	= { by lemma 17 R->L }
16.30/2.41	  mult(ld(rd(X, X), rd(Y, rd(X, X))), rd(X, X))
16.30/2.42	= { by lemma 17 R->L }
16.30/2.42	  rd(ld(rd(X, X), rd(Y, rd(X, X))), rd(X, X))
16.30/2.42	= { by axiom 4 (f02) }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(mult(Z, rd(X, X)), rd(ld(rd(X, X), rd(Y, rd(X, X))), rd(X, X))))
16.30/2.42	= { by lemma 20 }
16.30/2.42	  ld(mult(Z, rd(X, X)), rd(mult(Z, rd(Y, rd(X, X))), rd(X, X)))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(mult(Z, rd(Y, rd(X, X))), rd(X, X)))
16.30/2.42	= { by lemma 21 }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(rd(Z, rd(X, X)), mult(rd(X, X), Y)))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(mult(Z, rd(X, X)), mult(rd(X, X), Y)))
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  mult(rd(X, X), Y)
16.30/2.42	
16.30/2.42	Lemma 23: mult(ld(rd(X, X), Y), rd(X, X)) = mult(rd(X, X), mult(Y, rd(X, X))).
16.30/2.42	Proof:
16.30/2.42	  mult(ld(rd(X, X), Y), rd(X, X))
16.30/2.42	= { by axiom 2 (f03) R->L }
16.30/2.42	  mult(ld(rd(X, X), mult(rd(Y, rd(X, X)), rd(X, X))), rd(X, X))
16.30/2.42	= { by lemma 22 }
16.30/2.42	  mult(rd(X, X), rd(Y, rd(X, X)))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(rd(X, X), mult(Y, rd(X, X)))
16.30/2.42	
16.30/2.42	Lemma 24: ld(rd(X, Y), X) = Y.
16.30/2.42	Proof:
16.30/2.42	  ld(rd(X, Y), X)
16.30/2.42	= { by axiom 2 (f03) R->L }
16.30/2.42	  ld(rd(X, Y), mult(rd(X, Y), Y))
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  Y
16.30/2.42	
16.30/2.42	Lemma 25: mult(rd(X, X), mult(X, rd(X, X))) = mult(X, rd(X, X)).
16.30/2.42	Proof:
16.30/2.42	  mult(rd(X, X), mult(X, rd(X, X)))
16.30/2.42	= { by lemma 23 R->L }
16.30/2.42	  mult(ld(rd(X, X), X), rd(X, X))
16.30/2.42	= { by lemma 24 }
16.30/2.42	  mult(X, rd(X, X))
16.30/2.42	
16.30/2.42	Lemma 26: mult(rd(X, X), rd(X, X)) = rd(X, X).
16.30/2.42	Proof:
16.30/2.42	  mult(rd(X, X), rd(X, X))
16.30/2.42	= { by lemma 17 R->L }
16.30/2.42	  rd(rd(X, X), rd(X, X))
16.30/2.42	= { by lemma 13 R->L }
16.30/2.42	  rd(mult(X, rd(X, X)), ld(rd(rd(X, X), rd(X, X)), mult(X, rd(X, X))))
16.30/2.42	= { by lemma 12 }
16.30/2.42	  rd(mult(X, rd(X, X)), mult(rd(X, X), mult(X, rd(X, X))))
16.30/2.42	= { by lemma 25 }
16.30/2.42	  rd(mult(X, rd(X, X)), mult(X, rd(X, X)))
16.30/2.42	= { by lemma 25 R->L }
16.30/2.42	  rd(mult(rd(X, X), mult(X, rd(X, X))), mult(X, rd(X, X)))
16.30/2.42	= { by axiom 3 (f04) }
16.30/2.42	  rd(X, X)
16.30/2.42	
16.30/2.42	Lemma 27: ld(rd(X, Y), mult(Z, Y)) = mult(Y, mult(ld(X, Z), Y)).
16.30/2.42	Proof:
16.30/2.42	  ld(rd(X, Y), mult(Z, Y))
16.30/2.42	= { by axiom 1 (f01) }
16.30/2.42	  ld(rd(X, Y), mult(mult(X, ld(X, Z)), Y))
16.30/2.42	= { by lemma 11 }
16.30/2.42	  mult(Y, mult(ld(X, Z), Y))
16.30/2.42	
16.30/2.42	Lemma 28: mult(ld(X, rd(X, Y)), Y) = ld(Y, Y).
16.30/2.42	Proof:
16.30/2.42	  mult(ld(X, rd(X, Y)), Y)
16.30/2.42	= { by axiom 4 (f02) }
16.30/2.42	  ld(Y, mult(Y, mult(ld(X, rd(X, Y)), Y)))
16.30/2.42	= { by lemma 27 R->L }
16.30/2.42	  ld(Y, ld(rd(X, Y), mult(rd(X, Y), Y)))
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  ld(Y, Y)
16.30/2.42	
16.30/2.42	Lemma 29: rd(X, ld(Y, Z)) = mult(X, ld(Z, Y)).
16.30/2.42	Proof:
16.30/2.42	  rd(X, ld(Y, Z))
16.30/2.42	= { by axiom 1 (f01) }
16.30/2.42	  mult(X, ld(X, rd(X, ld(Y, Z))))
16.30/2.42	= { by axiom 3 (f04) R->L }
16.30/2.42	  mult(X, rd(mult(ld(X, rd(X, ld(Y, Z))), ld(Y, Z)), ld(Y, Z)))
16.30/2.42	= { by lemma 28 }
16.30/2.42	  mult(X, rd(ld(ld(Y, Z), ld(Y, Z)), ld(Y, Z)))
16.30/2.42	= { by lemma 28 R->L }
16.30/2.42	  mult(X, rd(mult(ld(Z, rd(Z, ld(Y, Z))), ld(Y, Z)), ld(Y, Z)))
16.30/2.42	= { by lemma 13 }
16.30/2.42	  mult(X, rd(mult(ld(Z, Y), ld(Y, Z)), ld(Y, Z)))
16.30/2.42	= { by axiom 3 (f04) }
16.30/2.42	  mult(X, ld(Z, Y))
16.30/2.42	
16.30/2.42	Lemma 30: mult(X, ld(Y, Y)) = mult(X, rd(Y, Y)).
16.30/2.42	Proof:
16.30/2.42	  mult(X, ld(Y, Y))
16.30/2.42	= { by axiom 3 (f04) R->L }
16.30/2.42	  rd(mult(mult(X, ld(Y, Y)), Y), Y)
16.30/2.42	= { by lemma 24 R->L }
16.30/2.42	  rd(mult(mult(X, ld(Y, Y)), ld(rd(X, Y), X)), Y)
16.30/2.42	= { by lemma 29 R->L }
16.30/2.42	  rd(rd(mult(X, ld(Y, Y)), ld(X, rd(X, Y))), Y)
16.30/2.42	= { by lemma 20 R->L }
16.30/2.42	  rd(mult(mult(X, ld(X, rd(X, Y))), rd(ld(ld(X, rd(X, Y)), ld(Y, Y)), ld(X, rd(X, Y)))), Y)
16.30/2.42	= { by lemma 28 R->L }
16.30/2.42	  rd(mult(mult(X, ld(X, rd(X, Y))), rd(ld(ld(X, rd(X, Y)), mult(ld(X, rd(X, Y)), Y)), ld(X, rd(X, Y)))), Y)
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  rd(mult(mult(X, ld(X, rd(X, Y))), rd(Y, ld(X, rd(X, Y)))), Y)
16.30/2.42	= { by axiom 1 (f01) R->L }
16.30/2.42	  rd(mult(rd(X, Y), rd(Y, ld(X, rd(X, Y)))), Y)
16.30/2.42	= { by lemma 29 }
16.30/2.42	  rd(mult(rd(X, Y), mult(Y, ld(rd(X, Y), X))), Y)
16.30/2.42	= { by lemma 24 }
16.30/2.42	  rd(mult(rd(X, Y), mult(Y, Y)), Y)
16.30/2.42	= { by lemma 20 R->L }
16.30/2.42	  mult(mult(rd(X, Y), Y), rd(ld(Y, mult(Y, Y)), Y))
16.30/2.42	= { by axiom 2 (f03) }
16.30/2.42	  mult(X, rd(ld(Y, mult(Y, Y)), Y))
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  mult(X, rd(Y, Y))
16.30/2.42	
16.30/2.42	Lemma 31: mult(mult(X, rd(Z, Z)), rd(mult(Y, rd(Z, Z)), mult(Y, rd(Z, Z)))) = mult(mult(X, rd(Y, Y)), rd(Z, Z)).
16.30/2.42	Proof:
16.30/2.42	  mult(mult(X, rd(Z, Z)), rd(mult(Y, rd(Z, Z)), mult(Y, rd(Z, Z))))
16.30/2.42	= { by lemma 30 R->L }
16.30/2.42	  mult(mult(X, rd(Z, Z)), ld(mult(Y, rd(Z, Z)), mult(Y, rd(Z, Z))))
16.30/2.42	= { by axiom 3 (f04) R->L }
16.30/2.42	  mult(mult(X, rd(Z, Z)), rd(mult(ld(mult(Y, rd(Z, Z)), mult(Y, rd(Z, Z))), rd(Z, Z)), rd(Z, Z)))
16.30/2.42	= { by lemma 17 R->L }
16.30/2.42	  mult(mult(X, rd(Z, Z)), rd(mult(ld(mult(Y, rd(Z, Z)), rd(Y, rd(Z, Z))), rd(Z, Z)), rd(Z, Z)))
16.30/2.42	= { by lemma 9 R->L }
16.30/2.42	  mult(mult(X, rd(Z, Z)), rd(ld(rd(Z, Z), ld(Y, mult(rd(Y, rd(Z, Z)), rd(Z, Z)))), rd(Z, Z)))
16.30/2.42	= { by axiom 2 (f03) }
16.30/2.42	  mult(mult(X, rd(Z, Z)), rd(ld(rd(Z, Z), ld(Y, Y)), rd(Z, Z)))
16.30/2.42	= { by lemma 20 }
16.30/2.42	  rd(mult(X, ld(Y, Y)), rd(Z, Z))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(mult(X, ld(Y, Y)), rd(Z, Z))
16.30/2.42	= { by lemma 30 }
16.30/2.42	  mult(mult(X, rd(Y, Y)), rd(Z, Z))
16.30/2.42	
16.30/2.42	Lemma 32: ld(rd(X, X), mult(Y, rd(X, X))) = mult(mult(rd(X, X), Y), rd(X, X)).
16.30/2.42	Proof:
16.30/2.42	  ld(rd(X, X), mult(Y, rd(X, X)))
16.30/2.42	= { by axiom 3 (f04) R->L }
16.30/2.42	  rd(mult(ld(rd(X, X), mult(Y, rd(X, X))), rd(X, X)), rd(X, X))
16.30/2.42	= { by lemma 22 }
16.30/2.42	  rd(mult(rd(X, X), Y), rd(X, X))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(mult(rd(X, X), Y), rd(X, X))
16.30/2.42	
16.30/2.42	Lemma 33: mult(rd(X, X), mult(rd(X, X), Y)) = ld(rd(X, X), Y).
16.30/2.42	Proof:
16.30/2.42	  mult(rd(X, X), mult(rd(X, X), Y))
16.30/2.42	= { by axiom 2 (f03) R->L }
16.30/2.42	  mult(rd(X, X), mult(rd(X, X), mult(rd(Y, rd(X, X)), rd(X, X))))
16.30/2.42	= { by lemma 23 R->L }
16.30/2.42	  mult(rd(X, X), mult(ld(rd(X, X), rd(Y, rd(X, X))), rd(X, X)))
16.30/2.42	= { by lemma 27 R->L }
16.30/2.42	  ld(rd(rd(X, X), rd(X, X)), mult(rd(Y, rd(X, X)), rd(X, X)))
16.30/2.42	= { by axiom 2 (f03) }
16.30/2.42	  ld(rd(rd(X, X), rd(X, X)), Y)
16.30/2.42	= { by lemma 17 }
16.30/2.42	  ld(mult(rd(X, X), rd(X, X)), Y)
16.30/2.42	= { by lemma 26 }
16.30/2.42	  ld(rd(X, X), Y)
16.30/2.42	
16.30/2.42	Lemma 34: mult(mult(X, Y), ld(Y, X)) = mult(Y, mult(ld(Y, X), X)).
16.30/2.42	Proof:
16.30/2.42	  mult(mult(X, Y), ld(Y, X))
16.30/2.42	= { by axiom 1 (f01) }
16.30/2.42	  mult(mult(mult(Y, ld(Y, X)), Y), ld(Y, X))
16.30/2.42	= { by axiom 5 (f05) }
16.30/2.42	  mult(Y, mult(ld(Y, X), mult(Y, ld(Y, X))))
16.30/2.42	= { by axiom 1 (f01) R->L }
16.30/2.42	  mult(Y, mult(ld(Y, X), X))
16.30/2.42	
16.30/2.42	Lemma 35: mult(rd(X, X), mult(mult(rd(X, X), Y), rd(X, X))) = mult(Y, rd(X, X)).
16.30/2.42	Proof:
16.30/2.42	  mult(rd(X, X), mult(mult(rd(X, X), Y), rd(X, X)))
16.30/2.42	= { by lemma 23 R->L }
16.30/2.42	  mult(ld(rd(X, X), mult(rd(X, X), Y)), rd(X, X))
16.30/2.42	= { by lemma 17 R->L }
16.30/2.42	  rd(ld(rd(X, X), mult(rd(X, X), Y)), rd(X, X))
16.30/2.42	= { by axiom 4 (f02) }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(mult(Z, rd(X, X)), rd(ld(rd(X, X), mult(rd(X, X), Y)), rd(X, X))))
16.30/2.42	= { by lemma 20 }
16.30/2.42	  ld(mult(Z, rd(X, X)), rd(mult(Z, mult(rd(X, X), Y)), rd(X, X)))
16.30/2.42	= { by lemma 19 }
16.30/2.42	  ld(mult(Z, rd(X, X)), mult(mult(Z, rd(X, X)), rd(Y, rd(X, X))))
16.30/2.42	= { by axiom 4 (f02) R->L }
16.30/2.42	  rd(Y, rd(X, X))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(Y, rd(X, X))
16.30/2.42	
16.30/2.42	Lemma 36: rd(Y, Y) = rd(X, X).
16.30/2.42	Proof:
16.30/2.42	  rd(Y, Y)
16.30/2.42	= { by lemma 16 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 26 R->L }
16.30/2.42	  mult(mult(mult(rd(Y, Y), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 31 }
16.30/2.42	  mult(mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 26 R->L }
16.30/2.42	  mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(X, X))), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 32 R->L }
16.30/2.42	  mult(ld(rd(Y, Y), mult(mult(rd(X, X), rd(X, X)), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by axiom 4 (f02) }
16.30/2.42	  mult(ld(rd(Y, Y), mult(mult(rd(X, X), ld(mult(W, rd(Y, Y)), mult(mult(W, rd(Y, Y)), rd(X, X)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 6 R->L }
16.30/2.42	  mult(ld(rd(Y, Y), mult(rd(rd(X, X), rd(Y, Y)), mult(rd(Y, Y), mult(ld(mult(W, rd(Y, Y)), mult(mult(W, rd(Y, Y)), rd(X, X))), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 8 }
16.30/2.42	  mult(ld(rd(Y, Y), mult(rd(rd(X, X), rd(Y, Y)), ld(W, mult(mult(mult(W, rd(Y, Y)), rd(X, X)), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(ld(rd(Y, Y), mult(mult(rd(X, X), rd(Y, Y)), ld(W, mult(mult(mult(W, rd(Y, Y)), rd(X, X)), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 7 }
16.30/2.42	  mult(ld(rd(Y, Y), mult(mult(rd(X, X), rd(Y, Y)), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by axiom 4 (f02) }
16.30/2.42	  mult(ld(rd(Y, Y), mult(ld(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 33 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(rd(Y, Y), mult(ld(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 34 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)), ld(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 33 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 18 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), rd(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), mult(rd(Y, Y), mult(mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(Y, Y)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 35 }
16.30/2.42	  mult(mult(rd(Y, Y), rd(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), mult(mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), rd(Y, Y))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 17 }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), mult(mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), rd(Y, Y))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 32 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), ld(rd(Y, Y), mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 17 R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by axiom 2 (f03) R->L }
16.30/2.42	  mult(mult(rd(Y, Y), mult(mult(mult(rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)), rd(Y, Y)), ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.42	= { by lemma 34 }
16.30/2.43	  mult(mult(rd(Y, Y), mult(mult(rd(Y, Y), mult(ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y))), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)))), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 35 }
16.30/2.43	  mult(mult(mult(ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y))), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y))), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 21 }
16.30/2.43	  mult(mult(rd(ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y))), rd(Y, Y)), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 17 }
16.30/2.43	  mult(mult(mult(ld(rd(Y, Y), rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y))), rd(Y, Y)), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 23 }
16.30/2.43	  mult(mult(mult(rd(Y, Y), mult(rd(mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), rd(Y, Y)), rd(Y, Y))), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by axiom 2 (f03) }
16.30/2.43	  mult(mult(mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 33 }
16.30/2.43	  mult(mult(ld(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), mult(rd(Y, Y), mult(rd(Y, Y), mult(rd(X, X), rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 33 }
16.30/2.43	  mult(mult(ld(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), ld(rd(Y, Y), mult(rd(X, X), rd(Y, Y)))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 32 }
16.30/2.43	  mult(mult(ld(rd(Y, Y), mult(rd(X, X), rd(Y, Y))), mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 32 }
16.30/2.43	  mult(mult(mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y)), mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 31 R->L }
16.30/2.43	  mult(mult(mult(mult(rd(Y, Y), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 26 }
16.30/2.43	  mult(mult(mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), mult(mult(rd(Y, Y), rd(X, X)), rd(Y, Y))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 31 R->L }
16.30/2.43	  mult(mult(mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), mult(mult(rd(Y, Y), rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by lemma 26 }
16.30/2.43	  mult(mult(mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))
16.30/2.43	= { by axiom 5 (f05) }
16.30/2.43	  mult(rd(Y, Y), mult(rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))), mult(mult(rd(Y, Y), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))))))
16.30/2.43	= { by lemma 16 }
16.30/2.43	  mult(rd(Y, Y), mult(rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y))), rd(Y, Y)))
16.30/2.43	= { by lemma 7 R->L }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(Y, Y)))
16.30/2.43	= { by lemma 30 R->L }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), ld(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(Y, Y)))
16.30/2.43	= { by lemma 17 R->L }
16.30/2.43	  ld(Z, rd(mult(mult(Z, rd(Y, Y)), ld(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(Y, Y)))
16.30/2.43	= { by lemma 20 R->L }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), rd(ld(rd(Y, Y), ld(mult(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(Y, Y))))
16.30/2.43	= { by lemma 17 R->L }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), rd(ld(rd(Y, Y), ld(rd(X, rd(Y, Y)), mult(X, rd(Y, Y)))), rd(Y, Y))))
16.30/2.43	= { by lemma 27 }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), rd(ld(rd(Y, Y), mult(rd(Y, Y), mult(ld(X, X), rd(Y, Y)))), rd(Y, Y))))
16.30/2.43	= { by axiom 4 (f02) R->L }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), rd(mult(ld(X, X), rd(Y, Y)), rd(Y, Y))))
16.30/2.43	= { by axiom 3 (f04) }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), ld(X, X)))
16.30/2.43	= { by lemma 30 }
16.30/2.43	  ld(Z, mult(mult(mult(Z, rd(Y, Y)), rd(Y, Y)), rd(X, X)))
16.30/2.43	= { by lemma 16 }
16.30/2.43	  ld(Z, mult(Z, rd(X, X)))
16.30/2.43	= { by axiom 4 (f02) R->L }
16.30/2.43	  rd(X, X)
16.30/2.43	
16.30/2.43	Lemma 37: mult(X, rd(Y, Y)) = X.
16.30/2.43	Proof:
16.30/2.43	  mult(X, rd(Y, Y))
16.30/2.43	= { by lemma 36 }
16.30/2.43	  mult(X, rd(X, X))
16.30/2.43	= { by lemma 30 R->L }
16.30/2.43	  mult(X, ld(X, X))
16.30/2.43	= { by axiom 1 (f01) R->L }
16.30/2.43	  X
16.30/2.43	
16.30/2.43	Lemma 38: rd(mult(X, rd(Y, Y)), mult(Z, rd(W, W))) = mult(rd(X, mult(rd(V, V), Z)), rd(U, U)).
16.30/2.43	Proof:
16.30/2.43	  rd(mult(X, rd(Y, Y)), mult(Z, rd(W, W)))
16.30/2.43	= { by lemma 36 }
16.30/2.43	  rd(mult(X, rd(V, V)), mult(Z, rd(W, W)))
16.30/2.43	= { by lemma 36 }
16.30/2.43	  rd(mult(X, rd(V, V)), mult(Z, rd(V, V)))
16.30/2.43	= { by axiom 2 (f03) R->L }
16.30/2.43	  rd(mult(mult(rd(X, mult(rd(V, V), Z)), mult(rd(V, V), Z)), rd(V, V)), mult(Z, rd(V, V)))
16.30/2.43	= { by lemma 17 R->L }
16.30/2.43	  rd(mult(mult(rd(X, mult(rd(V, V), Z)), mult(rd(V, V), Z)), rd(V, V)), rd(Z, rd(V, V)))
16.30/2.43	= { by lemma 17 R->L }
16.30/2.43	  rd(rd(mult(rd(X, mult(rd(V, V), Z)), mult(rd(V, V), Z)), rd(V, V)), rd(Z, rd(V, V)))
16.30/2.43	= { by lemma 19 }
16.30/2.43	  rd(mult(mult(rd(X, mult(rd(V, V), Z)), rd(V, V)), rd(Z, rd(V, V))), rd(Z, rd(V, V)))
16.30/2.43	= { by axiom 3 (f04) }
16.30/2.43	  mult(rd(X, mult(rd(V, V), Z)), rd(V, V))
16.30/2.43	= { by lemma 36 R->L }
16.30/2.43	  mult(rd(X, mult(rd(V, V), Z)), rd(U, U))
16.30/2.43	
16.30/2.43	Goal 1 (goals): tuple(x0, x0_2) = tuple(mult(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2))).
16.30/2.43	Proof:
16.30/2.43	  tuple(x0, x0_2)
16.30/2.43	= { by lemma 37 R->L }
16.30/2.43	  tuple(x0, mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 4 (f02) }
16.30/2.43	  tuple(ld(rd(x1, x1), mult(rd(x1, x1), x0)), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 33 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(rd(x1, x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 4 (f02) }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(X, mult(X, rd(x1, x1))), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 30 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(X, mult(X, ld(x1, x1))), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 4 (f02) R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(x1, x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 37 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(mult(x1, rd(Y, Y)), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 3 (f04) R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), rd(x0, mult(rd(x1, x1), x0))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 16 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(mult(rd(x0, mult(rd(x1, x1), x0)), rd(Y, Y)), rd(Y, Y))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 38 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(rd(mult(x0, rd(Z, Z)), mult(x0, rd(Z, Z))), rd(Y, Y))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 26 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(mult(rd(mult(x0, rd(Z, Z)), mult(x0, rd(Z, Z))), rd(mult(x0, rd(Z, Z)), mult(x0, rd(Z, Z)))), rd(Y, Y))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 38 }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(mult(mult(rd(x0, mult(rd(x1, x1), x0)), rd(Y, Y)), rd(mult(x0, rd(Z, Z)), mult(x0, rd(Z, Z)))), rd(Y, Y))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 38 }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(mult(mult(rd(x0, mult(rd(x1, x1), x0)), rd(Y, Y)), mult(rd(x0, mult(rd(x1, x1), x0)), rd(Y, Y))), rd(Y, Y))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 5 (f05) }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(rd(x0, mult(rd(x1, x1), x0)), mult(rd(Y, Y), mult(mult(rd(x0, mult(rd(x1, x1), x0)), rd(Y, Y)), rd(Y, Y))))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 16 }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), mult(rd(x0, mult(rd(x1, x1), x0)), mult(rd(Y, Y), rd(x0, mult(rd(x1, x1), x0))))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 11 R->L }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))), ld(rd(x1, rd(x0, mult(rd(x1, x1), x0))), mult(mult(x1, rd(Y, Y)), rd(x0, mult(rd(x1, x1), x0))))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 13 }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(ld(rd(x1, rd(x0, mult(rd(x1, x1), x0))), x1), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by lemma 24 }
16.30/2.43	  tuple(mult(rd(x1, x1), mult(rd(x0, mult(rd(x1, x1), x0)), mult(rd(x1, x1), x0))), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	= { by axiom 2 (f03) }
16.30/2.43	  tuple(mult(rd(x1, x1), x0), mult(x0_2, rd(x1_2, x1_2)))
16.30/2.43	% SZS output end Proof
16.30/2.43	
16.30/2.43	RESULT: Theorem (the conjecture is true).
16.30/2.44	EOF