0.00/0.04	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.25	% Computer   : n010.star.cs.uiowa.edu
0.03/0.25	% Model      : x86_64 x86_64
0.03/0.25	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.25	% Memory     : 32218.625MB
0.03/0.25	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.25	% CPULimit   : 300
0.03/0.25	% DateTime   : Sat Jul 14 04:18:39 CDT 2018
0.03/0.25	% CPUTime    : 
6.45/6.68	% SZS status Theorem
6.45/6.68	
6.45/6.72	% SZS output start Proof
6.45/6.72	Take the following subset of the input axioms:
6.54/6.77	  fof(goals, conjecture, ![X0]: bigC(a, b, X0)=bigC(c, c, X0)).
6.54/6.77	  fof(sos01, axiom, ![B, A]: B=difference(A, product(A, B))).
6.54/6.77	  fof(sos02, axiom, ![B, A]: B=product(A, difference(A, B))).
6.54/6.77	  fof(sos03, axiom, ![B, A]: quotient(product(A, B), B)=A).
6.54/6.77	  fof(sos04, axiom, ![B, A]: product(quotient(A, B), B)=A).
6.54/6.77	  fof(sos05, axiom,
6.54/6.77	      ![B, A, C, D]:
6.54/6.77	        product(product(A, C), product(B, D))=product(product(A, B),
6.54/6.77	                                                      product(C, D))).
6.54/6.77	  fof(sos06, axiom, ![A]: A=product(A, A)).
6.54/6.77	  fof(sos07, axiom,
6.54/6.77	      ![B, A]:
6.54/6.77	        B=product(product(product(A, B), B), product(B, product(B, A)))).
6.54/6.77	  fof(sos08, axiom,
6.54/6.77	      ![B, A, C]: product(product(A, B), product(C, A))=bigC(A, B, C)).
6.54/6.77	  fof(sos09, axiom,
6.54/6.77	      product(product(a, c), product(c, b))=product(a, b)).
6.54/6.77	
6.54/6.77	Now clausify the problem and encode Horn clauses using encoding 3 of
6.54/6.77	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
6.54/6.77	We repeatedly replace C & s=t => u=v by the two clauses:
6.54/6.77	  $$fresh(y, y, x1...xn) = u
6.54/6.77	  C => $$fresh(s, t, x1...xn) = v
6.54/6.77	where $$fresh is a fresh function symbol and x1..xn are the free
6.54/6.77	variables of u and v.
6.54/6.77	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
6.54/6.77	input problem has no model of domain size 1).
6.54/6.77	
6.54/6.77	The encoding turns the above axioms into the following unit equations and goals:
6.54/6.77	
6.54/6.77	Axiom 1 (sos01): X = difference(Y, product(Y, X)).
6.54/6.77	Axiom 2 (sos05): product(product(X, Y), product(Z, W)) = product(product(X, Z), product(Y, W)).
6.54/6.77	Axiom 3 (sos08): product(product(X, Y), product(Z, X)) = bigC(X, Y, Z).
6.54/6.77	Axiom 4 (sos09): product(product(a, c), product(c, b)) = product(a, b).
6.54/6.77	Axiom 5 (sos03): quotient(product(X, Y), Y) = X.
6.54/6.77	Axiom 6 (sos06): X = product(X, X).
6.54/6.77	Axiom 7 (sos07): X = product(product(product(Y, X), X), product(X, product(X, Y))).
6.54/6.77	Axiom 8 (sos02): X = product(Y, difference(Y, X)).
6.54/6.77	Axiom 9 (sos04): product(quotient(X, Y), Y) = X.
6.54/6.77	
6.54/6.77	Lemma 10: bigC(X, X, X) = X.
6.54/6.77	Proof:
6.54/6.77	  bigC(X, X, X)
6.54/6.77	= { by axiom 3 (sos08) }
6.54/6.77	  product(product(X, X), product(X, X))
6.54/6.77	= { by axiom 6 (sos06) }
6.54/6.77	  product(X, X)
6.54/6.77	= { by axiom 6 (sos06) }
6.54/6.77	  X
6.54/6.77	
6.54/6.77	Lemma 11: bigC(X, Y, Z) = bigC(X, Z, Y).
6.54/6.77	Proof:
6.54/6.77	  bigC(X, Y, Z)
6.54/6.77	= { by axiom 3 (sos08) }
6.54/6.77	  product(product(X, Y), product(Z, X))
6.54/6.77	= { by axiom 2 (sos05) }
6.54/6.77	  product(product(X, Z), product(Y, X))
6.54/6.77	= { by axiom 3 (sos08) }
6.54/6.77	  bigC(X, Z, Y)
6.54/6.77	
6.54/6.77	Lemma 12: product(X, product(Y, X)) = bigC(X, X, Y).
6.54/6.77	Proof:
6.54/6.77	  product(X, product(Y, X))
6.54/6.77	= { by axiom 6 (sos06) }
6.54/6.77	  product(product(X, X), product(Y, X))
6.54/6.77	= { by axiom 3 (sos08) }
6.54/6.77	  bigC(X, X, Y)
6.54/6.77	
6.54/6.77	Lemma 13: product(product(X, Y), X) = bigC(X, Y, X).
6.54/6.77	Proof:
6.54/6.77	  product(product(X, Y), X)
6.54/6.77	= { by axiom 6 (sos06) }
6.54/6.77	  product(product(X, Y), product(X, X))
6.54/6.77	= { by axiom 3 (sos08) }
6.54/6.78	  bigC(X, Y, X)
6.54/6.78	
6.54/6.78	Lemma 14: bigC(X, Z, difference(X, Y)) = product(Y, product(Z, X)).
6.54/6.78	Proof:
6.54/6.78	  bigC(X, Z, difference(X, Y))
6.54/6.78	= { by lemma 11 }
6.54/6.78	  bigC(X, difference(X, Y), Z)
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  product(product(X, difference(X, Y)), product(Z, X))
6.54/6.78	= { by axiom 8 (sos02) }
6.54/6.78	  product(Y, product(Z, X))
6.54/6.78	
6.54/6.78	Lemma 15: bigC(X, Y, quotient(Z, X)) = product(product(X, Y), Z).
6.54/6.78	Proof:
6.54/6.78	  bigC(X, Y, quotient(Z, X))
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  product(product(X, Y), product(quotient(Z, X), X))
6.54/6.78	= { by axiom 9 (sos04) }
6.54/6.78	  product(product(X, Y), Z)
6.54/6.78	
6.54/6.78	Lemma 16: quotient(product(X, Y), X) = product(X, quotient(Y, X)).
6.54/6.78	Proof:
6.54/6.78	  quotient(product(X, Y), X)
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  quotient(product(product(X, X), Y), X)
6.54/6.78	= { by lemma 15 }
6.54/6.78	  quotient(bigC(X, X, quotient(Y, X)), X)
6.54/6.78	= { by lemma 11 }
6.54/6.78	  quotient(bigC(X, quotient(Y, X), X), X)
6.54/6.78	= { by lemma 13 }
6.54/6.78	  quotient(product(product(X, quotient(Y, X)), X), X)
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(X, quotient(Y, X))
6.54/6.78	
6.54/6.78	Lemma 17: bigC(Z, difference(Z, X), Y) = product(X, product(Y, Z)).
6.54/6.78	Proof:
6.54/6.78	  bigC(Z, difference(Z, X), Y)
6.54/6.78	= { by lemma 11 }
6.54/6.78	  bigC(Z, Y, difference(Z, X))
6.54/6.78	= { by lemma 14 }
6.54/6.78	  product(X, product(Y, Z))
6.54/6.78	
6.54/6.78	Lemma 18: difference(X, product(Y, X)) = product(difference(X, Y), X).
6.54/6.78	Proof:
6.54/6.78	  difference(X, product(Y, X))
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  difference(X, product(Y, product(X, X)))
6.54/6.78	= { by lemma 17 }
6.54/6.78	  difference(X, bigC(X, difference(X, Y), X))
6.54/6.78	= { by lemma 11 }
6.54/6.78	  difference(X, bigC(X, X, difference(X, Y)))
6.54/6.78	= { by lemma 12 }
6.54/6.78	  difference(X, product(X, product(difference(X, Y), X)))
6.54/6.78	= { by axiom 1 (sos01) }
6.54/6.78	  product(difference(X, Y), X)
6.54/6.78	
6.54/6.78	Lemma 19: product(product(X, Y), product(X, Z)) = product(X, product(Y, Z)).
6.54/6.78	Proof:
6.54/6.78	  product(product(X, Y), product(X, Z))
6.54/6.78	= { by axiom 2 (sos05) }
6.54/6.78	  product(product(X, X), product(Y, Z))
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  product(X, product(Y, Z))
6.54/6.78	
6.54/6.78	Lemma 20: product(product(X, Y), product(Z, Y)) = product(product(X, Z), Y).
6.54/6.78	Proof:
6.54/6.78	  product(product(X, Y), product(Z, Y))
6.54/6.78	= { by axiom 2 (sos05) }
6.54/6.78	  product(product(X, Z), product(Y, Y))
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  product(product(X, Z), Y)
6.54/6.78	
6.54/6.78	Lemma 21: quotient(Y, product(Y, product(Y, X))) = product(product(X, Y), Y).
6.54/6.78	Proof:
6.54/6.78	  quotient(Y, product(Y, product(Y, X)))
6.54/6.78	= { by axiom 7 (sos07) }
6.54/6.78	  quotient(product(product(product(X, Y), Y), product(Y, product(Y, X))), product(Y, product(Y, X)))
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(product(X, Y), Y)
6.54/6.78	
6.54/6.78	Lemma 22: quotient(product(a, b), product(c, b)) = product(a, c).
6.54/6.78	Proof:
6.54/6.78	  quotient(product(a, b), product(c, b))
6.54/6.78	= { by axiom 4 (sos09) }
6.54/6.78	  quotient(product(product(a, c), product(c, b)), product(c, b))
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(a, c)
6.54/6.78	
6.54/6.78	Lemma 23: product(product(quotient(X, Y), Z), Y) = product(X, product(Z, Y)).
6.54/6.78	Proof:
6.54/6.78	  product(product(quotient(X, Y), Z), Y)
6.54/6.78	= { by lemma 20 }
6.54/6.78	  product(product(quotient(X, Y), Y), product(Z, Y))
6.54/6.78	= { by axiom 9 (sos04) }
6.54/6.78	  product(X, product(Z, Y))
6.54/6.78	
6.54/6.78	Lemma 24: product(product(difference(X, Y), X), X) = quotient(X, product(X, Y)).
6.54/6.78	Proof:
6.54/6.78	  product(product(difference(X, Y), X), X)
6.54/6.78	= { by lemma 21 }
6.54/6.78	  quotient(X, product(X, product(X, difference(X, Y))))
6.54/6.78	= { by axiom 8 (sos02) }
6.54/6.78	  quotient(X, product(X, Y))
6.54/6.78	
6.54/6.78	Lemma 25: difference(X, difference(product(Y, X), X)) = product(X, quotient(Y, X)).
6.54/6.78	Proof:
6.54/6.78	  difference(X, difference(product(Y, X), X))
6.54/6.78	= { by axiom 9 (sos04) }
6.54/6.78	  difference(X, difference(product(product(quotient(Y, X), X), X), X))
6.54/6.78	= { by axiom 7 (sos07) }
6.54/6.78	  difference(X, difference(product(product(quotient(Y, X), X), X), product(product(product(quotient(Y, X), X), X), product(X, product(X, quotient(Y, X))))))
6.54/6.78	= { by axiom 1 (sos01) }
6.54/6.78	  difference(X, product(X, product(X, quotient(Y, X))))
6.54/6.78	= { by axiom 1 (sos01) }
6.54/6.78	  product(X, quotient(Y, X))
6.54/6.78	
6.54/6.78	Lemma 26: quotient(X, product(X, Y)) = product(X, quotient(X, Y)).
6.54/6.78	Proof:
6.54/6.78	  quotient(X, product(X, Y))
6.54/6.78	= { by lemma 24 }
6.54/6.78	  product(product(difference(X, Y), X), X)
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  product(product(difference(X, Y), product(X, X)), X)
6.54/6.78	= { by lemma 14 }
6.54/6.78	  product(bigC(X, X, difference(X, difference(X, Y))), X)
6.54/6.78	= { by lemma 12 }
6.54/6.78	  product(product(X, product(difference(X, difference(X, Y)), X)), X)
6.54/6.78	= { by lemma 13 }
6.54/6.78	  bigC(X, product(difference(X, difference(X, Y)), X), X)
6.54/6.78	= { by lemma 11 }
6.54/6.78	  bigC(X, X, product(difference(X, difference(X, Y)), X))
6.54/6.78	= { by lemma 12 }
6.54/6.78	  product(X, product(product(difference(X, difference(X, Y)), X), X))
6.54/6.78	= { by lemma 24 }
6.54/6.78	  product(X, quotient(X, product(X, difference(X, Y))))
6.54/6.78	= { by axiom 8 (sos02) }
6.54/6.78	  product(X, quotient(X, Y))
6.54/6.78	
6.54/6.78	Lemma 27: product(product(X, quotient(Y, Z)), product(W, Z)) = product(product(X, W), Y).
6.54/6.78	Proof:
6.54/6.78	  product(product(X, quotient(Y, Z)), product(W, Z))
6.54/6.78	= { by axiom 2 (sos05) }
6.54/6.78	  product(product(X, W), product(quotient(Y, Z), Z))
6.54/6.78	= { by axiom 9 (sos04) }
6.54/6.78	  product(product(X, W), Y)
6.54/6.78	
6.54/6.78	Lemma 28: product(product(X, Y), bigC(X, X, Z)) = bigC(X, Y, product(X, Z)).
6.54/6.78	Proof:
6.54/6.78	  product(product(X, Y), bigC(X, X, Z))
6.54/6.78	= { by lemma 11 }
6.54/6.78	  product(product(X, Y), bigC(X, Z, X))
6.54/6.78	= { by lemma 13 }
6.54/6.78	  product(product(X, Y), product(product(X, Z), X))
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  bigC(X, Y, product(X, Z))
6.54/6.78	
6.54/6.78	Lemma 29: quotient(product(product(X, Z), Y), product(Z, Y)) = product(X, Y).
6.54/6.78	Proof:
6.54/6.78	  quotient(product(product(X, Z), Y), product(Z, Y))
6.54/6.78	= { by lemma 20 }
6.54/6.78	  quotient(product(product(X, Y), product(Z, Y)), product(Z, Y))
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(X, Y)
6.54/6.78	
6.54/6.78	Lemma 30: product(X, quotient(quotient(X, Y), X)) = product(difference(X, Y), X).
6.54/6.78	Proof:
6.54/6.78	  product(X, quotient(quotient(X, Y), X))
6.54/6.78	= { by lemma 16 }
6.54/6.78	  quotient(product(X, quotient(X, Y)), X)
6.54/6.78	= { by lemma 26 }
6.54/6.78	  quotient(quotient(X, product(X, Y)), X)
6.54/6.78	= { by lemma 24 }
6.54/6.78	  quotient(product(product(difference(X, Y), X), X), X)
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  quotient(product(product(difference(X, Y), X), X), product(X, X))
6.54/6.78	= { by lemma 29 }
6.54/6.78	  product(difference(X, Y), X)
6.54/6.78	
6.54/6.78	Lemma 31: bigC(X, Y, Z) = bigC(Z, X, Y).
6.54/6.78	Proof:
6.54/6.78	  bigC(X, Y, Z)
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  product(product(X, Y), product(Z, X))
6.54/6.78	= { by lemma 20 }
6.54/6.78	  product(product(X, product(Z, X)), product(Y, product(Z, X)))
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  product(product(X, product(product(Z, X), product(Z, X))), product(Y, product(Z, X)))
6.54/6.78	= { by lemma 14 }
6.54/6.78	  product(bigC(product(Z, X), product(Z, X), difference(product(Z, X), X)), product(Y, product(Z, X)))
6.54/6.78	= { by lemma 12 }
6.54/6.78	  product(product(product(Z, X), product(difference(product(Z, X), X), product(Z, X))), product(Y, product(Z, X)))
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  bigC(product(Z, X), product(difference(product(Z, X), X), product(Z, X)), Y)
6.54/6.78	= { by lemma 11 }
6.54/6.78	  bigC(product(Z, X), Y, product(difference(product(Z, X), X), product(Z, X)))
6.54/6.78	= { by lemma 30 }
6.54/6.78	  bigC(product(Z, X), Y, product(product(Z, X), quotient(quotient(product(Z, X), X), product(Z, X))))
6.54/6.78	= { by lemma 28 }
6.54/6.78	  product(product(product(Z, X), Y), bigC(product(Z, X), product(Z, X), quotient(quotient(product(Z, X), X), product(Z, X))))
6.54/6.78	= { by lemma 15 }
6.54/6.78	  product(product(product(Z, X), Y), product(product(product(Z, X), product(Z, X)), quotient(product(Z, X), X)))
6.54/6.78	= { by axiom 6 (sos06) }
6.54/6.78	  product(product(product(Z, X), Y), product(product(Z, X), quotient(product(Z, X), X)))
6.54/6.78	= { by lemma 19 }
6.54/6.78	  product(product(Z, X), product(Y, quotient(product(Z, X), X)))
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(product(Z, X), product(Y, Z))
6.54/6.78	= { by axiom 3 (sos08) }
6.54/6.78	  bigC(Z, X, Y)
6.54/6.78	
6.54/6.78	Lemma 32: bigC(X, Y, Z) = bigC(Y, X, Z).
6.54/6.78	Proof:
6.54/6.78	  bigC(X, Y, Z)
6.54/6.78	= { by lemma 31 }
6.54/6.78	  bigC(Y, Z, X)
6.54/6.78	= { by lemma 11 }
6.54/6.78	  bigC(Y, X, Z)
6.54/6.78	
6.54/6.78	Lemma 33: quotient(product(product(X, W), Y), product(W, Z)) = product(X, quotient(Y, Z)).
6.54/6.78	Proof:
6.54/6.78	  quotient(product(product(X, W), Y), product(W, Z))
6.54/6.78	= { by lemma 27 }
6.54/6.78	  quotient(product(product(X, quotient(Y, Z)), product(W, Z)), product(W, Z))
6.54/6.78	= { by axiom 5 (sos03) }
6.54/6.78	  product(X, quotient(Y, Z))
6.54/6.78	
6.54/6.78	Lemma 34: quotient(product(X, Z), product(Y, W)) = product(quotient(X, Y), quotient(Z, W)).
6.54/6.78	Proof:
6.54/6.78	  quotient(product(X, Z), product(Y, W))
6.54/6.78	= { by axiom 9 (sos04) }
6.54/6.78	  quotient(product(product(quotient(X, Y), Y), Z), product(Y, W))
6.54/6.78	= { by lemma 33 }
6.59/6.82	  product(quotient(X, Y), quotient(Z, W))
6.59/6.82	
6.59/6.82	Lemma 35: product(quotient(c, a), c) = product(c, b).
6.59/6.82	Proof:
6.59/6.82	  product(quotient(c, a), c)
6.59/6.82	= { by lemma 29 }
6.59/6.82	  quotient(product(product(quotient(c, a), a), c), product(a, c))
6.59/6.82	= { by axiom 9 (sos04) }
6.59/6.82	  quotient(product(c, c), product(a, c))
6.59/6.82	= { by lemma 10 }
6.59/6.82	  quotient(product(c, bigC(c, c, c)), product(a, c))
6.59/6.82	= { by lemma 13 }
6.59/6.82	  quotient(product(c, product(product(c, c), c)), product(a, c))
6.59/6.82	= { by lemma 12 }
6.59/6.82	  quotient(bigC(c, c, product(c, c)), product(a, c))
6.59/6.82	= { by lemma 31 }
6.59/6.82	  quotient(bigC(c, product(c, c), c), product(a, c))
6.59/6.82	= { by lemma 13 }
6.59/6.82	  quotient(product(product(c, product(c, c)), c), product(a, c))
6.59/6.82	= { by axiom 8 (sos02) }
6.59/6.82	  quotient(product(product(c, product(c, c)), c), product(product(c, c), difference(product(c, c), product(a, c))))
6.59/6.82	= { by lemma 33 }
6.59/6.82	  product(c, quotient(c, difference(product(c, c), product(a, c))))
6.59/6.82	= { by axiom 5 (sos03) }
6.59/6.82	  quotient(product(product(c, quotient(c, difference(product(c, c), product(a, c)))), difference(product(c, c), product(a, c))), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 20 }
6.59/6.82	  quotient(product(product(c, difference(product(c, c), product(a, c))), product(quotient(c, difference(product(c, c), product(a, c))), difference(product(c, c), product(a, c)))), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 9 (sos04) }
6.59/6.82	  quotient(product(product(c, difference(product(c, c), product(a, c))), c), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 13 }
6.59/6.82	  quotient(bigC(c, difference(product(c, c), product(a, c)), c), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 11 }
6.59/6.82	  quotient(bigC(c, c, difference(product(c, c), product(a, c))), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 3 (sos08) }
6.59/6.82	  quotient(product(product(c, c), product(difference(product(c, c), product(a, c)), c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 19 }
6.59/6.82	  quotient(product(product(product(c, c), difference(product(c, c), product(a, c))), product(product(c, c), c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 8 (sos02) }
6.59/6.82	  quotient(product(product(a, c), product(product(c, c), c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 13 }
6.59/6.82	  quotient(product(product(a, c), bigC(c, c, c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 10 }
6.59/6.82	  quotient(product(product(a, c), c), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 27 }
6.59/6.82	  quotient(product(product(a, quotient(c, b)), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 5 (sos03) }
6.59/6.82	  quotient(product(quotient(product(product(a, quotient(c, b)), product(a, b)), product(a, b)), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 19 }
6.59/6.82	  quotient(product(quotient(product(a, product(quotient(c, b), b)), product(a, b)), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 9 (sos04) }
6.59/6.82	  quotient(product(quotient(product(a, c), product(a, b)), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 22 }
6.59/6.82	  quotient(product(quotient(quotient(product(a, b), product(c, b)), product(a, b)), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 1 (sos01) }
6.59/6.82	  quotient(product(difference(quotient(product(a, b), product(c, b)), product(quotient(product(a, b), product(c, b)), quotient(quotient(product(a, b), product(c, b)), product(a, b)))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 26 }
6.59/6.82	  quotient(product(difference(quotient(product(a, b), product(c, b)), quotient(quotient(product(a, b), product(c, b)), product(quotient(product(a, b), product(c, b)), product(a, b)))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 9 (sos04) }
6.59/6.82	  quotient(product(difference(quotient(product(a, b), product(c, b)), quotient(quotient(product(a, b), product(c, b)), product(quotient(product(a, b), product(c, b)), product(quotient(product(a, b), product(c, b)), product(c, b))))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 21 }
6.59/6.82	  quotient(product(difference(quotient(product(a, b), product(c, b)), product(product(product(c, b), quotient(product(a, b), product(c, b))), quotient(product(a, b), product(c, b)))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 18 }
6.59/6.82	  quotient(product(product(difference(quotient(product(a, b), product(c, b)), product(product(c, b), quotient(product(a, b), product(c, b)))), quotient(product(a, b), product(c, b))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 18 }
6.59/6.82	  quotient(product(product(product(difference(quotient(product(a, b), product(c, b)), product(c, b)), quotient(product(a, b), product(c, b))), quotient(product(a, b), product(c, b))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 24 }
6.59/6.82	  quotient(product(quotient(quotient(product(a, b), product(c, b)), product(quotient(product(a, b), product(c, b)), product(c, b))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 26 }
6.59/6.82	  quotient(product(product(quotient(product(a, b), product(c, b)), quotient(quotient(product(a, b), product(c, b)), product(c, b))), product(c, b)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 23 }
6.59/6.82	  quotient(product(product(a, b), product(quotient(quotient(product(a, b), product(c, b)), product(c, b)), product(c, b))), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 9 (sos04) }
6.59/6.82	  quotient(product(product(a, b), quotient(product(a, b), product(c, b))), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 22 }
6.59/6.82	  quotient(product(product(a, b), product(a, c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by lemma 19 }
6.59/6.82	  quotient(product(a, product(b, c)), difference(product(c, c), product(a, c)))
6.59/6.82	= { by axiom 6 (sos06) }
6.59/6.82	  quotient(product(a, product(b, c)), difference(c, product(a, c)))
6.59/6.82	= { by lemma 18 }
6.59/6.82	  quotient(product(a, product(b, c)), product(difference(c, a), c))
6.59/6.82	= { by lemma 34 }
6.59/6.82	  product(quotient(a, difference(c, a)), quotient(product(b, c), c))
6.59/6.82	= { by axiom 8 (sos02) }
6.59/6.82	  product(quotient(product(c, difference(c, a)), difference(c, a)), quotient(product(b, c), c))
6.59/6.82	= { by axiom 5 (sos03) }
6.59/6.82	  product(c, quotient(product(b, c), c))
6.59/6.82	= { by axiom 5 (sos03) }
6.66/6.87	  product(c, b)
6.66/6.87	
6.66/6.87	Goal 1 (goals): bigC(a, b, sK1_goals_X0) = bigC(c, c, sK1_goals_X0).
6.66/6.87	Proof:
6.66/6.87	  bigC(a, b, sK1_goals_X0)
6.66/6.87	= { by lemma 11 }
6.66/6.87	  bigC(a, sK1_goals_X0, b)
6.66/6.87	= { by lemma 32 }
6.66/6.87	  bigC(sK1_goals_X0, a, b)
6.66/6.87	= { by lemma 31 }
6.66/6.87	  bigC(a, b, sK1_goals_X0)
6.66/6.87	= { by lemma 31 }
6.66/6.87	  bigC(b, sK1_goals_X0, a)
6.66/6.87	= { by axiom 3 (sos08) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(a, b))
6.66/6.87	= { by axiom 9 (sos04) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(quotient(product(a, b), c), c))
6.66/6.87	= { by axiom 9 (sos04) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(quotient(product(a, product(quotient(b, c), c)), c), c))
6.66/6.87	= { by lemma 23 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(quotient(product(product(quotient(a, c), quotient(b, c)), c), c), c))
6.66/6.87	= { by axiom 5 (sos03) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(quotient(a, c), quotient(b, c)), c))
6.66/6.87	= { by lemma 34 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(quotient(product(a, b), product(c, c)), c))
6.66/6.87	= { by axiom 4 (sos09) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(quotient(product(product(a, c), product(c, b)), product(c, c)), c))
6.66/6.87	= { by lemma 33 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(a, quotient(product(c, b), c)), c))
6.66/6.87	= { by axiom 6 (sos06) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(product(a, a), quotient(product(c, b), c)), c))
6.66/6.87	= { by lemma 35 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(product(a, a), quotient(product(quotient(c, a), c), c)), c))
6.66/6.87	= { by axiom 5 (sos03) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(product(a, a), quotient(c, a)), c))
6.66/6.87	= { by lemma 15 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(bigC(a, a, quotient(quotient(c, a), a)), c))
6.66/6.87	= { by lemma 12 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(a, product(quotient(quotient(c, a), a), a)), c))
6.66/6.87	= { by axiom 6 (sos06) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(a, product(product(quotient(quotient(c, a), a), a), product(quotient(quotient(c, a), a), a))), c))
6.66/6.87	= { by lemma 19 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(product(a, product(quotient(quotient(c, a), a), a)), product(a, product(quotient(quotient(c, a), a), a))), c))
6.66/6.87	= { by lemma 12 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(product(a, product(quotient(quotient(c, a), a), a)), bigC(a, a, quotient(quotient(c, a), a))), c))
6.66/6.87	= { by lemma 28 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(bigC(a, product(quotient(quotient(c, a), a), a), product(a, quotient(quotient(c, a), a))), c))
6.66/6.87	= { by lemma 11 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(bigC(a, product(a, quotient(quotient(c, a), a)), product(quotient(quotient(c, a), a), a)), c))
6.66/6.87	= { by lemma 25 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(bigC(a, difference(a, difference(product(quotient(c, a), a), a)), product(quotient(quotient(c, a), a), a)), c))
6.66/6.87	= { by axiom 9 (sos04) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(bigC(a, difference(a, difference(c, a)), product(quotient(quotient(c, a), a), a)), c))
6.66/6.87	= { by lemma 17 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(difference(c, a), product(product(quotient(quotient(c, a), a), a), a)), c))
6.66/6.87	= { by axiom 9 (sos04) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(difference(c, a), product(quotient(c, a), a)), c))
6.66/6.87	= { by axiom 9 (sos04) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(difference(c, a), c), c))
6.66/6.87	= { by lemma 30 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(product(c, quotient(quotient(c, a), c)), c))
6.66/6.87	= { by lemma 25 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, difference(product(quotient(c, a), c), c)), c))
6.66/6.87	= { by lemma 35 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, difference(product(c, b), c)), c))
6.66/6.87	= { by axiom 6 (sos06) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, difference(product(c, b), product(c, c))), c))
6.66/6.87	= { by axiom 8 (sos02) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, difference(product(c, b), product(c, product(b, difference(b, c))))), c))
6.66/6.87	= { by lemma 19 }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, difference(product(c, b), product(product(c, b), product(c, difference(b, c))))), c))
6.66/6.87	= { by axiom 1 (sos01) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(c, product(c, difference(b, c))), c))
6.66/6.87	= { by axiom 1 (sos01) }
6.66/6.87	  product(product(b, sK1_goals_X0), product(difference(b, c), c))
6.66/6.87	= { by axiom 2 (sos05) }
6.66/6.87	  product(product(b, difference(b, c)), product(sK1_goals_X0, c))
6.66/6.87	= { by axiom 8 (sos02) }
6.66/6.87	  product(c, product(sK1_goals_X0, c))
6.66/6.87	= { by lemma 12 }
6.66/6.87	  bigC(c, c, sK1_goals_X0)
6.66/6.87	% SZS output end Proof
6.66/6.87	
6.66/6.87	RESULT: Theorem (the conjecture is true).
6.66/6.88	EOF