0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.11/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.14/0.33	% Computer   : n014.cluster.edu
0.14/0.33	% Model      : x86_64 x86_64
0.14/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.33	% Memory     : 8042.1875MB
0.14/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 1200
0.14/0.34	% WCLimit    : 120
0.14/0.34	% DateTime   : Tue Jul 13 15:03:04 EDT 2021
0.14/0.34	% CPUTime    : 
40.46/5.49	% SZS status Theorem
40.46/5.49	
40.84/5.55	% SZS output start Proof
40.84/5.55	Take the following subset of the input axioms:
40.84/5.55	  fof(goals, conjecture, m(bigA, b1, bigC)).
40.84/5.55	  fof(sos01, axiom, ![B, A]: difference(A, product(A, B))=B).
40.84/5.55	  fof(sos02, axiom, ![B, A]: B=product(A, difference(A, B))).
40.84/5.55	  fof(sos03, axiom, ![B, A]: A=quotient(product(A, B), B)).
40.84/5.55	  fof(sos04, axiom, ![B, A]: A=product(quotient(A, B), B)).
40.84/5.55	  fof(sos05, axiom, ![B, A, D, C]: product(product(A, C), product(B, D))=product(product(A, B), product(C, D))).
40.84/5.55	  fof(sos06, axiom, ![A]: product(A, A)=A).
40.84/5.55	  fof(sos07, axiom, ![B, A]: B=product(product(product(A, B), B), product(B, product(B, A)))).
40.84/5.55	  fof(sos08, axiom, ![X0, X1, X2]: (product(X0, X1)=product(X1, X2) <=> d(X0, X1, X2))).
40.84/5.55	  fof(sos09, axiom, ![X3, X4, X5]: (product(X3, X5)=product(product(X3, X4), product(X4, X5)) <=> m(X3, X4, X5))).
40.84/5.55	  fof(sos10, axiom, d(b, bigA, c)).
40.84/5.55	  fof(sos11, axiom, d(bigB, a, c)).
40.84/5.55	  fof(sos12, axiom, d(b, a, bigC)).
40.84/5.55	  fof(sos13, axiom, d(a, b1, c1)).
40.84/5.55	  fof(sos14, axiom, d(a1, b, c1)).
40.84/5.55	  fof(sos15, axiom, d(a1, b1, c)).
40.84/5.55	
40.84/5.55	Now clausify the problem and encode Horn clauses using encoding 3 of
40.84/5.55	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
40.84/5.55	We repeatedly replace C & s=t => u=v by the two clauses:
40.84/5.55	  fresh(y, y, x1...xn) = u
40.84/5.55	  C => fresh(s, t, x1...xn) = v
40.84/5.55	where fresh is a fresh function symbol and x1..xn are the free
40.84/5.55	variables of u and v.
40.84/5.55	A predicate p(X) is encoded as p(X)=true (this is sound, because the
40.84/5.55	input problem has no model of domain size 1).
40.84/5.55	
40.84/5.55	The encoding turns the above axioms into the following unit equations and goals:
40.84/5.55	
40.84/5.55	Axiom 1 (sos06): product(X, X) = X.
40.84/5.55	Axiom 2 (sos11): d(bigB, a, c) = true.
40.84/5.55	Axiom 3 (sos14): d(a1, b, c1) = true.
40.84/5.55	Axiom 4 (sos15): d(a1, b1, c) = true.
40.84/5.55	Axiom 5 (sos10): d(b, bigA, c) = true.
40.84/5.55	Axiom 6 (sos12): d(b, a, bigC) = true.
40.84/5.55	Axiom 7 (sos13): d(a, b1, c1) = true.
40.84/5.55	Axiom 8 (sos01): difference(X, product(X, Y)) = Y.
40.84/5.55	Axiom 9 (sos03): X = quotient(product(X, Y), Y).
40.84/5.55	Axiom 10 (sos02): X = product(Y, difference(Y, X)).
40.84/5.55	Axiom 11 (sos04): X = product(quotient(X, Y), Y).
40.84/5.55	Axiom 12 (sos08_1): fresh4(X, X, Y, Z, W) = product(Z, W).
40.84/5.55	Axiom 13 (sos09): fresh2(X, X, Y, Z, W) = true.
40.84/5.55	Axiom 14 (sos05): product(product(X, Y), product(Z, W)) = product(product(X, Z), product(Y, W)).
40.84/5.55	Axiom 15 (sos08_1): fresh4(d(X, Y, Z), true, X, Y, Z) = product(X, Y).
40.84/5.55	Axiom 16 (sos07): X = product(product(product(Y, X), X), product(X, product(X, Y))).
40.84/5.55	Axiom 17 (sos09): fresh2(product(X, Y), product(product(X, Z), product(Z, Y)), X, Z, Y) = m(X, Z, Y).
40.84/5.55	
40.84/5.55	Lemma 18: d(a1, b, c1) = d(bigB, a, c).
40.84/5.55	Proof:
40.84/5.55	  d(a1, b, c1)
40.84/5.55	= { by axiom 3 (sos14) }
40.84/5.55	  true
40.84/5.55	= { by axiom 2 (sos11) R->L }
40.84/5.55	  d(bigB, a, c)
40.84/5.55	
40.84/5.55	Lemma 19: fresh4(d(X, Y, Z), d(a1, b, c1), X, Y, Z) = product(X, Y).
40.84/5.55	Proof:
40.84/5.55	  fresh4(d(X, Y, Z), d(a1, b, c1), X, Y, Z)
40.84/5.55	= { by lemma 18 }
40.84/5.55	  fresh4(d(X, Y, Z), d(bigB, a, c), X, Y, Z)
40.84/5.55	= { by axiom 2 (sos11) }
40.84/5.55	  fresh4(d(X, Y, Z), true, X, Y, Z)
40.84/5.55	= { by axiom 15 (sos08_1) }
40.84/5.55	  product(X, Y)
40.84/5.55	
40.84/5.55	Lemma 20: product(b, c1) = product(a1, b).
40.84/5.55	Proof:
40.84/5.55	  product(b, c1)
40.84/5.55	= { by axiom 12 (sos08_1) R->L }
40.84/5.55	  fresh4(d(a1, b, c1), d(a1, b, c1), a1, b, c1)
40.84/5.55	= { by lemma 19 }
40.84/5.55	  product(a1, b)
40.84/5.55	
40.84/5.55	Lemma 21: product(product(X, Y), product(X, Z)) = product(X, product(Y, Z)).
40.84/5.55	Proof:
40.84/5.55	  product(product(X, Y), product(X, Z))
40.84/5.55	= { by axiom 14 (sos05) R->L }
40.84/5.55	  product(product(X, X), product(Y, Z))
40.84/5.55	= { by axiom 1 (sos06) }
40.84/5.55	  product(X, product(Y, Z))
40.84/5.55	
40.84/5.55	Lemma 22: product(product(X, Y), product(Z, Y)) = product(product(X, Z), Y).
40.84/5.55	Proof:
40.84/5.55	  product(product(X, Y), product(Z, Y))
40.84/5.55	= { by axiom 14 (sos05) R->L }
40.84/5.55	  product(product(X, Z), product(Y, Y))
40.84/5.55	= { by axiom 1 (sos06) }
40.84/5.55	  product(product(X, Z), Y)
40.84/5.55	
40.84/5.55	Lemma 23: product(product(X, quotient(Y, Z)), Z) = product(product(X, Z), Y).
40.84/5.55	Proof:
40.84/5.55	  product(product(X, quotient(Y, Z)), Z)
40.84/5.55	= { by lemma 22 R->L }
40.84/5.55	  product(product(X, Z), product(quotient(Y, Z), Z))
40.84/5.55	= { by axiom 11 (sos04) R->L }
40.84/5.55	  product(product(X, Z), Y)
40.84/5.55	
40.84/5.55	Lemma 24: quotient(product(product(X, Y), Z), Y) = product(X, quotient(Z, Y)).
40.84/5.55	Proof:
40.84/5.55	  quotient(product(product(X, Y), Z), Y)
40.84/5.55	= { by lemma 23 R->L }
40.84/5.55	  quotient(product(product(X, quotient(Z, Y)), Y), Y)
40.84/5.55	= { by axiom 9 (sos03) R->L }
40.84/5.55	  product(X, quotient(Z, Y))
40.84/5.55	
40.84/5.55	Lemma 25: product(a, bigC) = product(b, a).
40.84/5.55	Proof:
40.84/5.55	  product(a, bigC)
40.84/5.55	= { by axiom 12 (sos08_1) R->L }
40.84/5.55	  fresh4(d(a1, b, c1), d(a1, b, c1), b, a, bigC)
40.84/5.55	= { by lemma 18 }
40.84/5.55	  fresh4(d(bigB, a, c), d(a1, b, c1), b, a, bigC)
40.84/5.55	= { by axiom 2 (sos11) }
40.84/5.55	  fresh4(true, d(a1, b, c1), b, a, bigC)
40.84/5.55	= { by axiom 6 (sos12) R->L }
40.84/5.55	  fresh4(d(b, a, bigC), d(a1, b, c1), b, a, bigC)
40.84/5.55	= { by lemma 19 }
40.84/5.55	  product(b, a)
40.84/5.55	
40.84/5.55	Lemma 26: product(a, c) = product(bigB, a).
40.84/5.55	Proof:
40.84/5.55	  product(a, c)
40.84/5.55	= { by axiom 12 (sos08_1) R->L }
40.84/5.55	  fresh4(d(a1, b, c1), d(a1, b, c1), bigB, a, c)
40.84/5.55	= { by lemma 18 }
40.84/5.55	  fresh4(d(bigB, a, c), d(a1, b, c1), bigB, a, c)
40.84/5.55	= { by lemma 19 }
40.84/5.55	  product(bigB, a)
40.84/5.55	
40.84/5.55	Lemma 27: product(product(a, X), c) = product(bigB, product(X, a)).
40.84/5.55	Proof:
40.84/5.55	  product(product(a, X), c)
40.84/5.55	= { by lemma 22 R->L }
40.84/5.55	  product(product(a, c), product(X, c))
40.84/5.55	= { by lemma 26 }
40.84/5.55	  product(product(bigB, a), product(X, c))
40.84/5.55	= { by axiom 14 (sos05) }
40.84/5.55	  product(product(bigB, X), product(a, c))
40.84/5.55	= { by lemma 26 }
40.84/5.55	  product(product(bigB, X), product(bigB, a))
40.84/5.55	= { by lemma 21 }
40.84/5.55	  product(bigB, product(X, a))
40.84/5.55	
40.84/5.55	Lemma 28: product(product(quotient(X, Y), Z), Y) = product(X, product(Z, Y)).
40.84/5.55	Proof:
40.84/5.55	  product(product(quotient(X, Y), Z), Y)
40.84/5.55	= { by lemma 22 R->L }
40.84/5.55	  product(product(quotient(X, Y), Y), product(Z, Y))
40.84/5.55	= { by axiom 11 (sos04) R->L }
40.84/5.55	  product(X, product(Z, Y))
40.84/5.55	
40.84/5.55	Lemma 29: product(b, c) = product(bigB, bigC).
40.84/5.55	Proof:
40.84/5.55	  product(b, c)
40.84/5.55	= { by axiom 11 (sos04) }
40.84/5.55	  product(b, product(quotient(c, a), a))
40.84/5.55	= { by lemma 21 R->L }
40.84/5.55	  product(product(b, quotient(c, a)), product(b, a))
40.84/5.55	= { by lemma 24 R->L }
40.84/5.55	  product(quotient(product(product(b, a), c), a), product(b, a))
40.84/5.55	= { by lemma 25 R->L }
40.84/5.55	  product(quotient(product(product(a, bigC), c), a), product(b, a))
40.84/5.55	= { by lemma 27 }
40.84/5.55	  product(quotient(product(bigB, product(bigC, a)), a), product(b, a))
40.84/5.55	= { by lemma 28 R->L }
40.84/5.55	  product(quotient(product(product(quotient(bigB, a), bigC), a), a), product(b, a))
40.84/5.55	= { by axiom 9 (sos03) R->L }
40.84/5.55	  product(product(quotient(bigB, a), bigC), product(b, a))
40.84/5.55	= { by lemma 25 R->L }
40.84/5.55	  product(product(quotient(bigB, a), bigC), product(a, bigC))
40.84/5.55	= { by lemma 22 }
40.84/5.55	  product(product(quotient(bigB, a), a), bigC)
40.84/5.55	= { by axiom 11 (sos04) R->L }
40.84/5.55	  product(bigB, bigC)
40.84/5.55	
40.84/5.55	Lemma 30: d(b, bigA, c) = d(a1, b, c1).
40.84/5.55	Proof:
40.84/5.55	  d(b, bigA, c)
40.84/5.55	= { by axiom 5 (sos10) }
40.84/5.55	  true
40.84/5.55	= { by axiom 2 (sos11) R->L }
40.84/5.55	  d(bigB, a, c)
40.84/5.55	= { by lemma 18 R->L }
40.84/5.55	  d(a1, b, c1)
40.84/5.55	
40.84/5.55	Lemma 31: product(b, bigA) = product(bigA, c).
40.84/5.55	Proof:
40.84/5.55	  product(b, bigA)
40.84/5.55	= { by lemma 19 R->L }
40.84/5.55	  fresh4(d(b, bigA, c), d(a1, b, c1), b, bigA, c)
40.84/5.55	= { by lemma 30 }
40.84/5.55	  fresh4(d(a1, b, c1), d(a1, b, c1), b, bigA, c)
40.84/5.55	= { by axiom 12 (sos08_1) }
40.84/5.55	  product(bigA, c)
40.84/5.55	
40.84/5.55	Lemma 32: product(a, b1) = product(b1, c1).
40.84/5.55	Proof:
40.84/5.55	  product(a, b1)
40.84/5.55	= { by lemma 19 R->L }
40.84/5.55	  fresh4(d(a, b1, c1), d(a1, b, c1), a, b1, c1)
40.84/5.55	= { by axiom 7 (sos13) }
40.84/5.55	  fresh4(true, d(a1, b, c1), a, b1, c1)
40.84/5.55	= { by axiom 2 (sos11) R->L }
40.84/5.55	  fresh4(d(bigB, a, c), d(a1, b, c1), a, b1, c1)
40.84/5.55	= { by lemma 18 R->L }
40.84/5.55	  fresh4(d(a1, b, c1), d(a1, b, c1), a, b1, c1)
40.84/5.55	= { by axiom 12 (sos08_1) }
40.84/5.55	  product(b1, c1)
40.84/5.55	
40.84/5.55	Lemma 33: difference(product(product(X, Y), Y), Y) = product(Y, product(Y, X)).
40.84/5.55	Proof:
40.84/5.55	  difference(product(product(X, Y), Y), Y)
40.84/5.55	= { by axiom 16 (sos07) }
40.84/5.55	  difference(product(product(X, Y), Y), product(product(product(X, Y), Y), product(Y, product(Y, X))))
40.84/5.55	= { by axiom 8 (sos01) }
40.84/5.55	  product(Y, product(Y, X))
40.84/5.55	
40.84/5.55	Lemma 34: product(product(X, Y), X) = product(X, product(Y, X)).
40.84/5.55	Proof:
40.84/5.56	  product(product(X, Y), X)
40.84/5.56	= { by axiom 1 (sos06) R->L }
40.84/5.56	  product(product(X, Y), product(X, X))
40.84/5.56	= { by lemma 21 }
40.84/5.56	  product(X, product(Y, X))
40.84/5.56	
40.84/5.56	Lemma 35: quotient(product(X, product(Y, X)), X) = product(X, Y).
40.84/5.56	Proof:
40.84/5.56	  quotient(product(X, product(Y, X)), X)
40.84/5.56	= { by lemma 34 R->L }
40.84/5.56	  quotient(product(product(X, Y), X), X)
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  product(X, Y)
40.84/5.56	
40.84/5.56	Lemma 36: difference(X, difference(product(Y, X), X)) = quotient(product(X, Y), X).
40.84/5.56	Proof:
40.84/5.56	  difference(X, difference(product(Y, X), X))
40.84/5.56	= { by axiom 11 (sos04) }
40.84/5.56	  difference(X, difference(product(product(quotient(Y, X), X), X), X))
40.84/5.56	= { by lemma 33 }
40.84/5.56	  difference(X, product(X, product(X, quotient(Y, X))))
40.84/5.56	= { by axiom 8 (sos01) }
40.84/5.56	  product(X, quotient(Y, X))
40.84/5.56	= { by lemma 35 R->L }
40.84/5.56	  quotient(product(X, product(quotient(Y, X), X)), X)
40.84/5.56	= { by axiom 11 (sos04) R->L }
40.84/5.56	  quotient(product(X, Y), X)
40.84/5.56	
40.84/5.56	Lemma 37: product(b1, c1) = product(c1, a).
40.84/5.56	Proof:
40.84/5.56	  product(b1, c1)
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  product(difference(c1, product(c1, b1)), c1)
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(a, c1), product(c1, b1)))), c1)
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(a, product(c1, b1)), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by lemma 21 R->L }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(product(a, c1), product(a, b1)), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by lemma 32 }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(product(a, c1), product(b1, c1)), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by axiom 14 (sos05) }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(product(a, b1), product(c1, c1)), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by lemma 32 }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(product(b1, c1), product(c1, c1)), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by lemma 22 }
40.84/5.56	  product(difference(c1, difference(product(a, c1), product(product(product(b1, c1), c1), product(c1, product(c1, b1))))), c1)
40.84/5.56	= { by axiom 16 (sos07) R->L }
40.84/5.56	  product(difference(c1, difference(product(a, c1), c1)), c1)
40.84/5.56	= { by lemma 36 }
40.84/5.56	  product(quotient(product(c1, a), c1), c1)
40.84/5.56	= { by axiom 11 (sos04) R->L }
40.84/5.56	  product(c1, a)
40.84/5.56	
40.84/5.56	Lemma 38: product(b1, c) = product(a1, b1).
40.84/5.56	Proof:
40.84/5.56	  product(b1, c)
40.84/5.56	= { by axiom 12 (sos08_1) R->L }
40.84/5.56	  fresh4(d(a1, b, c1), d(a1, b, c1), a1, b1, c)
40.84/5.56	= { by lemma 18 }
40.84/5.56	  fresh4(d(bigB, a, c), d(a1, b, c1), a1, b1, c)
40.84/5.56	= { by axiom 2 (sos11) }
40.84/5.56	  fresh4(true, d(a1, b, c1), a1, b1, c)
40.84/5.56	= { by axiom 4 (sos15) R->L }
40.84/5.56	  fresh4(d(a1, b1, c), d(a1, b, c1), a1, b1, c)
40.84/5.56	= { by lemma 19 }
40.84/5.56	  product(a1, b1)
40.84/5.56	
40.84/5.56	Lemma 39: product(difference(X, Y), X) = difference(X, product(Y, X)).
40.84/5.56	Proof:
40.84/5.56	  product(difference(X, Y), X)
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  difference(X, product(X, product(difference(X, Y), X)))
40.84/5.56	= { by lemma 34 R->L }
40.84/5.56	  difference(X, product(product(X, difference(X, Y)), X))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  difference(X, product(Y, X))
40.84/5.56	
40.84/5.56	Lemma 40: quotient(quotient(X, product(X, Y)), X) = difference(X, product(Y, X)).
40.84/5.56	Proof:
40.84/5.56	  quotient(quotient(X, product(X, Y)), X)
40.84/5.56	= { by axiom 10 (sos02) }
40.84/5.56	  quotient(quotient(X, product(X, product(X, difference(X, Y)))), X)
40.84/5.56	= { by axiom 16 (sos07) }
40.84/5.56	  quotient(quotient(product(product(product(difference(X, Y), X), X), product(X, product(X, difference(X, Y)))), product(X, product(X, difference(X, Y)))), X)
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  quotient(product(product(difference(X, Y), X), X), X)
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  product(difference(X, Y), X)
40.84/5.56	= { by lemma 39 }
40.84/5.56	  difference(X, product(Y, X))
40.84/5.56	
40.84/5.56	Lemma 41: product(product(a1, c), product(c, b1)) = c.
40.84/5.56	Proof:
40.84/5.56	  product(product(a1, c), product(c, b1))
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(product(a1, product(c, b1)), product(c, product(c, b1)))
40.84/5.56	= { by lemma 33 R->L }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(product(b1, c), c), c))
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(product(b1, c), product(c, c)), c))
40.84/5.56	= { by lemma 38 }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(product(a1, b1), product(c, c)), c))
40.84/5.56	= { by axiom 14 (sos05) }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(product(a1, c), product(b1, c)), c))
40.84/5.56	= { by lemma 38 }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(product(a1, c), product(a1, b1)), c))
40.84/5.56	= { by lemma 21 }
40.84/5.56	  product(product(a1, product(c, b1)), difference(product(a1, product(c, b1)), c))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  c
40.84/5.56	
40.84/5.56	Lemma 42: product(a1, quotient(product(c, b1), c)) = c.
40.84/5.56	Proof:
40.84/5.56	  product(a1, quotient(product(c, b1), c))
40.84/5.56	= { by lemma 24 R->L }
40.84/5.56	  quotient(product(product(a1, c), product(c, b1)), c)
40.84/5.56	= { by lemma 41 }
40.84/5.56	  quotient(c, c)
40.84/5.56	= { by axiom 1 (sos06) R->L }
40.84/5.56	  quotient(product(c, c), c)
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  c
40.84/5.56	
40.84/5.56	Lemma 43: product(bigA, c) = product(c1, b1).
40.84/5.56	Proof:
40.84/5.56	  product(bigA, c)
40.84/5.56	= { by axiom 10 (sos02) }
40.84/5.56	  product(c, difference(c, product(bigA, c)))
40.84/5.56	= { by lemma 40 R->L }
40.84/5.56	  product(c, quotient(quotient(c, product(c, bigA)), c))
40.84/5.56	= { by axiom 16 (sos07) }
40.84/5.56	  product(c, quotient(quotient(product(product(product(bigA, c), c), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(c, quotient(quotient(product(product(product(bigA, c), product(c, c)), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by lemma 31 R->L }
40.84/5.56	  product(c, quotient(quotient(product(product(product(b, bigA), product(c, c)), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by axiom 14 (sos05) R->L }
40.84/5.56	  product(c, quotient(quotient(product(product(product(b, c), product(bigA, c)), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by lemma 31 R->L }
40.84/5.56	  product(c, quotient(quotient(product(product(product(b, c), product(b, bigA)), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by lemma 21 }
40.84/5.56	  product(c, quotient(quotient(product(product(b, product(c, bigA)), product(c, product(c, bigA))), product(c, bigA)), c))
40.84/5.56	= { by lemma 22 }
40.84/5.56	  product(c, quotient(quotient(product(product(b, c), product(c, bigA)), product(c, bigA)), c))
40.84/5.56	= { by lemma 29 }
40.84/5.56	  product(c, quotient(quotient(product(product(bigB, bigC), product(c, bigA)), product(c, bigA)), c))
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  product(c, quotient(product(bigB, bigC), c))
40.84/5.56	= { by lemma 29 R->L }
40.84/5.56	  product(c, quotient(product(b, c), c))
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  product(c, b)
40.84/5.56	= { by lemma 42 R->L }
40.84/5.56	  product(product(a1, quotient(product(c, b1), c)), b)
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(product(a1, b), product(quotient(product(c, b1), c), b))
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  product(product(a1, b), product(quotient(product(c, b1), c), difference(a1, product(a1, b))))
40.84/5.56	= { by axiom 11 (sos04) }
40.84/5.56	  product(product(quotient(product(a1, b), a1), a1), product(quotient(product(c, b1), c), difference(a1, product(a1, b))))
40.84/5.56	= { by axiom 14 (sos05) R->L }
40.84/5.56	  product(product(quotient(product(a1, b), a1), quotient(product(c, b1), c)), product(a1, difference(a1, product(a1, b))))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  product(product(quotient(product(a1, b), a1), quotient(product(c, b1), c)), product(a1, b))
40.84/5.56	= { by axiom 11 (sos04) }
40.84/5.56	  product(product(quotient(product(a1, b), a1), quotient(product(c, b1), c)), product(quotient(product(a1, b), a1), a1))
40.84/5.56	= { by lemma 21 }
40.84/5.56	  product(quotient(product(a1, b), a1), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 9 (sos03) }
40.84/5.56	  product(quotient(product(a1, b), quotient(product(a1, c1), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 9 (sos03) }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, c1), product(c1, b)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), product(c1, product(c1, b))), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 33 R->L }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(product(b, c1), c1), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 22 R->L }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(product(b, c1), product(c1, c1)), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 20 }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(product(a1, b), product(c1, c1)), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 14 (sos05) }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(product(a1, c1), product(b, c1)), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 20 }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(product(a1, c1), product(a1, b)), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 21 }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(product(product(a1, product(c1, b)), difference(product(a1, product(c1, b)), c1)), product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  product(quotient(product(a1, b), quotient(quotient(c1, product(c1, b)), c1)), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 40 }
40.84/5.56	  product(quotient(product(a1, b), difference(c1, product(b, c1))), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by lemma 20 }
40.84/5.56	  product(quotient(product(a1, b), difference(c1, product(a1, b))), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 10 (sos02) }
40.84/5.56	  product(quotient(product(c1, difference(c1, product(a1, b))), difference(c1, product(a1, b))), product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  product(c1, product(quotient(product(c, b1), c), a1))
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  product(c1, product(difference(a1, product(a1, quotient(product(c, b1), c))), a1))
40.84/5.56	= { by lemma 42 }
40.84/5.56	  product(c1, product(difference(a1, c), a1))
40.84/5.56	= { by lemma 39 }
40.84/5.56	  product(c1, difference(a1, product(c, a1)))
40.84/5.56	= { by axiom 11 (sos04) }
40.84/5.56	  product(c1, difference(a1, product(quotient(product(c, a1), c), c)))
40.84/5.56	= { by lemma 36 R->L }
40.84/5.56	  product(c1, difference(a1, product(difference(c, difference(product(a1, c), c)), c)))
40.84/5.56	= { by lemma 41 R->L }
40.84/5.56	  product(c1, difference(a1, product(difference(c, difference(product(a1, c), product(product(a1, c), product(c, b1)))), c)))
40.84/5.56	= { by axiom 8 (sos01) }
40.84/5.56	  product(c1, difference(a1, product(difference(c, product(c, b1)), c)))
40.84/5.56	= { by axiom 8 (sos01) }
40.84/5.56	  product(c1, difference(a1, product(b1, c)))
40.84/5.56	= { by lemma 38 }
40.84/5.56	  product(c1, difference(a1, product(a1, b1)))
40.84/5.56	= { by axiom 8 (sos01) }
40.84/5.56	  product(c1, b1)
40.84/5.56	
40.84/5.56	Lemma 44: difference(a, product(X, a)) = difference(bigB, product(X, c)).
40.84/5.56	Proof:
40.84/5.56	  difference(a, product(X, a))
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  difference(bigB, product(bigB, difference(a, product(X, a))))
40.84/5.56	= { by lemma 39 R->L }
40.84/5.56	  difference(bigB, product(bigB, product(difference(a, X), a)))
40.84/5.56	= { by lemma 27 R->L }
40.84/5.56	  difference(bigB, product(product(a, difference(a, X)), c))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  difference(bigB, product(X, c))
40.84/5.56	
40.84/5.56	Lemma 45: quotient(product(b1, c1), b1) = a.
40.84/5.56	Proof:
40.84/5.56	  quotient(product(b1, c1), b1)
40.84/5.56	= { by lemma 32 R->L }
40.84/5.56	  quotient(product(a, b1), b1)
40.84/5.56	= { by axiom 9 (sos03) R->L }
40.84/5.56	  a
40.84/5.56	
40.84/5.56	Lemma 46: product(X, product(Y, difference(X, Z))) = product(product(X, Y), Z).
40.84/5.56	Proof:
40.84/5.56	  product(X, product(Y, difference(X, Z)))
40.84/5.56	= { by lemma 21 R->L }
40.84/5.56	  product(product(X, Y), product(X, difference(X, Z)))
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  product(product(X, Y), Z)
40.84/5.56	
40.84/5.56	Lemma 47: product(b, difference(bigB, product(X, c))) = product(X, bigC).
40.84/5.56	Proof:
40.84/5.56	  product(b, difference(bigB, product(X, c)))
40.84/5.56	= { by lemma 44 R->L }
40.84/5.56	  product(b, difference(a, product(X, a)))
40.84/5.56	= { by lemma 39 R->L }
40.84/5.56	  product(b, product(difference(a, X), a))
40.84/5.56	= { by lemma 21 R->L }
40.84/5.56	  product(product(b, difference(a, X)), product(b, a))
40.84/5.56	= { by lemma 25 R->L }
40.84/5.56	  product(product(b, difference(a, X)), product(a, bigC))
40.84/5.56	= { by axiom 14 (sos05) R->L }
40.84/5.56	  product(product(b, a), product(difference(a, X), bigC))
40.84/5.56	= { by lemma 25 R->L }
40.84/5.56	  product(product(a, bigC), product(difference(a, X), bigC))
40.84/5.56	= { by lemma 22 }
40.84/5.56	  product(product(a, difference(a, X)), bigC)
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  product(X, bigC)
40.84/5.56	
40.84/5.56	Lemma 48: product(c1, difference(bigB, product(X, c))) = product(X, b1).
40.84/5.56	Proof:
40.84/5.56	  product(c1, difference(bigB, product(X, c)))
40.84/5.56	= { by lemma 44 R->L }
40.84/5.56	  product(c1, difference(a, product(X, a)))
40.84/5.56	= { by lemma 39 R->L }
40.84/5.56	  product(c1, product(difference(a, X), a))
40.84/5.56	= { by lemma 21 R->L }
40.84/5.56	  product(product(c1, difference(a, X)), product(c1, a))
40.84/5.56	= { by lemma 37 R->L }
40.84/5.56	  product(product(c1, difference(a, X)), product(b1, c1))
40.84/5.56	= { by lemma 32 R->L }
40.84/5.56	  product(product(c1, difference(a, X)), product(a, b1))
40.84/5.56	= { by axiom 14 (sos05) }
40.84/5.56	  product(product(c1, a), product(difference(a, X), b1))
40.84/5.56	= { by lemma 37 R->L }
40.84/5.56	  product(product(b1, c1), product(difference(a, X), b1))
40.84/5.56	= { by lemma 28 R->L }
40.84/5.56	  product(product(quotient(product(b1, c1), b1), difference(a, X)), b1)
40.84/5.56	= { by lemma 45 }
40.84/5.56	  product(product(a, difference(a, X)), b1)
40.84/5.56	= { by axiom 10 (sos02) R->L }
40.84/5.56	  product(X, b1)
40.84/5.56	
40.84/5.56	Lemma 49: difference(X, product(product(X, Y), Z)) = product(Y, difference(X, Z)).
40.84/5.56	Proof:
40.84/5.56	  difference(X, product(product(X, Y), Z))
40.84/5.56	= { by lemma 46 R->L }
40.84/5.56	  difference(X, product(X, product(Y, difference(X, Z))))
40.84/5.56	= { by axiom 8 (sos01) }
40.84/5.56	  product(Y, difference(X, Z))
40.84/5.56	
40.84/5.56	Goal 1 (goals): m(bigA, b1, bigC) = true.
40.84/5.56	Proof:
40.84/5.56	  m(bigA, b1, bigC)
40.84/5.56	= { by axiom 17 (sos09) R->L }
40.84/5.56	  fresh2(product(bigA, bigC), product(product(bigA, b1), product(b1, bigC)), bigA, b1, bigC)
40.84/5.56	= { by lemma 35 R->L }
40.84/5.56	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), product(bigA, b1))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.56	= { by axiom 8 (sos01) R->L }
40.84/5.56	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(b, product(bigA, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.56	= { by lemma 21 R->L }
40.84/5.56	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(product(b, bigA), product(b, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.56	= { by lemma 19 R->L }
40.84/5.56	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(fresh4(d(b, bigA, c), d(a1, b, c1), b, bigA, c), product(b, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 30 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(fresh4(d(a1, b, c1), d(a1, b, c1), b, bigA, c), product(b, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 12 (sos08_1) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(product(bigA, c), product(b, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 43 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(product(c1, b1), product(b, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 14 (sos05) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(product(c1, b), product(b1, b1))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 1 (sos06) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(product(c1, b), b1)))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 46 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(c1, b1)))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 8 (sos01) R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, product(a, difference(c1, b1)))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 49 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(product(c1, a), b1)))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 45 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(product(c1, quotient(product(b1, c1), b1)), b1)))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 23 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(product(c1, b1), product(b1, c1))))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 37 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(product(c1, b1), product(c1, a))))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 14 (sos05) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(product(c1, c1), product(b1, a))))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 1 (sos06) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, difference(c1, product(c1, product(b1, a))))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 8 (sos01) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(a, product(b1, a))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 44 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b, difference(bigB, product(b1, c))))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 47 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(product(b1, bigC), difference(b, product(c1, product(b1, bigC))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 49 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), difference(b, product(product(b, product(b1, bigC)), product(c1, product(b1, bigC))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 22 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), difference(b, product(product(b, c1), product(b1, bigC)))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 49 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(c1, difference(b, product(b1, bigC)))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 47 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(c1, difference(b, product(b, difference(bigB, product(b1, c)))))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 8 (sos01) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(c1, difference(bigB, product(b1, c)))), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 48 }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), product(b1, b1)), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 1 (sos06) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(bigA, b1), b1), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 11 (sos04) }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(quotient(product(bigA, b1), bigA), bigA), b1), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by lemma 22 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), quotient(product(product(quotient(product(bigA, b1), bigA), b1), product(bigA, b1)), product(bigA, b1)), bigA, b1, bigC)
40.84/5.57	= { by axiom 9 (sos03) R->L }
40.84/5.57	  fresh2(product(bigA, bigC), product(quotient(product(bigA, b1), bigA), b1), bigA, b1, bigC)
40.84/5.57	= { by axiom 8 (sos01) R->L }
40.84/5.57	  fresh2(product(bigA, bigC), difference(c1, product(c1, product(quotient(product(bigA, b1), bigA), b1))), bigA, b1, bigC)
40.84/5.57	= { by lemma 21 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), difference(c1, product(product(c1, quotient(product(bigA, b1), bigA)), product(c1, b1))), bigA, b1, bigC)
40.84/5.57	= { by lemma 43 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), difference(c1, product(product(c1, quotient(product(bigA, b1), bigA)), product(bigA, c))), bigA, b1, bigC)
40.84/5.57	= { by lemma 31 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), difference(c1, product(product(c1, quotient(product(bigA, b1), bigA)), product(b, bigA))), bigA, b1, bigC)
40.84/5.57	= { by axiom 14 (sos05) }
40.84/5.57	  fresh2(product(bigA, bigC), difference(c1, product(product(c1, b), product(quotient(product(bigA, b1), bigA), bigA))), bigA, b1, bigC)
40.84/5.57	= { by lemma 49 }
40.84/5.57	  fresh2(product(bigA, bigC), product(b, difference(c1, product(quotient(product(bigA, b1), bigA), bigA))), bigA, b1, bigC)
40.84/5.57	= { by axiom 11 (sos04) R->L }
40.84/5.57	  fresh2(product(bigA, bigC), product(b, difference(c1, product(bigA, b1))), bigA, b1, bigC)
40.84/5.57	= { by lemma 48 R->L }
40.84/5.57	  fresh2(product(bigA, bigC), product(b, difference(c1, product(c1, difference(bigB, product(bigA, c))))), bigA, b1, bigC)
40.84/5.57	= { by axiom 8 (sos01) }
40.84/5.57	  fresh2(product(bigA, bigC), product(b, difference(bigB, product(bigA, c))), bigA, b1, bigC)
40.84/5.57	= { by lemma 47 }
40.84/5.57	  fresh2(product(bigA, bigC), product(bigA, bigC), bigA, b1, bigC)
40.84/5.57	= { by axiom 13 (sos09) }
40.84/5.57	  true
40.84/5.57	% SZS output end Proof
40.84/5.57	
40.84/5.57	RESULT: Theorem (the conjecture is true).
40.84/5.58	EOF