0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n027.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 09:56:09 EDT 2019
0.12/0.34	% CPUTime    : 
0.40/0.57	% SZS status Unsatisfiable
0.40/0.57	
0.40/0.57	% SZS output start Proof
0.40/0.57	Take the following subset of the input axioms:
0.40/0.57	  fof(intersection, axiom, ![Y, X]: multiply(X, Y)=difference(X, difference(X, Y))).
0.40/0.57	  fof(prove_associativity_of_multiply, negated_conjecture, multiply(a, multiply(b, c))!=multiply(multiply(a, b), c)).
0.40/0.57	  fof(set_difference_1, axiom, ![Y, X]: X=difference(X, difference(Y, X))).
0.40/0.57	  fof(set_difference_2, axiom, ![Y, X]: difference(Y, difference(Y, X))=difference(X, difference(X, Y))).
0.40/0.57	  fof(set_difference_3, axiom, ![Y, X, Z]: difference(difference(X, Y), Z)=difference(difference(X, Z), difference(Y, Z))).
0.40/0.57	
0.40/0.57	Now clausify the problem and encode Horn clauses using encoding 3 of
0.40/0.57	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.40/0.57	We repeatedly replace C & s=t => u=v by the two clauses:
0.40/0.57	  fresh(y, y, x1...xn) = u
0.40/0.57	  C => fresh(s, t, x1...xn) = v
0.40/0.57	where fresh is a fresh function symbol and x1..xn are the free
0.40/0.57	variables of u and v.
0.40/0.57	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.40/0.57	input problem has no model of domain size 1).
0.40/0.57	
0.40/0.57	The encoding turns the above axioms into the following unit equations and goals:
0.40/0.57	
0.40/0.57	Axiom 1 (intersection): multiply(X, Y) = difference(X, difference(X, Y)).
0.40/0.57	Axiom 2 (set_difference_1): X = difference(X, difference(Y, X)).
0.40/0.57	Axiom 3 (set_difference_3): difference(difference(X, Y), Z) = difference(difference(X, Z), difference(Y, Z)).
0.40/0.58	Axiom 4 (set_difference_2): difference(X, difference(X, Y)) = difference(Y, difference(Y, X)).
0.40/0.58	
0.40/0.58	Lemma 5: multiply(X, Y) = multiply(Y, X).
0.40/0.58	Proof:
0.40/0.58	  multiply(X, Y)
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(X, difference(X, Y))
0.40/0.58	= { by axiom 4 (set_difference_2) }
0.40/0.58	  difference(Y, difference(Y, X))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  multiply(Y, X)
0.40/0.58	
0.40/0.58	Lemma 6: multiply(X, difference(X, Y)) = difference(X, multiply(X, Y)).
0.40/0.58	Proof:
0.40/0.58	  multiply(X, difference(X, Y))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(X, difference(X, difference(X, Y)))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(X, multiply(X, Y))
0.40/0.58	
0.40/0.58	Lemma 7: multiply(X, difference(Y, X)) = difference(X, X).
0.40/0.58	Proof:
0.40/0.58	  multiply(X, difference(Y, X))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(X, difference(X, difference(Y, X)))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  difference(X, X)
0.40/0.58	
0.40/0.58	Lemma 8: difference(multiply(X, Y), X) = difference(X, X).
0.40/0.58	Proof:
0.40/0.58	  difference(multiply(X, Y), X)
0.40/0.58	= { by lemma 5 }
0.40/0.58	  difference(multiply(Y, X), X)
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(difference(Y, difference(Y, X)), X)
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(Y, X), difference(difference(Y, X), X))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  multiply(difference(Y, X), X)
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(X, difference(Y, X))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(X, X)
0.40/0.58	
0.40/0.58	Lemma 9: difference(multiply(Y, X), X) = difference(X, X).
0.40/0.58	Proof:
0.40/0.58	  difference(multiply(Y, X), X)
0.40/0.58	= { by lemma 5 }
0.40/0.58	  difference(multiply(X, Y), X)
0.40/0.58	= { by lemma 8 }
0.40/0.58	  difference(X, X)
0.40/0.58	
0.40/0.58	Lemma 10: difference(difference(X, X), Y) = difference(difference(?, ?), Y).
0.40/0.58	Proof:
0.40/0.58	  difference(difference(X, X), Y)
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(X, Y), difference(X, Y))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  multiply(difference(X, Y), difference(Y, difference(X, Y)))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  multiply(difference(X, Y), Y)
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(Y, difference(X, Y))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(Y, Y)
0.40/0.58	= { by lemma 7 }
0.40/0.58	  multiply(Y, difference(?, Y))
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(difference(?, Y), Y)
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  multiply(difference(?, Y), difference(Y, difference(?, Y)))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(difference(?, Y), difference(?, Y))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(?, ?), Y)
0.40/0.58	
0.40/0.58	Lemma 11: difference(difference(?, ?), Y) = difference(Y, Y).
0.40/0.58	Proof:
0.40/0.58	  difference(difference(?, ?), Y)
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(?, Y), difference(?, Y))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  multiply(difference(?, Y), difference(Y, difference(?, Y)))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  multiply(difference(?, Y), Y)
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(Y, difference(?, Y))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(Y, Y)
0.40/0.58	
0.40/0.58	Lemma 12: multiply(difference(X, Y), multiply(X, Y)) = difference(Y, Y).
0.40/0.58	Proof:
0.40/0.58	  multiply(difference(X, Y), multiply(X, Y))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  multiply(difference(X, Y), difference(X, difference(X, Y)))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(difference(X, Y), difference(X, Y))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(X, X), Y)
0.40/0.58	= { by lemma 10 }
0.40/0.58	  difference(difference(?, ?), Y)
0.40/0.58	= { by lemma 11 }
0.40/0.58	  difference(Y, Y)
0.40/0.58	
0.40/0.58	Lemma 13: difference(difference(X, Y), multiply(X, Y)) = difference(X, Y).
0.40/0.58	Proof:
0.40/0.58	  difference(difference(X, Y), multiply(X, Y))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(difference(X, Y), difference(X, difference(X, Y)))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  difference(X, Y)
0.40/0.58	
0.40/0.58	Lemma 14: multiply(difference(X, Y), difference(Z, Y)) = difference(multiply(X, Z), Y).
0.40/0.58	Proof:
0.40/0.58	  multiply(difference(X, Y), difference(Z, Y))
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(difference(X, Y), difference(difference(X, Y), difference(Z, Y)))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(X, Y), difference(difference(X, Z), Y))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(X, difference(X, Z)), Y)
0.40/0.58	= { by axiom 1 (intersection) }
0.40/0.58	  difference(multiply(X, Z), Y)
0.40/0.58	
0.40/0.58	Lemma 15: difference(difference(?, ?), multiply(X, Y)) = difference(difference(?, ?), multiply(X, ?)).
0.40/0.58	Proof:
0.40/0.58	  difference(difference(?, ?), multiply(X, Y))
0.40/0.58	= { by lemma 5 }
0.40/0.58	  difference(difference(?, ?), multiply(Y, X))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(?, multiply(Y, X)), difference(?, multiply(Y, X)))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  multiply(difference(?, multiply(Y, X)), difference(multiply(Y, X), difference(?, multiply(Y, X))))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  multiply(difference(?, multiply(Y, X)), multiply(Y, X))
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(multiply(Y, X), difference(?, multiply(Y, X)))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(multiply(Y, X), multiply(Y, X))
0.40/0.58	= { by lemma 12 }
0.40/0.58	  multiply(difference(difference(Y, X), multiply(Y, X)), multiply(difference(Y, X), multiply(Y, X)))
0.40/0.58	= { by lemma 13 }
0.40/0.58	  multiply(difference(Y, X), multiply(difference(Y, X), multiply(Y, X)))
0.40/0.58	= { by lemma 12 }
0.40/0.58	  multiply(difference(Y, X), difference(X, X))
0.40/0.58	= { by lemma 14 }
0.40/0.58	  difference(multiply(Y, X), X)
0.40/0.58	= { by lemma 9 }
0.40/0.58	  difference(X, X)
0.40/0.58	= { by lemma 9 }
0.40/0.58	  difference(multiply(?, X), X)
0.40/0.58	= { by lemma 14 }
0.40/0.58	  multiply(difference(?, X), difference(X, X))
0.40/0.58	= { by lemma 13 }
0.40/0.58	  multiply(difference(difference(?, X), multiply(?, X)), difference(X, X))
0.40/0.58	= { by lemma 12 }
0.40/0.58	  multiply(difference(difference(?, X), multiply(?, X)), multiply(difference(?, X), multiply(?, X)))
0.40/0.58	= { by lemma 12 }
0.40/0.58	  difference(multiply(?, X), multiply(?, X))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  multiply(multiply(?, X), difference(?, multiply(?, X)))
0.40/0.58	= { by lemma 5 }
0.40/0.58	  multiply(difference(?, multiply(?, X)), multiply(?, X))
0.40/0.58	= { by axiom 2 (set_difference_1) }
0.40/0.58	  multiply(difference(?, multiply(?, X)), difference(multiply(?, X), difference(?, multiply(?, X))))
0.40/0.58	= { by lemma 7 }
0.40/0.58	  difference(difference(?, multiply(?, X)), difference(?, multiply(?, X)))
0.40/0.58	= { by axiom 3 (set_difference_3) }
0.40/0.58	  difference(difference(?, ?), multiply(?, X))
0.40/0.58	= { by lemma 5 }
0.43/0.58	  difference(difference(?, ?), multiply(X, ?))
0.43/0.58	
0.43/0.58	Lemma 16: difference(difference(?, ?), X) = difference(?, ?).
0.43/0.58	Proof:
0.43/0.58	  difference(difference(?, ?), X)
0.43/0.58	= { by axiom 3 (set_difference_3) }
0.43/0.58	  difference(difference(?, X), difference(?, X))
0.43/0.58	= { by lemma 12 }
0.43/0.58	  multiply(difference(?, difference(?, X)), multiply(?, difference(?, X)))
0.43/0.58	= { by axiom 1 (intersection) }
0.43/0.58	  multiply(multiply(?, X), multiply(?, difference(?, X)))
0.43/0.58	= { by lemma 6 }
0.43/0.58	  multiply(multiply(?, X), difference(?, multiply(?, X)))
0.43/0.58	= { by lemma 5 }
0.43/0.58	  multiply(difference(?, multiply(?, X)), multiply(?, X))
0.43/0.58	= { by axiom 2 (set_difference_1) }
0.43/0.58	  multiply(difference(?, multiply(?, X)), difference(multiply(?, X), difference(?, multiply(?, X))))
0.43/0.58	= { by lemma 7 }
0.43/0.58	  difference(difference(?, multiply(?, X)), difference(?, multiply(?, X)))
0.43/0.58	= { by axiom 3 (set_difference_3) }
0.43/0.58	  difference(difference(?, ?), multiply(?, X))
0.43/0.58	= { by lemma 15 }
0.43/0.58	  difference(difference(?, ?), multiply(?, ?))
0.43/0.58	= { by axiom 3 (set_difference_3) }
0.43/0.58	  difference(difference(?, multiply(?, ?)), difference(?, multiply(?, ?)))
0.43/0.58	= { by lemma 7 }
0.43/0.58	  multiply(difference(?, multiply(?, ?)), difference(multiply(?, ?), difference(?, multiply(?, ?))))
0.43/0.58	= { by axiom 2 (set_difference_1) }
0.43/0.58	  multiply(difference(?, multiply(?, ?)), multiply(?, ?))
0.43/0.58	= { by lemma 5 }
0.43/0.58	  multiply(multiply(?, ?), difference(?, multiply(?, ?)))
0.43/0.58	= { by lemma 7 }
0.43/0.58	  difference(multiply(?, ?), multiply(?, ?))
0.43/0.58	= { by lemma 12 }
0.43/0.58	  multiply(difference(difference(?, ?), multiply(?, ?)), multiply(difference(?, ?), multiply(?, ?)))
0.43/0.58	= { by lemma 13 }
0.43/0.58	  multiply(difference(?, ?), multiply(difference(?, ?), multiply(?, ?)))
0.43/0.58	= { by lemma 12 }
0.43/0.58	  multiply(difference(?, ?), difference(?, ?))
0.43/0.58	= { by lemma 14 }
0.43/0.58	  difference(multiply(?, ?), ?)
0.43/0.58	= { by lemma 9 }
0.43/0.58	  difference(?, ?)
0.43/0.58	
0.43/0.58	Lemma 17: difference(X, X) = difference(?, ?).
0.43/0.58	Proof:
0.43/0.58	  difference(X, X)
0.43/0.58	= { by lemma 11 }
0.43/0.58	  difference(difference(?, ?), X)
0.43/0.58	= { by lemma 16 }
0.43/0.58	  difference(?, ?)
0.43/0.58	
0.43/0.58	Lemma 18: difference(X, difference(?, ?)) = X.
0.43/0.58	Proof:
0.43/0.58	  difference(X, difference(?, ?))
0.43/0.58	= { by lemma 16 }
0.43/0.58	  difference(X, difference(difference(?, ?), X))
0.43/0.58	= { by axiom 2 (set_difference_1) }
0.43/0.59	  X
0.43/0.59	
0.43/0.59	Lemma 19: difference(X, multiply(X, Y)) = difference(X, Y).
0.43/0.59	Proof:
0.43/0.59	  difference(X, multiply(X, Y))
0.43/0.59	= { by lemma 6 }
0.43/0.59	  multiply(X, difference(X, Y))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  multiply(difference(X, Y), X)
0.43/0.59	= { by axiom 1 (intersection) }
0.43/0.59	  difference(difference(X, Y), difference(difference(X, Y), X))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(difference(X, difference(difference(X, Y), X)), difference(Y, difference(difference(X, Y), X)))
0.43/0.59	= { by axiom 2 (set_difference_1) }
0.43/0.59	  difference(X, difference(Y, difference(difference(X, Y), X)))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(X, difference(Y, difference(difference(X, X), difference(Y, X))))
0.43/0.59	= { by lemma 7 }
0.43/0.59	  difference(X, difference(Y, difference(multiply(X, difference(Y, X)), difference(Y, X))))
0.43/0.59	= { by lemma 9 }
0.43/0.59	  difference(X, difference(Y, difference(difference(Y, X), difference(Y, X))))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(X, difference(Y, difference(difference(Y, Y), X)))
0.43/0.59	= { by lemma 10 }
0.43/0.59	  difference(X, difference(Y, difference(difference(?, ?), X)))
0.43/0.59	= { by lemma 11 }
0.43/0.59	  difference(X, difference(Y, difference(X, X)))
0.43/0.59	= { by lemma 17 }
0.43/0.59	  difference(X, difference(Y, difference(?, ?)))
0.43/0.59	= { by lemma 18 }
0.43/0.59	  difference(X, Y)
0.43/0.59	
0.43/0.59	Lemma 20: difference(multiply(X, Z), difference(Y, difference(X, Z))) = difference(difference(X, Y), difference(X, Z)).
0.43/0.59	Proof:
0.43/0.59	  difference(multiply(X, Z), difference(Y, difference(X, Z)))
0.43/0.59	= { by axiom 1 (intersection) }
0.43/0.59	  difference(difference(X, difference(X, Z)), difference(Y, difference(X, Z)))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(difference(X, Y), difference(X, Z))
0.43/0.59	
0.43/0.59	Lemma 21: difference(difference(X, Y), Y) = difference(X, Y).
0.43/0.59	Proof:
0.43/0.59	  difference(difference(X, Y), Y)
0.43/0.59	= { by axiom 2 (set_difference_1) }
0.43/0.59	  difference(difference(X, Y), difference(Y, difference(X, Y)))
0.43/0.59	= { by axiom 2 (set_difference_1) }
0.43/0.59	  difference(X, Y)
0.43/0.59	
0.43/0.59	Lemma 22: difference(X, multiply(Y, X)) = difference(X, Y).
0.43/0.59	Proof:
0.43/0.59	  difference(X, multiply(Y, X))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(X, multiply(X, Y))
0.43/0.59	= { by lemma 19 }
0.43/0.59	  difference(X, Y)
0.43/0.59	
0.43/0.59	Lemma 23: multiply(X, difference(Y, Z)) = difference(multiply(X, Y), Z).
0.43/0.59	Proof:
0.43/0.59	  multiply(X, difference(Y, Z))
0.43/0.59	= { by lemma 18 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(?, ?))
0.43/0.59	= { by lemma 16 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(?, ?), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 10 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(difference(Y, Z), difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 8 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(difference(Y, Z), Z), difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 14 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(multiply(difference(difference(Y, Z), difference(Y, Z)), difference(Z, difference(Y, Z))), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 8 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(multiply(difference(multiply(difference(Y, Z), X), difference(Y, Z)), difference(Z, difference(Y, Z))), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 14 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(multiply(difference(Y, Z), X), Z), difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(Z, multiply(difference(Y, Z), X)), difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(Z, multiply(X, difference(Y, Z))), difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(Z, multiply(X, difference(Y, Z))), difference(Z, multiply(X, difference(Y, Z)))), difference(difference(Y, Z), difference(Z, multiply(X, difference(Y, Z))))))
0.43/0.59	= { by axiom 3 (set_difference_3) }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(multiply(Z, multiply(X, difference(Y, Z))), difference(difference(Y, Z), difference(Z, multiply(X, difference(Y, Z))))), difference(difference(Z, multiply(X, difference(Y, Z))), difference(difference(Y, Z), difference(Z, multiply(X, difference(Y, Z)))))))
0.43/0.59	= { by lemma 20 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(difference(Z, difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))), difference(difference(Z, multiply(X, difference(Y, Z))), difference(difference(Y, Z), difference(Z, multiply(X, difference(Y, Z)))))))
0.43/0.59	= { by axiom 2 (set_difference_1) }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(difference(Z, difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 21 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(difference(Z, difference(Y, Z)), difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by axiom 2 (set_difference_1) }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(Z, difference(Z, multiply(X, difference(Y, Z)))))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), difference(Z, difference(Z, multiply(difference(Y, Z), X))))
0.43/0.59	= { by axiom 1 (intersection) }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), multiply(Z, multiply(difference(Y, Z), X)))
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), multiply(Z, multiply(X, difference(Y, Z))))
0.43/0.59	= { by lemma 22 }
0.43/0.59	  difference(multiply(X, difference(Y, Z)), Z)
0.43/0.59	= { by lemma 5 }
0.43/0.59	  difference(multiply(difference(Y, Z), X), Z)
0.43/0.59	= { by lemma 14 }
0.43/0.59	  multiply(difference(difference(Y, Z), Z), difference(X, Z))
0.43/0.59	= { by lemma 21 }
0.43/0.59	  multiply(difference(Y, Z), difference(X, Z))
0.43/0.59	= { by lemma 14 }
0.43/0.59	  difference(multiply(Y, X), Z)
0.43/0.59	= { by lemma 5 }
0.43/0.60	  difference(multiply(X, Y), Z)
0.43/0.60	
0.43/0.60	Lemma 24: difference(multiply(X, Y), difference(Y, Z)) = multiply(X, multiply(Y, Z)).
0.43/0.60	Proof:
0.43/0.60	  difference(multiply(X, Y), difference(Y, Z))
0.43/0.60	= { by axiom 1 (intersection) }
0.43/0.60	  difference(difference(X, difference(X, Y)), difference(Y, Z))
0.43/0.60	= { by lemma 19 }
0.43/0.60	  difference(difference(X, difference(X, Y)), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by axiom 3 (set_difference_3) }
0.43/0.60	  difference(difference(X, difference(Y, multiply(Y, Z))), difference(difference(X, Y), difference(Y, multiply(Y, Z))))
0.43/0.60	= { by axiom 3 (set_difference_3) }
0.43/0.60	  difference(difference(X, difference(Y, multiply(Y, Z))), difference(difference(X, difference(Y, multiply(Y, Z))), difference(Y, difference(Y, multiply(Y, Z)))))
0.43/0.60	= { by axiom 1 (intersection) }
0.43/0.60	  difference(difference(X, difference(Y, multiply(Y, Z))), difference(difference(X, difference(Y, multiply(Y, Z))), multiply(Y, multiply(Y, Z))))
0.43/0.60	= { by axiom 1 (intersection) }
0.43/0.60	  multiply(difference(X, difference(Y, multiply(Y, Z))), multiply(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  multiply(multiply(Y, multiply(Y, Z)), difference(X, difference(Y, multiply(Y, Z))))
0.43/0.60	= { by lemma 23 }
0.43/0.60	  difference(multiply(multiply(Y, multiply(Y, Z)), X), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  difference(multiply(X, multiply(Y, multiply(Y, Z))), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  difference(multiply(X, multiply(multiply(Y, Z), Y)), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by axiom 1 (intersection) }
0.43/0.60	  difference(multiply(X, difference(multiply(Y, Z), difference(multiply(Y, Z), Y))), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 8 }
0.43/0.60	  difference(multiply(X, difference(multiply(Y, Z), difference(Y, Y))), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 17 }
0.43/0.60	  difference(multiply(X, difference(multiply(Y, Z), difference(?, ?))), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 18 }
0.43/0.60	  difference(multiply(X, multiply(Y, Z)), difference(Y, multiply(Y, Z)))
0.43/0.60	= { by lemma 23 }
0.43/0.60	  multiply(X, difference(multiply(Y, Z), difference(Y, multiply(Y, Z))))
0.43/0.60	= { by axiom 2 (set_difference_1) }
0.43/0.60	  multiply(X, multiply(Y, Z))
0.43/0.60	
0.43/0.60	Lemma 25: difference(multiply(Y, X), difference(Y, Z)) = multiply(X, multiply(Y, Z)).
0.43/0.60	Proof:
0.43/0.60	  difference(multiply(Y, X), difference(Y, Z))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  difference(multiply(X, Y), difference(Y, Z))
0.43/0.60	= { by lemma 24 }
0.43/0.60	  multiply(X, multiply(Y, Z))
0.43/0.60	
0.43/0.60	Goal 1 (prove_associativity_of_multiply): multiply(a, multiply(b, c)) = multiply(multiply(a, b), c).
0.43/0.60	Proof:
0.43/0.60	  multiply(a, multiply(b, c))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  multiply(a, multiply(c, b))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  multiply(a, multiply(b, c))
0.43/0.60	= { by lemma 25 }
0.43/0.60	  difference(multiply(b, a), difference(b, c))
0.43/0.60	= { by axiom 1 (intersection) }
0.43/0.60	  difference(difference(b, difference(b, a)), difference(b, c))
0.43/0.60	= { by lemma 22 }
0.43/0.60	  difference(difference(b, multiply(difference(b, a), b)), difference(b, c))
0.43/0.60	= { by lemma 20 }
0.43/0.60	  difference(multiply(b, c), difference(multiply(difference(b, a), b), difference(b, c)))
0.43/0.60	= { by lemma 24 }
0.43/0.60	  difference(multiply(b, c), multiply(difference(b, a), multiply(b, c)))
0.43/0.60	= { by lemma 22 }
0.43/0.60	  difference(multiply(b, c), difference(b, a))
0.43/0.60	= { by lemma 25 }
0.43/0.60	  multiply(c, multiply(b, a))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  multiply(c, multiply(a, b))
0.43/0.60	= { by lemma 5 }
0.43/0.60	  multiply(multiply(a, b), c)
0.43/0.60	% SZS output end Proof
0.43/0.60	
0.43/0.60	RESULT: Unsatisfiable (the axioms are contradictory).
0.43/0.60	EOF