0.07/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n008.cluster.edu
0.14/0.35	% Model      : x86_64 x86_64
0.14/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.35	% Memory     : 8042.1875MB
0.14/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.35	% CPULimit   : 180
0.14/0.35	% DateTime   : Thu Aug 29 12:05:23 EDT 2019
0.14/0.35	% CPUTime    : 
0.85/1.03	% SZS status Unsatisfiable
0.85/1.03	
0.85/1.03	% SZS output start Proof
0.85/1.03	Take the following subset of the input axioms:
1.09/1.28	  fof(associativity, axiom, ![Y, Z, X]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
1.09/1.28	  fof(intersection_associative, axiom, ![Y, Z, X]: intersection(intersection(X, Y), Z)=intersection(X, intersection(Y, Z))).
1.09/1.28	  fof(intersection_commutative, axiom, ![Y, X]: intersection(Y, X)=intersection(X, Y)).
1.09/1.28	  fof(intersection_union_absorbtion, axiom, ![Y, X]: Y=intersection(union(X, Y), Y)).
1.09/1.28	  fof(inverse_involution, axiom, ![X]: inverse(inverse(X))=X).
1.09/1.28	  fof(inverse_of_identity, axiom, identity=inverse(identity)).
1.09/1.28	  fof(inverse_product_lemma, axiom, ![Y, X]: inverse(multiply(X, Y))=multiply(inverse(Y), inverse(X))).
1.09/1.28	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
1.09/1.28	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
1.09/1.28	  fof(multiply_intersection1, axiom, ![Y, Z, X]: multiply(X, intersection(Y, Z))=intersection(multiply(X, Y), multiply(X, Z))).
1.09/1.28	  fof(multiply_intersection2, axiom, ![Y, Z, X]: intersection(multiply(Y, X), multiply(Z, X))=multiply(intersection(Y, Z), X)).
1.09/1.28	  fof(multiply_union1, axiom, ![Y, Z, X]: union(multiply(X, Y), multiply(X, Z))=multiply(X, union(Y, Z))).
1.09/1.28	  fof(multiply_union2, axiom, ![Y, Z, X]: multiply(union(Y, Z), X)=union(multiply(Y, X), multiply(Z, X))).
1.09/1.28	  fof(negative_part, axiom, ![X]: intersection(X, identity)=negative_part(X)).
1.09/1.28	  fof(positive_part, axiom, ![X]: union(X, identity)=positive_part(X)).
1.09/1.28	  fof(prove_product, negated_conjecture, a!=multiply(positive_part(a), negative_part(a))).
1.09/1.28	  fof(union_associative, axiom, ![Y, Z, X]: union(union(X, Y), Z)=union(X, union(Y, Z))).
1.09/1.28	  fof(union_commutative, axiom, ![Y, X]: union(X, Y)=union(Y, X)).
1.09/1.28	  fof(union_intersection_absorbtion, axiom, ![Y, X]: Y=union(intersection(X, Y), Y)).
1.09/1.28	
1.09/1.28	Now clausify the problem and encode Horn clauses using encoding 3 of
1.09/1.28	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.09/1.28	We repeatedly replace C & s=t => u=v by the two clauses:
1.09/1.28	  fresh(y, y, x1...xn) = u
1.09/1.28	  C => fresh(s, t, x1...xn) = v
1.09/1.28	where fresh is a fresh function symbol and x1..xn are the free
1.09/1.28	variables of u and v.
1.09/1.28	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.09/1.28	input problem has no model of domain size 1).
1.09/1.28	
1.09/1.28	The encoding turns the above axioms into the following unit equations and goals:
1.09/1.28	
1.09/1.28	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
1.09/1.28	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
1.09/1.28	Axiom 3 (left_identity): X = multiply(identity, X).
1.09/1.28	Axiom 4 (intersection_commutative): intersection(X, Y) = intersection(Y, X).
1.09/1.28	Axiom 5 (union_commutative): union(X, Y) = union(Y, X).
1.09/1.28	Axiom 6 (intersection_union_absorbtion): X = intersection(union(Y, X), X).
1.09/1.28	Axiom 7 (inverse_involution): inverse(inverse(X)) = X.
1.09/1.28	Axiom 8 (inverse_product_lemma): inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
1.09/1.28	Axiom 9 (multiply_union1): union(multiply(X, Y), multiply(X, Z)) = multiply(X, union(Y, Z)).
1.09/1.28	Axiom 10 (intersection_associative): intersection(intersection(X, Y), Z) = intersection(X, intersection(Y, Z)).
1.09/1.28	Axiom 11 (multiply_intersection1): multiply(X, intersection(Y, Z)) = intersection(multiply(X, Y), multiply(X, Z)).
1.09/1.28	Axiom 12 (multiply_intersection2): intersection(multiply(X, Y), multiply(Z, Y)) = multiply(intersection(X, Z), Y).
1.09/1.28	Axiom 13 (positive_part): union(X, identity) = positive_part(X).
1.09/1.28	Axiom 14 (union_associative): union(union(X, Y), Z) = union(X, union(Y, Z)).
1.09/1.28	Axiom 15 (negative_part): intersection(X, identity) = negative_part(X).
1.09/1.28	Axiom 16 (inverse_of_identity): identity = inverse(identity).
1.09/1.28	Axiom 17 (union_intersection_absorbtion): X = union(intersection(Y, X), X).
1.09/1.28	Axiom 18 (multiply_union2): multiply(union(X, Y), Z) = union(multiply(X, Z), multiply(Y, Z)).
1.09/1.28	
1.09/1.28	Lemma 19: multiply(inverse(X), multiply(X, Y)) = Y.
1.09/1.28	Proof:
1.09/1.28	  multiply(inverse(X), multiply(X, Y))
1.09/1.28	= { by axiom 2 (associativity) }
1.09/1.28	  multiply(multiply(inverse(X), X), Y)
1.09/1.28	= { by axiom 1 (left_inverse) }
1.09/1.28	  multiply(identity, Y)
1.09/1.28	= { by axiom 3 (left_identity) }
1.09/1.28	  Y
1.09/1.28	
1.09/1.28	Lemma 20: union(Y, multiply(X, Y)) = multiply(positive_part(X), Y).
1.09/1.28	Proof:
1.09/1.28	  union(Y, multiply(X, Y))
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(multiply(X, Y), Y)
1.09/1.28	= { by axiom 3 (left_identity) }
1.09/1.28	  union(multiply(X, Y), multiply(identity, Y))
1.09/1.28	= { by axiom 18 (multiply_union2) }
1.09/1.28	  multiply(union(X, identity), Y)
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  multiply(positive_part(X), Y)
1.09/1.28	
1.09/1.28	Lemma 21: intersection(identity, X) = negative_part(X).
1.09/1.28	Proof:
1.09/1.28	  intersection(identity, X)
1.09/1.28	= { by axiom 4 (intersection_commutative) }
1.09/1.28	  intersection(X, identity)
1.09/1.28	= { by axiom 15 (negative_part) }
1.09/1.28	  negative_part(X)
1.09/1.28	
1.09/1.28	Lemma 22: multiply(inverse(X), intersection(X, Y)) = negative_part(multiply(inverse(X), Y)).
1.09/1.28	Proof:
1.09/1.28	  multiply(inverse(X), intersection(X, Y))
1.09/1.28	= { by axiom 11 (multiply_intersection1) }
1.09/1.28	  intersection(multiply(inverse(X), X), multiply(inverse(X), Y))
1.09/1.28	= { by axiom 1 (left_inverse) }
1.09/1.28	  intersection(identity, multiply(inverse(X), Y))
1.09/1.28	= { by lemma 21 }
1.09/1.28	  negative_part(multiply(inverse(X), Y))
1.09/1.28	
1.09/1.28	Lemma 23: multiply(X, identity) = X.
1.09/1.28	Proof:
1.09/1.28	  multiply(X, identity)
1.09/1.28	= { by axiom 7 (inverse_involution) }
1.09/1.28	  inverse(inverse(multiply(X, identity)))
1.09/1.28	= { by axiom 8 (inverse_product_lemma) }
1.09/1.28	  inverse(multiply(inverse(identity), inverse(X)))
1.09/1.28	= { by axiom 16 (inverse_of_identity) }
1.09/1.28	  inverse(multiply(identity, inverse(X)))
1.09/1.28	= { by axiom 3 (left_identity) }
1.09/1.28	  inverse(inverse(X))
1.09/1.28	= { by axiom 7 (inverse_involution) }
1.09/1.28	  X
1.09/1.28	
1.09/1.28	Lemma 24: multiply(inverse(X), negative_part(X)) = negative_part(inverse(X)).
1.09/1.28	Proof:
1.09/1.28	  multiply(inverse(X), negative_part(X))
1.09/1.28	= { by axiom 15 (negative_part) }
1.09/1.28	  multiply(inverse(X), intersection(X, identity))
1.09/1.28	= { by lemma 22 }
1.09/1.28	  negative_part(multiply(inverse(X), identity))
1.09/1.28	= { by lemma 23 }
1.09/1.28	  negative_part(inverse(X))
1.09/1.28	
1.09/1.28	Lemma 25: multiply(positive_part(inverse(X)), negative_part(X)) = multiply(positive_part(X), negative_part(inverse(X))).
1.09/1.28	Proof:
1.09/1.28	  multiply(positive_part(inverse(X)), negative_part(X))
1.09/1.28	= { by lemma 20 }
1.09/1.28	  union(negative_part(X), multiply(inverse(X), negative_part(X)))
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(multiply(inverse(X), negative_part(X)), negative_part(X))
1.09/1.28	= { by lemma 19 }
1.09/1.28	  union(multiply(inverse(X), negative_part(X)), multiply(inverse(inverse(X)), multiply(inverse(X), negative_part(X))))
1.09/1.28	= { by lemma 20 }
1.09/1.28	  multiply(positive_part(inverse(inverse(X))), multiply(inverse(X), negative_part(X)))
1.09/1.28	= { by axiom 7 (inverse_involution) }
1.09/1.28	  multiply(positive_part(X), multiply(inverse(X), negative_part(X)))
1.09/1.28	= { by lemma 24 }
1.09/1.28	  multiply(positive_part(X), negative_part(inverse(X)))
1.09/1.28	
1.09/1.28	Lemma 26: intersection(Y, multiply(X, Y)) = multiply(negative_part(X), Y).
1.09/1.28	Proof:
1.09/1.28	  intersection(Y, multiply(X, Y))
1.09/1.28	= { by axiom 3 (left_identity) }
1.09/1.28	  intersection(multiply(identity, Y), multiply(X, Y))
1.09/1.28	= { by axiom 12 (multiply_intersection2) }
1.09/1.28	  multiply(intersection(identity, X), Y)
1.09/1.28	= { by lemma 21 }
1.09/1.28	  multiply(negative_part(X), Y)
1.09/1.28	
1.09/1.28	Lemma 27: union(identity, X) = positive_part(X).
1.09/1.28	Proof:
1.09/1.28	  union(identity, X)
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(X, identity)
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  positive_part(X)
1.09/1.28	
1.09/1.28	Lemma 28: union(X, multiply(X, Y)) = multiply(X, positive_part(Y)).
1.09/1.28	Proof:
1.09/1.28	  union(X, multiply(X, Y))
1.09/1.28	= { by lemma 23 }
1.09/1.28	  union(multiply(X, identity), multiply(X, Y))
1.09/1.28	= { by axiom 9 (multiply_union1) }
1.09/1.28	  multiply(X, union(identity, Y))
1.09/1.28	= { by lemma 27 }
1.09/1.28	  multiply(X, positive_part(Y))
1.09/1.28	
1.09/1.28	Lemma 29: union(X, union(Z, multiply(Y, X))) = union(Z, multiply(positive_part(Y), X)).
1.09/1.28	Proof:
1.09/1.28	  union(X, union(Z, multiply(Y, X)))
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(X, union(multiply(Y, X), Z))
1.09/1.28	= { by axiom 14 (union_associative) }
1.09/1.28	  union(union(X, multiply(Y, X)), Z)
1.09/1.28	= { by lemma 20 }
1.09/1.28	  union(multiply(positive_part(Y), X), Z)
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(Z, multiply(positive_part(Y), X))
1.09/1.28	
1.09/1.28	Lemma 30: union(X, union(Y, Z)) = union(Z, union(X, Y)).
1.09/1.28	Proof:
1.09/1.28	  union(X, union(Y, Z))
1.09/1.28	= { by axiom 14 (union_associative) }
1.09/1.28	  union(union(X, Y), Z)
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(Z, union(X, Y))
1.09/1.28	
1.09/1.28	Lemma 31: union(X, intersection(Y, X)) = X.
1.09/1.28	Proof:
1.09/1.28	  union(X, intersection(Y, X))
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(intersection(Y, X), X)
1.09/1.28	= { by axiom 17 (union_intersection_absorbtion) }
1.09/1.28	  X
1.09/1.28	
1.09/1.28	Lemma 32: positive_part(negative_part(X)) = identity.
1.09/1.28	Proof:
1.09/1.28	  positive_part(negative_part(X))
1.09/1.28	= { by axiom 15 (negative_part) }
1.09/1.28	  positive_part(intersection(X, identity))
1.09/1.28	= { by lemma 27 }
1.09/1.28	  union(identity, intersection(X, identity))
1.09/1.28	= { by lemma 31 }
1.09/1.28	  identity
1.09/1.28	
1.09/1.28	Lemma 33: positive_part(union(X, Y)) = union(X, positive_part(Y)).
1.09/1.28	Proof:
1.09/1.28	  positive_part(union(X, Y))
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  union(union(X, Y), identity)
1.09/1.28	= { by axiom 14 (union_associative) }
1.09/1.28	  union(X, union(Y, identity))
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  union(X, positive_part(Y))
1.09/1.28	
1.09/1.28	Lemma 34: union(X, positive_part(multiply(Y, X))) = positive_part(multiply(positive_part(Y), X)).
1.09/1.28	Proof:
1.09/1.28	  union(X, positive_part(multiply(Y, X)))
1.09/1.28	= { by lemma 33 }
1.09/1.28	  positive_part(union(X, multiply(Y, X)))
1.09/1.28	= { by lemma 20 }
1.09/1.28	  positive_part(multiply(positive_part(Y), X))
1.09/1.28	
1.09/1.28	Lemma 35: union(negative_part(X), positive_part(Y)) = positive_part(Y).
1.09/1.28	Proof:
1.09/1.28	  union(negative_part(X), positive_part(Y))
1.09/1.28	= { by axiom 5 (union_commutative) }
1.09/1.28	  union(positive_part(Y), negative_part(X))
1.09/1.28	= { by lemma 27 }
1.09/1.28	  union(union(identity, Y), negative_part(X))
1.09/1.28	= { by axiom 14 (union_associative) }
1.09/1.28	  union(identity, union(Y, negative_part(X)))
1.09/1.28	= { by lemma 27 }
1.09/1.28	  positive_part(union(Y, negative_part(X)))
1.09/1.28	= { by lemma 33 }
1.09/1.28	  union(Y, positive_part(negative_part(X)))
1.09/1.28	= { by lemma 32 }
1.09/1.28	  union(Y, identity)
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  positive_part(Y)
1.09/1.28	
1.09/1.28	Lemma 36: negative_part(positive_part(X)) = identity.
1.09/1.28	Proof:
1.09/1.28	  negative_part(positive_part(X))
1.09/1.28	= { by axiom 13 (positive_part) }
1.09/1.28	  negative_part(union(X, identity))
1.09/1.28	= { by lemma 21 }
1.09/1.28	  intersection(identity, union(X, identity))
1.09/1.28	= { by axiom 4 (intersection_commutative) }
1.09/1.28	  intersection(union(X, identity), identity)
1.09/1.28	= { by axiom 6 (intersection_union_absorbtion) }
1.36/1.53	  identity
1.36/1.53	
1.36/1.53	Goal 1 (prove_product): a = multiply(positive_part(a), negative_part(a)).
1.36/1.53	Proof:
1.36/1.53	  a
1.36/1.53	= { by lemma 19 }
1.36/1.53	  multiply(inverse(negative_part(inverse(a))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 19 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), multiply(positive_part(a), negative_part(inverse(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 25 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), multiply(positive_part(inverse(a)), negative_part(a)))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 3 (left_identity) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), multiply(identity, multiply(positive_part(inverse(a)), negative_part(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 32 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), multiply(positive_part(negative_part(multiply(positive_part(inverse(a)), negative_part(a)))), multiply(positive_part(inverse(a)), negative_part(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 20 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(inverse(a)), negative_part(a)), multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 5 (union_commutative) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 31 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))), union(multiply(positive_part(inverse(a)), negative_part(a)), intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 21 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))), union(multiply(positive_part(inverse(a)), negative_part(a)), negative_part(multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 30 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), union(multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 30 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(inverse(a)), negative_part(a)), union(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 25 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), union(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 28 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(negative_part(multiply(positive_part(inverse(a)), negative_part(a))), positive_part(multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 26 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), intersection(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(multiply(positive_part(inverse(a)), negative_part(a)), positive_part(multiply(positive_part(inverse(a)), negative_part(a)))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 23 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), intersection(multiply(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), identity), multiply(multiply(positive_part(inverse(a)), negative_part(a)), positive_part(multiply(positive_part(inverse(a)), negative_part(a)))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 28 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), intersection(multiply(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), identity), union(multiply(positive_part(inverse(a)), negative_part(a)), multiply(multiply(positive_part(inverse(a)), negative_part(a)), multiply(positive_part(inverse(a)), negative_part(a)))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 20 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), intersection(multiply(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), identity), multiply(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 11 (multiply_intersection1) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(positive_part(multiply(positive_part(inverse(a)), negative_part(a))), intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 34 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(union(negative_part(a), positive_part(multiply(inverse(a), negative_part(a)))), intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 24 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(union(negative_part(a), positive_part(negative_part(inverse(a)))), intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 35 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(positive_part(negative_part(inverse(a))), intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 32 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(identity, intersection(identity, multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 21 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(identity, negative_part(multiply(positive_part(inverse(a)), negative_part(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 3 (left_identity) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(multiply(positive_part(inverse(a)), negative_part(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 22 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(inverse(positive_part(inverse(a))), negative_part(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 15 (negative_part) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(inverse(positive_part(inverse(a))), intersection(a, identity)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 10 (intersection_associative) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(intersection(inverse(positive_part(inverse(a))), a), identity))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 15 (negative_part) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), negative_part(intersection(inverse(positive_part(inverse(a))), a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 21 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(identity, intersection(inverse(positive_part(inverse(a))), a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 10 (intersection_associative) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(intersection(identity, inverse(positive_part(inverse(a)))), a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 21 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(negative_part(inverse(positive_part(inverse(a)))), a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 4 (intersection_commutative) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, negative_part(inverse(positive_part(inverse(a))))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 15 (negative_part) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, intersection(inverse(positive_part(inverse(a))), identity)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 1 (left_inverse) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, intersection(inverse(positive_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), inverse(positive_part(inverse(a)))))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, intersection(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), inverse(positive_part(inverse(a)))))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 26 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, multiply(negative_part(positive_part(inverse(a))), inverse(positive_part(inverse(a))))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 36 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, multiply(identity, inverse(positive_part(inverse(a))))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 3 (left_identity) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(a, inverse(positive_part(inverse(a)))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 4 (intersection_commutative) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), multiply(inverse(inverse(positive_part(inverse(a)))), intersection(inverse(positive_part(inverse(a))), a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 22 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(multiply(inverse(inverse(positive_part(inverse(a)))), a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(multiply(positive_part(inverse(a)), a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 20 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(union(a, multiply(inverse(a), a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 1 (left_inverse) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(union(a, identity))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 13 (positive_part) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), negative_part(positive_part(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 36 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(multiply(positive_part(a), negative_part(inverse(a))), identity))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 13 (positive_part) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), positive_part(multiply(positive_part(a), negative_part(inverse(a)))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 34 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(negative_part(inverse(a)), positive_part(multiply(a, negative_part(inverse(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(negative_part(inverse(a)), positive_part(multiply(inverse(inverse(a)), negative_part(inverse(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 24 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(negative_part(inverse(a)), positive_part(negative_part(inverse(inverse(a))))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), union(negative_part(inverse(a)), positive_part(negative_part(a))))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 35 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), positive_part(negative_part(a)))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 32 }
1.36/1.53	  multiply(inverse(multiply(inverse(positive_part(a)), identity)), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 23 }
1.36/1.53	  multiply(inverse(inverse(positive_part(a))), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by axiom 7 (inverse_involution) }
1.36/1.53	  multiply(positive_part(a), multiply(negative_part(inverse(a)), a))
1.36/1.53	= { by lemma 26 }
1.36/1.53	  multiply(positive_part(a), intersection(a, multiply(inverse(a), a)))
1.36/1.53	= { by axiom 1 (left_inverse) }
1.36/1.53	  multiply(positive_part(a), intersection(a, identity))
1.36/1.53	= { by axiom 15 (negative_part) }
1.36/1.53	  multiply(positive_part(a), negative_part(a))
1.36/1.53	% SZS output end Proof
1.36/1.53	
1.36/1.53	RESULT: Unsatisfiable (the axioms are contradictory).
1.36/1.53	EOF