TSTP Solution File: GRP114-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP114-1 : TPTP v8.1.2. Released v1.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n029.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:17:02 EDT 2023
% Result : Unsatisfiable 3.08s 0.73s
% Output : Proof 3.08s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
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%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.11 % Problem : GRP114-1 : TPTP v8.1.2. Released v1.2.0.
% 0.10/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.11/0.32 % Computer : n029.cluster.edu
% 0.11/0.32 % Model : x86_64 x86_64
% 0.11/0.32 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32 % Memory : 8042.1875MB
% 0.11/0.32 % OS : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32 % CPULimit : 300
% 0.11/0.32 % WCLimit : 300
% 0.11/0.32 % DateTime : Tue Aug 29 00:18:56 EDT 2023
% 0.11/0.32 % CPUTime :
% 3.08/0.73 Command-line arguments: --no-flatten-goal
% 3.08/0.73
% 3.08/0.73 % SZS status Unsatisfiable
% 3.08/0.73
% 3.08/0.75 % SZS output start Proof
% 3.08/0.75 Axiom 1 (inverse_of_identity): inverse(identity) = identity.
% 3.08/0.75 Axiom 2 (inverse_involution): inverse(inverse(X)) = X.
% 3.08/0.75 Axiom 3 (intersection_commutative): intersection(X, Y) = intersection(Y, X).
% 3.08/0.75 Axiom 4 (negative_part): negative_part(X) = intersection(X, identity).
% 3.08/0.75 Axiom 5 (union_commutative): union(X, Y) = union(Y, X).
% 3.08/0.75 Axiom 6 (positive_part): positive_part(X) = union(X, identity).
% 3.08/0.75 Axiom 7 (left_identity): multiply(identity, X) = X.
% 3.08/0.75 Axiom 8 (left_inverse): multiply(inverse(X), X) = identity.
% 3.08/0.75 Axiom 9 (intersection_associative): intersection(X, intersection(Y, Z)) = intersection(intersection(X, Y), Z).
% 3.08/0.75 Axiom 10 (intersection_union_absorbtion): intersection(union(X, Y), Y) = Y.
% 3.08/0.75 Axiom 11 (union_intersection_absorbtion): union(intersection(X, Y), Y) = Y.
% 3.08/0.75 Axiom 12 (union_associative): union(X, union(Y, Z)) = union(union(X, Y), Z).
% 3.08/0.75 Axiom 13 (inverse_product_lemma): inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
% 3.08/0.75 Axiom 14 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 3.08/0.75 Axiom 15 (multiply_intersection1): multiply(X, intersection(Y, Z)) = intersection(multiply(X, Y), multiply(X, Z)).
% 3.08/0.75 Axiom 16 (multiply_intersection2): multiply(intersection(X, Y), Z) = intersection(multiply(X, Z), multiply(Y, Z)).
% 3.08/0.75 Axiom 17 (multiply_union1): multiply(X, union(Y, Z)) = union(multiply(X, Y), multiply(X, Z)).
% 3.08/0.75 Axiom 18 (multiply_union2): multiply(union(X, Y), Z) = union(multiply(X, Z), multiply(Y, Z)).
% 3.08/0.75
% 3.08/0.75 Lemma 19: multiply(X, identity) = X.
% 3.08/0.75 Proof:
% 3.08/0.75 multiply(X, identity)
% 3.08/0.75 = { by axiom 2 (inverse_involution) R->L }
% 3.08/0.75 inverse(inverse(multiply(X, identity)))
% 3.08/0.75 = { by axiom 13 (inverse_product_lemma) }
% 3.08/0.75 inverse(multiply(inverse(identity), inverse(X)))
% 3.08/0.75 = { by axiom 1 (inverse_of_identity) }
% 3.08/0.75 inverse(multiply(identity, inverse(X)))
% 3.08/0.75 = { by axiom 7 (left_identity) }
% 3.08/0.75 inverse(inverse(X))
% 3.08/0.75 = { by axiom 2 (inverse_involution) }
% 3.08/0.75 X
% 3.08/0.75
% 3.08/0.75 Lemma 20: intersection(identity, X) = negative_part(X).
% 3.08/0.75 Proof:
% 3.08/0.75 intersection(identity, X)
% 3.08/0.75 = { by axiom 3 (intersection_commutative) R->L }
% 3.08/0.75 intersection(X, identity)
% 3.08/0.75 = { by axiom 4 (negative_part) R->L }
% 3.08/0.75 negative_part(X)
% 3.08/0.75
% 3.08/0.75 Lemma 21: union(identity, X) = positive_part(X).
% 3.08/0.75 Proof:
% 3.08/0.75 union(identity, X)
% 3.08/0.75 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.75 union(X, identity)
% 3.08/0.75 = { by axiom 6 (positive_part) R->L }
% 3.08/0.75 positive_part(X)
% 3.08/0.75
% 3.08/0.75 Lemma 22: union(X, intersection(Y, X)) = X.
% 3.08/0.75 Proof:
% 3.08/0.75 union(X, intersection(Y, X))
% 3.08/0.75 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.75 union(intersection(Y, X), X)
% 3.08/0.75 = { by axiom 11 (union_intersection_absorbtion) }
% 3.08/0.75 X
% 3.08/0.75
% 3.08/0.75 Lemma 23: positive_part(negative_part(X)) = identity.
% 3.08/0.75 Proof:
% 3.08/0.75 positive_part(negative_part(X))
% 3.08/0.75 = { by axiom 4 (negative_part) }
% 3.08/0.75 positive_part(intersection(X, identity))
% 3.08/0.75 = { by lemma 21 R->L }
% 3.08/0.75 union(identity, intersection(X, identity))
% 3.08/0.75 = { by lemma 22 }
% 3.08/0.75 identity
% 3.08/0.75
% 3.08/0.75 Lemma 24: negative_part(positive_part(X)) = identity.
% 3.08/0.75 Proof:
% 3.08/0.75 negative_part(positive_part(X))
% 3.08/0.75 = { by axiom 6 (positive_part) }
% 3.08/0.75 negative_part(union(X, identity))
% 3.08/0.75 = { by lemma 20 R->L }
% 3.08/0.75 intersection(identity, union(X, identity))
% 3.08/0.75 = { by axiom 3 (intersection_commutative) R->L }
% 3.08/0.75 intersection(union(X, identity), identity)
% 3.08/0.75 = { by axiom 10 (intersection_union_absorbtion) }
% 3.08/0.75 identity
% 3.08/0.75
% 3.08/0.75 Lemma 25: union(X, positive_part(Y)) = positive_part(union(X, Y)).
% 3.08/0.75 Proof:
% 3.08/0.75 union(X, positive_part(Y))
% 3.08/0.75 = { by axiom 6 (positive_part) }
% 3.08/0.75 union(X, union(Y, identity))
% 3.08/0.75 = { by axiom 12 (union_associative) }
% 3.08/0.75 union(union(X, Y), identity)
% 3.08/0.75 = { by axiom 6 (positive_part) R->L }
% 3.08/0.75 positive_part(union(X, Y))
% 3.08/0.75
% 3.08/0.75 Lemma 26: intersection(X, multiply(Y, X)) = multiply(negative_part(Y), X).
% 3.08/0.75 Proof:
% 3.08/0.75 intersection(X, multiply(Y, X))
% 3.08/0.75 = { by axiom 3 (intersection_commutative) R->L }
% 3.08/0.75 intersection(multiply(Y, X), X)
% 3.08/0.75 = { by axiom 7 (left_identity) R->L }
% 3.08/0.75 intersection(multiply(Y, X), multiply(identity, X))
% 3.08/0.75 = { by axiom 16 (multiply_intersection2) R->L }
% 3.08/0.75 multiply(intersection(Y, identity), X)
% 3.08/0.75 = { by axiom 4 (negative_part) R->L }
% 3.08/0.75 multiply(negative_part(Y), X)
% 3.08/0.75
% 3.08/0.75 Lemma 27: union(X, multiply(X, Y)) = multiply(X, positive_part(Y)).
% 3.08/0.75 Proof:
% 3.08/0.75 union(X, multiply(X, Y))
% 3.08/0.75 = { by lemma 19 R->L }
% 3.08/0.75 union(multiply(X, identity), multiply(X, Y))
% 3.08/0.75 = { by axiom 17 (multiply_union1) R->L }
% 3.08/0.75 multiply(X, union(identity, Y))
% 3.08/0.75 = { by lemma 21 }
% 3.08/0.75 multiply(X, positive_part(Y))
% 3.08/0.75
% 3.08/0.75 Lemma 28: union(X, multiply(Y, X)) = multiply(positive_part(Y), X).
% 3.08/0.75 Proof:
% 3.08/0.75 union(X, multiply(Y, X))
% 3.08/0.75 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.75 union(multiply(Y, X), X)
% 3.08/0.75 = { by axiom 7 (left_identity) R->L }
% 3.08/0.75 union(multiply(Y, X), multiply(identity, X))
% 3.08/0.75 = { by axiom 18 (multiply_union2) R->L }
% 3.08/0.75 multiply(union(Y, identity), X)
% 3.08/0.75 = { by axiom 6 (positive_part) R->L }
% 3.08/0.75 multiply(positive_part(Y), X)
% 3.08/0.75
% 3.08/0.75 Lemma 29: multiply(inverse(X), multiply(X, Y)) = Y.
% 3.08/0.75 Proof:
% 3.08/0.75 multiply(inverse(X), multiply(X, Y))
% 3.08/0.75 = { by axiom 14 (associativity) R->L }
% 3.08/0.75 multiply(multiply(inverse(X), X), Y)
% 3.08/0.75 = { by axiom 8 (left_inverse) }
% 3.08/0.75 multiply(identity, Y)
% 3.08/0.75 = { by axiom 7 (left_identity) }
% 3.08/0.75 Y
% 3.08/0.75
% 3.08/0.75 Lemma 30: union(negative_part(X), negative_part(inverse(X))) = intersection(positive_part(X), positive_part(inverse(X))).
% 3.08/0.75 Proof:
% 3.08/0.75 union(negative_part(X), negative_part(inverse(X)))
% 3.08/0.75 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.75 union(negative_part(inverse(X)), negative_part(X))
% 3.08/0.75 = { by axiom 4 (negative_part) }
% 3.08/0.75 union(negative_part(inverse(X)), intersection(X, identity))
% 3.08/0.75 = { by axiom 8 (left_inverse) R->L }
% 3.08/0.75 union(negative_part(inverse(X)), intersection(X, multiply(inverse(X), X)))
% 3.08/0.75 = { by lemma 26 }
% 3.08/0.75 union(negative_part(inverse(X)), multiply(negative_part(inverse(X)), X))
% 3.08/0.75 = { by lemma 27 }
% 3.08/0.75 multiply(negative_part(inverse(X)), positive_part(X))
% 3.08/0.75 = { by lemma 26 R->L }
% 3.08/0.75 intersection(positive_part(X), multiply(inverse(X), positive_part(X)))
% 3.08/0.75 = { by axiom 6 (positive_part) }
% 3.08/0.75 intersection(positive_part(X), multiply(inverse(X), union(X, identity)))
% 3.08/0.75 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.75 intersection(positive_part(X), multiply(inverse(X), union(identity, X)))
% 3.08/0.75 = { by axiom 17 (multiply_union1) }
% 3.08/0.75 intersection(positive_part(X), union(multiply(inverse(X), identity), multiply(inverse(X), X)))
% 3.08/0.75 = { by axiom 8 (left_inverse) }
% 3.08/0.75 intersection(positive_part(X), union(multiply(inverse(X), identity), identity))
% 3.08/0.75 = { by axiom 6 (positive_part) R->L }
% 3.08/0.75 intersection(positive_part(X), positive_part(multiply(inverse(X), identity)))
% 3.08/0.75 = { by lemma 19 }
% 3.08/0.76 intersection(positive_part(X), positive_part(inverse(X)))
% 3.08/0.76
% 3.08/0.76 Goal 1 (prove_product): multiply(positive_part(a), negative_part(a)) = a.
% 3.08/0.76 Proof:
% 3.08/0.76 multiply(positive_part(a), negative_part(a))
% 3.08/0.76 = { by lemma 28 R->L }
% 3.08/0.76 union(negative_part(a), multiply(a, negative_part(a)))
% 3.08/0.76 = { by lemma 20 R->L }
% 3.08/0.76 union(negative_part(a), multiply(a, intersection(identity, a)))
% 3.08/0.76 = { by axiom 15 (multiply_intersection1) }
% 3.08/0.76 union(negative_part(a), intersection(multiply(a, identity), multiply(a, a)))
% 3.08/0.76 = { by lemma 19 }
% 3.08/0.76 union(negative_part(a), intersection(a, multiply(a, a)))
% 3.08/0.76 = { by lemma 26 }
% 3.08/0.76 union(negative_part(a), multiply(negative_part(a), a))
% 3.08/0.76 = { by lemma 27 }
% 3.08/0.76 multiply(negative_part(a), positive_part(a))
% 3.08/0.76 = { by axiom 2 (inverse_involution) R->L }
% 3.08/0.76 multiply(negative_part(inverse(inverse(a))), positive_part(a))
% 3.08/0.76 = { by lemma 29 R->L }
% 3.08/0.76 multiply(negative_part(multiply(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by lemma 28 R->L }
% 3.08/0.76 multiply(negative_part(multiply(inverse(positive_part(inverse(a))), union(inverse(inverse(a)), multiply(inverse(a), inverse(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by axiom 2 (inverse_involution) R->L }
% 3.08/0.76 multiply(negative_part(multiply(inverse(positive_part(inverse(a))), union(inverse(inverse(a)), multiply(inverse(inverse(inverse(a))), inverse(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by axiom 8 (left_inverse) }
% 3.08/0.76 multiply(negative_part(multiply(inverse(positive_part(inverse(a))), union(inverse(inverse(a)), identity))), positive_part(a))
% 3.08/0.76 = { by axiom 6 (positive_part) R->L }
% 3.08/0.76 multiply(negative_part(multiply(inverse(positive_part(inverse(a))), positive_part(inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by axiom 4 (negative_part) }
% 3.08/0.76 multiply(intersection(multiply(inverse(positive_part(inverse(a))), positive_part(inverse(inverse(a)))), identity), positive_part(a))
% 3.08/0.76 = { by axiom 8 (left_inverse) R->L }
% 3.08/0.76 multiply(intersection(multiply(inverse(positive_part(inverse(a))), positive_part(inverse(inverse(a)))), multiply(inverse(positive_part(inverse(a))), positive_part(inverse(a)))), positive_part(a))
% 3.08/0.76 = { by axiom 15 (multiply_intersection1) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), intersection(positive_part(inverse(inverse(a))), positive_part(inverse(a)))), positive_part(a))
% 3.08/0.76 = { by axiom 3 (intersection_commutative) }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), intersection(positive_part(inverse(a)), positive_part(inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by lemma 30 R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), negative_part(inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(inverse(a))), negative_part(inverse(a)))), positive_part(a))
% 3.08/0.76 = { by lemma 22 R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(union(negative_part(inverse(inverse(a))), negative_part(inverse(a))), intersection(identity, union(negative_part(inverse(inverse(a))), negative_part(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by axiom 12 (union_associative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(inverse(a))), union(negative_part(inverse(a)), intersection(identity, union(negative_part(inverse(inverse(a))), negative_part(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by lemma 20 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(inverse(a))), union(negative_part(inverse(a)), negative_part(union(negative_part(inverse(inverse(a))), negative_part(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(union(negative_part(inverse(a)), negative_part(union(negative_part(inverse(inverse(a))), negative_part(inverse(a))))), negative_part(inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by axiom 12 (union_associative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(union(negative_part(inverse(inverse(a))), negative_part(inverse(a)))), negative_part(inverse(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by axiom 5 (union_commutative) }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), negative_part(union(negative_part(inverse(inverse(a))), negative_part(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by axiom 5 (union_commutative) }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), negative_part(union(negative_part(inverse(a)), negative_part(inverse(inverse(a)))))))), positive_part(a))
% 3.08/0.76 = { by lemma 30 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), negative_part(intersection(positive_part(inverse(a)), positive_part(inverse(inverse(a)))))))), positive_part(a))
% 3.08/0.76 = { by axiom 3 (intersection_commutative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), negative_part(intersection(positive_part(inverse(inverse(a))), positive_part(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by axiom 4 (negative_part) }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), intersection(intersection(positive_part(inverse(inverse(a))), positive_part(inverse(a))), identity)))), positive_part(a))
% 3.08/0.76 = { by axiom 9 (intersection_associative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), intersection(positive_part(inverse(inverse(a))), intersection(positive_part(inverse(a)), identity))))), positive_part(a))
% 3.08/0.76 = { by axiom 4 (negative_part) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), intersection(positive_part(inverse(inverse(a))), negative_part(positive_part(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by lemma 24 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), intersection(positive_part(inverse(inverse(a))), identity)))), positive_part(a))
% 3.08/0.76 = { by axiom 4 (negative_part) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), negative_part(positive_part(inverse(inverse(a))))))), positive_part(a))
% 3.08/0.76 = { by lemma 24 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), union(negative_part(inverse(inverse(a))), identity))), positive_part(a))
% 3.08/0.76 = { by axiom 6 (positive_part) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(a)), positive_part(negative_part(inverse(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by lemma 25 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), positive_part(union(negative_part(inverse(a)), negative_part(inverse(inverse(a)))))), positive_part(a))
% 3.08/0.76 = { by axiom 5 (union_commutative) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), positive_part(union(negative_part(inverse(inverse(a))), negative_part(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by lemma 25 R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(inverse(a))), positive_part(negative_part(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by lemma 23 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), union(negative_part(inverse(inverse(a))), identity)), positive_part(a))
% 3.08/0.76 = { by axiom 6 (positive_part) R->L }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), positive_part(negative_part(inverse(inverse(a))))), positive_part(a))
% 3.08/0.76 = { by lemma 23 }
% 3.08/0.76 multiply(multiply(inverse(positive_part(inverse(a))), identity), positive_part(a))
% 3.08/0.76 = { by lemma 19 }
% 3.08/0.76 multiply(inverse(positive_part(inverse(a))), positive_part(a))
% 3.08/0.76 = { by axiom 6 (positive_part) }
% 3.08/0.76 multiply(inverse(positive_part(inverse(a))), union(a, identity))
% 3.08/0.76 = { by axiom 8 (left_inverse) R->L }
% 3.08/0.76 multiply(inverse(positive_part(inverse(a))), union(a, multiply(inverse(a), a)))
% 3.08/0.76 = { by lemma 28 }
% 3.08/0.76 multiply(inverse(positive_part(inverse(a))), multiply(positive_part(inverse(a)), a))
% 3.08/0.76 = { by lemma 29 }
% 3.08/0.76 a
% 3.08/0.76 % SZS output end Proof
% 3.08/0.76
% 3.08/0.76 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------