TSTP Solution File: GRP183-4 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP183-4 : TPTP v8.1.2. Bugfixed v1.2.1.
% 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 : n007.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 01:17:38 EDT 2023
% Result : Unsatisfiable 4.29s 0.89s
% Output : Proof 4.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : GRP183-4 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n007.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Tue Aug 29 00:24:13 EDT 2023
% 0.12/0.34 % CPUTime :
% 4.29/0.89 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 4.29/0.89
% 4.29/0.89 % SZS status Unsatisfiable
% 4.29/0.89
% 4.56/0.94 % SZS output start Proof
% 4.56/0.94 Axiom 1 (p20x_1): inverse(identity) = identity.
% 4.56/0.94 Axiom 2 (left_identity): multiply(identity, X) = X.
% 4.56/0.94 Axiom 3 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 4.56/0.94 Axiom 4 (idempotence_of_gld): greatest_lower_bound(X, X) = X.
% 4.56/0.94 Axiom 5 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 4.56/0.94 Axiom 6 (p20x_2): inverse(inverse(X)) = X.
% 4.56/0.94 Axiom 7 (left_inverse): multiply(inverse(X), X) = identity.
% 4.56/0.94 Axiom 8 (p20x_3): inverse(multiply(X, Y)) = multiply(inverse(Y), inverse(X)).
% 4.56/0.94 Axiom 9 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 4.56/0.94 Axiom 10 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 4.56/0.94 Axiom 11 (associativity_of_lub): least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z).
% 4.56/0.94 Axiom 12 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 4.56/0.94 Axiom 13 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
% 4.56/0.94 Axiom 14 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 4.56/0.94 Axiom 15 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 4.56/0.94 Axiom 16 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 4.56/0.94 Axiom 17 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 4.56/0.94
% 4.56/0.94 Lemma 18: multiply(X, identity) = X.
% 4.56/0.94 Proof:
% 4.56/0.94 multiply(X, identity)
% 4.56/0.94 = { by axiom 6 (p20x_2) R->L }
% 4.56/0.94 inverse(inverse(multiply(X, identity)))
% 4.56/0.94 = { by axiom 8 (p20x_3) }
% 4.56/0.94 inverse(multiply(inverse(identity), inverse(X)))
% 4.56/0.94 = { by axiom 1 (p20x_1) }
% 4.56/0.94 inverse(multiply(identity, inverse(X)))
% 4.56/0.94 = { by axiom 2 (left_identity) }
% 4.56/0.94 inverse(inverse(X))
% 4.56/0.94 = { by axiom 6 (p20x_2) }
% 4.56/0.94 X
% 4.56/0.94
% 4.56/0.94 Lemma 19: multiply(X, inverse(X)) = identity.
% 4.56/0.94 Proof:
% 4.56/0.94 multiply(X, inverse(X))
% 4.56/0.94 = { by axiom 6 (p20x_2) R->L }
% 4.56/0.94 multiply(inverse(inverse(X)), inverse(X))
% 4.56/0.94 = { by axiom 7 (left_inverse) }
% 4.56/0.94 identity
% 4.56/0.94
% 4.56/0.94 Lemma 20: greatest_lower_bound(X, least_upper_bound(Y, X)) = X.
% 4.56/0.94 Proof:
% 4.56/0.94 greatest_lower_bound(X, least_upper_bound(Y, X))
% 4.56/0.94 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.94 greatest_lower_bound(X, least_upper_bound(X, Y))
% 4.56/0.94 = { by axiom 12 (glb_absorbtion) }
% 4.56/0.94 X
% 4.56/0.95
% 4.56/0.95 Lemma 21: least_upper_bound(X, greatest_lower_bound(Y, X)) = X.
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(X, greatest_lower_bound(Y, X))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) R->L }
% 4.56/0.95 least_upper_bound(X, greatest_lower_bound(X, Y))
% 4.56/0.95 = { by axiom 10 (lub_absorbtion) }
% 4.56/0.95 X
% 4.56/0.95
% 4.56/0.95 Lemma 22: least_upper_bound(Y, least_upper_bound(Z, X)) = least_upper_bound(X, least_upper_bound(Y, Z)).
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(Y, least_upper_bound(Z, X))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 least_upper_bound(least_upper_bound(Z, X), Y)
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 least_upper_bound(least_upper_bound(X, Z), Y)
% 4.56/0.95 = { by axiom 11 (associativity_of_lub) R->L }
% 4.56/0.95 least_upper_bound(X, least_upper_bound(Z, Y))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 least_upper_bound(X, least_upper_bound(Y, Z))
% 4.56/0.95
% 4.56/0.95 Lemma 23: least_upper_bound(Z, least_upper_bound(Y, X)) = least_upper_bound(X, least_upper_bound(Y, Z)).
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(Z, least_upper_bound(Y, X))
% 4.56/0.95 = { by lemma 22 }
% 4.56/0.95 least_upper_bound(X, least_upper_bound(Z, Y))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 least_upper_bound(X, least_upper_bound(Y, Z))
% 4.56/0.95
% 4.56/0.95 Lemma 24: multiply(X, multiply(inverse(X), Y)) = Y.
% 4.56/0.95 Proof:
% 4.56/0.95 multiply(X, multiply(inverse(X), Y))
% 4.56/0.95 = { by axiom 9 (associativity) R->L }
% 4.56/0.95 multiply(multiply(X, inverse(X)), Y)
% 4.56/0.95 = { by lemma 19 }
% 4.56/0.95 multiply(identity, Y)
% 4.56/0.95 = { by axiom 2 (left_identity) }
% 4.56/0.95 Y
% 4.56/0.95
% 4.56/0.95 Lemma 25: multiply(inverse(X), multiply(X, Y)) = Y.
% 4.56/0.95 Proof:
% 4.56/0.95 multiply(inverse(X), multiply(X, Y))
% 4.56/0.95 = { by axiom 9 (associativity) R->L }
% 4.56/0.95 multiply(multiply(inverse(X), X), Y)
% 4.56/0.95 = { by axiom 7 (left_inverse) }
% 4.56/0.95 multiply(identity, Y)
% 4.56/0.95 = { by axiom 2 (left_identity) }
% 4.56/0.95 Y
% 4.56/0.95
% 4.56/0.95 Lemma 26: least_upper_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), least_upper_bound(Y, X)).
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(identity, multiply(inverse(X), Y))
% 4.56/0.95 = { by axiom 7 (left_inverse) R->L }
% 4.56/0.95 least_upper_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
% 4.56/0.95 = { by axiom 14 (monotony_lub1) R->L }
% 4.56/0.95 multiply(inverse(X), least_upper_bound(X, Y))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 multiply(inverse(X), least_upper_bound(Y, X))
% 4.56/0.95
% 4.56/0.95 Lemma 27: multiply(inverse(X), least_upper_bound(X, identity)) = least_upper_bound(inverse(X), identity).
% 4.56/0.95 Proof:
% 4.56/0.95 multiply(inverse(X), least_upper_bound(X, identity))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 multiply(inverse(X), least_upper_bound(identity, X))
% 4.56/0.95 = { by lemma 26 R->L }
% 4.56/0.95 least_upper_bound(identity, multiply(inverse(X), identity))
% 4.56/0.95 = { by lemma 18 }
% 4.56/0.95 least_upper_bound(identity, inverse(X))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 least_upper_bound(inverse(X), identity)
% 4.56/0.95
% 4.56/0.95 Lemma 28: multiply(least_upper_bound(inverse(X), identity), X) = least_upper_bound(X, identity).
% 4.56/0.95 Proof:
% 4.56/0.95 multiply(least_upper_bound(inverse(X), identity), X)
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 multiply(least_upper_bound(identity, inverse(X)), X)
% 4.56/0.95 = { by axiom 15 (monotony_lub2) }
% 4.56/0.95 least_upper_bound(multiply(identity, X), multiply(inverse(X), X))
% 4.56/0.95 = { by axiom 2 (left_identity) }
% 4.56/0.95 least_upper_bound(X, multiply(inverse(X), X))
% 4.56/0.95 = { by axiom 7 (left_inverse) }
% 4.56/0.95 least_upper_bound(X, identity)
% 4.56/0.95
% 4.56/0.95 Lemma 29: least_upper_bound(identity, multiply(X, inverse(Y))) = multiply(least_upper_bound(X, Y), inverse(Y)).
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(identity, multiply(X, inverse(Y)))
% 4.56/0.95 = { by lemma 19 R->L }
% 4.56/0.95 least_upper_bound(multiply(Y, inverse(Y)), multiply(X, inverse(Y)))
% 4.56/0.95 = { by axiom 15 (monotony_lub2) R->L }
% 4.56/0.95 multiply(least_upper_bound(Y, X), inverse(Y))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 multiply(least_upper_bound(X, Y), inverse(Y))
% 4.56/0.95
% 4.56/0.95 Lemma 30: least_upper_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))) = greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)).
% 4.56/0.95 Proof:
% 4.56/0.95 least_upper_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), X)
% 4.56/0.95 = { by lemma 20 R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(X, least_upper_bound(Z, X)))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), X))
% 4.56/0.95 = { by lemma 20 R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), greatest_lower_bound(X, least_upper_bound(Y, X))))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(least_upper_bound(Z, X), greatest_lower_bound(least_upper_bound(Y, X), X)))
% 4.56/0.95 = { by axiom 13 (associativity_of_glb) }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(Z, X), least_upper_bound(Y, X)), X))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) R->L }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), X))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) }
% 4.56/0.95 least_upper_bound(greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X)), greatest_lower_bound(X, greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))))
% 4.56/0.95 = { by lemma 21 }
% 4.56/0.95 greatest_lower_bound(least_upper_bound(Y, X), least_upper_bound(Z, X))
% 4.56/0.95
% 4.56/0.95 Lemma 31: multiply(inverse(greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), identity))), greatest_lower_bound(identity, inverse(X))) = inverse(least_upper_bound(X, identity)).
% 4.56/0.95 Proof:
% 4.56/0.95 multiply(inverse(greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(inverse(X), identity))), greatest_lower_bound(identity, inverse(X)))
% 4.56/0.95 = { by lemma 27 R->L }
% 4.56/0.95 multiply(inverse(greatest_lower_bound(least_upper_bound(X, identity), multiply(inverse(X), least_upper_bound(X, identity)))), greatest_lower_bound(identity, inverse(X)))
% 4.56/0.95 = { by axiom 2 (left_identity) R->L }
% 4.56/0.95 multiply(inverse(greatest_lower_bound(multiply(identity, least_upper_bound(X, identity)), multiply(inverse(X), least_upper_bound(X, identity)))), greatest_lower_bound(identity, inverse(X)))
% 4.56/0.95 = { by axiom 17 (monotony_glb2) R->L }
% 4.56/0.95 multiply(inverse(multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity))), greatest_lower_bound(identity, inverse(X)))
% 4.56/0.95 = { by axiom 6 (p20x_2) R->L }
% 4.56/0.95 multiply(inverse(multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity))), inverse(inverse(greatest_lower_bound(identity, inverse(X)))))
% 4.56/0.95 = { by axiom 8 (p20x_3) R->L }
% 4.56/0.95 inverse(multiply(inverse(greatest_lower_bound(identity, inverse(X))), multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity))))
% 4.56/0.95 = { by lemma 25 }
% 4.56/0.95 inverse(least_upper_bound(X, identity))
% 4.56/0.95
% 4.56/0.95 Goal 1 (prove_20x): greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)) = identity.
% 4.56/0.95 Proof:
% 4.56/0.95 greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))
% 4.56/0.95 = { by axiom 6 (p20x_2) R->L }
% 4.56/0.95 inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))
% 4.56/0.95 = { by lemma 25 R->L }
% 4.56/0.95 inverse(multiply(inverse(inverse(least_upper_bound(a, identity))), multiply(inverse(least_upper_bound(a, identity)), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))
% 4.56/0.95 = { by axiom 8 (p20x_3) R->L }
% 4.56/0.95 inverse(multiply(inverse(inverse(least_upper_bound(a, identity))), inverse(multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity)))))
% 4.56/0.95 = { by axiom 6 (p20x_2) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(a, identity)))))
% 4.56/0.95 = { by axiom 6 (p20x_2) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, identity)))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(identity, a)))))
% 4.56/0.95 = { by axiom 10 (lub_absorbtion) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(identity, greatest_lower_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))), a)))))
% 4.56/0.95 = { by axiom 11 (associativity_of_lub) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(identity, least_upper_bound(greatest_lower_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), a))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(identity, least_upper_bound(a, greatest_lower_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))))
% 4.56/0.95 = { by lemma 23 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(greatest_lower_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, identity))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))))))))
% 4.56/0.95 = { by axiom 5 (symmetry_of_glb) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), identity))))))
% 4.56/0.95 = { by lemma 19 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))))
% 4.56/0.95 = { by lemma 30 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))))
% 4.56/0.95 = { by lemma 29 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), least_upper_bound(identity, multiply(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))))))))))
% 4.56/0.95 = { by lemma 18 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), least_upper_bound(multiply(identity, identity), multiply(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))))))))))
% 4.56/0.95 = { by axiom 14 (monotony_lub1) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(identity, least_upper_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))))))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(identity, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), identity))))))))
% 4.56/0.95 = { by axiom 2 (left_identity) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), identity)))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), least_upper_bound(identity, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))))
% 4.56/0.95 = { by lemma 20 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, identity), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 24 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(inverse(a), least_upper_bound(a, identity))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 27 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, least_upper_bound(inverse(a), identity)), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 18 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(inverse(a), identity), identity)), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 19 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(inverse(a), identity), multiply(least_upper_bound(a, identity), inverse(least_upper_bound(a, identity))))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 9 (associativity) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(multiply(least_upper_bound(inverse(a), identity), least_upper_bound(a, identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 14 (monotony_lub1) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(multiply(least_upper_bound(inverse(a), identity), a), multiply(least_upper_bound(inverse(a), identity), identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 28 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(a, identity), multiply(least_upper_bound(inverse(a), identity), identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 18 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(inverse(a), identity), least_upper_bound(a, identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 23 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(identity, least_upper_bound(a, least_upper_bound(inverse(a), identity))), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 22 }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(identity, least_upper_bound(identity, least_upper_bound(a, inverse(a)))), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by lemma 23 R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(identity, least_upper_bound(inverse(a), least_upper_bound(a, identity))), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(identity, least_upper_bound(least_upper_bound(a, identity), inverse(a))), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 11 (associativity_of_lub) }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(identity, least_upper_bound(a, identity)), inverse(a)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.95 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.95 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), identity), inverse(a)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 20 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(least_upper_bound(a, identity), greatest_lower_bound(identity, least_upper_bound(a, identity))), inverse(a)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 21 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(least_upper_bound(a, identity), inverse(a)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(least_upper_bound(inverse(a), least_upper_bound(a, identity)), inverse(least_upper_bound(a, identity)))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 29 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, least_upper_bound(identity, multiply(inverse(a), inverse(least_upper_bound(a, identity))))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 26 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(inverse(a), least_upper_bound(inverse(least_upper_bound(a, identity)), a))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(multiply(a, multiply(inverse(a), least_upper_bound(a, inverse(least_upper_bound(a, identity))))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 24 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(least_upper_bound(a, inverse(least_upper_bound(a, identity))), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by axiom 11 (associativity_of_lub) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(least_upper_bound(a, identity)), inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))))))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), inverse(least_upper_bound(a, identity))))))))
% 4.56/0.96 = { by lemma 31 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(identity, inverse(a)))))))))
% 4.56/0.96 = { by axiom 4 (idempotence_of_gld) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(greatest_lower_bound(identity, identity), inverse(a)))))))))
% 4.56/0.96 = { by axiom 13 (associativity_of_glb) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(identity, greatest_lower_bound(identity, inverse(a))))))))))
% 4.56/0.96 = { by axiom 16 (monotony_glb1) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), identity), multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(identity, inverse(a))))))))))
% 4.56/0.96 = { by lemma 18 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), multiply(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(identity, inverse(a))))))))))
% 4.56/0.96 = { by lemma 31 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, least_upper_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), greatest_lower_bound(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), inverse(least_upper_bound(a, identity)))))))))
% 4.56/0.96 = { by axiom 10 (lub_absorbtion) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), least_upper_bound(a, inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))))))
% 4.56/0.96 = { by lemma 26 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(least_upper_bound(identity, multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), a)))))
% 4.56/0.96 = { by axiom 7 (left_inverse) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(least_upper_bound(multiply(inverse(a), a), multiply(inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), a)))))
% 4.56/0.96 = { by axiom 15 (monotony_lub2) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(inverse(a), inverse(inverse(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))))), a))))
% 4.56/0.96 = { by axiom 6 (p20x_2) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(inverse(a), greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), a))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), inverse(a)), a))))
% 4.56/0.96 = { by lemma 30 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(least_upper_bound(identity, greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), inverse(a)), a))))
% 4.56/0.96 = { by axiom 11 (associativity_of_lub) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(identity, least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), inverse(a))), a))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(identity, least_upper_bound(inverse(a), greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)))), a))))
% 4.56/0.96 = { by lemma 23 R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity)), least_upper_bound(inverse(a), identity)), a))))
% 4.56/0.96 = { by axiom 3 (symmetry_of_lub) R->L }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(least_upper_bound(inverse(a), identity), greatest_lower_bound(least_upper_bound(a, identity), least_upper_bound(inverse(a), identity))), a))))
% 4.56/0.96 = { by axiom 5 (symmetry_of_glb) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(least_upper_bound(inverse(a), identity), greatest_lower_bound(least_upper_bound(inverse(a), identity), least_upper_bound(a, identity))), a))))
% 4.56/0.96 = { by axiom 10 (lub_absorbtion) }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(multiply(least_upper_bound(inverse(a), identity), a))))
% 4.56/0.96 = { by lemma 28 }
% 4.56/0.96 inverse(multiply(least_upper_bound(a, identity), inverse(least_upper_bound(a, identity))))
% 4.56/0.96 = { by lemma 19 }
% 4.56/0.96 inverse(identity)
% 4.56/0.96 = { by axiom 1 (p20x_1) }
% 4.56/0.96 identity
% 4.56/0.96 % SZS output end Proof
% 4.56/0.96
% 4.56/0.96 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------