TSTP Solution File: GRP187-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP187-1 : 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 : n011.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 01:17:41 EDT 2023
% Result : Unsatisfiable 75.43s 10.10s
% Output : Proof 76.28s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : GRP187-1 : TPTP v8.1.2. Bugfixed v1.2.1.
% 0.12/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n011.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 : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Tue Aug 29 01:11:11 EDT 2023
% 0.14/0.35 % CPUTime :
% 75.43/10.10 Command-line arguments: --ground-connectedness --complete-subsets
% 75.43/10.10
% 75.43/10.10 % SZS status Unsatisfiable
% 75.43/10.10
% 76.28/10.21 % SZS output start Proof
% 76.28/10.21 Axiom 1 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
% 76.28/10.21 Axiom 2 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
% 76.28/10.21 Axiom 3 (left_identity): multiply(identity, X) = X.
% 76.28/10.21 Axiom 4 (left_inverse): multiply(inverse(X), X) = identity.
% 76.28/10.21 Axiom 5 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
% 76.28/10.21 Axiom 6 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
% 76.28/10.21 Axiom 7 (lub_absorbtion): least_upper_bound(X, greatest_lower_bound(X, Y)) = X.
% 76.28/10.21 Axiom 8 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
% 76.28/10.21 Axiom 9 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
% 76.28/10.21 Axiom 10 (monotony_glb2): multiply(greatest_lower_bound(X, Y), Z) = greatest_lower_bound(multiply(X, Z), multiply(Y, Z)).
% 76.28/10.21 Axiom 11 (monotony_lub1): multiply(X, least_upper_bound(Y, Z)) = least_upper_bound(multiply(X, Y), multiply(X, Z)).
% 76.28/10.21 Axiom 12 (monotony_lub2): multiply(least_upper_bound(X, Y), Z) = least_upper_bound(multiply(X, Z), multiply(Y, Z)).
% 76.28/10.21 Axiom 13 (p33_1): greatest_lower_bound(least_upper_bound(a, inverse(a)), least_upper_bound(b, inverse(b))) = identity.
% 76.28/10.21
% 76.28/10.21 Lemma 14: multiply(inverse(X), multiply(X, Y)) = Y.
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(inverse(X), multiply(X, Y))
% 76.28/10.21 = { by axiom 8 (associativity) R->L }
% 76.28/10.21 multiply(multiply(inverse(X), X), Y)
% 76.28/10.21 = { by axiom 4 (left_inverse) }
% 76.28/10.21 multiply(identity, Y)
% 76.28/10.21 = { by axiom 3 (left_identity) }
% 76.28/10.21 Y
% 76.28/10.21
% 76.28/10.21 Lemma 15: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(inverse(inverse(X)), Y)
% 76.28/10.21 = { by lemma 14 R->L }
% 76.28/10.21 multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
% 76.28/10.21 = { by lemma 14 }
% 76.28/10.21 multiply(X, Y)
% 76.28/10.21
% 76.28/10.21 Lemma 16: multiply(X, identity) = X.
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(X, identity)
% 76.28/10.21 = { by lemma 15 R->L }
% 76.28/10.21 multiply(inverse(inverse(X)), identity)
% 76.28/10.21 = { by axiom 4 (left_inverse) R->L }
% 76.28/10.21 multiply(inverse(inverse(X)), multiply(inverse(X), X))
% 76.28/10.21 = { by lemma 14 }
% 76.28/10.21 X
% 76.28/10.21
% 76.28/10.21 Lemma 17: multiply(X, inverse(X)) = identity.
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(X, inverse(X))
% 76.28/10.21 = { by lemma 15 R->L }
% 76.28/10.21 multiply(inverse(inverse(X)), inverse(X))
% 76.28/10.21 = { by axiom 4 (left_inverse) }
% 76.28/10.21 identity
% 76.28/10.21
% 76.28/10.21 Lemma 18: multiply(X, greatest_lower_bound(Y, identity)) = greatest_lower_bound(X, multiply(X, Y)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(X, greatest_lower_bound(Y, identity))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 multiply(X, greatest_lower_bound(identity, Y))
% 76.28/10.21 = { by axiom 9 (monotony_glb1) }
% 76.28/10.21 greatest_lower_bound(multiply(X, identity), multiply(X, Y))
% 76.28/10.21 = { by lemma 16 }
% 76.28/10.21 greatest_lower_bound(X, multiply(X, Y))
% 76.28/10.21
% 76.28/10.21 Lemma 19: multiply(X, greatest_lower_bound(identity, Y)) = greatest_lower_bound(X, multiply(X, Y)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(X, greatest_lower_bound(identity, Y))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 multiply(X, greatest_lower_bound(Y, identity))
% 76.28/10.21 = { by lemma 18 }
% 76.28/10.21 greatest_lower_bound(X, multiply(X, Y))
% 76.28/10.21
% 76.28/10.21 Lemma 20: multiply(greatest_lower_bound(X, identity), Y) = greatest_lower_bound(Y, multiply(X, Y)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(greatest_lower_bound(X, identity), Y)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 multiply(greatest_lower_bound(identity, X), Y)
% 76.28/10.21 = { by axiom 10 (monotony_glb2) }
% 76.28/10.21 greatest_lower_bound(multiply(identity, Y), multiply(X, Y))
% 76.28/10.21 = { by axiom 3 (left_identity) }
% 76.28/10.21 greatest_lower_bound(Y, multiply(X, Y))
% 76.28/10.21
% 76.28/10.21 Lemma 21: multiply(greatest_lower_bound(identity, Y), X) = greatest_lower_bound(X, multiply(Y, X)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(greatest_lower_bound(identity, Y), X)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 multiply(greatest_lower_bound(Y, identity), X)
% 76.28/10.21 = { by lemma 20 }
% 76.28/10.21 greatest_lower_bound(X, multiply(Y, X))
% 76.28/10.21
% 76.28/10.21 Lemma 22: greatest_lower_bound(X, least_upper_bound(Y, X)) = X.
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(X, least_upper_bound(Y, X))
% 76.28/10.21 = { by axiom 2 (symmetry_of_lub) R->L }
% 76.28/10.21 greatest_lower_bound(X, least_upper_bound(X, Y))
% 76.28/10.21 = { by axiom 5 (glb_absorbtion) }
% 76.28/10.21 X
% 76.28/10.21
% 76.28/10.21 Lemma 23: greatest_lower_bound(Y, greatest_lower_bound(X, Z)) = greatest_lower_bound(X, greatest_lower_bound(Y, Z)).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(Y, greatest_lower_bound(X, Z))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 greatest_lower_bound(greatest_lower_bound(X, Z), Y)
% 76.28/10.21 = { by axiom 6 (associativity_of_glb) R->L }
% 76.28/10.21 greatest_lower_bound(X, greatest_lower_bound(Z, Y))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(X, greatest_lower_bound(Y, Z))
% 76.28/10.21
% 76.28/10.21 Lemma 24: greatest_lower_bound(X, greatest_lower_bound(Y, least_upper_bound(X, Z))) = greatest_lower_bound(X, Y).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(X, greatest_lower_bound(Y, least_upper_bound(X, Z)))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 greatest_lower_bound(X, greatest_lower_bound(least_upper_bound(X, Z), Y))
% 76.28/10.21 = { by axiom 6 (associativity_of_glb) }
% 76.28/10.21 greatest_lower_bound(greatest_lower_bound(X, least_upper_bound(X, Z)), Y)
% 76.28/10.21 = { by axiom 5 (glb_absorbtion) }
% 76.28/10.21 greatest_lower_bound(X, Y)
% 76.28/10.21
% 76.28/10.21 Lemma 25: greatest_lower_bound(b, least_upper_bound(a, inverse(a))) = greatest_lower_bound(identity, b).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(b, least_upper_bound(a, inverse(a)))
% 76.28/10.21 = { by lemma 24 R->L }
% 76.28/10.21 greatest_lower_bound(b, greatest_lower_bound(least_upper_bound(a, inverse(a)), least_upper_bound(b, inverse(b))))
% 76.28/10.21 = { by axiom 13 (p33_1) }
% 76.28/10.21 greatest_lower_bound(b, identity)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(identity, b)
% 76.28/10.21
% 76.28/10.21 Lemma 26: greatest_lower_bound(identity, greatest_lower_bound(a, b)) = greatest_lower_bound(a, b).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(identity, greatest_lower_bound(a, b))
% 76.28/10.21 = { by lemma 23 }
% 76.28/10.21 greatest_lower_bound(a, greatest_lower_bound(identity, b))
% 76.28/10.21 = { by lemma 25 R->L }
% 76.28/10.21 greatest_lower_bound(a, greatest_lower_bound(b, least_upper_bound(a, inverse(a))))
% 76.28/10.21 = { by lemma 24 }
% 76.28/10.21 greatest_lower_bound(a, b)
% 76.28/10.21
% 76.28/10.21 Lemma 27: greatest_lower_bound(X, greatest_lower_bound(Y, least_upper_bound(Z, X))) = greatest_lower_bound(X, Y).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(X, greatest_lower_bound(Y, least_upper_bound(Z, X)))
% 76.28/10.21 = { by lemma 23 }
% 76.28/10.21 greatest_lower_bound(Y, greatest_lower_bound(X, least_upper_bound(Z, X)))
% 76.28/10.21 = { by lemma 22 }
% 76.28/10.21 greatest_lower_bound(Y, X)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(X, Y)
% 76.28/10.21
% 76.28/10.21 Lemma 28: greatest_lower_bound(a, least_upper_bound(b, inverse(b))) = greatest_lower_bound(identity, a).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(a, least_upper_bound(b, inverse(b)))
% 76.28/10.21 = { by lemma 24 R->L }
% 76.28/10.21 greatest_lower_bound(a, greatest_lower_bound(least_upper_bound(b, inverse(b)), least_upper_bound(a, inverse(a))))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(a, greatest_lower_bound(least_upper_bound(a, inverse(a)), least_upper_bound(b, inverse(b))))
% 76.28/10.21 = { by axiom 13 (p33_1) }
% 76.28/10.21 greatest_lower_bound(a, identity)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(identity, a)
% 76.28/10.21
% 76.28/10.21 Lemma 29: multiply(greatest_lower_bound(multiply(X, Y), identity), inverse(Y)) = greatest_lower_bound(X, inverse(Y)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(greatest_lower_bound(multiply(X, Y), identity), inverse(Y))
% 76.28/10.21 = { by lemma 20 }
% 76.28/10.21 greatest_lower_bound(inverse(Y), multiply(multiply(X, Y), inverse(Y)))
% 76.28/10.21 = { by axiom 8 (associativity) }
% 76.28/10.21 greatest_lower_bound(inverse(Y), multiply(X, multiply(Y, inverse(Y))))
% 76.28/10.21 = { by lemma 17 }
% 76.28/10.21 greatest_lower_bound(inverse(Y), multiply(X, identity))
% 76.28/10.21 = { by lemma 16 }
% 76.28/10.21 greatest_lower_bound(inverse(Y), X)
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.21 greatest_lower_bound(X, inverse(Y))
% 76.28/10.21
% 76.28/10.21 Lemma 30: multiply(X, greatest_lower_bound(Y, inverse(X))) = greatest_lower_bound(identity, multiply(X, Y)).
% 76.28/10.21 Proof:
% 76.28/10.21 multiply(X, greatest_lower_bound(Y, inverse(X)))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.21 multiply(X, greatest_lower_bound(inverse(X), Y))
% 76.28/10.21 = { by lemma 15 R->L }
% 76.28/10.21 multiply(inverse(inverse(X)), greatest_lower_bound(inverse(X), Y))
% 76.28/10.21 = { by axiom 9 (monotony_glb1) }
% 76.28/10.21 greatest_lower_bound(multiply(inverse(inverse(X)), inverse(X)), multiply(inverse(inverse(X)), Y))
% 76.28/10.21 = { by axiom 4 (left_inverse) }
% 76.28/10.21 greatest_lower_bound(identity, multiply(inverse(inverse(X)), Y))
% 76.28/10.21 = { by lemma 15 }
% 76.28/10.21 greatest_lower_bound(identity, multiply(X, Y))
% 76.28/10.21
% 76.28/10.21 Lemma 31: greatest_lower_bound(identity, greatest_lower_bound(inverse(a), inverse(b))) = greatest_lower_bound(inverse(a), inverse(b)).
% 76.28/10.21 Proof:
% 76.28/10.21 greatest_lower_bound(identity, greatest_lower_bound(inverse(a), inverse(b)))
% 76.28/10.21 = { by lemma 23 }
% 76.28/10.21 greatest_lower_bound(inverse(a), greatest_lower_bound(identity, inverse(b)))
% 76.28/10.21 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(inverse(a), greatest_lower_bound(inverse(b), identity))
% 76.28/10.22 = { by axiom 13 (p33_1) R->L }
% 76.28/10.22 greatest_lower_bound(inverse(a), greatest_lower_bound(inverse(b), greatest_lower_bound(least_upper_bound(a, inverse(a)), least_upper_bound(b, inverse(b)))))
% 76.28/10.22 = { by lemma 27 }
% 76.28/10.22 greatest_lower_bound(inverse(a), greatest_lower_bound(inverse(b), least_upper_bound(a, inverse(a))))
% 76.28/10.22 = { by lemma 27 }
% 76.28/10.22 greatest_lower_bound(inverse(a), inverse(b))
% 76.28/10.22
% 76.28/10.22 Lemma 32: multiply(X, inverse(multiply(Y, X))) = inverse(Y).
% 76.28/10.22 Proof:
% 76.28/10.22 multiply(X, inverse(multiply(Y, X)))
% 76.28/10.22 = { by lemma 14 R->L }
% 76.28/10.22 multiply(inverse(Y), multiply(Y, multiply(X, inverse(multiply(Y, X)))))
% 76.28/10.22 = { by axiom 8 (associativity) R->L }
% 76.28/10.22 multiply(inverse(Y), multiply(multiply(Y, X), inverse(multiply(Y, X))))
% 76.28/10.22 = { by lemma 17 }
% 76.28/10.22 multiply(inverse(Y), identity)
% 76.28/10.22 = { by lemma 16 }
% 76.28/10.22 inverse(Y)
% 76.28/10.22
% 76.28/10.22 Lemma 33: multiply(inverse(X), greatest_lower_bound(Y, multiply(X, Z))) = greatest_lower_bound(Z, multiply(inverse(X), Y)).
% 76.28/10.22 Proof:
% 76.28/10.22 multiply(inverse(X), greatest_lower_bound(Y, multiply(X, Z)))
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 multiply(inverse(X), greatest_lower_bound(multiply(X, Z), Y))
% 76.28/10.22 = { by axiom 9 (monotony_glb1) }
% 76.28/10.22 greatest_lower_bound(multiply(inverse(X), multiply(X, Z)), multiply(inverse(X), Y))
% 76.28/10.22 = { by lemma 14 }
% 76.28/10.22 greatest_lower_bound(Z, multiply(inverse(X), Y))
% 76.28/10.22
% 76.28/10.22 Lemma 34: greatest_lower_bound(multiply(a, b), identity) = greatest_lower_bound(a, b).
% 76.28/10.22 Proof:
% 76.28/10.22 greatest_lower_bound(multiply(a, b), identity)
% 76.28/10.22 = { by axiom 4 (left_inverse) R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, b), multiply(inverse(b), b))
% 76.28/10.22 = { by axiom 10 (monotony_glb2) R->L }
% 76.28/10.22 multiply(greatest_lower_bound(a, inverse(b)), b)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 multiply(greatest_lower_bound(inverse(b), a), b)
% 76.28/10.22 = { by lemma 27 R->L }
% 76.28/10.22 multiply(greatest_lower_bound(inverse(b), greatest_lower_bound(a, least_upper_bound(b, inverse(b)))), b)
% 76.28/10.22 = { by lemma 28 }
% 76.28/10.22 multiply(greatest_lower_bound(inverse(b), greatest_lower_bound(identity, a)), b)
% 76.28/10.22 = { by lemma 23 R->L }
% 76.28/10.22 multiply(greatest_lower_bound(identity, greatest_lower_bound(inverse(b), a)), b)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.22 multiply(greatest_lower_bound(identity, greatest_lower_bound(a, inverse(b))), b)
% 76.28/10.22 = { by lemma 29 R->L }
% 76.28/10.22 multiply(greatest_lower_bound(identity, multiply(greatest_lower_bound(multiply(a, b), identity), inverse(b))), b)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 multiply(greatest_lower_bound(multiply(greatest_lower_bound(multiply(a, b), identity), inverse(b)), identity), b)
% 76.28/10.22 = { by lemma 20 }
% 76.28/10.22 greatest_lower_bound(b, multiply(multiply(greatest_lower_bound(multiply(a, b), identity), inverse(b)), b))
% 76.28/10.22 = { by axiom 8 (associativity) }
% 76.28/10.22 greatest_lower_bound(b, multiply(greatest_lower_bound(multiply(a, b), identity), multiply(inverse(b), b)))
% 76.28/10.22 = { by axiom 4 (left_inverse) }
% 76.28/10.22 greatest_lower_bound(b, multiply(greatest_lower_bound(multiply(a, b), identity), identity))
% 76.28/10.22 = { by lemma 16 }
% 76.28/10.22 greatest_lower_bound(b, greatest_lower_bound(multiply(a, b), identity))
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), b)
% 76.28/10.22 = { by lemma 17 R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(multiply(a, b), multiply(a, inverse(a))), b)
% 76.28/10.22 = { by axiom 9 (monotony_glb1) R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(b, inverse(a))), b)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(inverse(a), b)), b)
% 76.28/10.22 = { by lemma 27 R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(inverse(a), greatest_lower_bound(b, least_upper_bound(a, inverse(a))))), b)
% 76.28/10.22 = { by lemma 25 }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(inverse(a), greatest_lower_bound(identity, b))), b)
% 76.28/10.22 = { by lemma 23 R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(identity, greatest_lower_bound(inverse(a), b))), b)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(identity, greatest_lower_bound(b, inverse(a)))), b)
% 76.28/10.22 = { by lemma 23 }
% 76.28/10.22 greatest_lower_bound(multiply(a, greatest_lower_bound(b, greatest_lower_bound(identity, inverse(a)))), b)
% 76.28/10.22 = { by axiom 9 (monotony_glb1) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(multiply(a, b), multiply(a, greatest_lower_bound(identity, inverse(a)))), b)
% 76.28/10.22 = { by lemma 30 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(multiply(a, b), greatest_lower_bound(identity, multiply(a, identity))), b)
% 76.28/10.22 = { by lemma 16 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(multiply(a, b), greatest_lower_bound(identity, a)), b)
% 76.28/10.22 = { by axiom 6 (associativity_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, b), greatest_lower_bound(greatest_lower_bound(identity, a), b))
% 76.28/10.22 = { by axiom 6 (associativity_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(multiply(a, b), greatest_lower_bound(identity, greatest_lower_bound(a, b)))
% 76.28/10.22 = { by lemma 26 }
% 76.28/10.22 greatest_lower_bound(multiply(a, b), greatest_lower_bound(a, b))
% 76.28/10.22 = { by lemma 23 }
% 76.28/10.22 greatest_lower_bound(a, greatest_lower_bound(multiply(a, b), b))
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(a, greatest_lower_bound(b, multiply(a, b)))
% 76.28/10.22 = { by axiom 6 (associativity_of_glb) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, b))
% 76.28/10.22 = { by axiom 3 (left_identity) R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, multiply(identity, b)))
% 76.28/10.22 = { by axiom 7 (lub_absorbtion) R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, multiply(least_upper_bound(identity, greatest_lower_bound(identity, greatest_lower_bound(inverse(a), inverse(b)))), b)))
% 76.28/10.22 = { by lemma 31 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, multiply(least_upper_bound(identity, greatest_lower_bound(inverse(a), inverse(b))), b)))
% 76.28/10.22 = { by axiom 12 (monotony_lub2) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, least_upper_bound(multiply(identity, b), multiply(greatest_lower_bound(inverse(a), inverse(b)), b))))
% 76.28/10.22 = { by axiom 3 (left_identity) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, least_upper_bound(b, multiply(greatest_lower_bound(inverse(a), inverse(b)), b))))
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, least_upper_bound(b, multiply(greatest_lower_bound(inverse(b), inverse(a)), b))))
% 76.28/10.22 = { by axiom 10 (monotony_glb2) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, least_upper_bound(b, greatest_lower_bound(multiply(inverse(b), b), multiply(inverse(a), b)))))
% 76.28/10.22 = { by axiom 4 (left_inverse) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), multiply(a, least_upper_bound(b, greatest_lower_bound(identity, multiply(inverse(a), b)))))
% 76.28/10.22 = { by axiom 11 (monotony_lub1) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(a, greatest_lower_bound(identity, multiply(inverse(a), b)))))
% 76.28/10.22 = { by lemma 14 R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(a, multiply(inverse(X), multiply(X, greatest_lower_bound(identity, multiply(inverse(a), b)))))))
% 76.28/10.22 = { by axiom 8 (associativity) R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(multiply(a, inverse(X)), multiply(X, greatest_lower_bound(identity, multiply(inverse(a), b))))))
% 76.28/10.22 = { by lemma 32 R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(multiply(a, multiply(inverse(a), inverse(multiply(X, inverse(a))))), multiply(X, greatest_lower_bound(identity, multiply(inverse(a), b))))))
% 76.28/10.22 = { by lemma 15 R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(multiply(inverse(inverse(a)), multiply(inverse(a), inverse(multiply(X, inverse(a))))), multiply(X, greatest_lower_bound(identity, multiply(inverse(a), b))))))
% 76.28/10.22 = { by lemma 14 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(inverse(multiply(X, inverse(a))), multiply(X, greatest_lower_bound(identity, multiply(inverse(a), b))))))
% 76.28/10.22 = { by lemma 33 R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(inverse(multiply(X, inverse(a))), multiply(X, multiply(inverse(a), greatest_lower_bound(b, multiply(a, identity)))))))
% 76.28/10.22 = { by axiom 8 (associativity) R->L }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), multiply(inverse(multiply(X, inverse(a))), multiply(multiply(X, inverse(a)), greatest_lower_bound(b, multiply(a, identity))))))
% 76.28/10.22 = { by lemma 14 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), greatest_lower_bound(b, multiply(a, identity))))
% 76.28/10.22 = { by lemma 16 }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), greatest_lower_bound(b, a)))
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.22 greatest_lower_bound(greatest_lower_bound(a, b), least_upper_bound(multiply(a, b), greatest_lower_bound(a, b)))
% 76.28/10.22 = { by lemma 22 }
% 76.28/10.22 greatest_lower_bound(a, b)
% 76.28/10.22
% 76.28/10.22 Lemma 35: multiply(X, least_upper_bound(Y, inverse(X))) = least_upper_bound(identity, multiply(X, Y)).
% 76.28/10.22 Proof:
% 76.28/10.22 multiply(X, least_upper_bound(Y, inverse(X)))
% 76.28/10.22 = { by axiom 2 (symmetry_of_lub) R->L }
% 76.28/10.22 multiply(X, least_upper_bound(inverse(X), Y))
% 76.28/10.22 = { by lemma 15 R->L }
% 76.28/10.22 multiply(inverse(inverse(X)), least_upper_bound(inverse(X), Y))
% 76.28/10.22 = { by axiom 11 (monotony_lub1) }
% 76.28/10.22 least_upper_bound(multiply(inverse(inverse(X)), inverse(X)), multiply(inverse(inverse(X)), Y))
% 76.28/10.22 = { by axiom 4 (left_inverse) }
% 76.28/10.22 least_upper_bound(identity, multiply(inverse(inverse(X)), Y))
% 76.28/10.22 = { by lemma 15 }
% 76.28/10.22 least_upper_bound(identity, multiply(X, Y))
% 76.28/10.22
% 76.28/10.22 Lemma 36: greatest_lower_bound(multiply(Y, Z), multiply(X, Z)) = multiply(greatest_lower_bound(X, Y), Z).
% 76.28/10.22 Proof:
% 76.28/10.22 greatest_lower_bound(multiply(Y, Z), multiply(X, Z))
% 76.28/10.22 = { by axiom 10 (monotony_glb2) R->L }
% 76.28/10.22 multiply(greatest_lower_bound(Y, X), Z)
% 76.28/10.22 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.23 multiply(greatest_lower_bound(X, Y), Z)
% 76.28/10.23
% 76.28/10.23 Lemma 37: greatest_lower_bound(multiply(b, a), least_upper_bound(identity, multiply(a, a))) = greatest_lower_bound(multiply(b, a), a).
% 76.28/10.23 Proof:
% 76.28/10.23 greatest_lower_bound(multiply(b, a), least_upper_bound(identity, multiply(a, a)))
% 76.28/10.23 = { by axiom 4 (left_inverse) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), least_upper_bound(multiply(inverse(a), a), multiply(a, a)))
% 76.28/10.23 = { by axiom 12 (monotony_lub2) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), multiply(least_upper_bound(inverse(a), a), a))
% 76.28/10.23 = { by axiom 2 (symmetry_of_lub) }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), multiply(least_upper_bound(a, inverse(a)), a))
% 76.28/10.23 = { by axiom 10 (monotony_glb2) R->L }
% 76.28/10.23 multiply(greatest_lower_bound(b, least_upper_bound(a, inverse(a))), a)
% 76.28/10.23 = { by lemma 25 }
% 76.28/10.23 multiply(greatest_lower_bound(identity, b), a)
% 76.28/10.23 = { by lemma 36 R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), multiply(identity, a))
% 76.28/10.23 = { by axiom 3 (left_identity) }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), a)
% 76.28/10.23
% 76.28/10.23 Lemma 38: greatest_lower_bound(multiply(X, multiply(Y, Z)), multiply(W, Z)) = multiply(greatest_lower_bound(W, multiply(X, Y)), Z).
% 76.28/10.23 Proof:
% 76.28/10.23 greatest_lower_bound(multiply(X, multiply(Y, Z)), multiply(W, Z))
% 76.28/10.23 = { by axiom 8 (associativity) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(multiply(X, Y), Z), multiply(W, Z))
% 76.28/10.23 = { by axiom 10 (monotony_glb2) R->L }
% 76.28/10.23 multiply(greatest_lower_bound(multiply(X, Y), W), Z)
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.23 multiply(greatest_lower_bound(W, multiply(X, Y)), Z)
% 76.28/10.23
% 76.28/10.23 Lemma 39: greatest_lower_bound(multiply(b, a), identity) = greatest_lower_bound(multiply(a, b), identity).
% 76.28/10.23 Proof:
% 76.28/10.23 greatest_lower_bound(multiply(b, a), identity)
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(identity, multiply(b, a))
% 76.28/10.23 = { by lemma 24 R->L }
% 76.28/10.23 greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), least_upper_bound(identity, multiply(b, b))))
% 76.28/10.23 = { by lemma 35 R->L }
% 76.28/10.23 greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), multiply(b, least_upper_bound(b, inverse(b)))))
% 76.28/10.23 = { by axiom 9 (monotony_glb1) R->L }
% 76.28/10.23 greatest_lower_bound(identity, multiply(b, greatest_lower_bound(a, least_upper_bound(b, inverse(b)))))
% 76.28/10.23 = { by lemma 28 }
% 76.28/10.23 greatest_lower_bound(identity, multiply(b, greatest_lower_bound(identity, a)))
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(identity, multiply(b, greatest_lower_bound(a, identity)))
% 76.28/10.23 = { by axiom 9 (monotony_glb1) }
% 76.28/10.23 greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), multiply(b, identity)))
% 76.28/10.23 = { by lemma 16 }
% 76.28/10.23 greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), b))
% 76.28/10.23 = { by lemma 23 R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), greatest_lower_bound(identity, b))
% 76.28/10.23 = { by axiom 6 (associativity_of_glb) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(b, a), identity), b)
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(identity, multiply(b, a)), b)
% 76.28/10.23 = { by lemma 24 R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), least_upper_bound(identity, multiply(a, a)))), b)
% 76.28/10.23 = { by lemma 37 }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(identity, greatest_lower_bound(multiply(b, a), a)), b)
% 76.28/10.23 = { by lemma 23 R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(b, a), greatest_lower_bound(identity, a)), b)
% 76.28/10.23 = { by axiom 6 (associativity_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), greatest_lower_bound(greatest_lower_bound(identity, a), b))
% 76.28/10.23 = { by axiom 6 (associativity_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), greatest_lower_bound(identity, greatest_lower_bound(a, b)))
% 76.28/10.23 = { by lemma 26 }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), greatest_lower_bound(a, b))
% 76.28/10.23 = { by lemma 34 R->L }
% 76.28/10.23 greatest_lower_bound(multiply(b, a), greatest_lower_bound(multiply(a, b), identity))
% 76.28/10.23 = { by lemma 23 R->L }
% 76.28/10.23 greatest_lower_bound(multiply(a, b), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.23 greatest_lower_bound(multiply(a, b), greatest_lower_bound(identity, multiply(b, a)))
% 76.28/10.23 = { by axiom 6 (associativity_of_glb) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), multiply(b, a))
% 76.28/10.23 = { by axiom 7 (lub_absorbtion) R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), multiply(b, least_upper_bound(a, greatest_lower_bound(a, multiply(greatest_lower_bound(inverse(a), inverse(b)), a)))))
% 76.28/10.23 = { by lemma 21 R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), multiply(b, least_upper_bound(a, multiply(greatest_lower_bound(identity, greatest_lower_bound(inverse(a), inverse(b))), a))))
% 76.28/10.23 = { by lemma 31 }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), multiply(b, least_upper_bound(a, multiply(greatest_lower_bound(inverse(a), inverse(b)), a))))
% 76.28/10.23 = { by axiom 11 (monotony_lub1) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), multiply(b, multiply(greatest_lower_bound(inverse(a), inverse(b)), a))))
% 76.28/10.23 = { by axiom 8 (associativity) R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), multiply(multiply(b, greatest_lower_bound(inverse(a), inverse(b))), a)))
% 76.28/10.23 = { by lemma 30 }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), multiply(greatest_lower_bound(identity, multiply(b, inverse(a))), a)))
% 76.28/10.23 = { by lemma 38 R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(multiply(b, multiply(inverse(a), a)), multiply(identity, a))))
% 76.28/10.23 = { by axiom 4 (left_inverse) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(multiply(b, identity), multiply(identity, a))))
% 76.28/10.23 = { by lemma 16 }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(b, multiply(identity, a))))
% 76.28/10.23 = { by axiom 3 (left_identity) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(b, a)))
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(a, b)))
% 76.28/10.23 = { by lemma 34 R->L }
% 76.28/10.23 greatest_lower_bound(greatest_lower_bound(multiply(a, b), identity), least_upper_bound(multiply(b, a), greatest_lower_bound(multiply(a, b), identity)))
% 76.28/10.23 = { by lemma 22 }
% 76.28/10.23 greatest_lower_bound(multiply(a, b), identity)
% 76.28/10.23
% 76.28/10.23 Lemma 40: multiply(greatest_lower_bound(Y, greatest_lower_bound(identity, Z)), X) = greatest_lower_bound(X, multiply(greatest_lower_bound(Y, Z), X)).
% 76.28/10.23 Proof:
% 76.28/10.23 multiply(greatest_lower_bound(Y, greatest_lower_bound(identity, Z)), X)
% 76.28/10.23 = { by lemma 23 }
% 76.28/10.23 multiply(greatest_lower_bound(identity, greatest_lower_bound(Y, Z)), X)
% 76.28/10.23 = { by lemma 21 }
% 76.28/10.23 greatest_lower_bound(X, multiply(greatest_lower_bound(Y, Z), X))
% 76.28/10.23
% 76.28/10.23 Lemma 41: multiply(inverse(greatest_lower_bound(identity, inverse(X))), greatest_lower_bound(X, identity)) = X.
% 76.28/10.23 Proof:
% 76.28/10.23 multiply(inverse(greatest_lower_bound(identity, inverse(X))), greatest_lower_bound(X, identity))
% 76.28/10.23 = { by axiom 4 (left_inverse) R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(identity, inverse(X))), greatest_lower_bound(X, multiply(inverse(X), X)))
% 76.28/10.23 = { by lemma 21 R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(identity, inverse(X))), multiply(greatest_lower_bound(identity, inverse(X)), X))
% 76.28/10.23 = { by lemma 14 }
% 76.28/10.23 X
% 76.28/10.23
% 76.28/10.23 Lemma 42: greatest_lower_bound(inverse(multiply(Z, X)), multiply(inverse(X), Y)) = multiply(inverse(X), greatest_lower_bound(Y, inverse(Z))).
% 76.28/10.23 Proof:
% 76.28/10.23 greatest_lower_bound(inverse(multiply(Z, X)), multiply(inverse(X), Y))
% 76.28/10.23 = { by lemma 33 R->L }
% 76.28/10.23 multiply(inverse(X), greatest_lower_bound(Y, multiply(X, inverse(multiply(Z, X)))))
% 76.28/10.23 = { by lemma 32 }
% 76.28/10.23 multiply(inverse(X), greatest_lower_bound(Y, inverse(Z)))
% 76.28/10.23
% 76.28/10.23 Goal 1 (prove_p33): multiply(a, b) = multiply(b, a).
% 76.28/10.23 Proof:
% 76.28/10.23 multiply(a, b)
% 76.28/10.23 = { by lemma 41 R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(identity, inverse(multiply(a, b)))), greatest_lower_bound(multiply(a, b), identity))
% 76.28/10.23 = { by lemma 39 R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(identity, inverse(multiply(a, b)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(inverse(multiply(a, b)), identity)), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by axiom 4 (left_inverse) R->L }
% 76.28/10.23 multiply(inverse(greatest_lower_bound(inverse(multiply(a, b)), multiply(inverse(b), b))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by lemma 42 }
% 76.28/10.23 multiply(inverse(multiply(inverse(b), greatest_lower_bound(b, inverse(a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by lemma 29 R->L }
% 76.28/10.23 multiply(inverse(multiply(inverse(b), multiply(greatest_lower_bound(multiply(b, a), identity), inverse(a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.23 = { by lemma 39 }
% 76.28/10.23 multiply(inverse(multiply(inverse(b), multiply(greatest_lower_bound(multiply(a, b), identity), inverse(a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 34 }
% 76.28/10.24 multiply(inverse(multiply(inverse(b), multiply(greatest_lower_bound(a, b), inverse(a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 10 (monotony_glb2) }
% 76.28/10.24 multiply(inverse(multiply(inverse(b), greatest_lower_bound(multiply(a, inverse(a)), multiply(b, inverse(a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 33 }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(a), multiply(inverse(b), multiply(a, inverse(a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 17 }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(a), multiply(inverse(b), identity))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 16 }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(a), inverse(b))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 32 R->L }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(a), multiply(a, inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 16 R->L }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(multiply(inverse(a), identity), multiply(a, inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 17 R->L }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(multiply(inverse(a), multiply(multiply(b, a), inverse(multiply(b, a)))), multiply(a, inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 38 }
% 76.28/10.24 multiply(inverse(multiply(greatest_lower_bound(a, multiply(inverse(a), multiply(b, a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 33 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(multiply(b, a), multiply(a, a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 36 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(greatest_lower_bound(a, b), a)), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 34 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(greatest_lower_bound(multiply(a, b), identity), a)), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 5 (glb_absorbtion) R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(greatest_lower_bound(multiply(a, b), greatest_lower_bound(identity, least_upper_bound(identity, multiply(a, a)))), a)), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 40 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(greatest_lower_bound(multiply(a, b), least_upper_bound(identity, multiply(a, a))), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 35 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(greatest_lower_bound(multiply(a, b), multiply(a, least_upper_bound(a, inverse(a)))), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 9 (monotony_glb1) R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(multiply(a, greatest_lower_bound(b, least_upper_bound(a, inverse(a)))), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 25 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(multiply(a, greatest_lower_bound(identity, b)), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 19 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(greatest_lower_bound(a, multiply(a, b)), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(greatest_lower_bound(multiply(a, b), a), a))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 10 (monotony_glb2) }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, greatest_lower_bound(multiply(multiply(a, b), a), multiply(a, a)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 8 (associativity) }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, greatest_lower_bound(multiply(a, multiply(b, a)), multiply(a, a)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 9 (monotony_glb1) R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(a, greatest_lower_bound(multiply(b, a), a)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 16 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(a, multiply(greatest_lower_bound(multiply(b, a), a), identity)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 19 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(identity, multiply(greatest_lower_bound(multiply(b, a), a), identity)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 37 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(identity, multiply(greatest_lower_bound(multiply(b, a), least_upper_bound(identity, multiply(a, a))), identity)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 40 R->L }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, multiply(greatest_lower_bound(multiply(b, a), greatest_lower_bound(identity, least_upper_bound(identity, multiply(a, a)))), identity))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 5 (glb_absorbtion) }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, multiply(greatest_lower_bound(multiply(b, a), identity), identity))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 20 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(identity, multiply(multiply(b, a), identity)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 16 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(identity, multiply(b, a)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(multiply(b, a), identity))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 39 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), multiply(a, greatest_lower_bound(multiply(a, b), identity))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 18 }
% 76.28/10.24 multiply(inverse(multiply(multiply(inverse(a), greatest_lower_bound(a, multiply(a, multiply(a, b)))), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 33 }
% 76.28/10.24 multiply(inverse(multiply(greatest_lower_bound(multiply(a, b), multiply(inverse(a), a)), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 4 (left_inverse) }
% 76.28/10.24 multiply(inverse(multiply(greatest_lower_bound(multiply(a, b), identity), inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 14 R->L }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(a, multiply(greatest_lower_bound(multiply(a, b), identity), inverse(multiply(b, a)))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 39 R->L }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(a, multiply(greatest_lower_bound(multiply(b, a), identity), inverse(multiply(b, a)))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 20 }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(a, greatest_lower_bound(inverse(multiply(b, a)), multiply(multiply(b, a), inverse(multiply(b, a))))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 1 (symmetry_of_glb) R->L }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(a, greatest_lower_bound(multiply(multiply(b, a), inverse(multiply(b, a))), inverse(multiply(b, a)))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 9 (monotony_glb1) }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), greatest_lower_bound(multiply(a, multiply(multiply(b, a), inverse(multiply(b, a)))), multiply(a, inverse(multiply(b, a)))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 38 }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(greatest_lower_bound(a, multiply(a, multiply(b, a))), inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 8 (associativity) R->L }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(greatest_lower_bound(a, multiply(multiply(a, b), a)), inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 20 R->L }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(multiply(greatest_lower_bound(multiply(a, b), identity), a), inverse(multiply(b, a))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 8 (associativity) }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(greatest_lower_bound(multiply(a, b), identity), multiply(a, inverse(multiply(b, a)))))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 32 }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), multiply(greatest_lower_bound(multiply(a, b), identity), inverse(b)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 29 }
% 76.28/10.24 multiply(inverse(multiply(inverse(a), greatest_lower_bound(a, inverse(b)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 42 R->L }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(multiply(b, a)), multiply(inverse(a), a))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 4 (left_inverse) }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(inverse(multiply(b, a)), identity)), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by axiom 1 (symmetry_of_glb) }
% 76.28/10.24 multiply(inverse(greatest_lower_bound(identity, inverse(multiply(b, a)))), greatest_lower_bound(multiply(b, a), identity))
% 76.28/10.24 = { by lemma 41 }
% 76.28/10.24 multiply(b, a)
% 76.28/10.24 % SZS output end Proof
% 76.28/10.24
% 76.28/10.24 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------