0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n005.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 14:17:18 EDT 2019
0.12/0.34	% CPUTime    : 
54.44/54.71	% SZS status Unsatisfiable
54.44/54.71	
54.44/54.72	% SZS output start Proof
54.44/54.72	Take the following subset of the input axioms:
54.44/54.72	  fof(associativity, axiom, ![X, Y, Z]: multiply(multiply(X, Y), Z)=multiply(X, multiply(Y, Z))).
54.44/54.72	  fof(associativity_of_glb, axiom, ![X, Y, Z]: greatest_lower_bound(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(greatest_lower_bound(X, Y), Z)).
54.44/54.72	  fof(associativity_of_lub, axiom, ![X, Y, Z]: least_upper_bound(X, least_upper_bound(Y, Z))=least_upper_bound(least_upper_bound(X, Y), Z)).
54.44/54.72	  fof(glb_absorbtion, axiom, ![X, Y]: greatest_lower_bound(X, least_upper_bound(X, Y))=X).
54.44/54.72	  fof(idempotence_of_lub, axiom, ![X]: least_upper_bound(X, X)=X).
54.44/54.72	  fof(left_identity, axiom, ![X]: X=multiply(identity, X)).
54.44/54.72	  fof(left_inverse, axiom, ![X]: multiply(inverse(X), X)=identity).
54.44/54.72	  fof(lub_absorbtion, axiom, ![X, Y]: X=least_upper_bound(X, greatest_lower_bound(X, Y))).
54.44/54.72	  fof(monotony_glb1, axiom, ![X, Y, Z]: multiply(X, greatest_lower_bound(Y, Z))=greatest_lower_bound(multiply(X, Y), multiply(X, Z))).
54.44/54.72	  fof(monotony_glb2, axiom, ![X, Y, Z]: greatest_lower_bound(multiply(Y, X), multiply(Z, X))=multiply(greatest_lower_bound(Y, Z), X)).
54.44/54.72	  fof(monotony_lub1, axiom, ![X, Y, Z]: least_upper_bound(multiply(X, Y), multiply(X, Z))=multiply(X, least_upper_bound(Y, Z))).
54.44/54.72	  fof(monotony_lub2, axiom, ![X, Y, Z]: least_upper_bound(multiply(Y, X), multiply(Z, X))=multiply(least_upper_bound(Y, Z), X)).
54.44/54.72	  fof(prove_distrun, negated_conjecture, greatest_lower_bound(a, least_upper_bound(b, c))!=least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c))).
54.44/54.72	  fof(symmetry_of_glb, axiom, ![X, Y]: greatest_lower_bound(Y, X)=greatest_lower_bound(X, Y)).
54.44/54.72	  fof(symmetry_of_lub, axiom, ![X, Y]: least_upper_bound(X, Y)=least_upper_bound(Y, X)).
54.44/54.72	
54.44/54.72	Now clausify the problem and encode Horn clauses using encoding 3 of
54.44/54.72	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
54.44/54.72	We repeatedly replace C & s=t => u=v by the two clauses:
54.44/54.72	  fresh(y, y, x1...xn) = u
54.44/54.72	  C => fresh(s, t, x1...xn) = v
54.44/54.72	where fresh is a fresh function symbol and x1..xn are the free
54.44/54.72	variables of u and v.
54.44/54.72	A predicate p(X) is encoded as p(X)=true (this is sound, because the
54.44/54.72	input problem has no model of domain size 1).
54.44/54.72	
54.44/54.72	The encoding turns the above axioms into the following unit equations and goals:
54.44/54.72	
54.44/54.72	Axiom 1 (left_inverse): multiply(inverse(X), X) = identity.
54.44/54.72	Axiom 2 (associativity): multiply(multiply(X, Y), Z) = multiply(X, multiply(Y, Z)).
54.44/54.72	Axiom 3 (left_identity): X = multiply(identity, X).
54.44/54.72	Axiom 4 (symmetry_of_lub): least_upper_bound(X, Y) = least_upper_bound(Y, X).
54.44/54.72	Axiom 5 (glb_absorbtion): greatest_lower_bound(X, least_upper_bound(X, Y)) = X.
54.44/54.72	Axiom 6 (lub_absorbtion): X = least_upper_bound(X, greatest_lower_bound(X, Y)).
54.44/54.72	Axiom 7 (monotony_glb1): multiply(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(multiply(X, Y), multiply(X, Z)).
54.44/54.72	Axiom 8 (symmetry_of_glb): greatest_lower_bound(X, Y) = greatest_lower_bound(Y, X).
54.44/54.72	Axiom 9 (idempotence_of_lub): least_upper_bound(X, X) = X.
54.44/54.72	Axiom 10 (monotony_lub2): least_upper_bound(multiply(X, Y), multiply(Z, Y)) = multiply(least_upper_bound(X, Z), Y).
54.44/54.72	Axiom 11 (monotony_glb2): greatest_lower_bound(multiply(X, Y), multiply(Z, Y)) = multiply(greatest_lower_bound(X, Z), Y).
54.44/54.72	Axiom 12 (monotony_lub1): least_upper_bound(multiply(X, Y), multiply(X, Z)) = multiply(X, least_upper_bound(Y, Z)).
54.44/54.72	Axiom 13 (associativity_of_glb): greatest_lower_bound(X, greatest_lower_bound(Y, Z)) = greatest_lower_bound(greatest_lower_bound(X, Y), Z).
54.65/54.91	Axiom 14 (associativity_of_lub): least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(least_upper_bound(X, Y), Z).
54.65/54.91	
54.65/54.91	Lemma 15: least_upper_bound(X, greatest_lower_bound(Y, X)) = X.
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(X, greatest_lower_bound(Y, X))
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  least_upper_bound(X, greatest_lower_bound(X, Y))
54.65/54.91	= { by axiom 6 (lub_absorbtion) }
54.65/54.91	  X
54.65/54.91	
54.65/54.91	Lemma 16: multiply(inverse(X), multiply(X, Y)) = Y.
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(X), multiply(X, Y))
54.65/54.91	= { by axiom 2 (associativity) }
54.65/54.91	  multiply(multiply(inverse(X), X), Y)
54.65/54.91	= { by axiom 1 (left_inverse) }
54.65/54.91	  multiply(identity, Y)
54.65/54.91	= { by axiom 3 (left_identity) }
54.65/54.91	  Y
54.65/54.91	
54.65/54.91	Lemma 17: multiply(inverse(inverse(X)), Y) = multiply(X, Y).
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(inverse(X)), Y)
54.65/54.91	= { by lemma 16 }
54.65/54.91	  multiply(inverse(inverse(X)), multiply(inverse(X), multiply(X, Y)))
54.65/54.91	= { by lemma 16 }
54.65/54.91	  multiply(X, Y)
54.65/54.91	
54.65/54.91	Lemma 18: multiply(X, multiply(inverse(X), Y)) = Y.
54.65/54.91	Proof:
54.65/54.91	  multiply(X, multiply(inverse(X), Y))
54.65/54.91	= { by lemma 17 }
54.65/54.91	  multiply(inverse(inverse(X)), multiply(inverse(X), Y))
54.65/54.91	= { by lemma 16 }
54.65/54.91	  Y
54.65/54.91	
54.65/54.91	Lemma 19: multiply(inverse(inverse(X)), identity) = X.
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(inverse(X)), identity)
54.65/54.91	= { by axiom 1 (left_inverse) }
54.65/54.91	  multiply(inverse(inverse(X)), multiply(inverse(X), X))
54.65/54.91	= { by lemma 16 }
54.65/54.91	  X
54.65/54.91	
54.65/54.91	Lemma 20: multiply(X, identity) = X.
54.65/54.91	Proof:
54.65/54.91	  multiply(X, identity)
54.65/54.91	= { by lemma 17 }
54.65/54.91	  multiply(inverse(inverse(X)), identity)
54.65/54.91	= { by lemma 19 }
54.65/54.91	  X
54.65/54.91	
54.65/54.91	Lemma 21: inverse(inverse(X)) = X.
54.65/54.91	Proof:
54.65/54.91	  inverse(inverse(X))
54.65/54.91	= { by lemma 20 }
54.65/54.91	  multiply(inverse(inverse(X)), identity)
54.65/54.91	= { by lemma 19 }
54.65/54.91	  X
54.65/54.91	
54.65/54.91	Lemma 22: multiply(inverse(X), least_upper_bound(Z, multiply(X, Y))) = least_upper_bound(Y, multiply(inverse(X), Z)).
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(X), least_upper_bound(Z, multiply(X, Y)))
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  multiply(inverse(X), least_upper_bound(multiply(X, Y), Z))
54.65/54.91	= { by axiom 12 (monotony_lub1) }
54.65/54.91	  least_upper_bound(multiply(inverse(X), multiply(X, Y)), multiply(inverse(X), Z))
54.65/54.91	= { by lemma 16 }
54.65/54.91	  least_upper_bound(Y, multiply(inverse(X), Z))
54.65/54.91	
54.65/54.91	Lemma 23: least_upper_bound(Y, multiply(X, Y)) = multiply(least_upper_bound(X, identity), Y).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(Y, multiply(X, Y))
54.65/54.91	= { by axiom 3 (left_identity) }
54.65/54.91	  least_upper_bound(multiply(identity, Y), multiply(X, Y))
54.65/54.91	= { by axiom 10 (monotony_lub2) }
54.65/54.91	  multiply(least_upper_bound(identity, X), Y)
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  multiply(least_upper_bound(X, identity), Y)
54.65/54.91	
54.65/54.91	Lemma 24: least_upper_bound(X, least_upper_bound(Y, Z)) = least_upper_bound(Z, least_upper_bound(X, Y)).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(X, least_upper_bound(Y, Z))
54.65/54.91	= { by axiom 14 (associativity_of_lub) }
54.65/54.91	  least_upper_bound(least_upper_bound(X, Y), Z)
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  least_upper_bound(Z, least_upper_bound(X, Y))
54.65/54.91	
54.65/54.91	Lemma 25: least_upper_bound(identity, multiply(inverse(X), Y)) = multiply(inverse(X), least_upper_bound(X, Y)).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(identity, multiply(inverse(X), Y))
54.65/54.91	= { by axiom 1 (left_inverse) }
54.65/54.91	  least_upper_bound(multiply(inverse(X), X), multiply(inverse(X), Y))
54.65/54.91	= { by axiom 12 (monotony_lub1) }
54.65/54.91	  multiply(inverse(X), least_upper_bound(X, Y))
54.65/54.91	
54.65/54.91	Lemma 26: multiply(inverse(X), least_upper_bound(X, identity)) = least_upper_bound(identity, inverse(X)).
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(X), least_upper_bound(X, identity))
54.65/54.91	= { by lemma 25 }
54.65/54.91	  least_upper_bound(identity, multiply(inverse(X), identity))
54.65/54.91	= { by lemma 23 }
54.65/54.91	  multiply(least_upper_bound(inverse(X), identity), identity)
54.65/54.91	= { by lemma 20 }
54.65/54.91	  least_upper_bound(inverse(X), identity)
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  least_upper_bound(identity, inverse(X))
54.65/54.91	
54.65/54.91	Lemma 27: greatest_lower_bound(Y, multiply(X, Y)) = multiply(greatest_lower_bound(X, identity), Y).
54.65/54.91	Proof:
54.65/54.91	  greatest_lower_bound(Y, multiply(X, Y))
54.65/54.91	= { by axiom 3 (left_identity) }
54.65/54.91	  greatest_lower_bound(multiply(identity, Y), multiply(X, Y))
54.65/54.91	= { by axiom 11 (monotony_glb2) }
54.65/54.91	  multiply(greatest_lower_bound(identity, X), Y)
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  multiply(greatest_lower_bound(X, identity), Y)
54.65/54.91	
54.65/54.91	Lemma 28: greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(identity, inverse(X))) = multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity)).
54.65/54.91	Proof:
54.65/54.91	  greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(identity, inverse(X)))
54.65/54.91	= { by lemma 26 }
54.65/54.91	  greatest_lower_bound(least_upper_bound(X, identity), multiply(inverse(X), least_upper_bound(X, identity)))
54.65/54.91	= { by lemma 27 }
54.65/54.91	  multiply(greatest_lower_bound(inverse(X), identity), least_upper_bound(X, identity))
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity))
54.65/54.91	
54.65/54.91	Lemma 29: greatest_lower_bound(least_upper_bound(identity, X), least_upper_bound(identity, inverse(X))) = multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity)).
54.65/54.91	Proof:
54.65/54.91	  greatest_lower_bound(least_upper_bound(identity, X), least_upper_bound(identity, inverse(X)))
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  greatest_lower_bound(least_upper_bound(X, identity), least_upper_bound(identity, inverse(X)))
54.65/54.91	= { by lemma 28 }
54.65/54.91	  multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, identity))
54.65/54.91	
54.65/54.91	Lemma 30: greatest_lower_bound(X, greatest_lower_bound(Z, least_upper_bound(X, Y))) = greatest_lower_bound(X, Z).
54.65/54.91	Proof:
54.65/54.91	  greatest_lower_bound(X, greatest_lower_bound(Z, least_upper_bound(X, Y)))
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  greatest_lower_bound(X, greatest_lower_bound(least_upper_bound(X, Y), Z))
54.65/54.91	= { by axiom 13 (associativity_of_glb) }
54.65/54.91	  greatest_lower_bound(greatest_lower_bound(X, least_upper_bound(X, Y)), Z)
54.65/54.91	= { by axiom 5 (glb_absorbtion) }
54.65/54.91	  greatest_lower_bound(X, Z)
54.65/54.91	
54.65/54.91	Lemma 31: least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, least_upper_bound(Y, Z))) = greatest_lower_bound(X, least_upper_bound(Y, Z)).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(greatest_lower_bound(X, Y), greatest_lower_bound(X, least_upper_bound(Y, Z)))
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  least_upper_bound(greatest_lower_bound(Y, X), greatest_lower_bound(X, least_upper_bound(Y, Z)))
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  least_upper_bound(greatest_lower_bound(X, least_upper_bound(Y, Z)), greatest_lower_bound(Y, X))
54.65/54.91	= { by lemma 30 }
54.65/54.91	  least_upper_bound(greatest_lower_bound(X, least_upper_bound(Y, Z)), greatest_lower_bound(Y, greatest_lower_bound(X, least_upper_bound(Y, Z))))
54.65/54.91	= { by lemma 15 }
54.65/54.91	  greatest_lower_bound(X, least_upper_bound(Y, Z))
54.65/54.91	
54.65/54.91	Lemma 32: least_upper_bound(X, multiply(X, Y)) = multiply(X, least_upper_bound(Y, identity)).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(X, multiply(X, Y))
54.65/54.91	= { by lemma 20 }
54.65/54.91	  least_upper_bound(multiply(X, identity), multiply(X, Y))
54.65/54.91	= { by axiom 12 (monotony_lub1) }
54.65/54.91	  multiply(X, least_upper_bound(identity, Y))
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  multiply(X, least_upper_bound(Y, identity))
54.65/54.91	
54.65/54.91	Lemma 33: least_upper_bound(X, least_upper_bound(X, Y)) = least_upper_bound(X, Y).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(X, least_upper_bound(X, Y))
54.65/54.91	= { by axiom 14 (associativity_of_lub) }
54.65/54.91	  least_upper_bound(least_upper_bound(X, X), Y)
54.65/54.91	= { by axiom 9 (idempotence_of_lub) }
54.65/54.91	  least_upper_bound(X, Y)
54.65/54.91	
54.65/54.91	Lemma 34: multiply(inverse(X), greatest_lower_bound(Z, multiply(X, Y))) = greatest_lower_bound(Y, multiply(inverse(X), Z)).
54.65/54.91	Proof:
54.65/54.91	  multiply(inverse(X), greatest_lower_bound(Z, multiply(X, Y)))
54.65/54.91	= { by axiom 8 (symmetry_of_glb) }
54.65/54.91	  multiply(inverse(X), greatest_lower_bound(multiply(X, Y), Z))
54.65/54.91	= { by axiom 7 (monotony_glb1) }
54.65/54.91	  greatest_lower_bound(multiply(inverse(X), multiply(X, Y)), multiply(inverse(X), Z))
54.65/54.91	= { by lemma 16 }
54.65/54.91	  greatest_lower_bound(Y, multiply(inverse(X), Z))
54.65/54.91	
54.65/54.91	Lemma 35: least_upper_bound(X, least_upper_bound(Z, greatest_lower_bound(X, Y))) = least_upper_bound(X, Z).
54.65/54.91	Proof:
54.65/54.91	  least_upper_bound(X, least_upper_bound(Z, greatest_lower_bound(X, Y)))
54.65/54.91	= { by axiom 4 (symmetry_of_lub) }
54.65/54.91	  least_upper_bound(X, least_upper_bound(greatest_lower_bound(X, Y), Z))
54.65/54.91	= { by axiom 14 (associativity_of_lub) }
54.65/54.91	  least_upper_bound(least_upper_bound(X, greatest_lower_bound(X, Y)), Z)
54.65/54.91	= { by axiom 6 (lub_absorbtion) }
54.78/55.03	  least_upper_bound(X, Z)
54.78/55.03	
54.78/55.03	Lemma 36: greatest_lower_bound(identity, least_upper_bound(X, multiply(least_upper_bound(X, identity), Y))) = greatest_lower_bound(identity, least_upper_bound(Y, greatest_lower_bound(X, identity))).
54.78/55.03	Proof:
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(least_upper_bound(X, identity), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(least_upper_bound(identity, X), Y)))
54.78/55.03	= { by lemma 21 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(least_upper_bound(identity, inverse(inverse(X))), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(least_upper_bound(inverse(inverse(X)), identity), Y)))
54.78/55.03	= { by lemma 16 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), multiply(greatest_lower_bound(identity, inverse(inverse(inverse(X)))), least_upper_bound(inverse(inverse(X)), identity))), Y)))
54.78/55.03	= { by lemma 29 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), greatest_lower_bound(least_upper_bound(identity, inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X)))))), Y)))
54.78/55.03	= { by lemma 31 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(greatest_lower_bound(least_upper_bound(identity, inverse(inverse(X))), identity), greatest_lower_bound(least_upper_bound(identity, inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 8 (symmetry_of_glb) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(greatest_lower_bound(identity, least_upper_bound(identity, inverse(inverse(X)))), greatest_lower_bound(least_upper_bound(identity, inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 5 (glb_absorbtion) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(identity, inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by lemma 29 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, multiply(greatest_lower_bound(identity, inverse(inverse(inverse(X)))), least_upper_bound(inverse(inverse(X)), identity)))), Y)))
54.78/55.03	= { by lemma 28 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(inverse(inverse(X)), identity), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 3 (left_identity) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(identity, least_upper_bound(inverse(inverse(X)), identity)), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by lemma 32 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(identity, multiply(identity, inverse(inverse(X)))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 1 (left_inverse) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(identity, inverse(inverse(X)))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 10 (monotony_lub2) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(least_upper_bound(inverse(inverse(inverse(X))), identity), inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), inverse(inverse(X))), least_upper_bound(identity, inverse(inverse(inverse(X))))))), Y)))
54.78/55.03	= { by lemma 33 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), inverse(inverse(X))), least_upper_bound(identity, least_upper_bound(identity, inverse(inverse(inverse(X)))))))), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), inverse(inverse(X))), least_upper_bound(identity, least_upper_bound(inverse(inverse(inverse(X))), identity))))), Y)))
54.78/55.03	= { by axiom 14 (associativity_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), inverse(inverse(X))), least_upper_bound(least_upper_bound(identity, inverse(inverse(inverse(X)))), identity)))), Y)))
54.78/55.03	= { by axiom 8 (symmetry_of_glb) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(least_upper_bound(identity, inverse(inverse(inverse(X)))), identity), multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), inverse(inverse(X)))))), Y)))
54.78/55.03	= { by lemma 17 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(least_upper_bound(least_upper_bound(identity, inverse(inverse(inverse(X)))), identity), multiply(inverse(inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))), inverse(inverse(X)))))), Y)))
54.78/55.03	= { by lemma 34 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, multiply(inverse(inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))), greatest_lower_bound(inverse(inverse(X)), multiply(inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(least_upper_bound(identity, inverse(inverse(inverse(X)))), identity)))))), Y)))
54.78/55.03	= { by lemma 26 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, multiply(inverse(inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))), greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))))))), Y)))
54.78/55.03	= { by lemma 17 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, multiply(least_upper_bound(identity, inverse(inverse(inverse(X)))), greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))))))), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, multiply(least_upper_bound(inverse(inverse(inverse(X))), identity), greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X)))))))))), Y)))
54.78/55.03	= { by lemma 23 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, least_upper_bound(greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))), multiply(inverse(inverse(inverse(X))), greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))))))), Y)))
54.78/55.03	= { by axiom 7 (monotony_glb1) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, least_upper_bound(greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))), greatest_lower_bound(multiply(inverse(inverse(inverse(X))), inverse(inverse(X))), multiply(inverse(inverse(inverse(X))), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))))))), Y)))
54.78/55.03	= { by axiom 1 (left_inverse) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, least_upper_bound(greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))), greatest_lower_bound(identity, multiply(inverse(inverse(inverse(X))), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))))))), Y)))
54.78/55.03	= { by lemma 35 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(inverse(inverse(X)), least_upper_bound(identity, inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))))))), Y)))
54.78/55.03	= { by lemma 26 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(inverse(inverse(X)), multiply(inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(least_upper_bound(identity, inverse(inverse(inverse(X)))), identity))))), Y)))
54.78/55.03	= { by axiom 4 (symmetry_of_lub) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(inverse(inverse(X)), multiply(inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, least_upper_bound(identity, inverse(inverse(inverse(X))))))))), Y)))
54.78/55.03	= { by lemma 33 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(inverse(inverse(X)), multiply(inverse(least_upper_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, inverse(inverse(inverse(X)))))))), Y)))
54.78/55.03	= { by axiom 1 (left_inverse) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), least_upper_bound(identity, greatest_lower_bound(inverse(inverse(X)), identity))), Y)))
54.78/55.03	= { by lemma 15 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), identity), Y)))
54.78/55.03	= { by lemma 20 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(inverse(inverse(X))))), Y)))
54.78/55.03	= { by lemma 21 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y)))
54.78/55.03	= { by lemma 30 }
54.78/55.03	  greatest_lower_bound(identity, greatest_lower_bound(least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y)), least_upper_bound(identity, multiply(inverse(X), multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y)))))
54.78/55.03	= { by lemma 25 }
54.78/55.03	  greatest_lower_bound(identity, greatest_lower_bound(least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y)), multiply(inverse(X), least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y)))))
54.78/55.03	= { by lemma 27 }
54.78/55.03	  greatest_lower_bound(identity, multiply(greatest_lower_bound(inverse(X), identity), least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y))))
54.78/55.03	= { by axiom 8 (symmetry_of_glb) }
54.78/55.03	  greatest_lower_bound(identity, multiply(greatest_lower_bound(identity, inverse(X)), least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y))))
54.78/55.03	= { by lemma 17 }
54.78/55.03	  greatest_lower_bound(identity, multiply(inverse(inverse(greatest_lower_bound(identity, inverse(X)))), least_upper_bound(X, multiply(inverse(greatest_lower_bound(identity, inverse(X))), Y))))
54.78/55.03	= { by lemma 22 }
54.78/55.03	  greatest_lower_bound(identity, multiply(inverse(inverse(greatest_lower_bound(identity, inverse(X)))), multiply(inverse(greatest_lower_bound(identity, inverse(X))), least_upper_bound(Y, multiply(greatest_lower_bound(identity, inverse(X)), X)))))
54.78/55.03	= { by lemma 16 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, multiply(greatest_lower_bound(identity, inverse(X)), X)))
54.78/55.03	= { by axiom 8 (symmetry_of_glb) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, multiply(greatest_lower_bound(inverse(X), identity), X)))
54.78/55.03	= { by axiom 11 (monotony_glb2) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, greatest_lower_bound(multiply(inverse(X), X), multiply(identity, X))))
54.78/55.03	= { by axiom 1 (left_inverse) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, greatest_lower_bound(identity, multiply(identity, X))))
54.78/55.03	= { by lemma 20 }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, greatest_lower_bound(multiply(identity, identity), multiply(identity, X))))
54.78/55.03	= { by axiom 7 (monotony_glb1) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, multiply(identity, greatest_lower_bound(identity, X))))
54.78/55.03	= { by axiom 8 (symmetry_of_glb) }
54.78/55.03	  greatest_lower_bound(identity, least_upper_bound(Y, multiply(identity, greatest_lower_bound(X, identity))))
54.78/55.03	= { by axiom 3 (left_identity) }
54.84/55.10	  greatest_lower_bound(identity, least_upper_bound(Y, greatest_lower_bound(X, identity)))
54.84/55.10	
54.84/55.10	Lemma 37: greatest_lower_bound(X, least_upper_bound(Z, greatest_lower_bound(X, Y))) = greatest_lower_bound(X, least_upper_bound(Y, Z)).
54.84/55.10	Proof:
54.84/55.10	  greatest_lower_bound(X, least_upper_bound(Z, greatest_lower_bound(X, Y)))
54.84/55.10	= { by lemma 21 }
54.84/55.10	  greatest_lower_bound(inverse(inverse(X)), least_upper_bound(Z, greatest_lower_bound(X, Y)))
54.84/55.10	= { by lemma 20 }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), identity), least_upper_bound(Z, greatest_lower_bound(X, Y)))
54.84/55.10	= { by lemma 18 }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), identity), least_upper_bound(multiply(X, multiply(inverse(X), Z)), greatest_lower_bound(X, Y)))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), identity), least_upper_bound(multiply(X, multiply(inverse(X), Z)), greatest_lower_bound(Y, X)))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), identity), least_upper_bound(greatest_lower_bound(Y, X), multiply(X, multiply(inverse(X), Z))))
54.84/55.10	= { by lemma 21 }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), identity), least_upper_bound(greatest_lower_bound(Y, X), multiply(inverse(inverse(X)), multiply(inverse(X), Z))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  greatest_lower_bound(least_upper_bound(greatest_lower_bound(Y, X), multiply(inverse(inverse(X)), multiply(inverse(X), Z))), multiply(inverse(inverse(X)), identity))
54.84/55.10	= { by lemma 22 }
54.84/55.10	  greatest_lower_bound(multiply(inverse(inverse(X)), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), greatest_lower_bound(Y, X)))), multiply(inverse(inverse(X)), identity))
54.84/55.10	= { by axiom 7 (monotony_glb1) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), greatest_lower_bound(Y, X))), identity))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), greatest_lower_bound(Y, X)))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), greatest_lower_bound(X, Y)))))
54.84/55.10	= { by axiom 7 (monotony_glb1) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), greatest_lower_bound(multiply(inverse(X), X), multiply(inverse(X), Y)))))
54.84/55.10	= { by axiom 1 (left_inverse) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), greatest_lower_bound(identity, multiply(inverse(X), Y)))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), greatest_lower_bound(multiply(inverse(X), Y), identity))))
54.84/55.10	= { by lemma 36 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Y), multiply(least_upper_bound(multiply(inverse(X), Y), identity), multiply(inverse(X), Z)))))
54.84/55.10	= { by lemma 20 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(multiply(inverse(X), Y), identity), multiply(least_upper_bound(multiply(inverse(X), Y), identity), multiply(inverse(X), Z)))))
54.84/55.10	= { by lemma 23 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(multiply(inverse(X), Y), identity), least_upper_bound(multiply(inverse(X), Z), multiply(multiply(inverse(X), Y), multiply(inverse(X), Z))))))
54.84/55.10	= { by lemma 24 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(multiply(inverse(X), Y), identity)))))
54.84/55.10	= { by axiom 12 (monotony_lub1) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), multiply(multiply(inverse(X), Y), least_upper_bound(multiply(inverse(X), Z), identity)))))
54.84/55.10	= { by lemma 32 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), least_upper_bound(multiply(inverse(X), Y), multiply(multiply(inverse(X), Y), multiply(inverse(X), Z))))))
54.84/55.10	= { by axiom 14 (associativity_of_lub) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)))))
54.84/55.10	= { by axiom 5 (glb_absorbtion) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(multiply(inverse(X), Z), multiply(inverse(X), Z)), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y))))))))
54.84/55.10	= { by lemma 20 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), identity), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(multiply(inverse(X), Z), multiply(inverse(X), Z)), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y))))))))
54.84/55.10	= { by axiom 1 (left_inverse) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(multiply(inverse(X), Z), multiply(inverse(X), Z)), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y))))))))
54.84/55.10	= { by axiom 2 (associativity) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(multiply(inverse(X), Z), multiply(inverse(X), Z)), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y))))))))
54.84/55.10	= { by lemma 24 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y)), least_upper_bound(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(multiply(inverse(X), Z), multiply(inverse(X), Z))))))))
54.84/55.10	= { by axiom 10 (monotony_lub2) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Y)), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), multiply(inverse(X), Z)))))))
54.84/55.10	= { by axiom 12 (monotony_lub1) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), multiply(least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)), least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)))))))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), multiply(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), least_upper_bound(multiply(inverse(X), Y), multiply(inverse(X), Z)))))))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), multiply(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))))))
54.84/55.10	= { by axiom 11 (monotony_glb2) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(greatest_lower_bound(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))))
54.84/55.10	= { by lemma 35 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), least_upper_bound(multiply(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))))))))
54.84/55.10	= { by lemma 24 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))), least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))))
54.84/55.10	= { by lemma 24 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))))))
54.84/55.10	= { by lemma 23 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), multiply(least_upper_bound(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), identity), least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))))
54.84/55.10	= { by lemma 36 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))), identity))))
54.84/55.10	= { by axiom 13 (associativity_of_glb) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)), greatest_lower_bound(multiply(multiply(multiply(inverse(X), Y), multiply(inverse(X), Z)), inverse(least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y)))), identity)))))
54.84/55.10	= { by axiom 6 (lub_absorbtion) }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, least_upper_bound(multiply(inverse(X), Z), multiply(inverse(X), Y))))
54.84/55.10	= { by lemma 22 }
54.84/55.10	  multiply(inverse(inverse(X)), greatest_lower_bound(identity, multiply(inverse(X), least_upper_bound(Y, multiply(X, multiply(inverse(X), Z))))))
54.84/55.10	= { by lemma 34 }
54.84/55.10	  greatest_lower_bound(least_upper_bound(Y, multiply(X, multiply(inverse(X), Z))), multiply(inverse(inverse(X)), identity))
54.84/55.10	= { by lemma 21 }
54.84/55.10	  greatest_lower_bound(least_upper_bound(Y, multiply(X, multiply(inverse(X), Z))), multiply(X, identity))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  greatest_lower_bound(multiply(X, identity), least_upper_bound(Y, multiply(X, multiply(inverse(X), Z))))
54.84/55.10	= { by lemma 20 }
54.84/55.10	  greatest_lower_bound(X, least_upper_bound(Y, multiply(X, multiply(inverse(X), Z))))
54.84/55.10	= { by lemma 18 }
54.84/55.10	  greatest_lower_bound(X, least_upper_bound(Y, Z))
54.84/55.10	
54.84/55.10	Goal 1 (prove_distrun): greatest_lower_bound(a, least_upper_bound(b, c)) = least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c)).
54.84/55.10	Proof:
54.84/55.10	  greatest_lower_bound(a, least_upper_bound(b, c))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  greatest_lower_bound(a, least_upper_bound(c, b))
54.84/55.10	= { by lemma 37 }
54.84/55.10	  greatest_lower_bound(a, least_upper_bound(b, greatest_lower_bound(a, c)))
54.84/55.10	= { by lemma 31 }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, least_upper_bound(b, greatest_lower_bound(a, c))))
54.84/55.10	= { by lemma 37 }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, c), greatest_lower_bound(a, b))))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))))
54.84/55.10	= { by lemma 15 }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))), greatest_lower_bound(c, greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))))))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))), greatest_lower_bound(c, greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(c, a), greatest_lower_bound(a, b))))))
54.84/55.10	= { by axiom 13 (associativity_of_glb) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))), greatest_lower_bound(greatest_lower_bound(c, a), least_upper_bound(greatest_lower_bound(c, a), greatest_lower_bound(a, b)))))
54.84/55.10	= { by axiom 5 (glb_absorbtion) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a))), greatest_lower_bound(c, a)))
54.84/55.10	= { by axiom 4 (symmetry_of_lub) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(c, a), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a)))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, c), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(c, a)))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, c), greatest_lower_bound(a, least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c)))))
54.84/55.10	= { by axiom 8 (symmetry_of_glb) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), least_upper_bound(greatest_lower_bound(a, c), greatest_lower_bound(least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c)), a)))
54.84/55.10	= { by axiom 14 (associativity_of_lub) }
54.84/55.10	  least_upper_bound(least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c)), greatest_lower_bound(least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c)), a))
54.84/55.10	= { by axiom 6 (lub_absorbtion) }
54.84/55.10	  least_upper_bound(greatest_lower_bound(a, b), greatest_lower_bound(a, c))
54.84/55.10	% SZS output end Proof
54.84/55.10	
54.84/55.10	RESULT: Unsatisfiable (the axioms are contradictory).
54.90/55.11	EOF