0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.13/0.34	% Computer   : n005.cluster.edu
0.13/0.34	% Model      : x86_64 x86_64
0.13/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.34	% Memory     : 8042.1875MB
0.13/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.34	% CPULimit   : 180
0.13/0.34	% DateTime   : Thu Aug 29 14:11:18 EDT 2019
0.13/0.34	% CPUTime    : 
117.72/117.94	% SZS status Unsatisfiable
117.72/117.94	
117.72/117.94	% SZS output start Proof
117.72/117.94	Take the following subset of the input axioms:
117.72/117.94	  fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
117.72/117.94	  fof(absorption2, axiom, ![X, Y]: join(X, meet(X, Y))=X).
117.72/117.94	  fof(associativity_of_join, axiom, ![X, Y, Z]: join(join(X, Y), Z)=join(X, join(Y, Z))).
117.72/117.94	  fof(associativity_of_meet, axiom, ![X, Y, Z]: meet(X, meet(Y, Z))=meet(meet(X, Y), Z)).
117.72/117.94	  fof(commutativity_of_join, axiom, ![X, Y]: join(X, Y)=join(Y, X)).
117.72/117.94	  fof(commutativity_of_meet, axiom, ![X, Y]: meet(Y, X)=meet(X, Y)).
117.72/117.94	  fof(complement_join, axiom, ![X]: one=join(X, complement(X))).
117.72/117.94	  fof(complement_meet, axiom, ![X]: meet(X, complement(X))=zero).
117.72/117.94	  fof(equation_H11, axiom, ![X, Y, Z]: meet(X, join(Y, meet(Z, join(X, meet(Y, join(Z, meet(X, Y)))))))=meet(X, join(Y, meet(X, Z)))).
117.72/117.94	  fof(idempotence_of_meet, axiom, ![X]: X=meet(X, X)).
117.72/117.94	  fof(ifeq_axiom, axiom, ![B, A, C]: B=ifeq(A, A, B, C)).
117.72/117.94	  fof(meet_join_complement, axiom, ![X, Y]: ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y)=Y).
117.72/117.94	  fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
117.72/117.94	
117.72/117.94	Now clausify the problem and encode Horn clauses using encoding 3 of
117.72/117.94	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
117.72/117.94	We repeatedly replace C & s=t => u=v by the two clauses:
117.72/117.94	  fresh(y, y, x1...xn) = u
117.72/117.94	  C => fresh(s, t, x1...xn) = v
117.72/117.94	where fresh is a fresh function symbol and x1..xn are the free
117.72/117.94	variables of u and v.
117.72/117.94	A predicate p(X) is encoded as p(X)=true (this is sound, because the
117.72/117.94	input problem has no model of domain size 1).
117.72/117.94	
117.72/117.94	The encoding turns the above axioms into the following unit equations and goals:
117.72/117.94	
117.72/117.94	Axiom 1 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
117.72/117.94	Axiom 2 (idempotence_of_meet): X = meet(X, X).
117.72/117.94	Axiom 3 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
117.72/117.94	Axiom 4 (absorption2): join(X, meet(X, Y)) = X.
117.72/117.94	Axiom 5 (complement_join): one = join(X, complement(X)).
117.72/117.94	Axiom 6 (associativity_of_meet): meet(X, meet(Y, Z)) = meet(meet(X, Y), Z).
117.72/117.94	Axiom 7 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
117.72/117.94	Axiom 8 (commutativity_of_join): join(X, Y) = join(Y, X).
117.72/117.94	Axiom 9 (complement_meet): meet(X, complement(X)) = zero.
117.72/117.94	Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
117.72/117.94	Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
117.92/118.14	Axiom 12 (equation_H11): meet(X, join(Y, meet(Z, join(X, meet(Y, join(Z, meet(X, Y))))))) = meet(X, join(Y, meet(X, Z))).
117.92/118.14	
117.92/118.14	Lemma 13: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
117.92/118.14	Proof:
117.92/118.14	  meet(X, meet(Y, join(Z, meet(X, Y))))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  meet(X, meet(Y, join(meet(X, Y), Z)))
117.92/118.14	= { by axiom 6 (associativity_of_meet) }
117.92/118.14	  meet(meet(X, Y), join(meet(X, Y), Z))
117.92/118.14	= { by axiom 10 (absorption1) }
117.92/118.14	  meet(X, Y)
117.92/118.14	
117.92/118.14	Lemma 14: join(X, meet(Y, X)) = X.
117.92/118.14	Proof:
117.92/118.14	  join(X, meet(Y, X))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(X, meet(X, Y))
117.92/118.14	= { by axiom 4 (absorption2) }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 15: join(meet(X, Z), meet(X, join(Y, meet(X, Z)))) = meet(X, join(Y, meet(X, Z))).
117.92/118.14	Proof:
117.92/118.14	  join(meet(X, Z), meet(X, join(Y, meet(X, Z))))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(meet(Z, X), meet(X, join(Y, meet(X, Z))))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(meet(Z, X), meet(X, join(Y, meet(Z, X))))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  join(meet(X, join(Y, meet(Z, X))), meet(Z, X))
117.92/118.14	= { by lemma 13 }
117.92/118.14	  join(meet(X, join(Y, meet(Z, X))), meet(Z, meet(X, join(Y, meet(Z, X)))))
117.92/118.14	= { by lemma 14 }
117.92/118.14	  meet(X, join(Y, meet(Z, X)))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  meet(X, join(Y, meet(X, Z)))
117.92/118.14	
117.92/118.14	Lemma 16: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
117.92/118.14	Proof:
117.92/118.14	  join(X, join(Y, meet(Z, join(X, Y))))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(X, join(Y, meet(join(X, Y), Z)))
117.92/118.14	= { by axiom 7 (associativity_of_join) }
117.92/118.14	  join(join(X, Y), meet(join(X, Y), Z))
117.92/118.14	= { by axiom 4 (absorption2) }
117.92/118.14	  join(X, Y)
117.92/118.14	
117.92/118.14	Lemma 17: complement(complement(X)) = X.
117.92/118.14	Proof:
117.92/118.14	  complement(complement(X))
117.92/118.14	= { by axiom 1 (ifeq_axiom) }
117.92/118.14	  ifeq(one, one, complement(complement(X)), X)
117.92/118.14	= { by axiom 5 (complement_join) }
117.92/118.14	  ifeq(join(X, complement(X)), one, complement(complement(X)), X)
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  ifeq(join(complement(X), X), one, complement(complement(X)), X)
117.92/118.14	= { by axiom 1 (ifeq_axiom) }
117.92/118.14	  ifeq(join(complement(X), X), one, ifeq(zero, zero, complement(complement(X)), X), X)
117.92/118.14	= { by axiom 9 (complement_meet) }
117.92/118.14	  ifeq(join(complement(X), X), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), X), X)
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  ifeq(join(complement(X), X), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
117.92/118.14	= { by axiom 11 (meet_join_complement) }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 18: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
117.92/118.14	Proof:
117.92/118.14	  meet(X, meet(Y, Z))
117.92/118.14	= { by axiom 6 (associativity_of_meet) }
117.92/118.14	  meet(meet(X, Y), Z)
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  meet(Z, meet(X, Y))
117.92/118.14	
117.92/118.14	Lemma 19: meet(X, join(Y, X)) = X.
117.92/118.14	Proof:
117.92/118.14	  meet(X, join(Y, X))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  meet(X, join(X, Y))
117.92/118.14	= { by axiom 10 (absorption1) }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 20: meet(Y, meet(join(X, Y), Z)) = meet(Y, Z).
117.92/118.14	Proof:
117.92/118.14	  meet(Y, meet(join(X, Y), Z))
117.92/118.14	= { by lemma 18 }
117.92/118.14	  meet(join(X, Y), meet(Z, Y))
117.92/118.14	= { by lemma 18 }
117.92/118.14	  meet(Z, meet(Y, join(X, Y)))
117.92/118.14	= { by lemma 19 }
117.92/118.14	  meet(Z, Y)
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  meet(Y, Z)
117.92/118.14	
117.92/118.14	Lemma 21: join(X, join(Z, meet(X, Y))) = join(X, Z).
117.92/118.14	Proof:
117.92/118.14	  join(X, join(Z, meet(X, Y)))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  join(X, join(meet(X, Y), Z))
117.92/118.14	= { by axiom 7 (associativity_of_join) }
117.92/118.14	  join(join(X, meet(X, Y)), Z)
117.92/118.14	= { by axiom 4 (absorption2) }
117.92/118.14	  join(X, Z)
117.92/118.14	
117.92/118.14	Lemma 22: join(X, join(meet(X, Z), Y)) = join(X, Y).
117.92/118.14	Proof:
117.92/118.14	  join(X, join(meet(X, Z), Y))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  join(X, join(Y, meet(X, Z)))
117.92/118.14	= { by lemma 21 }
117.92/118.14	  join(X, Y)
117.92/118.14	
117.92/118.14	Lemma 23: join(X, meet(Z, join(meet(X, Y), meet(X, W)))) = X.
117.92/118.14	Proof:
117.92/118.14	  join(X, meet(Z, join(meet(X, Y), meet(X, W))))
117.92/118.14	= { by lemma 22 }
117.92/118.14	  join(X, join(meet(X, Y), meet(Z, join(meet(X, Y), meet(X, W)))))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.14	  join(X, join(meet(X, Y), meet(Z, join(meet(X, W), meet(X, Y)))))
117.92/118.14	= { by lemma 22 }
117.92/118.14	  join(X, join(meet(X, W), join(meet(X, Y), meet(Z, join(meet(X, W), meet(X, Y))))))
117.92/118.14	= { by lemma 16 }
117.92/118.14	  join(X, join(meet(X, W), meet(X, Y)))
117.92/118.14	= { by lemma 22 }
117.92/118.14	  join(X, meet(X, Y))
117.92/118.14	= { by axiom 4 (absorption2) }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 24: join(X, meet(join(meet(Y, meet(Z, X)), meet(X, W)), V)) = X.
117.92/118.14	Proof:
117.92/118.14	  join(X, meet(join(meet(Y, meet(Z, X)), meet(X, W)), V))
117.92/118.14	= { by axiom 6 (associativity_of_meet) }
117.92/118.14	  join(X, meet(join(meet(meet(Y, Z), X), meet(X, W)), V))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(X, meet(join(meet(X, meet(Y, Z)), meet(X, W)), V))
117.92/118.14	= { by axiom 3 (commutativity_of_meet) }
117.92/118.14	  join(X, meet(V, join(meet(X, meet(Y, Z)), meet(X, W))))
117.92/118.14	= { by lemma 23 }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 25: meet(X, one) = X.
117.92/118.14	Proof:
117.92/118.14	  meet(X, one)
117.92/118.14	= { by axiom 5 (complement_join) }
117.92/118.14	  meet(X, join(X, complement(X)))
117.92/118.14	= { by axiom 10 (absorption1) }
117.92/118.14	  X
117.92/118.14	
117.92/118.14	Lemma 26: meet(X, join(Y, meet(join(Y, X), complement(meet(X, Y))))) = meet(X, join(Y, meet(X, complement(meet(X, Y))))).
117.92/118.14	Proof:
117.92/118.14	  meet(X, join(Y, meet(join(Y, X), complement(meet(X, Y)))))
117.92/118.14	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  meet(X, join(Y, meet(join(X, Y), complement(meet(X, Y)))))
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.15	  meet(X, join(Y, meet(complement(meet(X, Y)), join(X, Y))))
117.92/118.15	= { by lemma 25 }
117.92/118.15	  meet(X, join(Y, meet(complement(meet(X, Y)), join(X, meet(Y, one)))))
117.92/118.15	= { by axiom 5 (complement_join) }
117.92/118.15	  meet(X, join(Y, meet(complement(meet(X, Y)), join(X, meet(Y, join(meet(X, Y), complement(meet(X, Y))))))))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  meet(X, join(Y, meet(complement(meet(X, Y)), join(X, meet(Y, join(complement(meet(X, Y)), meet(X, Y)))))))
117.92/118.15	= { by axiom 12 (equation_H11) }
117.92/118.15	  meet(X, join(Y, meet(X, complement(meet(X, Y)))))
117.92/118.15	
117.92/118.15	Lemma 27: meet(one, X) = X.
117.92/118.15	Proof:
117.92/118.15	  meet(one, X)
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.15	  meet(X, one)
117.92/118.15	= { by lemma 25 }
117.92/118.15	  X
117.92/118.15	
117.92/118.15	Lemma 28: join(X, one) = one.
117.92/118.15	Proof:
117.92/118.15	  join(X, one)
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  join(one, X)
117.92/118.15	= { by lemma 27 }
117.92/118.15	  join(one, meet(one, X))
117.92/118.15	= { by axiom 4 (absorption2) }
117.92/118.15	  one
117.92/118.15	
117.92/118.15	Lemma 29: join(one, X) = one.
117.92/118.15	Proof:
117.92/118.15	  join(one, X)
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  join(X, one)
117.92/118.15	= { by lemma 28 }
117.92/118.15	  one
117.92/118.15	
117.92/118.15	Lemma 30: join(X, join(Y, complement(X))) = one.
117.92/118.15	Proof:
117.92/118.15	  join(X, join(Y, complement(X)))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  join(X, join(complement(X), Y))
117.92/118.15	= { by axiom 7 (associativity_of_join) }
117.92/118.15	  join(join(X, complement(X)), Y)
117.92/118.15	= { by axiom 5 (complement_join) }
117.92/118.15	  join(one, Y)
117.92/118.15	= { by lemma 29 }
117.92/118.15	  one
117.92/118.15	
117.92/118.15	Lemma 31: join(X, complement(meet(X, Y))) = one.
117.92/118.15	Proof:
117.92/118.15	  join(X, complement(meet(X, Y)))
117.92/118.15	= { by lemma 22 }
117.92/118.15	  join(X, join(meet(X, Y), complement(meet(X, Y))))
117.92/118.15	= { by axiom 5 (complement_join) }
117.92/118.15	  join(X, one)
117.92/118.15	= { by lemma 28 }
117.92/118.15	  one
117.92/118.15	
117.92/118.15	Lemma 32: join(X, complement(meet(Y, X))) = one.
117.92/118.15	Proof:
117.92/118.15	  join(X, complement(meet(Y, X)))
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.15	  join(X, complement(meet(X, Y)))
117.92/118.15	= { by lemma 31 }
117.92/118.15	  one
117.92/118.15	
117.92/118.15	Lemma 33: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
117.92/118.15	Proof:
117.92/118.15	  meet(X, meet(Y, complement(meet(X, Y))))
117.92/118.15	= { by axiom 6 (associativity_of_meet) }
117.92/118.15	  meet(meet(X, Y), complement(meet(X, Y)))
117.92/118.15	= { by axiom 9 (complement_meet) }
117.92/118.15	  zero
117.92/118.15	
117.92/118.15	Lemma 34: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
117.92/118.15	Proof:
117.92/118.15	  meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
117.92/118.15	= { by axiom 11 (meet_join_complement) }
117.92/118.15	  ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 14 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), join(Y, meet(join(X, complement(Y)), complement(meet(join(X, complement(Y)), Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 26 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), join(Y, meet(join(Y, join(X, complement(Y))), complement(meet(join(X, complement(Y)), Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 30 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), join(Y, meet(one, complement(meet(join(X, complement(Y)), Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 27 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), join(Y, complement(meet(join(X, complement(Y)), Y))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 32 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), one)), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 25 }
117.92/118.15	  ifeq(join(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), join(X, complement(Y))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by axiom 7 (associativity_of_join) }
117.92/118.15	  ifeq(join(Y, join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), join(X, complement(Y)))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by axiom 7 (associativity_of_join) }
117.92/118.15	  ifeq(join(Y, join(join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), X), complement(Y))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 30 }
117.92/118.15	  ifeq(one, one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by axiom 1 (ifeq_axiom) }
117.92/118.15	  ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by lemma 33 }
117.92/118.15	  ifeq(zero, zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
117.92/118.15	= { by axiom 1 (ifeq_axiom) }
117.92/118.15	  complement(Y)
117.92/118.15	
117.92/118.15	Lemma 35: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
117.92/118.15	Proof:
117.92/118.15	  meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by lemma 17 }
117.92/118.15	  meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, X))))
117.92/118.15	= { by lemma 17 }
117.92/118.15	  meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
117.92/118.15	= { by lemma 34 }
117.92/118.15	  complement(complement(X))
117.92/118.15	= { by lemma 17 }
117.92/118.15	  X
117.92/118.15	
117.92/118.15	Lemma 36: join(X, complement(meet(complement(X), join(X, Y)))) = complement(meet(complement(X), join(X, Y))).
117.92/118.15	Proof:
117.92/118.15	  join(X, complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  join(complement(meet(complement(X), join(X, Y))), X)
117.92/118.15	= { by lemma 35 }
117.92/118.15	  join(complement(meet(complement(X), join(X, Y))), meet(join(X, Y), complement(meet(complement(X), join(X, Y)))))
117.92/118.15	= { by lemma 14 }
117.92/118.15	  complement(meet(complement(X), join(X, Y)))
117.92/118.15	
117.92/118.15	Lemma 37: meet(join(Y, X), complement(meet(complement(X), join(X, Y)))) = X.
117.92/118.15	Proof:
117.92/118.15	  meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by axiom 8 (commutativity_of_join) }
117.92/118.15	  meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
117.92/118.15	= { by lemma 35 }
117.92/118.15	  X
117.92/118.15	
117.92/118.15	Lemma 38: meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))) = complement(X).
117.92/118.15	Proof:
117.92/118.15	  meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))
117.92/118.15	= { by axiom 11 (meet_join_complement) }
117.92/118.15	  ifeq(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.15	= { by lemma 14 }
117.92/118.15	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.15	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.15	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(X, Y)))))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(Y, X)))))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(complement(meet(Y, X)), complement(meet(complement(meet(Y, X)), X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 26 }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(join(X, complement(meet(Y, X))), complement(meet(complement(meet(Y, X)), X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 32 }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, meet(one, complement(meet(complement(meet(Y, X)), X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 27 }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), join(X, complement(meet(complement(meet(Y, X)), X))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 32 }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(Y, X)), one)), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 25 }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), complement(meet(Y, X))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  ifeq(join(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), complement(meet(X, Y))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  ifeq(join(X, join(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))), complement(meet(X, Y)))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  ifeq(join(X, join(complement(meet(X, Y)), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  ifeq(join(join(X, complement(meet(X, Y))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 31 }
117.92/118.16	  ifeq(join(one, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 29 }
117.92/118.16	  ifeq(one, one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 1 (ifeq_axiom) }
117.92/118.16	  ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by lemma 33 }
117.92/118.16	  ifeq(zero, zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 1 (ifeq_axiom) }
117.92/118.16	  complement(X)
117.92/118.16	
117.92/118.16	Lemma 39: meet(complement(X), complement(meet(complement(X), join(X, Y)))) = complement(join(X, Y)).
117.92/118.16	Proof:
117.92/118.16	  meet(complement(X), complement(meet(complement(X), join(X, Y))))
117.92/118.16	= { by lemma 37 }
117.92/118.16	  meet(complement(meet(join(Y, X), complement(meet(complement(X), join(X, Y))))), complement(meet(complement(X), join(X, Y))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(complement(meet(join(Y, X), complement(meet(complement(X), join(X, Y))))), complement(meet(complement(X), join(Y, X))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(complement(meet(join(Y, X), complement(meet(complement(X), join(X, Y))))), complement(meet(join(Y, X), complement(X))))
117.92/118.16	= { by lemma 37 }
117.92/118.16	  meet(complement(meet(join(Y, X), complement(meet(complement(X), join(X, Y))))), complement(meet(join(Y, X), complement(meet(join(Y, X), complement(meet(complement(X), join(X, Y))))))))
117.92/118.16	= { by lemma 38 }
117.92/118.16	  complement(join(Y, X))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  complement(join(X, Y))
117.92/118.16	
117.92/118.16	Lemma 40: complement(join(X, meet(complement(X), join(X, Y)))) = complement(join(X, Y)).
117.92/118.16	Proof:
117.92/118.16	  complement(join(X, meet(complement(X), join(X, Y))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  complement(join(X, meet(join(X, Y), complement(X))))
117.92/118.16	= { by lemma 39 }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), join(X, meet(join(X, Y), complement(X))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), join(X, meet(complement(X), join(X, Y))))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), join(X, meet(complement(X), join(Y, X))))))
117.92/118.16	= { by lemma 19 }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), meet(join(X, meet(complement(X), join(Y, X))), join(Y, join(X, meet(complement(X), join(Y, X))))))))
117.92/118.16	= { by lemma 16 }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), meet(join(X, meet(complement(X), join(Y, X))), join(Y, X)))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), meet(join(Y, X), join(X, meet(complement(X), join(Y, X)))))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), meet(join(X, Y), join(X, meet(complement(X), join(Y, X)))))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), meet(join(X, Y), join(X, meet(complement(X), join(X, Y)))))))
117.92/118.16	= { by lemma 13 }
117.92/118.16	  meet(complement(X), complement(meet(complement(X), join(X, Y))))
117.92/118.16	= { by lemma 39 }
117.92/118.16	  complement(join(X, Y))
117.92/118.16	
117.92/118.16	Lemma 41: join(X, meet(complement(X), join(X, Y))) = join(X, Y).
117.92/118.16	Proof:
117.92/118.16	  join(X, meet(complement(X), join(X, Y)))
117.92/118.16	= { by lemma 17 }
117.92/118.16	  complement(complement(join(X, meet(complement(X), join(X, Y)))))
117.92/118.16	= { by lemma 40 }
117.92/118.16	  complement(complement(join(X, Y)))
117.92/118.16	= { by lemma 17 }
117.92/118.16	  join(X, Y)
117.92/118.16	
117.92/118.16	Lemma 42: complement(meet(complement(X), join(X, Y))) = join(X, complement(join(X, Y))).
117.92/118.16	Proof:
117.92/118.16	  complement(meet(complement(X), join(X, Y)))
117.92/118.16	= { by lemma 36 }
117.92/118.16	  join(X, complement(meet(complement(X), join(X, Y))))
117.92/118.16	= { by lemma 41 }
117.92/118.16	  join(X, meet(complement(X), join(X, complement(meet(complement(X), join(X, Y))))))
117.92/118.16	= { by lemma 36 }
117.92/118.16	  join(X, meet(complement(X), complement(meet(complement(X), join(X, Y)))))
117.92/118.16	= { by lemma 39 }
117.92/118.16	  join(X, complement(join(X, Y)))
117.92/118.16	
117.92/118.16	Lemma 43: join(X, join(Y, Z)) = join(Y, join(X, Z)).
117.92/118.16	Proof:
117.92/118.16	  join(X, join(Y, Z))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  join(join(Y, Z), X)
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  join(Y, join(Z, X))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  join(Y, join(X, Z))
117.92/118.16	
117.92/118.16	Lemma 44: meet(join(X, Y), join(X, complement(join(X, Y)))) = X.
117.92/118.16	Proof:
117.92/118.16	  meet(join(X, Y), join(X, complement(join(X, Y))))
117.92/118.16	= { by lemma 24 }
117.92/118.16	  meet(join(X, join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?))), join(X, complement(join(X, Y))))
117.92/118.16	= { by lemma 43 }
117.92/118.16	  meet(join(Y, join(X, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?))), join(X, complement(join(X, Y))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(join(Y, join(meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?), X)), join(X, complement(join(X, Y))))
117.92/118.16	= { by lemma 17 }
117.92/118.16	  meet(join(Y, join(meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?), complement(complement(X)))), join(X, complement(join(X, Y))))
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), join(X, complement(join(X, Y))))
117.92/118.16	= { by lemma 24 }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), join(X, complement(join(X, join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?))))))
117.92/118.16	= { by lemma 42 }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), complement(meet(complement(X), join(X, join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?))))))
117.92/118.16	= { by lemma 43 }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), complement(meet(complement(X), join(Y, join(X, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?))))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), complement(meet(complement(X), join(Y, join(meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?), X)))))
117.92/118.16	= { by lemma 17 }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), complement(meet(complement(X), join(Y, join(meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?), complement(complement(X)))))))
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  meet(join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))), complement(meet(complement(X), join(join(Y, meet(join(meet(?, meet(?, Y)), meet(Y, ?)), ?)), complement(complement(X))))))
117.92/118.16	= { by lemma 34 }
117.92/118.16	  complement(complement(X))
117.92/118.16	= { by lemma 17 }
117.92/118.16	  X
117.92/118.16	
117.92/118.16	Lemma 45: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
117.92/118.16	Proof:
117.92/118.16	  meet(X, join(Y, complement(join(X, Y))))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(X, join(Y, complement(join(Y, X))))
117.92/118.16	= { by lemma 20 }
117.92/118.16	  meet(X, meet(join(Y, X), join(Y, complement(join(Y, X)))))
117.92/118.16	= { by lemma 44 }
117.92/118.16	  meet(X, Y)
117.92/118.16	
117.92/118.16	Lemma 46: join(X, join(Y, complement(join(X, Y)))) = one.
117.92/118.16	Proof:
117.92/118.16	  join(X, join(Y, complement(join(X, Y))))
117.92/118.16	= { by axiom 7 (associativity_of_join) }
117.92/118.16	  join(join(X, Y), complement(join(X, Y)))
117.92/118.16	= { by axiom 5 (complement_join) }
117.92/118.16	  one
117.92/118.16	
117.92/118.16	Lemma 47: join(meet(X, complement(meet(X, Y))), complement(join(Y, meet(X, complement(meet(X, Y)))))) = complement(Y).
117.92/118.16	Proof:
117.92/118.16	  join(meet(X, complement(meet(X, Y))), complement(join(Y, meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(X, Y))))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))
117.92/118.16	= { by axiom 11 (meet_join_complement) }
117.92/118.16	  ifeq(join(Y, join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), one, ifeq(meet(Y, join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), zero, complement(Y), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X)))))))
117.92/118.16	= { by lemma 46 }
117.92/118.16	  ifeq(one, one, ifeq(meet(Y, join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), zero, complement(Y), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X)))))))
117.92/118.16	= { by axiom 1 (ifeq_axiom) }
117.92/118.16	  ifeq(meet(Y, join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X))))))), zero, complement(Y), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X)))))))
117.92/118.16	= { by lemma 45 }
117.92/118.16	  ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(Y), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X)))))))
117.92/118.16	= { by lemma 33 }
117.92/118.16	  ifeq(zero, zero, complement(Y), join(meet(X, complement(meet(Y, X))), complement(join(Y, meet(X, complement(meet(Y, X)))))))
117.92/118.16	= { by axiom 1 (ifeq_axiom) }
117.92/118.16	  complement(Y)
117.92/118.16	
117.92/118.16	Lemma 48: meet(join(Y, X), join(X, meet(Y, Z))) = join(X, meet(Y, Z)).
117.92/118.16	Proof:
117.92/118.16	  meet(join(Y, X), join(X, meet(Y, Z)))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(join(X, Y), join(X, meet(Y, Z)))
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  meet(join(Y, X), join(X, meet(Y, Z)))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(join(X, meet(Y, Z)), join(Y, X))
117.92/118.16	= { by lemma 21 }
117.92/118.16	  meet(join(X, meet(Y, Z)), join(Y, join(X, meet(Y, Z))))
117.92/118.16	= { by lemma 19 }
117.92/118.16	  join(X, meet(Y, Z))
117.92/118.16	
117.92/118.16	Lemma 49: meet(X, meet(Y, X)) = meet(X, Y).
117.92/118.16	Proof:
117.92/118.16	  meet(X, meet(Y, X))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(X, meet(X, Y))
117.92/118.16	= { by axiom 6 (associativity_of_meet) }
117.92/118.16	  meet(meet(X, X), Y)
117.92/118.16	= { by axiom 2 (idempotence_of_meet) }
117.92/118.16	  meet(X, Y)
117.92/118.16	
117.92/118.16	Lemma 50: join(X, zero) = X.
117.92/118.16	Proof:
117.92/118.16	  join(X, zero)
117.92/118.16	= { by axiom 9 (complement_meet) }
117.92/118.16	  join(X, meet(X, complement(X)))
117.92/118.16	= { by axiom 4 (absorption2) }
117.92/118.16	  X
117.92/118.16	
117.92/118.16	Lemma 51: join(zero, X) = X.
117.92/118.16	Proof:
117.92/118.16	  join(zero, X)
117.92/118.16	= { by axiom 8 (commutativity_of_join) }
117.92/118.16	  join(X, zero)
117.92/118.16	= { by lemma 50 }
117.92/118.16	  X
117.92/118.16	
117.92/118.16	Lemma 52: meet(X, join(meet(Y, complement(X)), meet(Z, complement(X)))) = zero.
117.92/118.16	Proof:
117.92/118.16	  meet(X, join(meet(Y, complement(X)), meet(Z, complement(X))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(X, join(meet(complement(X), Y), meet(Z, complement(X))))
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(X, join(meet(complement(X), Y), meet(complement(X), Z)))
117.92/118.16	= { by lemma 48 }
117.92/118.16	  meet(X, meet(join(complement(X), meet(complement(X), Y)), join(meet(complement(X), Y), meet(complement(X), Z))))
117.92/118.16	= { by axiom 4 (absorption2) }
117.92/118.16	  meet(X, meet(complement(X), join(meet(complement(X), Y), meet(complement(X), Z))))
117.92/118.16	= { by lemma 18 }
117.92/118.16	  meet(complement(X), meet(join(meet(complement(X), Y), meet(complement(X), Z)), X))
117.92/118.16	= { by lemma 18 }
117.92/118.16	  meet(join(meet(complement(X), Y), meet(complement(X), Z)), meet(X, complement(X)))
117.92/118.16	= { by axiom 9 (complement_meet) }
117.92/118.16	  meet(join(meet(complement(X), Y), meet(complement(X), Z)), zero)
117.92/118.16	= { by axiom 3 (commutativity_of_meet) }
117.92/118.16	  meet(zero, join(meet(complement(X), Y), meet(complement(X), Z)))
117.92/118.16	= { by lemma 51 }
117.92/118.16	  join(zero, meet(zero, join(meet(complement(X), Y), meet(complement(X), Z))))
117.92/118.16	= { by axiom 4 (absorption2) }
117.92/118.18	  zero
117.92/118.18	
117.92/118.18	Lemma 53: join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))) = complement(Y).
117.92/118.18	Proof:
117.92/118.18	  join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), Y)))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
117.92/118.18	= { by axiom 11 (meet_join_complement) }
117.92/118.18	  ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 46 }
117.92/118.18	  ifeq(one, one, ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 1 (ifeq_axiom) }
117.92/118.18	  ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  ifeq(meet(Y, join(complement(join(meet(X, complement(Y)), Y)), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  ifeq(meet(Y, join(complement(join(meet(X, complement(Y)), complement(complement(Y)))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 38 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))))), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 34 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(complement(complement(Y))), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))), join(meet(X, complement(Y)), complement(complement(Y)))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 14 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))), join(join(meet(X, complement(Y)), complement(complement(Y))), meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(meet(complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))), join(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))), join(meet(X, complement(Y)), complement(complement(Y))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 20 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), meet(join(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))), join(meet(X, complement(Y)), complement(complement(Y)))), complement(meet(complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))), join(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))), join(meet(X, complement(Y)), complement(complement(Y)))))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 35 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 49 }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(join(meet(X, complement(Y)), complement(complement(Y))), complement(Y)))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(Y), complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y)))))), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  ifeq(meet(Y, join(meet(complement(meet(complement(Y), join(meet(X, complement(Y)), complement(complement(Y))))), complement(Y)), meet(X, complement(Y)))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by lemma 52 }
117.92/118.18	  ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
117.92/118.18	= { by axiom 1 (ifeq_axiom) }
117.92/118.18	  complement(Y)
117.92/118.18	
117.92/118.18	Lemma 54: join(meet(X, Y), complement(join(complement(X), meet(X, Y)))) = X.
117.92/118.18	Proof:
117.92/118.18	  join(meet(X, Y), complement(join(complement(X), meet(X, Y))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(meet(Y, X), complement(join(complement(X), meet(X, Y))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(X, Y))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, X))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))
117.92/118.18	= { by lemma 53 }
117.92/118.18	  complement(complement(X))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  X
117.92/118.18	
117.92/118.18	Lemma 55: complement(meet(X, complement(meet(X, Y)))) = join(complement(X), meet(X, Y)).
117.92/118.18	Proof:
117.92/118.18	  complement(meet(X, complement(meet(X, Y))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  complement(meet(complement(meet(X, Y)), X))
117.92/118.18	= { by axiom 4 (absorption2) }
117.92/118.18	  join(complement(meet(complement(meet(X, Y)), X)), meet(complement(meet(complement(meet(X, Y)), X)), complement(meet(complement(meet(X, Y)), complement(meet(complement(meet(X, Y)), X))))))
117.92/118.18	= { by lemma 38 }
117.92/118.18	  join(complement(meet(complement(meet(X, Y)), X)), complement(complement(meet(X, Y))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(complement(complement(meet(X, Y))), complement(meet(complement(meet(X, Y)), X)))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), X)))
117.92/118.18	= { by lemma 13 }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), meet(X, join(meet(X, Y), meet(complement(meet(X, Y)), X))))))
117.92/118.18	= { by lemma 14 }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), meet(join(X, meet(complement(meet(X, Y)), X)), join(meet(X, Y), meet(complement(meet(X, Y)), X))))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), meet(join(X, meet(complement(meet(X, Y)), X)), join(meet(complement(meet(X, Y)), X), meet(X, Y))))))
117.92/118.18	= { by lemma 48 }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), join(meet(complement(meet(X, Y)), X), meet(X, Y)))))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(X, Y), complement(meet(complement(meet(X, Y)), join(meet(X, Y), meet(complement(meet(X, Y)), X)))))
117.92/118.18	= { by lemma 36 }
117.92/118.18	  complement(meet(complement(meet(X, Y)), join(meet(X, Y), meet(complement(meet(X, Y)), X))))
117.92/118.18	= { by lemma 42 }
117.92/118.18	  join(meet(X, Y), complement(join(meet(X, Y), meet(complement(meet(X, Y)), X))))
117.92/118.18	= { by lemma 54 }
117.92/118.18	  join(meet(X, Y), complement(join(meet(X, Y), meet(complement(meet(X, Y)), join(meet(X, Y), complement(join(complement(X), meet(X, Y))))))))
117.92/118.18	= { by lemma 41 }
117.92/118.18	  join(meet(X, Y), complement(join(meet(X, Y), complement(join(complement(X), meet(X, Y))))))
117.92/118.18	= { by lemma 54 }
117.92/118.18	  join(meet(X, Y), complement(X))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(complement(X), meet(X, Y))
117.92/118.18	
117.92/118.18	Lemma 56: join(complement(X), meet(Y, meet(Z, complement(X)))) = complement(X).
117.92/118.18	Proof:
117.92/118.18	  join(complement(X), meet(Y, meet(Z, complement(X))))
117.92/118.18	= { by lemma 50 }
117.92/118.18	  join(complement(X), meet(Y, join(meet(Z, complement(X)), zero)))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), zero)))
117.92/118.18	= { by lemma 52 }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), meet(complement(X), join(meet(?, complement(complement(X))), meet(?, complement(complement(X))))))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), meet(complement(X), join(meet(?, X), meet(?, complement(complement(X))))))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), meet(complement(X), join(meet(X, ?), meet(?, complement(complement(X))))))))
117.92/118.18	= { by lemma 17 }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), meet(complement(X), join(meet(X, ?), meet(?, X))))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(complement(X), meet(Y, join(meet(complement(X), Z), meet(complement(X), join(meet(X, ?), meet(X, ?))))))
117.92/118.18	= { by lemma 23 }
117.92/118.18	  complement(X)
117.92/118.18	
117.92/118.18	Lemma 57: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
117.92/118.18	Proof:
117.92/118.18	  meet(X, meet(Y, Z))
117.92/118.18	= { by axiom 6 (associativity_of_meet) }
117.92/118.18	  meet(meet(X, Y), Z)
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  meet(meet(Y, X), Z)
117.92/118.18	= { by axiom 6 (associativity_of_meet) }
117.92/118.18	  meet(Y, meet(X, Z))
117.92/118.18	
117.92/118.18	Lemma 58: join(meet(X, meet(Y, W)), meet(W, join(Y, Z))) = meet(W, join(Y, Z)).
117.92/118.18	Proof:
117.92/118.18	  join(meet(X, meet(Y, W)), meet(W, join(Y, Z)))
117.92/118.18	= { by lemma 57 }
117.92/118.18	  join(meet(Y, meet(X, W)), meet(W, join(Y, Z)))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  join(meet(Y, meet(W, X)), meet(W, join(Y, Z)))
117.92/118.18	= { by lemma 57 }
117.92/118.18	  join(meet(W, meet(Y, X)), meet(W, join(Y, Z)))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(W, meet(Y, X)), meet(W, join(Z, Y)))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  join(meet(W, join(Z, Y)), meet(W, meet(Y, X)))
117.92/118.18	= { by lemma 20 }
117.92/118.18	  join(meet(W, join(Z, Y)), meet(W, meet(Y, meet(join(Z, Y), X))))
117.92/118.18	= { by lemma 18 }
117.92/118.18	  join(meet(W, join(Z, Y)), meet(W, meet(join(Z, Y), meet(X, Y))))
117.92/118.18	= { by axiom 6 (associativity_of_meet) }
117.92/118.18	  join(meet(W, join(Z, Y)), meet(meet(W, join(Z, Y)), meet(X, Y)))
117.92/118.18	= { by axiom 4 (absorption2) }
117.92/118.18	  meet(W, join(Z, Y))
117.92/118.18	= { by axiom 8 (commutativity_of_join) }
117.92/118.18	  meet(W, join(Y, Z))
117.92/118.18	
117.92/118.18	Lemma 59: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
117.92/118.18	Proof:
117.92/118.18	  meet(X, complement(meet(X, Y)))
117.92/118.18	= { by axiom 10 (absorption1) }
117.92/118.18	  meet(meet(X, complement(meet(X, Y))), join(meet(X, complement(meet(X, Y))), complement(join(Y, meet(X, complement(meet(X, Y)))))))
117.92/118.18	= { by lemma 47 }
117.92/118.18	  meet(meet(X, complement(meet(X, Y))), complement(Y))
117.92/118.18	= { by axiom 6 (associativity_of_meet) }
117.92/118.18	  meet(X, meet(complement(meet(X, Y)), complement(Y)))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  meet(X, meet(complement(Y), complement(meet(X, Y))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  meet(X, meet(complement(Y), complement(meet(Y, X))))
117.92/118.18	= { by axiom 3 (commutativity_of_meet) }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), complement(Y)))
117.92/118.18	= { by lemma 38 }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), meet(complement(meet(Y, X)), complement(meet(Y, complement(meet(Y, X)))))))
117.92/118.18	= { by lemma 56 }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X))))))))))
117.92/118.18	= { by lemma 58 }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), join(meet(?, meet(complement(meet(Y, complement(meet(Y, X)))), complement(meet(Y, X)))), meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X)))))))))))
117.92/118.18	= { by lemma 15 }
117.92/118.18	  meet(X, join(meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X)))))))), meet(complement(meet(Y, X)), join(meet(?, meet(complement(meet(Y, complement(meet(Y, X)))), complement(meet(Y, X)))), meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X))))))))))))
117.92/118.18	= { by lemma 58 }
117.92/118.18	  meet(X, join(meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X)))))))), meet(complement(meet(Y, X)), meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X)))))))))))
117.92/118.18	= { by lemma 14 }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), join(complement(meet(Y, complement(meet(Y, X)))), meet(?, meet(?, complement(meet(Y, complement(meet(Y, X)))))))))
117.92/118.18	= { by lemma 56 }
117.92/118.18	  meet(X, meet(complement(meet(Y, X)), complement(meet(Y, complement(meet(Y, X))))))
117.92/118.18	= { by lemma 38 }
118.08/118.36	  meet(X, complement(Y))
118.08/118.36	
118.08/118.36	Lemma 60: complement(meet(Y, complement(X))) = join(X, complement(Y)).
118.08/118.36	Proof:
118.08/118.36	  complement(meet(Y, complement(X)))
118.08/118.36	= { by lemma 50 }
118.08/118.36	  complement(join(meet(Y, complement(X)), zero))
118.08/118.36	= { by lemma 59 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, X))), zero))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(X, Y))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(X, complement(complement(Y))))), zero))
118.08/118.36	= { by lemma 44 }
118.08/118.36	  complement(join(meet(Y, complement(meet(join(meet(X, complement(complement(Y))), complement(join(complement(Y), meet(X, complement(complement(Y)))))), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), complement(join(complement(Y), meet(X, complement(complement(Y))))))))))), zero))
118.08/118.36	= { by lemma 53 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), complement(join(complement(Y), meet(X, complement(complement(Y))))))))))), zero))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), complement(Y))))))))), zero))
118.08/118.36	= { by lemma 39 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), meet(complement(meet(X, complement(complement(Y)))), complement(meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(Y))))))))))), zero))
118.08/118.36	= { by lemma 36 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(Y)))))))))))), zero))
118.08/118.36	= { by lemma 40 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(join(meet(X, complement(complement(Y))), complement(meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(Y)))))))))), zero))
118.08/118.36	= { by lemma 36 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(complement(meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(Y))))))))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(meet(X, complement(complement(Y)))), join(meet(X, complement(complement(Y))), complement(Y))))))), zero))
118.08/118.36	= { by lemma 41 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(meet(X, complement(complement(Y))), complement(Y))))), zero))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(complement(Y), meet(X, complement(complement(Y))))))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(X, complement(complement(Y))))))), zero))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(complement(complement(Y)), X))))), zero))
118.08/118.36	= { by axiom 12 (equation_H11) }
118.08/118.36	  complement(join(meet(Y, complement(meet(complement(complement(Y)), join(complement(complement(complement(Y))), meet(X, join(complement(complement(Y)), meet(complement(complement(complement(Y))), join(X, meet(complement(complement(Y)), complement(complement(complement(Y)))))))))))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(complement(complement(Y))), meet(X, join(complement(complement(Y)), meet(complement(complement(complement(Y))), join(X, meet(complement(complement(Y)), complement(complement(complement(Y)))))))))))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(complement(complement(complement(Y))), join(X, meet(complement(complement(Y)), complement(complement(complement(Y)))))))))))), zero))
118.08/118.36	= { by axiom 9 (complement_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(complement(complement(complement(Y))), join(X, zero)))))))), zero))
118.08/118.36	= { by lemma 50 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(complement(complement(complement(Y))), X))))))), zero))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(X, complement(complement(complement(Y)))))))))), zero))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(X, complement(Y)))))))), zero))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(complement(complement(Y)), meet(complement(Y), X))))))), zero))
118.08/118.36	= { by lemma 55 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, complement(meet(complement(Y), complement(meet(complement(Y), X))))))))), zero))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, complement(meet(complement(Y), complement(meet(X, complement(Y)))))))))), zero))
118.08/118.36	= { by lemma 59 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, complement(meet(X, meet(complement(Y), complement(meet(X, complement(Y))))))))))), zero))
118.08/118.36	= { by lemma 33 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, complement(zero)))))), zero))
118.08/118.36	= { by lemma 51 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, join(zero, complement(zero))))))), zero))
118.08/118.36	= { by axiom 5 (complement_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), meet(X, one))))), zero))
118.08/118.36	= { by lemma 25 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(complement(Y), X)))), zero))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), zero))
118.08/118.36	= { by axiom 9 (complement_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), meet(one, complement(one))))
118.08/118.36	= { by lemma 27 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(one)))
118.08/118.36	= { by lemma 31 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(join(X, complement(Y)), complement(meet(join(X, complement(Y)), Y))))))
118.08/118.36	= { by axiom 7 (associativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, join(complement(Y), complement(meet(join(X, complement(Y)), Y)))))))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, join(complement(Y), complement(meet(Y, join(X, complement(Y)))))))))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, join(complement(meet(Y, join(X, complement(Y)))), complement(Y))))))
118.08/118.36	= { by lemma 34 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, join(complement(meet(Y, join(X, complement(Y)))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))))))
118.08/118.36	= { by lemma 14 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, join(X, complement(Y))))))))
118.08/118.36	= { by lemma 49 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, meet(join(X, complement(Y)), Y)))))))
118.08/118.36	= { by lemma 45 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, join(meet(join(X, complement(Y)), Y), complement(join(Y, meet(join(X, complement(Y)), Y))))))))))
118.08/118.36	= { by lemma 14 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, join(meet(join(X, complement(Y)), Y), complement(Y))))))))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, join(complement(Y), meet(join(X, complement(Y)), Y))))))))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, join(complement(Y), meet(Y, join(X, complement(Y))))))))))
118.08/118.36	= { by lemma 55 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, complement(meet(Y, complement(meet(Y, complement(meet(Y, join(X, complement(Y))))))))))))
118.08/118.36	= { by lemma 55 }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(X, join(complement(Y), meet(Y, complement(meet(Y, join(X, complement(Y))))))))))
118.08/118.36	= { by axiom 7 (associativity_of_join) }
118.08/118.36	  complement(join(meet(Y, complement(meet(Y, join(X, complement(Y))))), complement(join(join(X, complement(Y)), meet(Y, complement(meet(Y, join(X, complement(Y)))))))))
118.08/118.36	= { by lemma 47 }
118.08/118.36	  complement(complement(join(X, complement(Y))))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  join(X, complement(Y))
118.08/118.36	
118.08/118.36	Lemma 61: meet(X, join(Z, meet(X, Y))) = meet(X, join(Y, Z)).
118.08/118.36	Proof:
118.08/118.36	  meet(X, join(Z, meet(X, Y)))
118.08/118.36	= { by lemma 17 }
118.08/118.36	  meet(X, join(complement(complement(Z)), meet(X, Y)))
118.08/118.36	= { by axiom 8 (commutativity_of_join) }
118.08/118.36	  meet(X, join(meet(X, Y), complement(complement(Z))))
118.08/118.36	= { by lemma 60 }
118.08/118.36	  meet(X, complement(meet(complement(Z), complement(meet(X, Y)))))
118.08/118.36	= { by lemma 59 }
118.08/118.36	  meet(X, complement(meet(X, meet(complement(Z), complement(meet(X, Y))))))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  meet(X, complement(meet(X, meet(complement(meet(X, Y)), complement(Z)))))
118.08/118.36	= { by axiom 6 (associativity_of_meet) }
118.08/118.36	  meet(X, complement(meet(meet(X, complement(meet(X, Y))), complement(Z))))
118.08/118.36	= { by lemma 59 }
118.08/118.36	  meet(X, complement(meet(meet(X, complement(Y)), complement(Z))))
118.08/118.36	= { by axiom 6 (associativity_of_meet) }
118.08/118.36	  meet(X, complement(meet(X, meet(complement(Y), complement(Z)))))
118.08/118.36	= { by axiom 3 (commutativity_of_meet) }
118.08/118.36	  meet(X, complement(meet(X, meet(complement(Z), complement(Y)))))
118.08/118.36	= { by lemma 59 }
118.08/118.36	  meet(X, complement(meet(complement(Z), complement(Y))))
118.08/118.36	= { by lemma 60 }
118.08/118.36	  meet(X, join(Y, complement(complement(Z))))
118.08/118.36	= { by lemma 17 }
118.08/118.37	  meet(X, join(Y, Z))
118.08/118.37	
118.08/118.37	Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
118.08/118.37	Proof:
118.08/118.37	  meet(a, join(b, c))
118.08/118.37	= { by axiom 8 (commutativity_of_join) }
118.08/118.37	  meet(a, join(c, b))
118.08/118.37	= { by lemma 61 }
118.08/118.37	  meet(a, join(b, meet(a, c)))
118.08/118.37	= { by axiom 8 (commutativity_of_join) }
118.08/118.37	  meet(a, join(meet(a, c), b))
118.08/118.37	= { by lemma 14 }
118.08/118.37	  join(meet(a, join(meet(a, c), b)), meet(b, meet(a, join(meet(a, c), b))))
118.08/118.37	= { by lemma 57 }
118.08/118.37	  join(meet(a, join(meet(a, c), b)), meet(a, meet(b, join(meet(a, c), b))))
118.08/118.37	= { by lemma 19 }
118.08/118.37	  join(meet(a, join(meet(a, c), b)), meet(a, b))
118.08/118.37	= { by axiom 3 (commutativity_of_meet) }
118.08/118.37	  join(meet(a, join(meet(a, c), b)), meet(b, a))
118.08/118.37	= { by axiom 8 (commutativity_of_join) }
118.08/118.37	  join(meet(b, a), meet(a, join(meet(a, c), b)))
118.08/118.37	= { by axiom 3 (commutativity_of_meet) }
118.08/118.37	  join(meet(a, b), meet(a, join(meet(a, c), b)))
118.08/118.37	= { by axiom 8 (commutativity_of_join) }
118.08/118.37	  join(meet(a, b), meet(a, join(b, meet(a, c))))
118.08/118.37	= { by lemma 61 }
118.08/118.37	  join(meet(a, b), meet(a, join(meet(a, c), meet(a, b))))
118.08/118.37	= { by axiom 8 (commutativity_of_join) }
118.08/118.37	  join(meet(a, b), meet(a, join(meet(a, b), meet(a, c))))
118.08/118.37	= { by lemma 15 }
118.08/118.37	  join(meet(a, b), join(meet(a, c), meet(a, join(meet(a, b), meet(a, c)))))
118.08/118.37	= { by lemma 16 }
118.08/118.37	  join(meet(a, b), meet(a, c))
118.08/118.37	% SZS output end Proof
118.08/118.37	
118.08/118.37	RESULT: Unsatisfiable (the axioms are contradictory).
118.16/118.38	EOF