TSTP Solution File: LAT188-1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT188-1 : TPTP v8.1.2. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n011.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:41 EDT 2023
% Result : Unsatisfiable 72.97s 9.65s
% Output : Proof 74.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : LAT188-1 : TPTP v8.1.2. Released v3.1.0.
% 0.11/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n011.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Thu Aug 24 10:14:37 EDT 2023
% 0.12/0.34 % CPUTime :
% 72.97/9.65 Command-line arguments: --no-flatten-goal
% 72.97/9.65
% 72.97/9.65 % SZS status Unsatisfiable
% 72.97/9.65
% 73.41/9.74 % SZS output start Proof
% 73.41/9.74 Take the following subset of the input axioms:
% 73.41/9.74 fof(absorption1, axiom, ![X, Y]: meet(X, join(X, Y))=X).
% 73.41/9.74 fof(absorption2, axiom, ![X2, Y2]: join(X2, meet(X2, Y2))=X2).
% 73.41/9.74 fof(associativity_of_join, axiom, ![Z, X2, Y2]: join(join(X2, Y2), Z)=join(X2, join(Y2, Z))).
% 73.41/9.74 fof(associativity_of_meet, axiom, ![X2, Y2, Z2]: meet(meet(X2, Y2), Z2)=meet(X2, meet(Y2, Z2))).
% 73.41/9.74 fof(commutativity_of_join, axiom, ![X2, Y2]: join(X2, Y2)=join(Y2, X2)).
% 73.41/9.74 fof(commutativity_of_meet, axiom, ![X2, Y2]: meet(X2, Y2)=meet(Y2, X2)).
% 73.41/9.74 fof(complement_join, axiom, ![X2]: join(X2, complement(X2))=one).
% 73.41/9.74 fof(complement_meet, axiom, ![X2]: meet(X2, complement(X2))=zero).
% 73.41/9.74 fof(equation_H16, axiom, ![X2, Y2, Z2]: meet(X2, join(meet(X2, Y2), meet(X2, Z2)))=meet(X2, join(meet(X2, Y2), meet(Z2, join(Y2, meet(Z2, join(X2, Y2))))))).
% 73.41/9.74 fof(idempotence_of_join, axiom, ![X2]: join(X2, X2)=X2).
% 73.41/9.74 fof(meet_join_complement, axiom, ![X2, Y2]: (meet(X2, Y2)!=zero | (join(X2, Y2)!=one | complement(X2)=Y2))).
% 73.41/9.74 fof(prove_distributivity, negated_conjecture, meet(a, join(b, c))!=join(meet(a, b), meet(a, c))).
% 73.41/9.74
% 73.41/9.74 Now clausify the problem and encode Horn clauses using encoding 3 of
% 73.41/9.74 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 73.41/9.74 We repeatedly replace C & s=t => u=v by the two clauses:
% 73.41/9.74 fresh(y, y, x1...xn) = u
% 73.41/9.74 C => fresh(s, t, x1...xn) = v
% 73.41/9.74 where fresh is a fresh function symbol and x1..xn are the free
% 73.41/9.74 variables of u and v.
% 73.41/9.74 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 73.41/9.74 input problem has no model of domain size 1).
% 73.41/9.74
% 73.41/9.74 The encoding turns the above axioms into the following unit equations and goals:
% 73.41/9.74
% 73.41/9.74 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 73.41/9.74 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 73.41/9.74 Axiom 3 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 73.41/9.74 Axiom 4 (complement_join): join(X, complement(X)) = one.
% 73.41/9.74 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 73.41/9.74 Axiom 6 (meet_join_complement): fresh(X, X, Y, Z) = Z.
% 73.41/9.74 Axiom 7 (meet_join_complement): fresh2(X, X, Y, Z) = complement(Y).
% 73.41/9.74 Axiom 8 (absorption2): join(X, meet(X, Y)) = X.
% 73.41/9.74 Axiom 9 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 73.41/9.74 Axiom 10 (absorption1): meet(X, join(X, Y)) = X.
% 73.41/9.74 Axiom 11 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 73.41/9.74 Axiom 12 (meet_join_complement): fresh2(join(X, Y), one, X, Y) = fresh(meet(X, Y), zero, X, Y).
% 73.41/9.74 Axiom 13 (equation_H16): meet(X, join(meet(X, Y), meet(X, Z))) = meet(X, join(meet(X, Y), meet(Z, join(Y, meet(Z, join(X, Y)))))).
% 73.41/9.74
% 73.41/9.74 Lemma 14: fresh2(join(X, Y), one, Y, X) = fresh(meet(X, Y), zero, Y, X).
% 73.41/9.74 Proof:
% 73.41/9.74 fresh2(join(X, Y), one, Y, X)
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 fresh2(join(Y, X), one, Y, X)
% 73.41/9.74 = { by axiom 12 (meet_join_complement) }
% 73.41/9.74 fresh(meet(Y, X), zero, Y, X)
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.74 fresh(meet(X, Y), zero, Y, X)
% 73.41/9.74
% 73.41/9.74 Lemma 15: complement(complement(X)) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 complement(complement(X))
% 73.41/9.74 = { by axiom 7 (meet_join_complement) R->L }
% 73.41/9.74 fresh2(one, one, complement(X), X)
% 73.41/9.74 = { by axiom 4 (complement_join) R->L }
% 73.41/9.74 fresh2(join(X, complement(X)), one, complement(X), X)
% 73.41/9.74 = { by lemma 14 }
% 73.41/9.74 fresh(meet(X, complement(X)), zero, complement(X), X)
% 73.41/9.74 = { by axiom 5 (complement_meet) }
% 73.41/9.74 fresh(zero, zero, complement(X), X)
% 73.41/9.74 = { by axiom 6 (meet_join_complement) }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 16: meet(X, one) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, one)
% 73.41/9.74 = { by axiom 4 (complement_join) R->L }
% 73.41/9.74 meet(X, join(X, complement(X)))
% 73.41/9.74 = { by axiom 10 (absorption1) }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 17: join(X, zero) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 join(X, zero)
% 73.41/9.74 = { by axiom 5 (complement_meet) R->L }
% 73.41/9.74 join(X, meet(X, complement(X)))
% 73.41/9.74 = { by axiom 8 (absorption2) }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 18: join(zero, X) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 join(zero, X)
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(X, zero)
% 73.41/9.74 = { by lemma 17 }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 19: meet(X, zero) = zero.
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, zero)
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.74 meet(zero, X)
% 73.41/9.74 = { by lemma 18 R->L }
% 73.41/9.74 join(zero, meet(zero, X))
% 73.41/9.74 = { by axiom 8 (absorption2) }
% 73.41/9.74 zero
% 73.41/9.74
% 73.41/9.74 Lemma 20: meet(one, X) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 meet(one, X)
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.74 meet(X, one)
% 73.41/9.74 = { by lemma 16 }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 21: join(X, one) = one.
% 73.41/9.74 Proof:
% 73.41/9.74 join(X, one)
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(one, X)
% 73.41/9.74 = { by lemma 20 R->L }
% 73.41/9.74 join(one, meet(one, X))
% 73.41/9.74 = { by axiom 8 (absorption2) }
% 73.41/9.74 one
% 73.41/9.74
% 73.41/9.74 Lemma 22: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, meet(Y, complement(meet(X, Y))))
% 73.41/9.74 = { by axiom 11 (associativity_of_meet) R->L }
% 73.41/9.74 meet(meet(X, Y), complement(meet(X, Y)))
% 73.41/9.74 = { by axiom 5 (complement_meet) }
% 73.41/9.74 zero
% 73.41/9.74
% 73.41/9.74 Lemma 23: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 73.41/9.74 Proof:
% 73.41/9.74 join(X, join(Y, meet(X, Z)))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(X, join(meet(X, Z), Y))
% 73.41/9.74 = { by axiom 9 (associativity_of_join) R->L }
% 73.41/9.74 join(join(X, meet(X, Z)), Y)
% 73.41/9.74 = { by axiom 8 (absorption2) }
% 73.41/9.74 join(X, Y)
% 73.41/9.74
% 73.41/9.74 Lemma 24: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 73.41/9.74 Proof:
% 73.41/9.74 join(X, join(meet(X, Y), Z))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(X, join(Z, meet(X, Y)))
% 73.41/9.74 = { by lemma 23 }
% 73.41/9.74 join(X, Z)
% 73.41/9.74
% 73.41/9.74 Lemma 25: meet(X, join(Y, X)) = X.
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, join(Y, X))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 meet(X, join(X, Y))
% 73.41/9.74 = { by axiom 10 (absorption1) }
% 73.41/9.74 X
% 73.41/9.74
% 73.41/9.74 Lemma 26: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, meet(Y, join(Z, meet(X, Y))))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 meet(X, meet(Y, join(meet(X, Y), Z)))
% 73.41/9.74 = { by axiom 11 (associativity_of_meet) R->L }
% 73.41/9.74 meet(meet(X, Y), join(meet(X, Y), Z))
% 73.41/9.74 = { by axiom 10 (absorption1) }
% 73.41/9.74 meet(X, Y)
% 73.41/9.74
% 73.41/9.74 Lemma 27: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, join(Y, meet(X, join(Y, Z))))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 meet(X, join(Y, meet(X, join(Z, Y))))
% 73.41/9.74 = { by lemma 25 R->L }
% 73.41/9.74 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.74 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(join(Z, Y), X)))))
% 73.41/9.74 = { by axiom 9 (associativity_of_join) R->L }
% 73.41/9.74 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(join(Z, Y), meet(join(Z, Y), X))))
% 73.41/9.74 = { by axiom 8 (absorption2) }
% 73.41/9.74 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.74 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.74 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.74 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 73.41/9.74 = { by lemma 26 }
% 73.41/9.74 meet(X, join(Y, Z))
% 73.41/9.74
% 73.41/9.74 Lemma 28: meet(X, join(Y, meet(X, join(Z, Y)))) = meet(X, join(Y, Z)).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, join(Y, meet(X, join(Z, Y))))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 meet(X, join(Y, meet(X, join(Y, Z))))
% 73.41/9.74 = { by lemma 27 }
% 73.41/9.74 meet(X, join(Y, Z))
% 73.41/9.74
% 73.41/9.74 Lemma 29: meet(join(X, Y), join(Y, meet(X, Z))) = join(Y, meet(X, Z)).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(join(X, Y), join(Y, meet(X, Z)))
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.74 meet(join(Y, meet(X, Z)), join(X, Y))
% 73.41/9.74 = { by lemma 23 R->L }
% 73.41/9.74 meet(join(Y, meet(X, Z)), join(X, join(Y, meet(X, Z))))
% 73.41/9.74 = { by lemma 25 }
% 73.41/9.74 join(Y, meet(X, Z))
% 73.41/9.74
% 73.41/9.74 Lemma 30: meet(X, join(meet(X, Y), meet(Z, join(Y, X)))) = join(meet(X, Y), meet(X, Z)).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(X, join(meet(X, Y), meet(Z, join(Y, X))))
% 73.41/9.74 = { by lemma 28 R->L }
% 73.41/9.74 meet(X, join(meet(X, Y), meet(Z, join(Y, meet(Z, join(X, Y))))))
% 73.41/9.74 = { by axiom 13 (equation_H16) R->L }
% 73.41/9.74 meet(X, join(meet(X, Y), meet(X, Z)))
% 73.41/9.74 = { by axiom 8 (absorption2) R->L }
% 73.41/9.74 meet(join(X, meet(X, Y)), join(meet(X, Y), meet(X, Z)))
% 73.41/9.74 = { by lemma 29 }
% 73.41/9.74 join(meet(X, Y), meet(X, Z))
% 73.41/9.74
% 73.41/9.74 Lemma 31: join(X, join(Y, complement(X))) = one.
% 73.41/9.74 Proof:
% 73.41/9.74 join(X, join(Y, complement(X)))
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(X, join(complement(X), Y))
% 73.41/9.74 = { by axiom 9 (associativity_of_join) R->L }
% 73.41/9.74 join(join(X, complement(X)), Y)
% 73.41/9.74 = { by axiom 4 (complement_join) }
% 73.41/9.74 join(one, Y)
% 73.41/9.74 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.74 join(Y, one)
% 73.41/9.74 = { by lemma 21 }
% 73.41/9.74 one
% 73.41/9.74
% 73.41/9.74 Lemma 32: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 73.41/9.74 Proof:
% 73.41/9.74 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 73.41/9.74 = { by axiom 6 (meet_join_complement) R->L }
% 73.41/9.74 fresh(zero, zero, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by lemma 22 R->L }
% 73.41/9.74 fresh(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by axiom 12 (meet_join_complement) R->L }
% 73.41/9.74 fresh2(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by lemma 24 R->L }
% 73.41/9.74 fresh2(join(Y, join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.74 fresh2(join(Y, join(meet(join(X, complement(Y)), Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by lemma 30 R->L }
% 73.41/9.74 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), join(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by lemma 31 }
% 73.41/9.74 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), one)))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by lemma 16 }
% 73.41/9.74 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.74 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 fresh2(join(Y, meet(join(X, complement(Y)), join(meet(Y, join(X, complement(Y))), complement(meet(Y, join(X, complement(Y))))))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.75 = { by axiom 4 (complement_join) }
% 73.41/9.75 fresh2(join(Y, meet(join(X, complement(Y)), one)), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.75 = { by lemma 16 }
% 73.41/9.75 fresh2(join(Y, join(X, complement(Y))), one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.75 = { by lemma 31 }
% 73.41/9.75 fresh2(one, one, Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 73.41/9.75 = { by axiom 7 (meet_join_complement) }
% 73.41/9.75 complement(Y)
% 73.41/9.75
% 73.41/9.75 Lemma 33: join(X, meet(Y, X)) = X.
% 73.41/9.75 Proof:
% 73.41/9.75 join(X, meet(Y, X))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.75 join(X, meet(X, Y))
% 73.41/9.75 = { by axiom 8 (absorption2) }
% 73.41/9.75 X
% 73.41/9.75
% 73.41/9.75 Lemma 34: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(X, meet(Y, join(X, Z)))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.75 meet(X, meet(join(X, Z), Y))
% 73.41/9.75 = { by axiom 11 (associativity_of_meet) R->L }
% 73.41/9.75 meet(meet(X, join(X, Z)), Y)
% 73.41/9.75 = { by axiom 10 (absorption1) }
% 73.41/9.75 meet(X, Y)
% 73.41/9.75
% 73.41/9.75 Lemma 35: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
% 73.41/9.75 Proof:
% 73.41/9.75 meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(join(Y, X), complement(meet(complement(X), join(Y, X))))
% 73.41/9.75 = { by lemma 15 R->L }
% 73.41/9.75 meet(join(Y, X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 73.41/9.75 = { by lemma 15 R->L }
% 73.41/9.75 meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 73.41/9.75 = { by lemma 32 }
% 73.41/9.75 complement(complement(X))
% 73.41/9.75 = { by lemma 15 }
% 73.41/9.75 X
% 73.41/9.75
% 73.41/9.75 Lemma 36: meet(join(X, Y), join(X, Z)) = join(X, meet(Z, join(X, Y))).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(join(X, Y), join(X, Z))
% 73.41/9.75 = { by lemma 16 R->L }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, one)))
% 73.41/9.75 = { by lemma 21 R->L }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), one))))
% 73.41/9.75 = { by axiom 4 (complement_join) R->L }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), join(meet(join(X, Y), complement(X)), complement(meet(join(X, Y), complement(X))))))))
% 73.41/9.75 = { by lemma 24 }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(join(X, Y), complement(X)))))))
% 73.41/9.75 = { by axiom 9 (associativity_of_join) }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(join(X, Y), complement(X))))))))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(complement(X), join(X, Y))))))))
% 73.41/9.75 = { by axiom 9 (associativity_of_join) R->L }
% 73.41/9.75 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 73.41/9.75 = { by lemma 35 R->L }
% 73.41/9.75 meet(join(X, Y), join(meet(join(X, Y), complement(meet(complement(X), join(X, Y)))), meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(join(X, Y), join(meet(join(X, Y), complement(meet(complement(X), join(X, Y)))), meet(Z, join(complement(meet(complement(X), join(X, Y))), join(X, Y)))))
% 73.41/9.75 = { by lemma 30 }
% 73.41/9.75 join(meet(join(X, Y), complement(meet(complement(X), join(X, Y)))), meet(join(X, Y), Z))
% 73.41/9.75 = { by lemma 35 }
% 73.41/9.75 join(X, meet(join(X, Y), Z))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 join(X, meet(Z, join(X, Y)))
% 73.41/9.75
% 73.41/9.75 Lemma 37: meet(join(X, Y), join(Y, Z)) = join(Y, meet(Z, join(X, Y))).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(join(X, Y), join(Y, Z))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(join(Y, X), join(Y, Z))
% 73.41/9.75 = { by lemma 36 }
% 73.41/9.75 join(Y, meet(Z, join(Y, X)))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 join(Y, meet(Z, join(X, Y)))
% 73.41/9.75
% 73.41/9.75 Lemma 38: meet(join(X, Y), join(Y, Z)) = join(Y, meet(X, join(Y, Z))).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(join(X, Y), join(Y, Z))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(join(Y, X), join(Y, Z))
% 73.41/9.75 = { by lemma 33 R->L }
% 73.41/9.75 join(meet(join(Y, X), join(Y, Z)), meet(Y, meet(join(Y, X), join(Y, Z))))
% 73.41/9.75 = { by lemma 34 }
% 73.41/9.75 join(meet(join(Y, X), join(Y, Z)), meet(Y, join(Y, X)))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 join(meet(Y, join(Y, X)), meet(join(Y, X), join(Y, Z)))
% 73.41/9.75 = { by axiom 10 (absorption1) }
% 73.41/9.75 join(Y, meet(join(Y, X), join(Y, Z)))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 join(Y, meet(join(Y, X), join(Z, Y)))
% 73.41/9.75 = { by lemma 25 R->L }
% 73.41/9.75 join(Y, meet(join(Y, X), join(Z, meet(Y, join(Z, Y)))))
% 73.41/9.75 = { by lemma 36 R->L }
% 73.41/9.75 join(Y, meet(join(Y, X), meet(join(Z, Y), join(Z, Y))))
% 73.41/9.75 = { by axiom 11 (associativity_of_meet) R->L }
% 73.41/9.75 join(Y, meet(meet(join(Y, X), join(Z, Y)), join(Z, Y)))
% 73.41/9.75 = { by lemma 37 R->L }
% 73.41/9.75 meet(join(Z, Y), join(Y, meet(join(Y, X), join(Z, Y))))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.75 meet(join(Z, Y), join(Y, meet(join(Z, Y), join(Y, X))))
% 73.41/9.75 = { by lemma 27 }
% 73.41/9.75 meet(join(Z, Y), join(Y, X))
% 73.41/9.75 = { by lemma 37 }
% 73.41/9.75 join(Y, meet(X, join(Z, Y)))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 join(Y, meet(X, join(Y, Z)))
% 73.41/9.75
% 73.41/9.75 Lemma 39: meet(X, join(Y, meet(Z, join(X, Y)))) = meet(X, join(Y, Z)).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(X, join(Y, meet(Z, join(X, Y))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(X, join(Y, meet(Z, join(Y, X))))
% 73.41/9.75 = { by lemma 36 R->L }
% 73.41/9.75 meet(X, meet(join(Y, X), join(Y, Z)))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.75 meet(X, meet(join(Y, Z), join(Y, X)))
% 73.41/9.75 = { by lemma 36 }
% 73.41/9.75 meet(X, join(Y, meet(X, join(Y, Z))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 meet(X, join(Y, meet(X, join(Z, Y))))
% 73.41/9.75 = { by lemma 28 }
% 73.41/9.75 meet(X, join(Y, Z))
% 73.41/9.75
% 73.41/9.75 Lemma 40: meet(X, join(Y, complement(join(X, Y)))) = meet(X, Y).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 meet(X, join(Y, complement(join(Y, X))))
% 73.41/9.75 = { by lemma 39 R->L }
% 73.41/9.75 meet(X, join(Y, meet(complement(join(Y, X)), join(X, Y))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 meet(X, join(Y, meet(complement(join(Y, X)), join(Y, X))))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 meet(X, join(Y, meet(join(Y, X), complement(join(Y, X)))))
% 73.41/9.75 = { by axiom 5 (complement_meet) }
% 73.41/9.75 meet(X, join(Y, zero))
% 73.41/9.75 = { by lemma 17 }
% 73.41/9.75 meet(X, Y)
% 73.41/9.75
% 73.41/9.75 Lemma 41: join(X, join(Y, complement(join(X, Y)))) = one.
% 73.41/9.75 Proof:
% 73.41/9.75 join(X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by axiom 9 (associativity_of_join) R->L }
% 73.41/9.75 join(join(X, Y), complement(join(X, Y)))
% 73.41/9.75 = { by axiom 4 (complement_join) }
% 73.41/9.75 one
% 73.41/9.75
% 73.41/9.75 Lemma 42: fresh(meet(X, Y), zero, X, join(Y, complement(join(X, Y)))) = complement(X).
% 73.41/9.75 Proof:
% 73.41/9.75 fresh(meet(X, Y), zero, X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by lemma 40 R->L }
% 73.41/9.75 fresh(meet(X, join(Y, complement(join(X, Y)))), zero, X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by axiom 12 (meet_join_complement) R->L }
% 73.41/9.75 fresh2(join(X, join(Y, complement(join(X, Y)))), one, X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by lemma 41 }
% 73.41/9.75 fresh2(one, one, X, join(Y, complement(join(X, Y))))
% 73.41/9.75 = { by axiom 7 (meet_join_complement) }
% 73.41/9.75 complement(X)
% 73.41/9.75
% 73.41/9.75 Lemma 43: join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))) = complement(Y).
% 73.41/9.75 Proof:
% 73.41/9.75 join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
% 73.41/9.75 = { by axiom 6 (meet_join_complement) R->L }
% 73.41/9.75 fresh(zero, zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by lemma 19 R->L }
% 73.41/9.75 fresh(meet(X, zero), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 fresh(meet(zero, X), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by axiom 5 (complement_meet) R->L }
% 73.41/9.75 fresh(meet(meet(Y, complement(Y)), X), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by axiom 11 (associativity_of_meet) }
% 73.41/9.75 fresh(meet(Y, meet(complement(Y), X)), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) }
% 73.41/9.75 fresh(meet(Y, meet(X, complement(Y))), zero, Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 73.41/9.75 = { by lemma 42 }
% 73.41/9.75 complement(Y)
% 73.41/9.75
% 73.41/9.75 Lemma 44: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 73.41/9.75 Proof:
% 73.41/9.75 meet(X, meet(join(X, Y), Z))
% 73.41/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 73.41/9.75 meet(X, meet(Z, join(X, Y)))
% 73.41/9.75 = { by lemma 34 }
% 73.41/9.75 meet(X, Z)
% 73.41/9.75
% 73.41/9.75 Lemma 45: meet(X, complement(join(X, Y))) = zero.
% 73.41/9.75 Proof:
% 73.41/9.75 meet(X, complement(join(X, Y)))
% 73.41/9.75 = { by lemma 44 R->L }
% 73.41/9.75 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 73.41/9.75 = { by axiom 5 (complement_meet) }
% 73.41/9.75 meet(X, zero)
% 73.41/9.75 = { by lemma 19 }
% 73.41/9.75 zero
% 73.41/9.75
% 73.41/9.75 Lemma 46: join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))) = complement(X).
% 73.41/9.75 Proof:
% 73.41/9.75 join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 join(complement(join(Y, X)), complement(join(X, complement(join(X, Y)))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 73.41/9.75 join(complement(join(Y, X)), complement(join(X, complement(join(Y, X)))))
% 73.41/9.75 = { by axiom 6 (meet_join_complement) R->L }
% 73.41/9.75 fresh(zero, zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 73.41/9.75 = { by lemma 45 R->L }
% 73.41/9.75 fresh(meet(X, complement(join(X, Y))), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 73.41/9.75 = { by axiom 2 (commutativity_of_join) }
% 73.41/9.75 fresh(meet(X, complement(join(Y, X))), zero, X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X))))))
% 73.41/9.75 = { by lemma 42 }
% 73.41/9.75 complement(X)
% 73.41/9.75
% 73.41/9.75 Lemma 47: complement(join(X, complement(join(X, Y)))) = meet(complement(X), join(X, Y)).
% 73.41/9.75 Proof:
% 73.41/9.75 complement(join(X, complement(join(X, Y))))
% 73.41/9.75 = { by lemma 32 R->L }
% 73.41/9.75 meet(join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))), complement(meet(join(X, complement(join(X, Y))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))))
% 73.41/9.75 = { by lemma 46 }
% 74.03/9.75 meet(complement(X), complement(meet(join(X, complement(join(X, Y))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))))
% 74.03/9.75 = { by lemma 38 }
% 74.03/9.75 meet(complement(X), complement(join(complement(join(X, Y)), meet(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))))))
% 74.03/9.75 = { by lemma 46 }
% 74.03/9.75 meet(complement(X), complement(join(complement(join(X, Y)), meet(X, complement(X)))))
% 74.03/9.75 = { by axiom 5 (complement_meet) }
% 74.03/9.75 meet(complement(X), complement(join(complement(join(X, Y)), zero)))
% 74.03/9.75 = { by lemma 17 }
% 74.03/9.75 meet(complement(X), complement(complement(join(X, Y))))
% 74.03/9.75 = { by lemma 15 }
% 74.03/9.75 meet(complement(X), join(X, Y))
% 74.03/9.75
% 74.03/9.75 Lemma 48: fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y) = complement(join(X, complement(join(X, Y)))).
% 74.03/9.75 Proof:
% 74.03/9.75 fresh(meet(X, Y), zero, join(X, complement(join(X, Y))), Y)
% 74.03/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.75 fresh(meet(X, Y), zero, join(X, complement(join(Y, X))), Y)
% 74.03/9.75 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.75 fresh(meet(Y, X), zero, join(X, complement(join(Y, X))), Y)
% 74.03/9.75 = { by lemma 40 R->L }
% 74.03/9.75 fresh(meet(Y, join(X, complement(join(Y, X)))), zero, join(X, complement(join(Y, X))), Y)
% 74.03/9.75 = { by lemma 14 R->L }
% 74.03/9.75 fresh2(join(Y, join(X, complement(join(Y, X)))), one, join(X, complement(join(Y, X))), Y)
% 74.03/9.75 = { by lemma 41 }
% 74.03/9.75 fresh2(one, one, join(X, complement(join(Y, X))), Y)
% 74.03/9.75 = { by axiom 7 (meet_join_complement) }
% 74.03/9.75 complement(join(X, complement(join(Y, X))))
% 74.03/9.75 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.75 complement(join(X, complement(join(X, Y))))
% 74.03/9.75
% 74.03/9.75 Lemma 49: meet(X, meet(Y, complement(join(Z, meet(X, Y))))) = zero.
% 74.03/9.75 Proof:
% 74.03/9.75 meet(X, meet(Y, complement(join(Z, meet(X, Y)))))
% 74.03/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.75 meet(X, meet(Y, complement(join(meet(X, Y), Z))))
% 74.03/9.75 = { by axiom 11 (associativity_of_meet) R->L }
% 74.03/9.75 meet(meet(X, Y), complement(join(meet(X, Y), Z)))
% 74.03/9.75 = { by lemma 45 }
% 74.03/9.75 zero
% 74.03/9.75
% 74.03/9.75 Lemma 50: join(complement(X), complement(join(complement(X), meet(X, Y)))) = complement(meet(X, Y)).
% 74.03/9.75 Proof:
% 74.03/9.75 join(complement(X), complement(join(complement(X), meet(X, Y))))
% 74.03/9.75 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.75 join(complement(join(complement(X), meet(X, Y))), complement(X))
% 74.03/9.75 = { by axiom 6 (meet_join_complement) R->L }
% 74.03/9.75 fresh(zero, zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(X)))
% 74.03/9.75 = { by lemma 49 R->L }
% 74.03/9.75 fresh(meet(X, meet(Y, complement(join(complement(X), meet(X, Y))))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(X)))
% 74.03/9.76 = { by axiom 11 (associativity_of_meet) R->L }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(X)))
% 74.03/9.76 = { by lemma 15 R->L }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(complement(complement(X)))))
% 74.03/9.76 = { by lemma 43 R->L }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))))
% 74.03/9.76 = { by lemma 15 }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(join(meet(Y, X), complement(join(complement(X), meet(Y, complement(complement(X)))))))))
% 74.03/9.76 = { by lemma 15 }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(join(meet(Y, X), complement(join(complement(X), meet(Y, X)))))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(join(meet(Y, X), complement(join(complement(X), meet(X, Y)))))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 fresh(meet(meet(X, Y), complement(join(complement(X), meet(X, Y)))), zero, meet(X, Y), join(complement(join(complement(X), meet(X, Y))), complement(join(meet(X, Y), complement(join(complement(X), meet(X, Y)))))))
% 74.03/9.76 = { by lemma 42 }
% 74.03/9.76 complement(meet(X, Y))
% 74.03/9.76
% 74.03/9.76 Lemma 51: meet(complement(Y), join(Y, X)) = meet(X, complement(meet(X, Y))).
% 74.03/9.76 Proof:
% 74.03/9.76 meet(complement(Y), join(Y, X))
% 74.03/9.76 = { by lemma 15 R->L }
% 74.03/9.76 meet(complement(Y), join(Y, complement(complement(X))))
% 74.03/9.76 = { by lemma 43 R->L }
% 74.03/9.76 meet(complement(Y), join(Y, join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X))))))))
% 74.03/9.76 = { by lemma 24 }
% 74.03/9.76 meet(complement(Y), join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))
% 74.03/9.76 = { by lemma 47 R->L }
% 74.03/9.76 complement(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))))
% 74.03/9.76 = { by lemma 48 R->L }
% 74.03/9.76 fresh(meet(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))), zero, join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 74.03/9.76 = { by lemma 25 R->L }
% 74.03/9.76 fresh(meet(Y, meet(complement(join(complement(X), meet(Y, complement(complement(X))))), join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))), zero, join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 74.03/9.76 = { by lemma 43 }
% 74.03/9.76 fresh(meet(Y, meet(complement(join(complement(X), meet(Y, complement(complement(X))))), complement(complement(X)))), zero, join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 fresh(meet(Y, meet(complement(complement(X)), complement(join(complement(X), meet(Y, complement(complement(X))))))), zero, join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 74.03/9.76 = { by lemma 49 }
% 74.03/9.76 fresh(zero, zero, join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))))), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 74.03/9.76 = { by axiom 6 (meet_join_complement) }
% 74.03/9.76 complement(join(complement(X), meet(Y, complement(complement(X)))))
% 74.03/9.76 = { by lemma 15 }
% 74.03/9.76 complement(join(complement(X), meet(Y, X)))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.76 complement(join(complement(X), meet(X, Y)))
% 74.03/9.76 = { by axiom 6 (meet_join_complement) R->L }
% 74.03/9.76 fresh(zero, zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 19 R->L }
% 74.03/9.76 fresh(meet(complement(X), zero), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 22 R->L }
% 74.03/9.76 fresh(meet(complement(X), meet(join(complement(X), meet(X, Y)), meet(join(complement(X), X), complement(meet(join(complement(X), meet(X, Y)), join(complement(X), X)))))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 44 }
% 74.03/9.76 fresh(meet(complement(X), meet(join(complement(X), X), complement(meet(join(complement(X), meet(X, Y)), join(complement(X), X))))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 fresh(meet(complement(X), meet(join(complement(X), X), complement(meet(join(complement(X), X), join(complement(X), meet(X, Y)))))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 44 }
% 74.03/9.76 fresh(meet(complement(X), complement(meet(join(complement(X), X), join(complement(X), meet(X, Y))))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 fresh(meet(complement(X), complement(meet(join(X, complement(X)), join(complement(X), meet(X, Y))))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 29 }
% 74.03/9.76 fresh(meet(complement(X), complement(join(complement(X), meet(X, Y)))), zero, join(complement(X), complement(complement(meet(X, Y)))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 50 R->L }
% 74.03/9.76 fresh(meet(complement(X), complement(join(complement(X), meet(X, Y)))), zero, join(complement(X), complement(join(complement(X), complement(join(complement(X), meet(X, Y)))))), complement(join(complement(X), meet(X, Y))))
% 74.03/9.76 = { by lemma 48 }
% 74.03/9.76 complement(join(complement(X), complement(join(complement(X), complement(join(complement(X), meet(X, Y)))))))
% 74.03/9.76 = { by lemma 47 }
% 74.03/9.76 meet(complement(complement(X)), join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 74.03/9.76 = { by lemma 15 }
% 74.03/9.76 meet(X, join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 74.03/9.76 = { by lemma 50 }
% 74.03/9.76 meet(X, complement(meet(X, Y)))
% 74.03/9.76
% 74.03/9.76 Lemma 52: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 74.03/9.76 Proof:
% 74.03/9.76 meet(X, complement(meet(X, Y)))
% 74.03/9.76 = { by lemma 26 R->L }
% 74.03/9.76 meet(X, meet(complement(meet(X, Y)), join(meet(complement(meet(X, Y)), meet(X, Y)), meet(X, complement(meet(X, Y))))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(X, meet(complement(meet(X, Y)), join(meet(X, complement(meet(X, Y))), meet(complement(meet(X, Y)), meet(X, Y)))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.76 meet(X, meet(complement(meet(X, Y)), join(meet(X, complement(meet(X, Y))), meet(meet(X, Y), complement(meet(X, Y))))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.76 meet(X, meet(complement(meet(X, Y)), join(meet(complement(meet(X, Y)), X), meet(meet(X, Y), complement(meet(X, Y))))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 meet(X, meet(complement(meet(X, Y)), join(meet(meet(X, Y), complement(meet(X, Y))), meet(complement(meet(X, Y)), X))))
% 74.03/9.76 = { by lemma 33 R->L }
% 74.03/9.76 meet(X, meet(join(complement(meet(X, Y)), meet(meet(X, Y), complement(meet(X, Y)))), join(meet(meet(X, Y), complement(meet(X, Y))), meet(complement(meet(X, Y)), X))))
% 74.03/9.76 = { by lemma 29 }
% 74.03/9.76 meet(X, join(meet(meet(X, Y), complement(meet(X, Y))), meet(complement(meet(X, Y)), X)))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(X, join(meet(complement(meet(X, Y)), X), meet(meet(X, Y), complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(X, join(meet(X, complement(meet(X, Y))), meet(meet(X, Y), complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(X, join(meet(X, complement(meet(X, Y))), meet(complement(meet(X, Y)), meet(X, Y))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(X, join(meet(complement(meet(X, Y)), meet(X, Y)), meet(X, complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(X, join(meet(meet(X, Y), complement(meet(X, Y))), meet(X, complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 5 (complement_meet) }
% 74.03/9.76 meet(X, join(zero, meet(X, complement(meet(X, Y)))))
% 74.03/9.76 = { by lemma 18 }
% 74.03/9.76 meet(X, meet(X, complement(meet(X, Y))))
% 74.03/9.76 = { by lemma 51 R->L }
% 74.03/9.76 meet(X, meet(complement(Y), join(Y, X)))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.76 meet(X, meet(join(Y, X), complement(Y)))
% 74.03/9.76 = { by axiom 11 (associativity_of_meet) R->L }
% 74.03/9.76 meet(meet(X, join(Y, X)), complement(Y))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(complement(Y), meet(X, join(Y, X)))
% 74.03/9.76 = { by lemma 25 }
% 74.03/9.76 meet(complement(Y), X)
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(X, complement(Y))
% 74.03/9.76
% 74.03/9.76 Lemma 53: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 74.03/9.76 Proof:
% 74.03/9.76 complement(join(X, complement(Y)))
% 74.03/9.76 = { by axiom 10 (absorption1) R->L }
% 74.03/9.76 meet(complement(join(X, complement(Y))), join(complement(join(X, complement(Y))), complement(meet(join(X, complement(Y)), join(complement(Y), complement(join(X, complement(Y))))))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 meet(complement(join(X, complement(Y))), join(complement(meet(join(X, complement(Y)), join(complement(Y), complement(join(X, complement(Y)))))), complement(join(X, complement(Y)))))
% 74.03/9.76 = { by lemma 32 R->L }
% 74.03/9.76 meet(complement(join(X, complement(Y))), join(complement(meet(join(X, complement(Y)), join(complement(Y), complement(join(X, complement(Y)))))), meet(join(complement(Y), complement(join(X, complement(Y)))), complement(meet(join(X, complement(Y)), join(complement(Y), complement(join(X, complement(Y)))))))))
% 74.03/9.76 = { by lemma 33 }
% 74.03/9.76 meet(complement(join(X, complement(Y))), complement(meet(join(X, complement(Y)), join(complement(Y), complement(join(X, complement(Y)))))))
% 74.03/9.76 = { by lemma 38 }
% 74.03/9.76 meet(complement(join(X, complement(Y))), complement(join(complement(Y), meet(X, join(complement(Y), complement(join(X, complement(Y))))))))
% 74.03/9.76 = { by lemma 40 }
% 74.03/9.76 meet(complement(join(X, complement(Y))), complement(join(complement(Y), meet(X, complement(Y)))))
% 74.03/9.76 = { by lemma 33 }
% 74.03/9.76 meet(complement(join(X, complement(Y))), complement(complement(Y)))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 meet(complement(complement(Y)), complement(join(X, complement(Y))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(complement(complement(Y)), complement(join(complement(Y), X)))
% 74.03/9.76 = { by lemma 15 }
% 74.03/9.76 meet(Y, complement(join(complement(Y), X)))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(Y, complement(join(X, complement(Y))))
% 74.03/9.76 = { by lemma 52 R->L }
% 74.03/9.76 meet(Y, complement(meet(Y, join(X, complement(Y)))))
% 74.03/9.76 = { by lemma 39 R->L }
% 74.03/9.76 meet(Y, complement(meet(Y, join(X, meet(complement(Y), join(Y, X))))))
% 74.03/9.76 = { by lemma 51 }
% 74.03/9.76 meet(Y, complement(meet(Y, join(X, meet(X, complement(meet(X, Y)))))))
% 74.03/9.76 = { by axiom 8 (absorption2) }
% 74.03/9.76 meet(Y, complement(meet(Y, X)))
% 74.03/9.76 = { by lemma 52 }
% 74.03/9.76 meet(Y, complement(X))
% 74.03/9.76
% 74.03/9.76 Lemma 54: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 74.03/9.76 Proof:
% 74.03/9.76 join(Y, join(X, Z))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 join(join(X, Z), Y)
% 74.03/9.76 = { by axiom 9 (associativity_of_join) }
% 74.03/9.76 join(X, join(Z, Y))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 join(X, join(Y, Z))
% 74.03/9.76
% 74.03/9.76 Lemma 55: join(X, join(meet(Y, X), Z)) = join(X, Z).
% 74.03/9.76 Proof:
% 74.03/9.76 join(X, join(meet(Y, X), Z))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 join(X, join(Z, meet(Y, X)))
% 74.03/9.76 = { by lemma 54 }
% 74.03/9.76 join(Z, join(X, meet(Y, X)))
% 74.03/9.76 = { by lemma 33 }
% 74.03/9.76 join(Z, X)
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 join(X, Z)
% 74.03/9.76
% 74.03/9.76 Lemma 56: meet(X, join(Y, meet(X, Z))) = join(meet(X, Z), meet(X, Y)).
% 74.03/9.76 Proof:
% 74.03/9.76 meet(X, join(Y, meet(X, Z)))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 meet(X, join(meet(X, Z), Y))
% 74.03/9.76 = { by axiom 8 (absorption2) R->L }
% 74.03/9.76 meet(join(X, meet(X, Z)), join(meet(X, Z), Y))
% 74.03/9.76 = { by lemma 37 }
% 74.03/9.76 join(meet(X, Z), meet(Y, join(X, meet(X, Z))))
% 74.03/9.76 = { by axiom 8 (absorption2) }
% 74.03/9.76 join(meet(X, Z), meet(Y, X))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 join(meet(X, Z), meet(X, Y))
% 74.03/9.76
% 74.03/9.76 Lemma 57: join(meet(X, Y), meet(X, complement(meet(X, Y)))) = X.
% 74.03/9.76 Proof:
% 74.03/9.76 join(meet(X, Y), meet(X, complement(meet(X, Y))))
% 74.03/9.76 = { by lemma 56 R->L }
% 74.03/9.76 meet(X, join(complement(meet(X, Y)), meet(X, Y)))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.76 meet(X, join(complement(meet(X, Y)), meet(Y, X)))
% 74.03/9.76 = { by lemma 33 R->L }
% 74.03/9.76 join(meet(X, join(complement(meet(X, Y)), meet(Y, X))), meet(Y, meet(X, join(complement(meet(X, Y)), meet(Y, X)))))
% 74.03/9.76 = { by lemma 26 }
% 74.03/9.76 join(meet(X, join(complement(meet(X, Y)), meet(Y, X))), meet(Y, X))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 join(meet(Y, X), meet(X, join(complement(meet(X, Y)), meet(Y, X))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 join(meet(Y, X), meet(X, join(complement(meet(X, Y)), meet(X, Y))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 join(meet(X, Y), meet(X, join(complement(meet(X, Y)), meet(X, Y))))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 join(meet(X, Y), meet(X, join(meet(X, Y), complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.76 join(meet(Y, X), meet(X, join(meet(X, Y), complement(meet(X, Y)))))
% 74.03/9.76 = { by axiom 4 (complement_join) }
% 74.03/9.76 join(meet(Y, X), meet(X, one))
% 74.03/9.76 = { by lemma 16 }
% 74.03/9.76 join(meet(Y, X), X)
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 join(X, meet(Y, X))
% 74.03/9.76 = { by lemma 33 }
% 74.03/9.76 X
% 74.03/9.76
% 74.03/9.76 Lemma 58: join(X, meet(Y, complement(X))) = join(X, Y).
% 74.03/9.76 Proof:
% 74.03/9.76 join(X, meet(Y, complement(X)))
% 74.03/9.76 = { by lemma 52 R->L }
% 74.03/9.76 join(X, meet(Y, complement(meet(Y, X))))
% 74.03/9.76 = { by lemma 55 R->L }
% 74.03/9.76 join(X, join(meet(Y, X), meet(Y, complement(meet(Y, X)))))
% 74.03/9.76 = { by lemma 57 }
% 74.03/9.76 join(X, Y)
% 74.03/9.76
% 74.03/9.76 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 74.03/9.76 Proof:
% 74.03/9.76 meet(a, join(b, c))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.76 meet(a, join(c, b))
% 74.03/9.76 = { by lemma 57 R->L }
% 74.03/9.76 meet(a, join(join(meet(c, complement(a)), meet(c, complement(meet(c, complement(a))))), b))
% 74.03/9.76 = { by axiom 9 (associativity_of_join) }
% 74.03/9.76 meet(a, join(meet(c, complement(a)), join(meet(c, complement(meet(c, complement(a)))), b)))
% 74.03/9.76 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.76 meet(a, join(meet(c, complement(a)), join(b, meet(c, complement(meet(c, complement(a)))))))
% 74.03/9.77 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.77 meet(a, join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.77 meet(a, join(join(b, meet(c, complement(meet(c, complement(a))))), meet(complement(a), c)))
% 74.03/9.77 = { by lemma 15 R->L }
% 74.03/9.77 meet(complement(complement(a)), join(join(b, meet(c, complement(meet(c, complement(a))))), meet(complement(a), c)))
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.77 meet(complement(complement(a)), join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.77 meet(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))), complement(complement(a)))
% 74.03/9.77 = { by lemma 53 R->L }
% 74.03/9.77 complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))
% 74.03/9.77 = { by axiom 10 (absorption1) R->L }
% 74.03/9.77 complement(meet(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), join(b, meet(c, complement(meet(c, complement(a))))))))
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.77 complement(meet(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), join(b, meet(c, complement(meet(c, complement(a)))))), join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))
% 74.03/9.77 = { by lemma 58 R->L }
% 74.03/9.77 complement(meet(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))), join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))
% 74.03/9.77 = { by lemma 15 R->L }
% 74.03/9.77 complement(meet(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))), complement(complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))))
% 74.03/9.77 = { by lemma 53 R->L }
% 74.03/9.77 complement(complement(join(complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))), complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))))))
% 74.03/9.77 = { by lemma 15 }
% 74.03/9.77 join(complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))), complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))))
% 74.03/9.77 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.77 join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))
% 74.03/9.77 = { by lemma 43 R->L }
% 74.03/9.77 join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), join(meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))), complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))))))
% 74.03/9.77 = { by axiom 2 (commutativity_of_join) R->L }
% 74.03/9.77 join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))))
% 74.03/9.77 = { by axiom 9 (associativity_of_join) R->L }
% 74.03/9.77 join(join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by axiom 1 (idempotence_of_join) }
% 74.03/9.77 join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 58 }
% 74.03/9.77 join(complement(join(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))), join(b, meet(c, complement(meet(c, complement(a))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.77 join(complement(join(join(b, meet(c, complement(meet(c, complement(a))))), join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 55 R->L }
% 74.03/9.77 join(complement(join(join(b, meet(c, complement(meet(c, complement(a))))), join(complement(a), join(meet(c, complement(a)), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 54 }
% 74.03/9.77 join(complement(join(complement(a), join(join(b, meet(c, complement(meet(c, complement(a))))), join(meet(c, complement(a)), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 41 }
% 74.03/9.77 join(complement(join(complement(a), one)), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 21 }
% 74.03/9.77 join(complement(one), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 20 R->L }
% 74.03/9.77 join(meet(one, complement(one)), meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by axiom 5 (complement_meet) }
% 74.03/9.77 join(zero, meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))))))))
% 74.03/9.77 = { by lemma 18 }
% 74.03/9.77 meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(join(complement(a), complement(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a)))))))
% 74.03/9.77 = { by lemma 53 }
% 74.03/9.77 meet(join(b, meet(c, complement(meet(c, complement(a))))), meet(join(join(b, meet(c, complement(meet(c, complement(a))))), meet(c, complement(a))), complement(complement(a))))
% 74.03/9.77 = { by lemma 44 }
% 74.03/9.77 meet(join(b, meet(c, complement(meet(c, complement(a))))), complement(complement(a)))
% 74.03/9.77 = { by lemma 15 }
% 74.03/9.77 meet(join(b, meet(c, complement(meet(c, complement(a))))), a)
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) }
% 74.03/9.77 meet(a, join(b, meet(c, complement(meet(c, complement(a))))))
% 74.03/9.77 = { by lemma 52 }
% 74.03/9.77 meet(a, join(b, meet(c, complement(complement(a)))))
% 74.03/9.77 = { by lemma 15 }
% 74.03/9.77 meet(a, join(b, meet(c, a)))
% 74.03/9.77 = { by axiom 3 (commutativity_of_meet) R->L }
% 74.03/9.77 meet(a, join(b, meet(a, c)))
% 74.03/9.77 = { by lemma 56 }
% 74.03/9.77 join(meet(a, c), meet(a, b))
% 74.03/9.77 = { by axiom 2 (commutativity_of_join) }
% 74.03/9.77 join(meet(a, b), meet(a, c))
% 74.03/9.77 % SZS output end Proof
% 74.03/9.77
% 74.03/9.77 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------