%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT064-1 : TPTP v9.3.1. Released v2.5.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:42:56 AM UTC 2026
% Result : Unsatisfiable 154.70s 19.96s
% Output : Proof 156.05s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LAT064-1 : TPTP v9.3.1. Released v2.5.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.37 % Computer : n019.cluster.edu
% 0.11/0.37 % Model : x86_64 x86_64
% 0.11/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.37 % Memory : 8046.5625MB
% 0.11/0.37 % OS : Linux 6.8.0-71-generic
% 0.11/0.37 % CPULimit : 300
% 0.11/0.37 % WCLimit : 300
% 0.11/0.37 % DateTime : Sun Sep 27 13:59:18 UTC 2026
% 0.11/0.37 % CPUTime :
% 0.11/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 154.70/19.96 Command-line arguments: --no-flatten-goal
% 154.70/19.96
% 154.70/19.96 % SZS status Unsatisfiable
% 154.70/19.96
% 156.05/20.11 % SZS output start Proof
% 156.05/20.11 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 156.05/20.11 Axiom 2 (top): join(X, complement(X)) = n1.
% 156.05/20.11 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 156.05/20.11 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 156.05/20.11 Axiom 5 (bottom): meet(X, complement(X)) = n0.
% 156.05/20.11 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 156.05/20.11 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 156.05/20.11 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 156.05/20.11 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 156.05/20.11 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 156.05/20.11 Axiom 11 (c94_6): meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))) = meet(X, join(Y, meet(X, Z))).
% 156.05/20.11 Axiom 12 (complements_are_unique): ifeq(join(X, Y), n1, ifeq(meet(X, Y), n0, complement(X), Y), Y) = Y.
% 156.05/20.11
% 156.05/20.11 Lemma 13: meet(X, n1) = X.
% 156.05/20.11 Proof:
% 156.05/20.11 meet(X, n1)
% 156.05/20.11 = { by axiom 2 (top) R->L }
% 156.05/20.11 meet(X, join(X, complement(X)))
% 156.05/20.11 = { by axiom 9 (absorption1) }
% 156.05/20.11 X
% 156.05/20.11
% 156.05/20.11 Lemma 14: meet(n1, X) = X.
% 156.05/20.11 Proof:
% 156.05/20.11 meet(n1, X)
% 156.05/20.11 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.11 meet(X, n1)
% 156.05/20.11 = { by lemma 13 }
% 156.05/20.11 X
% 156.05/20.11
% 156.05/20.11 Lemma 15: join(X, n1) = n1.
% 156.05/20.11 Proof:
% 156.05/20.11 join(X, n1)
% 156.05/20.11 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.11 join(n1, X)
% 156.05/20.11 = { by lemma 14 R->L }
% 156.05/20.11 join(n1, meet(n1, X))
% 156.05/20.11 = { by axiom 7 (absorption2) }
% 156.05/20.11 n1
% 156.05/20.11
% 156.05/20.11 Lemma 16: ifeq(join(X, Y), n1, ifeq(meet(Y, X), n0, complement(Y), X), X) = X.
% 156.05/20.11 Proof:
% 156.05/20.11 ifeq(join(X, Y), n1, ifeq(meet(Y, X), n0, complement(Y), X), X)
% 156.05/20.11 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.11 ifeq(join(Y, X), n1, ifeq(meet(Y, X), n0, complement(Y), X), X)
% 156.05/20.11 = { by axiom 12 (complements_are_unique) }
% 156.05/20.11 X
% 156.05/20.11
% 156.05/20.12 Lemma 17: complement(complement(X)) = X.
% 156.05/20.12 Proof:
% 156.05/20.12 complement(complement(X))
% 156.05/20.12 = { by axiom 6 (ifeq_axiom) R->L }
% 156.05/20.12 ifeq(n0, n0, complement(complement(X)), X)
% 156.05/20.12 = { by axiom 5 (bottom) R->L }
% 156.05/20.12 ifeq(meet(X, complement(X)), n0, complement(complement(X)), X)
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.12 ifeq(meet(complement(X), X), n0, complement(complement(X)), X)
% 156.05/20.12 = { by axiom 6 (ifeq_axiom) R->L }
% 156.05/20.12 ifeq(n1, n1, ifeq(meet(complement(X), X), n0, complement(complement(X)), X), X)
% 156.05/20.12 = { by axiom 2 (top) R->L }
% 156.05/20.12 ifeq(join(X, complement(X)), n1, ifeq(meet(complement(X), X), n0, complement(complement(X)), X), X)
% 156.05/20.12 = { by lemma 16 }
% 156.05/20.12 X
% 156.05/20.12
% 156.05/20.12 Lemma 18: join(X, meet(Y, X)) = X.
% 156.05/20.12 Proof:
% 156.05/20.12 join(X, meet(Y, X))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.12 join(X, meet(X, Y))
% 156.05/20.12 = { by axiom 7 (absorption2) }
% 156.05/20.12 X
% 156.05/20.12
% 156.05/20.12 Lemma 19: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 156.05/20.12 Proof:
% 156.05/20.12 meet(X, meet(Y, Z))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.12 meet(meet(Y, Z), X)
% 156.05/20.12 = { by axiom 10 (associativity_of_meet) }
% 156.05/20.12 meet(Y, meet(Z, X))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.12 meet(Y, meet(X, Z))
% 156.05/20.12
% 156.05/20.12 Lemma 20: meet(X, join(Y, X)) = X.
% 156.05/20.12 Proof:
% 156.05/20.12 meet(X, join(Y, X))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 meet(X, join(X, Y))
% 156.05/20.12 = { by axiom 9 (absorption1) }
% 156.05/20.12 X
% 156.05/20.12
% 156.05/20.12 Lemma 21: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 156.05/20.12 Proof:
% 156.05/20.12 meet(X, meet(Y, join(Z, X)))
% 156.05/20.12 = { by lemma 19 R->L }
% 156.05/20.12 meet(Y, meet(X, join(Z, X)))
% 156.05/20.12 = { by lemma 20 }
% 156.05/20.12 meet(Y, X)
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.12 meet(X, Y)
% 156.05/20.12
% 156.05/20.12 Lemma 22: join(X, meet(join(X, Y), complement(meet(X, Y)))) = join(X, meet(Y, complement(meet(X, Y)))).
% 156.05/20.12 Proof:
% 156.05/20.12 join(X, meet(join(X, Y), complement(meet(X, Y))))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(X, meet(join(Y, X), complement(meet(X, Y))))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.12 join(X, meet(complement(meet(X, Y)), join(Y, X)))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(X, meet(complement(meet(X, Y)), join(X, Y)))
% 156.05/20.12 = { by lemma 18 R->L }
% 156.05/20.12 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, meet(complement(meet(X, Y)), join(X, Y)))))
% 156.05/20.12 = { by lemma 21 }
% 156.05/20.12 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, complement(meet(X, Y)))))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.12 join(X, join(meet(Y, complement(meet(X, Y))), meet(complement(meet(X, Y)), join(X, Y))))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.12 join(X, join(meet(Y, complement(meet(X, Y))), meet(join(X, Y), complement(meet(X, Y)))))
% 156.05/20.12 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(join(X, Y), complement(meet(X, Y))))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, Y)))
% 156.05/20.12 = { by lemma 13 R->L }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, n1))))
% 156.05/20.12 = { by axiom 2 (top) R->L }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, join(meet(X, Y), complement(meet(X, Y)))))))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, join(complement(meet(X, Y)), meet(X, Y))))))
% 156.05/20.12 = { by axiom 11 (c94_6) }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(complement(meet(X, Y)), Y))))
% 156.05/20.12 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.12 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, complement(meet(X, Y))))))
% 156.05/20.12 = { by lemma 18 }
% 156.05/20.12 join(X, meet(Y, complement(meet(X, Y))))
% 156.05/20.12
% 156.05/20.12 Lemma 23: join(X, join(Y, complement(X))) = n1.
% 156.05/20.12 Proof:
% 156.05/20.12 join(X, join(Y, complement(X)))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(X, join(complement(X), Y))
% 156.05/20.12 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.12 join(join(X, complement(X)), Y)
% 156.05/20.12 = { by axiom 2 (top) }
% 156.05/20.12 join(n1, Y)
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(Y, n1)
% 156.05/20.12 = { by lemma 15 }
% 156.05/20.12 n1
% 156.05/20.12
% 156.05/20.12 Lemma 24: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 156.05/20.12 Proof:
% 156.05/20.12 join(X, join(Y, meet(X, Z)))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.12 join(X, join(meet(X, Z), Y))
% 156.05/20.12 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.12 join(join(X, meet(X, Z)), Y)
% 156.05/20.12 = { by axiom 7 (absorption2) }
% 156.05/20.12 join(X, Y)
% 156.05/20.12
% 156.05/20.12 Lemma 25: join(X, complement(meet(X, Y))) = n1.
% 156.05/20.12 Proof:
% 156.05/20.12 join(X, complement(meet(X, Y)))
% 156.05/20.12 = { by lemma 24 R->L }
% 156.05/20.12 join(X, join(complement(meet(X, Y)), meet(X, Y)))
% 156.05/20.12 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.12 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 156.05/20.12 = { by axiom 2 (top) }
% 156.05/20.12 join(X, n1)
% 156.05/20.12 = { by lemma 15 }
% 156.05/20.12 n1
% 156.05/20.12
% 156.05/20.12 Lemma 26: meet(X, meet(Y, complement(meet(X, Y)))) = n0.
% 156.05/20.12 Proof:
% 156.05/20.12 meet(X, meet(Y, complement(meet(X, Y))))
% 156.05/20.12 = { by axiom 10 (associativity_of_meet) R->L }
% 156.05/20.12 meet(meet(X, Y), complement(meet(X, Y)))
% 156.05/20.12 = { by axiom 5 (bottom) }
% 156.05/20.12 n0
% 156.05/20.12
% 156.05/20.12 Lemma 27: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 156.05/20.12 Proof:
% 156.05/20.12 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 156.05/20.12 = { by axiom 12 (complements_are_unique) R->L }
% 156.05/20.12 ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n1, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, 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))))))
% 156.05/20.12 = { by lemma 22 R->L }
% 156.05/20.12 ifeq(join(Y, meet(join(Y, join(X, complement(Y))), complement(meet(Y, join(X, complement(Y)))))), n1, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, 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))))))
% 156.05/20.12 = { by lemma 23 }
% 156.05/20.12 ifeq(join(Y, meet(n1, complement(meet(Y, join(X, complement(Y)))))), n1, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, 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))))))
% 156.05/20.12 = { by lemma 14 }
% 156.05/20.12 ifeq(join(Y, complement(meet(Y, join(X, complement(Y))))), n1, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, 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))))))
% 156.05/20.12 = { by lemma 25 }
% 156.05/20.12 ifeq(n1, n1, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, 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))))))
% 156.05/20.12 = { by axiom 6 (ifeq_axiom) }
% 156.05/20.12 ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), n0, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 156.05/20.12 = { by lemma 26 }
% 156.05/20.12 ifeq(n0, n0, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 156.05/20.12 = { by axiom 6 (ifeq_axiom) }
% 156.05/20.12 complement(Y)
% 156.05/20.12
% 156.05/20.13 Lemma 28: meet(join(X, Y), complement(meet(complement(Y), join(X, Y)))) = Y.
% 156.05/20.13 Proof:
% 156.05/20.13 meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))
% 156.05/20.13 = { by lemma 17 R->L }
% 156.05/20.13 meet(join(X, Y), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 156.05/20.13 = { by lemma 17 R->L }
% 156.05/20.13 meet(join(X, complement(complement(Y))), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 156.05/20.13 = { by lemma 27 }
% 156.05/20.13 complement(complement(Y))
% 156.05/20.13 = { by lemma 17 }
% 156.05/20.13 Y
% 156.05/20.13
% 156.05/20.13 Lemma 29: meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))) = complement(X).
% 156.05/20.13 Proof:
% 156.05/20.13 meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))
% 156.05/20.13 = { by axiom 12 (complements_are_unique) R->L }
% 156.05/20.13 ifeq(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.13 ifeq(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(X, Y)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.13 ifeq(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(Y, X)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by lemma 22 R->L }
% 156.05/20.13 ifeq(join(X, meet(join(X, complement(meet(Y, X))), complement(meet(X, complement(meet(Y, X)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.13 ifeq(join(X, meet(join(X, complement(meet(X, Y))), complement(meet(X, complement(meet(Y, X)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by lemma 25 }
% 156.05/20.13 ifeq(join(X, meet(n1, complement(meet(X, complement(meet(Y, X)))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by lemma 14 }
% 156.05/20.13 ifeq(join(X, complement(meet(X, complement(meet(Y, X))))), n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by lemma 25 }
% 156.05/20.13 ifeq(n1, n1, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, 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))))))
% 156.05/20.13 = { by axiom 6 (ifeq_axiom) }
% 156.05/20.13 ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), n0, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 156.05/20.13 = { by lemma 26 }
% 156.05/20.13 ifeq(n0, n0, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 156.05/20.13 = { by axiom 6 (ifeq_axiom) }
% 156.05/20.13 complement(X)
% 156.05/20.13
% 156.05/20.13 Lemma 30: meet(complement(X), complement(meet(complement(X), join(X, Y)))) = complement(join(X, Y)).
% 156.05/20.13 Proof:
% 156.05/20.13 meet(complement(X), complement(meet(complement(X), join(X, Y))))
% 156.05/20.13 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.13 meet(complement(X), complement(meet(complement(X), join(Y, X))))
% 156.05/20.13 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.13 meet(complement(X), complement(meet(join(Y, X), complement(X))))
% 156.05/20.13 = { by lemma 28 R->L }
% 156.05/20.13 meet(complement(X), complement(meet(join(Y, X), complement(meet(join(Y, X), complement(meet(complement(X), join(Y, X))))))))
% 156.05/20.13 = { by lemma 28 R->L }
% 156.05/20.13 meet(complement(meet(join(Y, X), complement(meet(complement(X), join(Y, X))))), complement(meet(join(Y, X), complement(meet(join(Y, X), complement(meet(complement(X), join(Y, X))))))))
% 156.05/20.13 = { by lemma 29 }
% 156.05/20.13 complement(join(Y, X))
% 156.05/20.13 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.13 complement(join(X, Y))
% 156.05/20.13
% 156.05/20.13 Lemma 31: meet(X, meet(X, Y)) = meet(X, Y).
% 156.05/20.13 Proof:
% 156.05/20.13 meet(X, meet(X, Y))
% 156.05/20.13 = { by axiom 10 (associativity_of_meet) R->L }
% 156.05/20.13 meet(meet(X, X), Y)
% 156.05/20.13 = { by axiom 3 (idempotence_of_meet) }
% 156.05/20.13 meet(X, Y)
% 156.05/20.14
% 156.05/20.14 Lemma 32: join(complement(X), complement(meet(X, Y))) = complement(meet(X, Y)).
% 156.05/20.14 Proof:
% 156.05/20.14 join(complement(X), complement(meet(X, Y)))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 join(complement(meet(X, Y)), complement(X))
% 156.05/20.14 = { by lemma 29 R->L }
% 156.05/20.14 join(complement(meet(X, Y)), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 156.05/20.14 = { by axiom 7 (absorption2) }
% 156.05/20.14 complement(meet(X, Y))
% 156.05/20.14
% 156.05/20.14 Lemma 33: join(complement(complement(X)), complement(complement(meet(X, Y)))) = complement(complement(X)).
% 156.05/20.14 Proof:
% 156.05/20.14 join(complement(complement(X)), complement(complement(meet(X, Y))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 join(complement(complement(meet(X, Y))), complement(complement(X)))
% 156.05/20.14 = { by lemma 29 R->L }
% 156.05/20.14 join(complement(complement(meet(X, Y))), complement(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))))
% 156.05/20.14 = { by lemma 32 }
% 156.05/20.14 complement(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 156.05/20.14 = { by lemma 29 }
% 156.05/20.14 complement(complement(X))
% 156.05/20.14
% 156.05/20.14 Lemma 34: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 156.05/20.14 Proof:
% 156.05/20.14 join(Y, join(X, Z))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 join(join(X, Z), Y)
% 156.05/20.14 = { by axiom 8 (associativity_of_join) }
% 156.05/20.14 join(X, join(Z, Y))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 join(X, join(Y, Z))
% 156.05/20.14
% 156.05/20.14 Lemma 35: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 156.05/20.14 Proof:
% 156.05/20.14 meet(X, join(Y, meet(X, join(Y, Z))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 meet(X, join(Y, meet(X, join(Z, Y))))
% 156.05/20.14 = { by lemma 20 R->L }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Y, Z))))))
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(join(Y, Z), X)))))
% 156.05/20.14 = { by lemma 34 }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Y, join(Z, meet(join(Y, Z), X)))))
% 156.05/20.14 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(join(Y, Z), meet(join(Y, Z), X))))
% 156.05/20.14 = { by axiom 7 (absorption2) }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Y, Z)))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.14 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 meet(X, meet(join(Y, Z), join(Y, meet(join(Y, Z), X))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 meet(X, meet(join(Y, Z), join(meet(join(Y, Z), X), Y)))
% 156.05/20.14 = { by lemma 19 R->L }
% 156.05/20.14 meet(join(Y, Z), meet(X, join(meet(join(Y, Z), X), Y)))
% 156.05/20.14 = { by axiom 10 (associativity_of_meet) R->L }
% 156.05/20.14 meet(meet(join(Y, Z), X), join(meet(join(Y, Z), X), Y))
% 156.05/20.14 = { by axiom 9 (absorption1) }
% 156.05/20.14 meet(join(Y, Z), X)
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.14 meet(X, join(Y, Z))
% 156.05/20.14
% 156.05/20.14 Lemma 36: ifeq(join(X, Y), n1, complement(meet(X, complement(meet(X, Y)))), Y) = Y.
% 156.05/20.14 Proof:
% 156.05/20.14 ifeq(join(X, Y), n1, complement(meet(X, complement(meet(X, Y)))), Y)
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 ifeq(join(X, Y), n1, complement(meet(X, complement(meet(Y, X)))), Y)
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 ifeq(join(Y, X), n1, complement(meet(X, complement(meet(Y, X)))), Y)
% 156.05/20.14 = { by axiom 6 (ifeq_axiom) R->L }
% 156.05/20.14 ifeq(join(Y, X), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 R->L }
% 156.05/20.14 ifeq(join(Y, complement(complement(X))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 33 R->L }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(X)), complement(complement(meet(X, Y))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(meet(X, Y))), complement(complement(X)))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 R->L }
% 156.05/20.14 ifeq(join(Y, complement(complement(join(complement(complement(meet(X, Y))), complement(complement(X)))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 30 R->L }
% 156.05/20.14 ifeq(join(Y, complement(meet(complement(complement(complement(meet(X, Y)))), complement(meet(complement(complement(complement(meet(X, Y)))), join(complement(complement(meet(X, Y))), complement(complement(X)))))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 35 R->L }
% 156.05/20.14 ifeq(join(Y, complement(meet(complement(complement(complement(meet(X, Y)))), complement(meet(complement(complement(complement(meet(X, Y)))), join(complement(complement(meet(X, Y))), meet(complement(complement(complement(meet(X, Y)))), join(complement(complement(meet(X, Y))), complement(complement(X)))))))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 30 }
% 156.05/20.14 ifeq(join(Y, complement(complement(join(complement(complement(meet(X, Y))), meet(complement(complement(complement(meet(X, Y)))), join(complement(complement(meet(X, Y))), complement(complement(X)))))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(complement(complement(meet(X, Y)))), join(complement(complement(meet(X, Y))), complement(complement(X)))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), join(complement(complement(meet(X, Y))), complement(complement(X)))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), join(complement(complement(X)), complement(complement(meet(X, Y))))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 33 }
% 156.05/20.14 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), complement(complement(X))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 }
% 156.05/20.14 ifeq(join(Y, join(meet(X, Y), meet(complement(meet(X, Y)), complement(complement(X))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 17 }
% 156.05/20.14 ifeq(join(Y, join(meet(X, Y), meet(complement(meet(X, Y)), X))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.14 ifeq(join(Y, join(meet(X, Y), meet(X, complement(meet(X, Y))))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 ifeq(join(Y, join(meet(X, complement(meet(X, Y))), meet(X, Y))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 34 }
% 156.05/20.14 ifeq(join(meet(X, complement(meet(X, Y))), join(Y, meet(X, Y))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 18 }
% 156.05/20.14 ifeq(join(meet(X, complement(meet(X, Y))), Y), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.14 ifeq(join(Y, meet(X, complement(meet(X, Y)))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.14 ifeq(join(Y, meet(X, complement(meet(Y, X)))), n1, ifeq(n0, n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 26 R->L }
% 156.05/20.14 ifeq(join(Y, meet(X, complement(meet(Y, X)))), n1, ifeq(meet(Y, meet(X, complement(meet(Y, X)))), n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 ifeq(join(Y, meet(X, complement(meet(Y, X)))), n1, ifeq(meet(meet(X, complement(meet(Y, X))), Y), n0, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 156.05/20.14 = { by lemma 16 }
% 156.05/20.14 Y
% 156.05/20.14
% 156.05/20.14 Lemma 37: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 156.05/20.14 Proof:
% 156.05/20.14 meet(X, meet(Y, join(X, Z)))
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 meet(X, meet(join(X, Z), Y))
% 156.05/20.14 = { by axiom 10 (associativity_of_meet) R->L }
% 156.05/20.14 meet(meet(X, join(X, Z)), Y)
% 156.05/20.14 = { by axiom 9 (absorption1) }
% 156.05/20.14 meet(X, Y)
% 156.05/20.14
% 156.05/20.14 Lemma 38: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 156.05/20.14 Proof:
% 156.05/20.14 meet(X, meet(join(X, Y), Z))
% 156.05/20.14 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.14 meet(X, meet(Z, join(X, Y)))
% 156.05/20.14 = { by lemma 37 }
% 156.05/20.14 meet(X, Z)
% 156.05/20.14
% 156.05/20.14 Lemma 39: meet(complement(X), join(X, Y)) = meet(Y, complement(meet(X, Y))).
% 156.05/20.14 Proof:
% 156.05/20.14 meet(complement(X), join(X, Y))
% 156.05/20.14 = { by lemma 17 R->L }
% 156.05/20.14 complement(complement(meet(complement(X), join(X, Y))))
% 156.05/20.14 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.14 complement(complement(meet(complement(X), join(Y, X))))
% 156.05/20.14 = { by lemma 36 R->L }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), n1, complement(meet(Y, complement(meet(Y, complement(meet(complement(X), join(Y, X))))))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 17 R->L }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), n1, complement(meet(Y, complement(meet(Y, complement(meet(complement(X), join(Y, complement(complement(X))))))))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 38 R->L }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), n1, complement(meet(Y, complement(meet(Y, meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 27 }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), n1, complement(meet(Y, complement(meet(Y, complement(complement(X)))))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 17 }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 17 R->L }
% 156.05/20.15 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, complement(complement(X)))))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 32 R->L }
% 156.05/20.15 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(complement(X), join(Y, complement(complement(X))))))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.15 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(complement(X), join(complement(complement(X)), Y))))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.15 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(join(complement(complement(X)), Y), complement(X))))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 34 }
% 156.05/20.15 complement(ifeq(join(complement(complement(X)), join(Y, complement(meet(join(complement(complement(X)), Y), complement(X))))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.15 complement(ifeq(join(join(complement(complement(X)), Y), complement(meet(join(complement(complement(X)), Y), complement(X)))), n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by lemma 25 }
% 156.05/20.15 complement(ifeq(n1, n1, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 156.05/20.15 = { by axiom 6 (ifeq_axiom) }
% 156.05/20.15 complement(complement(meet(Y, complement(meet(Y, X)))))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.15 complement(complement(meet(Y, complement(meet(X, Y)))))
% 156.05/20.15 = { by lemma 17 }
% 156.05/20.15 meet(Y, complement(meet(X, Y)))
% 156.05/20.15
% 156.05/20.15 Lemma 40: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 156.05/20.15 Proof:
% 156.05/20.15 meet(complement(X), join(X, Y))
% 156.05/20.15 = { by lemma 31 R->L }
% 156.05/20.15 meet(complement(X), meet(complement(X), join(X, Y)))
% 156.05/20.15 = { by lemma 39 }
% 156.05/20.15 meet(complement(X), meet(Y, complement(meet(X, Y))))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.15 meet(complement(X), meet(complement(meet(X, Y)), Y))
% 156.05/20.15 = { by axiom 10 (associativity_of_meet) R->L }
% 156.05/20.15 meet(meet(complement(X), complement(meet(X, Y))), Y)
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.15 meet(meet(complement(meet(X, Y)), complement(X)), Y)
% 156.05/20.15 = { by lemma 29 R->L }
% 156.05/20.15 meet(meet(complement(meet(X, Y)), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), Y)
% 156.05/20.15 = { by lemma 31 }
% 156.05/20.15 meet(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))), Y)
% 156.05/20.15 = { by lemma 29 }
% 156.05/20.15 meet(complement(X), Y)
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.15 meet(Y, complement(X))
% 156.05/20.15
% 156.05/20.15 Lemma 41: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 156.05/20.15 Proof:
% 156.05/20.15 complement(join(X, Y))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.15 complement(join(Y, X))
% 156.05/20.15 = { by lemma 30 R->L }
% 156.05/20.15 meet(complement(Y), complement(meet(complement(Y), join(Y, X))))
% 156.05/20.15 = { by lemma 40 }
% 156.05/20.15 meet(complement(Y), complement(meet(X, complement(Y))))
% 156.05/20.15 = { by lemma 39 R->L }
% 156.05/20.15 meet(complement(X), join(X, complement(Y)))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.15 meet(complement(X), join(complement(Y), X))
% 156.05/20.15 = { by lemma 35 R->L }
% 156.05/20.15 meet(complement(X), join(complement(Y), meet(complement(X), join(complement(Y), X))))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.15 meet(complement(X), join(complement(Y), meet(complement(X), join(X, complement(Y)))))
% 156.05/20.15 = { by lemma 39 }
% 156.05/20.15 meet(complement(X), join(complement(Y), meet(complement(Y), complement(meet(X, complement(Y))))))
% 156.05/20.15 = { by axiom 7 (absorption2) }
% 156.05/20.15 meet(complement(X), complement(Y))
% 156.05/20.15
% 156.05/20.15 Lemma 42: meet(X, join(Y, complement(X))) = meet(X, Y).
% 156.05/20.15 Proof:
% 156.05/20.15 meet(X, join(Y, complement(X)))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.15 meet(X, join(complement(X), Y))
% 156.05/20.15 = { by lemma 17 R->L }
% 156.05/20.15 meet(complement(complement(X)), join(complement(X), Y))
% 156.05/20.15 = { by lemma 40 }
% 156.05/20.15 meet(Y, complement(complement(X)))
% 156.05/20.15 = { by lemma 17 }
% 156.05/20.15 meet(Y, X)
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.15 meet(X, Y)
% 156.05/20.15
% 156.05/20.15 Lemma 43: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 156.05/20.15 Proof:
% 156.05/20.15 complement(meet(X, complement(Y)))
% 156.05/20.15 = { by lemma 21 R->L }
% 156.05/20.15 complement(meet(X, meet(complement(Y), join(Y, X))))
% 156.05/20.15 = { by lemma 39 }
% 156.05/20.15 complement(meet(X, meet(X, complement(meet(Y, X)))))
% 156.05/20.15 = { by lemma 31 }
% 156.05/20.15 complement(meet(X, complement(meet(Y, X))))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.15 complement(meet(X, complement(meet(X, Y))))
% 156.05/20.15 = { by axiom 6 (ifeq_axiom) R->L }
% 156.05/20.15 ifeq(n1, n1, complement(meet(X, complement(meet(X, Y)))), join(Y, complement(X)))
% 156.05/20.15 = { by lemma 23 R->L }
% 156.05/20.15 ifeq(join(X, join(Y, complement(X))), n1, complement(meet(X, complement(meet(X, Y)))), join(Y, complement(X)))
% 156.05/20.15 = { by lemma 42 R->L }
% 156.05/20.15 ifeq(join(X, join(Y, complement(X))), n1, complement(meet(X, complement(meet(X, join(Y, complement(X)))))), join(Y, complement(X)))
% 156.05/20.15 = { by lemma 36 }
% 156.05/20.15 join(Y, complement(X))
% 156.05/20.15
% 156.05/20.15 Lemma 44: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 156.05/20.15 Proof:
% 156.05/20.15 complement(meet(complement(X), Y))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.15 complement(meet(Y, complement(X)))
% 156.05/20.15 = { by lemma 43 }
% 156.05/20.15 join(X, complement(Y))
% 156.05/20.15
% 156.05/20.15 Lemma 45: meet(X, join(complement(X), Y)) = meet(X, Y).
% 156.05/20.15 Proof:
% 156.05/20.15 meet(X, join(complement(X), Y))
% 156.05/20.15 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.15 meet(X, join(Y, complement(X)))
% 156.05/20.15 = { by lemma 42 }
% 156.05/20.15 meet(X, Y)
% 156.05/20.15
% 156.05/20.15 Lemma 46: join(X, complement(meet(complement(X), Y))) = complement(meet(Y, complement(X))).
% 156.05/20.15 Proof:
% 156.05/20.15 join(X, complement(meet(complement(X), Y)))
% 156.05/20.15 = { by lemma 17 R->L }
% 156.05/20.15 join(complement(complement(X)), complement(meet(complement(X), Y)))
% 156.05/20.15 = { by lemma 32 }
% 156.05/20.15 complement(meet(complement(X), Y))
% 156.05/20.15 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.15 complement(meet(Y, complement(X)))
% 156.05/20.15
% 156.05/20.15 Lemma 47: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 156.05/20.15 Proof:
% 156.05/20.15 meet(X, join(Y, meet(X, Z)))
% 156.05/20.15 = { by lemma 45 R->L }
% 156.05/20.15 meet(X, join(complement(X), join(Y, meet(X, Z))))
% 156.05/20.15 = { by axiom 8 (associativity_of_join) R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(X, Z)))
% 156.05/20.16 = { by axiom 11 (c94_6) R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(join(complement(X), Y), Z)))))
% 156.05/20.16 = { by axiom 4 (commutativity_of_meet) }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, join(complement(X), Y))))))
% 156.05/20.16 = { by lemma 17 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, join(complement(X), complement(complement(Y))))))))
% 156.05/20.16 = { by lemma 44 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, complement(meet(complement(complement(X)), complement(Y))))))))
% 156.05/20.16 = { by lemma 17 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(complement(complement(Z)), complement(meet(complement(complement(X)), complement(Y))))))))
% 156.05/20.16 = { by lemma 41 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(join(complement(Z), meet(complement(complement(X)), complement(Y))))))))
% 156.05/20.16 = { by lemma 43 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, complement(meet(join(complement(Z), meet(complement(complement(X)), complement(Y))), complement(X))))))
% 156.05/20.16 = { by lemma 46 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(Z), meet(complement(complement(X)), complement(Y)))))))))
% 156.05/20.16 = { by lemma 41 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(Z), complement(join(complement(X), Y)))))))))
% 156.05/20.16 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(join(complement(X), Y)), complement(Z))))))))
% 156.05/20.16 = { by lemma 43 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(meet(Z, complement(complement(join(complement(X), Y)))))))))))
% 156.05/20.16 = { by lemma 17 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(meet(Z, join(complement(X), Y)))))))))
% 156.05/20.16 = { by lemma 38 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), meet(join(complement(X), Y), complement(meet(Z, join(complement(X), Y))))))))))
% 156.05/20.16 = { by lemma 39 R->L }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), meet(complement(Z), join(Z, join(complement(X), Y)))))))))
% 156.05/20.16 = { by lemma 34 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), meet(complement(Z), join(complement(X), join(Z, Y)))))))))
% 156.05/20.16 = { by lemma 37 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(Z)))))))
% 156.05/20.16 = { by lemma 46 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, complement(meet(complement(Z), complement(X))))))
% 156.05/20.16 = { by lemma 44 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(Z, complement(complement(X))))))
% 156.05/20.16 = { by lemma 17 }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(Z, X))))
% 156.05/20.16 = { by axiom 1 (commutativity_of_join) }
% 156.05/20.16 meet(X, join(join(complement(X), Y), meet(Z, join(X, Z))))
% 156.05/20.16 = { by axiom 8 (associativity_of_join) }
% 156.05/20.16 meet(X, join(complement(X), join(Y, meet(Z, join(X, Z)))))
% 156.05/20.16 = { by lemma 45 }
% 156.05/20.16 meet(X, join(Y, meet(Z, join(X, Z))))
% 156.05/20.16 = { by lemma 20 }
% 156.05/20.16 meet(X, join(Y, Z))
% 156.05/20.16
% 156.05/20.16 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 156.05/20.16 Proof:
% 156.05/20.16 meet(a, join(b, c))
% 156.05/20.16 = { by axiom 3 (idempotence_of_meet) R->L }
% 156.05/20.16 meet(a, join(b, meet(c, c)))
% 156.05/20.16 = { by lemma 47 R->L }
% 156.05/20.16 meet(a, join(b, meet(a, meet(c, c))))
% 156.05/20.16 = { by lemma 19 R->L }
% 156.05/20.16 meet(a, join(b, meet(c, meet(a, c))))
% 156.05/20.16 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.16 meet(a, join(meet(c, meet(a, c)), b))
% 156.05/20.16 = { by lemma 47 R->L }
% 156.05/20.16 meet(a, join(meet(c, meet(a, c)), meet(a, b)))
% 156.05/20.16 = { by axiom 7 (absorption2) R->L }
% 156.05/20.16 meet(join(a, meet(a, b)), join(meet(c, meet(a, c)), meet(a, b)))
% 156.05/20.16 = { by lemma 19 R->L }
% 156.05/20.16 meet(join(a, meet(a, b)), join(meet(a, meet(c, c)), meet(a, b)))
% 156.05/20.16 = { by axiom 1 (commutativity_of_join) R->L }
% 156.05/20.16 meet(join(a, meet(a, b)), join(meet(a, b), meet(a, meet(c, c))))
% 156.05/20.16 = { by axiom 4 (commutativity_of_meet) R->L }
% 156.05/20.16 meet(join(meet(a, b), meet(a, meet(c, c))), join(a, meet(a, b)))
% 156.05/20.16 = { by lemma 24 R->L }
% 156.05/20.16 meet(join(meet(a, b), meet(a, meet(c, c))), join(a, join(meet(a, b), meet(a, meet(c, c)))))
% 156.05/20.16 = { by lemma 20 }
% 156.05/20.16 join(meet(a, b), meet(a, meet(c, c)))
% 156.05/20.16 = { by axiom 3 (idempotence_of_meet) }
% 156.05/20.16 join(meet(a, b), meet(a, c))
% 156.05/20.16 % SZS output end Proof
% 156.05/20.16
% 156.05/20.16 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------