%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT184-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n012.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 11:44:30 AM UTC 2026
% Result : Unsatisfiable 148.58s 19.07s
% Output : Proof 149.86s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LAT184-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.07/0.33 % Computer : n012.cluster.edu
% 0.07/0.33 % Model : x86_64 x86_64
% 0.07/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.07/0.33 % Memory : 8046.5625MB
% 0.07/0.33 % OS : Linux 6.8.0-71-generic
% 0.07/0.33 % CPULimit : 300
% 0.07/0.33 % WCLimit : 300
% 0.07/0.33 % DateTime : Sun Sep 27 14:04:34 UTC 2026
% 0.10/0.33 % CPUTime :
% 0.10/0.33 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 148.58/19.07 Command-line arguments: --no-flatten-goal
% 148.58/19.07
% 148.58/19.07 % SZS status Unsatisfiable
% 148.58/19.07
% 149.39/19.18 % SZS output start Proof
% 149.39/19.18 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 149.39/19.18 Axiom 2 (complement_join): join(X, complement(X)) = one.
% 149.39/19.18 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 149.39/19.18 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 149.39/19.18 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 149.39/19.18 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 149.39/19.18 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 149.39/19.18 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 149.39/19.18 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 149.39/19.18 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 149.39/19.18 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 149.39/19.18 Axiom 12 (equation_H8): meet(X, join(Y, meet(X, Z))) = join(meet(X, Y), meet(X, join(Y, meet(Z, join(X, meet(Y, Z)))))).
% 149.39/19.18
% 149.39/19.18 Lemma 13: meet(X, one) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 meet(X, one)
% 149.39/19.18 = { by axiom 2 (complement_join) R->L }
% 149.39/19.18 meet(X, join(X, complement(X)))
% 149.39/19.18 = { by axiom 9 (absorption1) }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 14: meet(one, X) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 meet(one, X)
% 149.39/19.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.18 meet(X, one)
% 149.39/19.18 = { by lemma 13 }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 15: join(X, one) = one.
% 149.39/19.18 Proof:
% 149.39/19.18 join(X, one)
% 149.39/19.18 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.18 join(one, X)
% 149.39/19.18 = { by lemma 14 R->L }
% 149.39/19.18 join(one, meet(one, X))
% 149.39/19.18 = { by axiom 7 (absorption2) }
% 149.39/19.18 one
% 149.39/19.18
% 149.39/19.18 Lemma 16: ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 149.39/19.18 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.18 ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 149.39/19.18 = { by axiom 11 (meet_join_complement) }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 17: complement(complement(X)) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 complement(complement(X))
% 149.39/19.18 = { by axiom 6 (ifeq_axiom) R->L }
% 149.39/19.18 ifeq(zero, zero, complement(complement(X)), X)
% 149.39/19.18 = { by axiom 5 (complement_meet) R->L }
% 149.39/19.18 ifeq(meet(X, complement(X)), zero, complement(complement(X)), X)
% 149.39/19.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.18 ifeq(meet(complement(X), X), zero, complement(complement(X)), X)
% 149.39/19.18 = { by axiom 6 (ifeq_axiom) R->L }
% 149.39/19.18 ifeq(one, one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 149.39/19.18 = { by axiom 2 (complement_join) R->L }
% 149.39/19.18 ifeq(join(X, complement(X)), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 149.39/19.18 = { by lemma 16 }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 18: join(X, meet(Y, X)) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 join(X, meet(Y, X))
% 149.39/19.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.18 join(X, meet(X, Y))
% 149.39/19.18 = { by axiom 7 (absorption2) }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 19: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 149.39/19.18 Proof:
% 149.39/19.18 meet(X, meet(Y, Z))
% 149.39/19.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.18 meet(meet(Y, Z), X)
% 149.39/19.18 = { by axiom 10 (associativity_of_meet) }
% 149.39/19.18 meet(Y, meet(Z, X))
% 149.39/19.18 = { by axiom 4 (commutativity_of_meet) }
% 149.39/19.18 meet(Y, meet(X, Z))
% 149.39/19.18
% 149.39/19.18 Lemma 20: meet(X, join(Y, X)) = X.
% 149.39/19.18 Proof:
% 149.39/19.18 meet(X, join(Y, X))
% 149.39/19.18 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.18 meet(X, join(X, Y))
% 149.39/19.18 = { by axiom 9 (absorption1) }
% 149.39/19.18 X
% 149.39/19.18
% 149.39/19.18 Lemma 21: meet(X, meet(Y, join(Z, X))) = meet(X, Y).
% 149.39/19.18 Proof:
% 149.39/19.18 meet(X, meet(Y, join(Z, X)))
% 149.39/19.19 = { by lemma 19 R->L }
% 149.39/19.19 meet(Y, meet(X, join(Z, X)))
% 149.39/19.19 = { by lemma 20 }
% 149.39/19.19 meet(Y, X)
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) }
% 149.39/19.19 meet(X, Y)
% 149.39/19.19
% 149.39/19.19 Lemma 22: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 149.39/19.19 Proof:
% 149.39/19.19 meet(X, meet(Y, join(X, Z)))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.19 meet(X, meet(join(X, Z), Y))
% 149.39/19.19 = { by axiom 10 (associativity_of_meet) R->L }
% 149.39/19.19 meet(meet(X, join(X, Z)), Y)
% 149.39/19.19 = { by axiom 9 (absorption1) }
% 149.39/19.19 meet(X, Y)
% 149.39/19.19
% 149.39/19.19 Lemma 23: meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))) = meet(X, join(Y, meet(X, Z))).
% 149.39/19.19 Proof:
% 149.39/19.19 meet(X, join(Y, meet(Z, join(X, meet(Y, Z)))))
% 149.39/19.19 = { by lemma 18 R->L }
% 149.39/19.19 join(meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))), meet(Y, meet(X, join(Y, meet(Z, join(X, meet(Y, Z)))))))
% 149.39/19.19 = { by lemma 22 }
% 149.39/19.19 join(meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))), meet(Y, X))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) }
% 149.39/19.19 join(meet(Y, X), meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) }
% 149.39/19.19 join(meet(X, Y), meet(X, join(Y, meet(Z, join(X, meet(Y, Z))))))
% 149.39/19.19 = { by axiom 12 (equation_H8) R->L }
% 149.39/19.19 meet(X, join(Y, meet(X, Z)))
% 149.39/19.19
% 149.39/19.19 Lemma 24: join(X, meet(join(X, Y), complement(meet(X, Y)))) = join(X, meet(Y, complement(meet(X, Y)))).
% 149.39/19.19 Proof:
% 149.39/19.19 join(X, meet(join(X, Y), complement(meet(X, Y))))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 join(X, meet(join(Y, X), complement(meet(X, Y))))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.19 join(X, meet(complement(meet(X, Y)), join(Y, X)))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 join(X, meet(complement(meet(X, Y)), join(X, Y)))
% 149.39/19.19 = { by lemma 18 R->L }
% 149.39/19.19 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, meet(complement(meet(X, Y)), join(X, Y)))))
% 149.39/19.19 = { by lemma 21 }
% 149.39/19.19 join(X, join(meet(complement(meet(X, Y)), join(X, Y)), meet(Y, complement(meet(X, Y)))))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) }
% 149.39/19.19 join(X, join(meet(Y, complement(meet(X, Y))), meet(complement(meet(X, Y)), join(X, Y))))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) }
% 149.39/19.19 join(X, join(meet(Y, complement(meet(X, Y))), meet(join(X, Y), complement(meet(X, Y)))))
% 149.39/19.19 = { by axiom 8 (associativity_of_join) R->L }
% 149.39/19.19 join(join(X, meet(Y, complement(meet(X, Y)))), meet(join(X, Y), complement(meet(X, Y))))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.19 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, Y)))
% 149.39/19.19 = { by lemma 13 R->L }
% 149.39/19.19 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, one))))
% 149.39/19.19 = { by axiom 2 (complement_join) R->L }
% 149.39/19.19 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)))))))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 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))))))
% 149.39/19.19 = { by lemma 23 }
% 149.39/19.19 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(complement(meet(X, Y)), Y))))
% 149.39/19.19 = { by axiom 4 (commutativity_of_meet) }
% 149.39/19.19 join(join(X, meet(Y, complement(meet(X, Y)))), meet(complement(meet(X, Y)), join(X, meet(Y, complement(meet(X, Y))))))
% 149.39/19.19 = { by lemma 18 }
% 149.39/19.19 join(X, meet(Y, complement(meet(X, Y))))
% 149.39/19.19
% 149.39/19.19 Lemma 25: join(X, join(Y, complement(X))) = one.
% 149.39/19.19 Proof:
% 149.39/19.19 join(X, join(Y, complement(X)))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 join(X, join(complement(X), Y))
% 149.39/19.19 = { by axiom 8 (associativity_of_join) R->L }
% 149.39/19.19 join(join(X, complement(X)), Y)
% 149.39/19.19 = { by axiom 2 (complement_join) }
% 149.39/19.19 join(one, Y)
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 join(Y, one)
% 149.39/19.19 = { by lemma 15 }
% 149.39/19.19 one
% 149.39/19.19
% 149.39/19.19 Lemma 26: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 149.39/19.19 Proof:
% 149.39/19.19 join(X, join(Y, meet(X, Z)))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.19 join(X, join(meet(X, Z), Y))
% 149.39/19.19 = { by axiom 8 (associativity_of_join) R->L }
% 149.39/19.19 join(join(X, meet(X, Z)), Y)
% 149.39/19.19 = { by axiom 7 (absorption2) }
% 149.39/19.19 join(X, Y)
% 149.39/19.19
% 149.39/19.19 Lemma 27: join(X, complement(meet(X, Y))) = one.
% 149.39/19.19 Proof:
% 149.39/19.19 join(X, complement(meet(X, Y)))
% 149.39/19.19 = { by lemma 26 R->L }
% 149.39/19.19 join(X, join(complement(meet(X, Y)), meet(X, Y)))
% 149.39/19.19 = { by axiom 1 (commutativity_of_join) }
% 149.39/19.19 join(X, join(meet(X, Y), complement(meet(X, Y))))
% 149.39/19.19 = { by axiom 2 (complement_join) }
% 149.39/19.19 join(X, one)
% 149.39/19.19 = { by lemma 15 }
% 149.39/19.19 one
% 149.39/19.19
% 149.39/19.19 Lemma 28: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 149.39/19.19 Proof:
% 149.39/19.19 meet(X, meet(Y, complement(meet(X, Y))))
% 149.39/19.19 = { by axiom 10 (associativity_of_meet) R->L }
% 149.39/19.19 meet(meet(X, Y), complement(meet(X, Y)))
% 149.39/19.19 = { by axiom 5 (complement_meet) }
% 149.39/19.19 zero
% 149.39/19.19
% 149.39/19.19 Lemma 29: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 149.39/19.19 Proof:
% 149.39/19.19 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 149.39/19.19 = { by axiom 11 (meet_join_complement) R->L }
% 149.39/19.19 ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by lemma 24 R->L }
% 149.39/19.19 ifeq(join(Y, meet(join(Y, join(X, complement(Y))), complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by lemma 25 }
% 149.39/19.19 ifeq(join(Y, meet(one, complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by lemma 14 }
% 149.39/19.19 ifeq(join(Y, complement(meet(Y, join(X, complement(Y))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by lemma 27 }
% 149.39/19.19 ifeq(one, one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by axiom 6 (ifeq_axiom) }
% 149.39/19.19 ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by lemma 28 }
% 149.39/19.19 ifeq(zero, zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 149.39/19.19 = { by axiom 6 (ifeq_axiom) }
% 149.39/19.19 complement(Y)
% 149.39/19.19
% 149.39/19.19 Lemma 30: meet(join(X, Y), complement(meet(complement(Y), join(X, Y)))) = Y.
% 149.39/19.19 Proof:
% 149.39/19.19 meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))
% 149.39/19.19 = { by lemma 17 R->L }
% 149.39/19.19 meet(join(X, Y), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 149.39/19.19 = { by lemma 17 R->L }
% 149.39/19.19 meet(join(X, complement(complement(Y))), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 149.39/19.19 = { by lemma 29 }
% 149.39/19.19 complement(complement(Y))
% 149.39/19.19 = { by lemma 17 }
% 149.39/19.19 Y
% 149.39/19.20
% 149.39/19.20 Lemma 31: meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))) = complement(X).
% 149.39/19.20 Proof:
% 149.39/19.20 meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))
% 149.39/19.20 = { by axiom 11 (meet_join_complement) R->L }
% 149.39/19.20 ifeq(join(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.20 ifeq(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.20 ifeq(join(X, meet(complement(meet(Y, X)), complement(meet(X, complement(meet(Y, X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 24 R->L }
% 149.39/19.20 ifeq(join(X, meet(join(X, complement(meet(Y, X))), complement(meet(X, complement(meet(Y, X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.20 ifeq(join(X, meet(join(X, complement(meet(X, Y))), complement(meet(X, complement(meet(Y, X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 27 }
% 149.39/19.20 ifeq(join(X, meet(one, complement(meet(X, complement(meet(Y, X)))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 14 }
% 149.39/19.20 ifeq(join(X, complement(meet(X, complement(meet(Y, X))))), one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 27 }
% 149.39/19.20 ifeq(one, one, ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 6 (ifeq_axiom) }
% 149.39/19.20 ifeq(meet(X, meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 28 }
% 149.39/19.20 ifeq(zero, zero, complement(X), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 6 (ifeq_axiom) }
% 149.39/19.20 complement(X)
% 149.39/19.20
% 149.39/19.20 Lemma 32: meet(complement(X), complement(meet(complement(X), join(X, Y)))) = complement(join(X, Y)).
% 149.39/19.20 Proof:
% 149.39/19.20 meet(complement(X), complement(meet(complement(X), join(X, Y))))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.20 meet(complement(X), complement(meet(complement(X), join(Y, X))))
% 149.39/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.39/19.20 meet(complement(X), complement(meet(join(Y, X), complement(X))))
% 149.39/19.20 = { by lemma 30 R->L }
% 149.39/19.20 meet(complement(X), complement(meet(join(Y, X), complement(meet(join(Y, X), complement(meet(complement(X), join(Y, X))))))))
% 149.39/19.20 = { by lemma 30 R->L }
% 149.39/19.20 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))))))))
% 149.39/19.20 = { by lemma 31 }
% 149.39/19.20 complement(join(Y, X))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) }
% 149.39/19.20 complement(join(X, Y))
% 149.39/19.20
% 149.39/19.20 Lemma 33: meet(X, meet(X, Y)) = meet(X, Y).
% 149.39/19.20 Proof:
% 149.39/19.20 meet(X, meet(X, Y))
% 149.39/19.20 = { by axiom 10 (associativity_of_meet) R->L }
% 149.39/19.20 meet(meet(X, X), Y)
% 149.39/19.20 = { by axiom 3 (idempotence_of_meet) }
% 149.39/19.20 meet(X, Y)
% 149.39/19.20
% 149.39/19.20 Lemma 34: join(complement(X), complement(meet(X, Y))) = complement(meet(X, Y)).
% 149.39/19.20 Proof:
% 149.39/19.20 join(complement(X), complement(meet(X, Y)))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.20 join(complement(meet(X, Y)), complement(X))
% 149.39/19.20 = { by lemma 31 R->L }
% 149.39/19.20 join(complement(meet(X, Y)), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by axiom 7 (absorption2) }
% 149.39/19.20 complement(meet(X, Y))
% 149.39/19.20
% 149.39/19.20 Lemma 35: join(complement(complement(X)), complement(complement(meet(X, Y)))) = complement(complement(X)).
% 149.39/19.20 Proof:
% 149.39/19.20 join(complement(complement(X)), complement(complement(meet(X, Y))))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.20 join(complement(complement(meet(X, Y))), complement(complement(X)))
% 149.39/19.20 = { by lemma 31 R->L }
% 149.39/19.20 join(complement(complement(meet(X, Y))), complement(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))))
% 149.39/19.20 = { by lemma 34 }
% 149.39/19.20 complement(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))))
% 149.39/19.20 = { by lemma 31 }
% 149.39/19.20 complement(complement(X))
% 149.39/19.20
% 149.39/19.20 Lemma 36: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 149.39/19.20 Proof:
% 149.39/19.20 join(Y, join(X, Z))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.39/19.20 join(join(X, Z), Y)
% 149.39/19.20 = { by axiom 8 (associativity_of_join) }
% 149.39/19.20 join(X, join(Z, Y))
% 149.39/19.20 = { by axiom 1 (commutativity_of_join) }
% 149.39/19.20 join(X, join(Y, Z))
% 149.39/19.20
% 149.39/19.20 Lemma 37: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 149.39/19.20 Proof:
% 149.39/19.20 meet(X, join(Y, meet(X, join(Y, Z))))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.20 meet(X, join(Y, meet(X, join(Z, Y))))
% 149.86/19.20 = { by lemma 20 R->L }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Y, Z))))))
% 149.86/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(join(Y, Z), X)))))
% 149.86/19.20 = { by lemma 36 }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Y, join(Z, meet(join(Y, Z), X)))))
% 149.86/19.20 = { by axiom 8 (associativity_of_join) R->L }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(join(Y, Z), meet(join(Y, Z), X))))
% 149.86/19.20 = { by axiom 7 (absorption2) }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Y, Z)))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.20 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 149.86/19.20 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.20 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.20 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.20 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 149.86/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.20 meet(X, meet(join(Y, Z), join(Y, meet(join(Y, Z), X))))
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.20 meet(X, meet(join(Y, Z), join(meet(join(Y, Z), X), Y)))
% 149.86/19.20 = { by lemma 19 R->L }
% 149.86/19.20 meet(join(Y, Z), meet(X, join(meet(join(Y, Z), X), Y)))
% 149.86/19.20 = { by axiom 10 (associativity_of_meet) R->L }
% 149.86/19.20 meet(meet(join(Y, Z), X), join(meet(join(Y, Z), X), Y))
% 149.86/19.20 = { by axiom 9 (absorption1) }
% 149.86/19.20 meet(join(Y, Z), X)
% 149.86/19.20 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.20 meet(X, join(Y, Z))
% 149.86/19.20
% 149.86/19.20 Lemma 38: ifeq(join(X, Y), one, complement(meet(X, complement(meet(X, Y)))), Y) = Y.
% 149.86/19.20 Proof:
% 149.86/19.20 ifeq(join(X, Y), one, complement(meet(X, complement(meet(X, Y)))), Y)
% 149.86/19.20 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.20 ifeq(join(X, Y), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.20 ifeq(join(Y, X), one, complement(meet(X, complement(meet(Y, X)))), Y)
% 149.86/19.20 = { by axiom 6 (ifeq_axiom) R->L }
% 149.86/19.20 ifeq(join(Y, X), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.20 = { by lemma 17 R->L }
% 149.86/19.20 ifeq(join(Y, complement(complement(X))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.20 = { by lemma 35 R->L }
% 149.86/19.20 ifeq(join(Y, join(complement(complement(X)), complement(complement(meet(X, Y))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.20 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.20 ifeq(join(Y, join(complement(complement(meet(X, Y))), complement(complement(X)))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.20 = { by lemma 17 R->L }
% 149.86/19.20 ifeq(join(Y, complement(complement(join(complement(complement(meet(X, Y))), complement(complement(X)))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.20 = { by lemma 32 R->L }
% 149.86/19.20 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)))))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 37 R->L }
% 149.86/19.21 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)))))))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 32 }
% 149.86/19.21 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)))))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 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)))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), join(complement(complement(meet(X, Y))), complement(complement(X)))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.21 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), join(complement(complement(X)), complement(complement(meet(X, Y))))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 35 }
% 149.86/19.21 ifeq(join(Y, join(complement(complement(meet(X, Y))), meet(complement(meet(X, Y)), complement(complement(X))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 ifeq(join(Y, join(meet(X, Y), meet(complement(meet(X, Y)), complement(complement(X))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 ifeq(join(Y, join(meet(X, Y), meet(complement(meet(X, Y)), X))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.21 ifeq(join(Y, join(meet(X, Y), meet(X, complement(meet(X, Y))))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.21 ifeq(join(Y, join(meet(X, complement(meet(X, Y))), meet(X, Y))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 36 }
% 149.86/19.21 ifeq(join(meet(X, complement(meet(X, Y))), join(Y, meet(X, Y))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 18 }
% 149.86/19.21 ifeq(join(meet(X, complement(meet(X, Y))), Y), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.21 ifeq(join(Y, meet(X, complement(meet(X, Y)))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.21 ifeq(join(Y, meet(X, complement(meet(Y, X)))), one, ifeq(zero, zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 28 R->L }
% 149.86/19.21 ifeq(join(Y, meet(X, complement(meet(Y, X)))), one, ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.21 ifeq(join(Y, meet(X, complement(meet(Y, X)))), one, ifeq(meet(meet(X, complement(meet(Y, X))), Y), zero, complement(meet(X, complement(meet(Y, X)))), Y), Y)
% 149.86/19.21 = { by lemma 16 }
% 149.86/19.21 Y
% 149.86/19.21
% 149.86/19.21 Lemma 39: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 149.86/19.21 Proof:
% 149.86/19.21 meet(X, meet(join(X, Y), Z))
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.21 meet(X, meet(Z, join(X, Y)))
% 149.86/19.21 = { by lemma 22 }
% 149.86/19.21 meet(X, Z)
% 149.86/19.21
% 149.86/19.21 Lemma 40: meet(complement(X), join(X, Y)) = meet(Y, complement(meet(X, Y))).
% 149.86/19.21 Proof:
% 149.86/19.21 meet(complement(X), join(X, Y))
% 149.86/19.21 = { by lemma 17 R->L }
% 149.86/19.21 complement(complement(meet(complement(X), join(X, Y))))
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.21 complement(complement(meet(complement(X), join(Y, X))))
% 149.86/19.21 = { by lemma 38 R->L }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), one, complement(meet(Y, complement(meet(Y, complement(meet(complement(X), join(Y, X))))))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 17 R->L }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), one, complement(meet(Y, complement(meet(Y, complement(meet(complement(X), join(Y, complement(complement(X))))))))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 39 R->L }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), one, 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)))))
% 149.86/19.21 = { by lemma 29 }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), one, complement(meet(Y, complement(meet(Y, complement(complement(X)))))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, X)))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 17 R->L }
% 149.86/19.21 complement(ifeq(join(Y, complement(meet(complement(X), join(Y, complement(complement(X)))))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 34 R->L }
% 149.86/19.21 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(complement(X), join(Y, complement(complement(X))))))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.21 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(complement(X), join(complement(complement(X)), Y))))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.21 complement(ifeq(join(Y, join(complement(complement(X)), complement(meet(join(complement(complement(X)), Y), complement(X))))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 36 }
% 149.86/19.21 complement(ifeq(join(complement(complement(X)), join(Y, complement(meet(join(complement(complement(X)), Y), complement(X))))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by axiom 8 (associativity_of_join) R->L }
% 149.86/19.21 complement(ifeq(join(join(complement(complement(X)), Y), complement(meet(join(complement(complement(X)), Y), complement(X)))), one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by lemma 27 }
% 149.86/19.21 complement(ifeq(one, one, complement(meet(Y, complement(meet(Y, X)))), complement(meet(complement(X), join(Y, X)))))
% 149.86/19.21 = { by axiom 6 (ifeq_axiom) }
% 149.86/19.21 complement(complement(meet(Y, complement(meet(Y, X)))))
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.21 complement(complement(meet(Y, complement(meet(X, Y)))))
% 149.86/19.21 = { by lemma 17 }
% 149.86/19.21 meet(Y, complement(meet(X, Y)))
% 149.86/19.21
% 149.86/19.21 Lemma 41: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 149.86/19.21 Proof:
% 149.86/19.21 meet(complement(X), join(X, Y))
% 149.86/19.21 = { by lemma 33 R->L }
% 149.86/19.21 meet(complement(X), meet(complement(X), join(X, Y)))
% 149.86/19.21 = { by lemma 40 }
% 149.86/19.21 meet(complement(X), meet(Y, complement(meet(X, Y))))
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.21 meet(complement(X), meet(complement(meet(X, Y)), Y))
% 149.86/19.21 = { by axiom 10 (associativity_of_meet) R->L }
% 149.86/19.21 meet(meet(complement(X), complement(meet(X, Y))), Y)
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.21 meet(meet(complement(meet(X, Y)), complement(X)), Y)
% 149.86/19.21 = { by lemma 31 R->L }
% 149.86/19.21 meet(meet(complement(meet(X, Y)), meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))), Y)
% 149.86/19.21 = { by lemma 33 }
% 149.86/19.21 meet(meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y))))), Y)
% 149.86/19.21 = { by lemma 31 }
% 149.86/19.21 meet(complement(X), Y)
% 149.86/19.21 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.21 meet(Y, complement(X))
% 149.86/19.21
% 149.86/19.21 Lemma 42: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 149.86/19.21 Proof:
% 149.86/19.21 complement(join(X, Y))
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.21 complement(join(Y, X))
% 149.86/19.21 = { by lemma 32 R->L }
% 149.86/19.21 meet(complement(Y), complement(meet(complement(Y), join(Y, X))))
% 149.86/19.21 = { by lemma 41 }
% 149.86/19.21 meet(complement(Y), complement(meet(X, complement(Y))))
% 149.86/19.21 = { by lemma 40 R->L }
% 149.86/19.21 meet(complement(X), join(X, complement(Y)))
% 149.86/19.21 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.21 meet(complement(X), join(complement(Y), X))
% 149.86/19.21 = { by lemma 37 R->L }
% 149.86/19.21 meet(complement(X), join(complement(Y), meet(complement(X), join(complement(Y), X))))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.22 meet(complement(X), join(complement(Y), meet(complement(X), join(X, complement(Y)))))
% 149.86/19.22 = { by lemma 40 }
% 149.86/19.22 meet(complement(X), join(complement(Y), meet(complement(Y), complement(meet(X, complement(Y))))))
% 149.86/19.22 = { by axiom 7 (absorption2) }
% 149.86/19.22 meet(complement(X), complement(Y))
% 149.86/19.22
% 149.86/19.22 Lemma 43: meet(X, join(Y, complement(X))) = meet(X, Y).
% 149.86/19.22 Proof:
% 149.86/19.22 meet(X, join(Y, complement(X)))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.22 meet(X, join(complement(X), Y))
% 149.86/19.22 = { by lemma 17 R->L }
% 149.86/19.22 meet(complement(complement(X)), join(complement(X), Y))
% 149.86/19.22 = { by lemma 41 }
% 149.86/19.22 meet(Y, complement(complement(X)))
% 149.86/19.22 = { by lemma 17 }
% 149.86/19.22 meet(Y, X)
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.22 meet(X, Y)
% 149.86/19.22
% 149.86/19.22 Lemma 44: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 149.86/19.22 Proof:
% 149.86/19.22 complement(meet(X, complement(Y)))
% 149.86/19.22 = { by lemma 21 R->L }
% 149.86/19.22 complement(meet(X, meet(complement(Y), join(Y, X))))
% 149.86/19.22 = { by lemma 40 }
% 149.86/19.22 complement(meet(X, meet(X, complement(meet(Y, X)))))
% 149.86/19.22 = { by lemma 33 }
% 149.86/19.22 complement(meet(X, complement(meet(Y, X))))
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.22 complement(meet(X, complement(meet(X, Y))))
% 149.86/19.22 = { by axiom 6 (ifeq_axiom) R->L }
% 149.86/19.22 ifeq(one, one, complement(meet(X, complement(meet(X, Y)))), join(Y, complement(X)))
% 149.86/19.22 = { by lemma 25 R->L }
% 149.86/19.22 ifeq(join(X, join(Y, complement(X))), one, complement(meet(X, complement(meet(X, Y)))), join(Y, complement(X)))
% 149.86/19.22 = { by lemma 43 R->L }
% 149.86/19.22 ifeq(join(X, join(Y, complement(X))), one, complement(meet(X, complement(meet(X, join(Y, complement(X)))))), join(Y, complement(X)))
% 149.86/19.22 = { by lemma 38 }
% 149.86/19.22 join(Y, complement(X))
% 149.86/19.22
% 149.86/19.22 Lemma 45: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 149.86/19.22 Proof:
% 149.86/19.22 complement(meet(complement(X), Y))
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.22 complement(meet(Y, complement(X)))
% 149.86/19.22 = { by lemma 44 }
% 149.86/19.22 join(X, complement(Y))
% 149.86/19.22
% 149.86/19.22 Lemma 46: meet(X, join(complement(X), Y)) = meet(X, Y).
% 149.86/19.22 Proof:
% 149.86/19.22 meet(X, join(complement(X), Y))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.22 meet(X, join(Y, complement(X)))
% 149.86/19.22 = { by lemma 43 }
% 149.86/19.22 meet(X, Y)
% 149.86/19.22
% 149.86/19.22 Lemma 47: join(X, complement(meet(complement(X), Y))) = complement(meet(Y, complement(X))).
% 149.86/19.22 Proof:
% 149.86/19.22 join(X, complement(meet(complement(X), Y)))
% 149.86/19.22 = { by lemma 17 R->L }
% 149.86/19.22 join(complement(complement(X)), complement(meet(complement(X), Y)))
% 149.86/19.22 = { by lemma 34 }
% 149.86/19.22 complement(meet(complement(X), Y))
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.22 complement(meet(Y, complement(X)))
% 149.86/19.22
% 149.86/19.22 Lemma 48: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 149.86/19.22 Proof:
% 149.86/19.22 meet(X, join(Y, meet(X, Z)))
% 149.86/19.22 = { by lemma 46 R->L }
% 149.86/19.22 meet(X, join(complement(X), join(Y, meet(X, Z))))
% 149.86/19.22 = { by axiom 8 (associativity_of_join) R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(X, Z)))
% 149.86/19.22 = { by lemma 23 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(join(complement(X), Y), Z)))))
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, join(complement(X), Y))))))
% 149.86/19.22 = { by lemma 17 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, join(complement(X), complement(complement(Y))))))))
% 149.86/19.22 = { by lemma 45 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(Z, complement(meet(complement(complement(X)), complement(Y))))))))
% 149.86/19.22 = { by lemma 17 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, meet(complement(complement(Z)), complement(meet(complement(complement(X)), complement(Y))))))))
% 149.86/19.22 = { by lemma 42 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(join(complement(Z), meet(complement(complement(X)), complement(Y))))))))
% 149.86/19.22 = { by lemma 44 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, complement(meet(join(complement(Z), meet(complement(complement(X)), complement(Y))), complement(X))))))
% 149.86/19.22 = { by lemma 47 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(Z), meet(complement(complement(X)), complement(Y)))))))))
% 149.86/19.22 = { by lemma 42 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(Z), complement(join(complement(X), Y)))))))))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), join(complement(join(complement(X), Y)), complement(Z))))))))
% 149.86/19.22 = { by lemma 44 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(meet(Z, complement(complement(join(complement(X), Y)))))))))))
% 149.86/19.22 = { by lemma 17 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(meet(Z, join(complement(X), Y)))))))))
% 149.86/19.22 = { by lemma 39 R->L }
% 149.86/19.22 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))))))))))
% 149.86/19.22 = { by lemma 40 R->L }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), meet(complement(Z), join(Z, join(complement(X), Y)))))))))
% 149.86/19.22 = { by lemma 36 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), meet(complement(Z), join(complement(X), join(Z, Y)))))))))
% 149.86/19.22 = { by lemma 22 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, complement(meet(complement(X), complement(Z)))))))
% 149.86/19.22 = { by lemma 47 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, complement(meet(complement(Z), complement(X))))))
% 149.86/19.22 = { by lemma 45 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(Z, complement(complement(X))))))
% 149.86/19.22 = { by lemma 17 }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(Z, X))))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) }
% 149.86/19.22 meet(X, join(join(complement(X), Y), meet(Z, join(X, Z))))
% 149.86/19.22 = { by axiom 8 (associativity_of_join) }
% 149.86/19.22 meet(X, join(complement(X), join(Y, meet(Z, join(X, Z)))))
% 149.86/19.22 = { by lemma 46 }
% 149.86/19.22 meet(X, join(Y, meet(Z, join(X, Z))))
% 149.86/19.22 = { by lemma 20 }
% 149.86/19.22 meet(X, join(Y, Z))
% 149.86/19.22
% 149.86/19.22 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 149.86/19.22 Proof:
% 149.86/19.22 meet(a, join(b, c))
% 149.86/19.22 = { by axiom 3 (idempotence_of_meet) R->L }
% 149.86/19.22 meet(a, join(b, meet(c, c)))
% 149.86/19.22 = { by lemma 48 R->L }
% 149.86/19.22 meet(a, join(b, meet(a, meet(c, c))))
% 149.86/19.22 = { by lemma 19 R->L }
% 149.86/19.22 meet(a, join(b, meet(c, meet(a, c))))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.22 meet(a, join(meet(c, meet(a, c)), b))
% 149.86/19.22 = { by lemma 48 R->L }
% 149.86/19.22 meet(a, join(meet(c, meet(a, c)), meet(a, b)))
% 149.86/19.22 = { by axiom 7 (absorption2) R->L }
% 149.86/19.22 meet(join(a, meet(a, b)), join(meet(c, meet(a, c)), meet(a, b)))
% 149.86/19.22 = { by lemma 19 R->L }
% 149.86/19.22 meet(join(a, meet(a, b)), join(meet(a, meet(c, c)), meet(a, b)))
% 149.86/19.22 = { by axiom 1 (commutativity_of_join) R->L }
% 149.86/19.22 meet(join(a, meet(a, b)), join(meet(a, b), meet(a, meet(c, c))))
% 149.86/19.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 149.86/19.22 meet(join(meet(a, b), meet(a, meet(c, c))), join(a, meet(a, b)))
% 149.86/19.22 = { by lemma 26 R->L }
% 149.86/19.22 meet(join(meet(a, b), meet(a, meet(c, c))), join(a, join(meet(a, b), meet(a, meet(c, c)))))
% 149.86/19.22 = { by lemma 20 }
% 149.86/19.22 join(meet(a, b), meet(a, meet(c, c)))
% 149.86/19.22 = { by axiom 3 (idempotence_of_meet) }
% 149.86/19.22 join(meet(a, b), meet(a, c))
% 149.86/19.23 % SZS output end Proof
% 149.86/19.23
% 149.86/19.23 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------