%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT202-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n006.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 11:44:32 AM UTC 2026
% Result : Timeout 292.74s 37.48s
% Output : None
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : LAT202-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.11/0.38 % Computer : n006.cluster.edu
% 0.11/0.38 % Model : x86_64 x86_64
% 0.11/0.38 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.38 % Memory : 8046.5625MB
% 0.11/0.38 % OS : Linux 6.8.0-71-generic
% 0.11/0.38 % CPULimit : 300
% 0.11/0.38 % WCLimit : 300
% 0.11/0.38 % DateTime : Sun Sep 27 14:05:41 UTC 2026
% 0.11/0.38 % CPUTime :
% 0.11/0.38 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 292.74/37.48 Command-line arguments: --no-flatten-goal
% 292.74/37.48
% 292.74/37.48 % SZS status Unsatisfiable
% 292.74/37.48
% 294.30/37.64 % SZS output start Proof
% 294.30/37.64 Axiom 1 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 294.30/37.64 Axiom 2 (complement_meet): meet(X, complement(X)) = zero.
% 294.30/37.64 Axiom 3 (idempotence_of_join): join(X, X) = X.
% 294.30/37.64 Axiom 4 (commutativity_of_join): join(X, Y) = join(Y, X).
% 294.30/37.65 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 294.30/37.65 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 294.30/37.65 Axiom 7 (absorption1): meet(X, join(X, Y)) = X.
% 294.30/37.65 Axiom 8 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 294.30/37.65 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 294.30/37.65 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 294.30/37.65 Axiom 11 (equation_H55): join(X, meet(Y, join(X, Z))) = join(X, meet(Y, join(Z, meet(X, join(Z, Y))))).
% 294.30/37.65 Axiom 12 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 294.30/37.65
% 294.30/37.65 Lemma 13: join(X, zero) = X.
% 294.30/37.65 Proof:
% 294.30/37.65 join(X, zero)
% 294.30/37.65 = { by axiom 2 (complement_meet) R->L }
% 294.30/37.65 join(X, meet(X, complement(X)))
% 294.30/37.65 = { by axiom 9 (absorption2) }
% 294.30/37.65 X
% 294.30/37.65
% 294.30/37.65 Lemma 14: join(zero, X) = X.
% 294.30/37.65 Proof:
% 294.30/37.65 join(zero, X)
% 294.30/37.65 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.65 join(X, zero)
% 294.30/37.65 = { by lemma 13 }
% 294.30/37.65 X
% 294.30/37.65
% 294.30/37.65 Lemma 15: meet(X, zero) = zero.
% 294.30/37.65 Proof:
% 294.30/37.65 meet(X, zero)
% 294.30/37.65 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.65 meet(zero, X)
% 294.30/37.65 = { by lemma 14 R->L }
% 294.30/37.65 meet(zero, join(zero, X))
% 294.30/37.65 = { by axiom 7 (absorption1) }
% 294.30/37.65 zero
% 294.30/37.65
% 294.30/37.65 Lemma 16: complement(complement(X)) = X.
% 294.30/37.65 Proof:
% 294.30/37.65 complement(complement(X))
% 294.30/37.65 = { by axiom 6 (ifeq_axiom) R->L }
% 294.30/37.65 ifeq(one, one, complement(complement(X)), X)
% 294.30/37.65 = { by axiom 5 (complement_join) R->L }
% 294.30/37.65 ifeq(join(X, complement(X)), one, complement(complement(X)), X)
% 294.30/37.65 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.65 ifeq(join(complement(X), X), one, complement(complement(X)), X)
% 294.30/37.65 = { by axiom 6 (ifeq_axiom) R->L }
% 294.30/37.65 ifeq(join(complement(X), X), one, ifeq(zero, zero, complement(complement(X)), X), X)
% 294.30/37.65 = { by axiom 2 (complement_meet) R->L }
% 294.30/37.65 ifeq(join(complement(X), X), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), X), X)
% 294.30/37.65 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.65 ifeq(join(complement(X), X), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 294.30/37.65 = { by axiom 12 (meet_join_complement) }
% 294.30/37.65 X
% 294.30/37.65
% 294.30/37.65 Lemma 17: meet(X, join(Y, X)) = X.
% 294.30/37.65 Proof:
% 294.30/37.65 meet(X, join(Y, X))
% 294.30/37.65 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.65 meet(X, join(X, Y))
% 294.30/37.65 = { by axiom 7 (absorption1) }
% 294.30/37.65 X
% 294.30/37.65
% 294.30/37.65 Lemma 18: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 294.30/37.65 Proof:
% 294.30/37.65 meet(Y, meet(X, Z))
% 294.30/37.65 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.65 meet(meet(X, Z), Y)
% 294.30/37.66 = { by axiom 8 (associativity_of_meet) }
% 294.30/37.66 meet(X, meet(Z, Y))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.66 meet(X, meet(Y, Z))
% 294.30/37.66
% 294.30/37.66 Lemma 19: join(X, meet(Y, X)) = X.
% 294.30/37.66 Proof:
% 294.30/37.66 join(X, meet(Y, X))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.66 join(X, meet(X, Y))
% 294.30/37.66 = { by axiom 9 (absorption2) }
% 294.30/37.66 X
% 294.30/37.66
% 294.30/37.66 Lemma 20: join(X, join(X, Y)) = join(X, Y).
% 294.30/37.66 Proof:
% 294.30/37.66 join(X, join(X, Y))
% 294.30/37.66 = { by axiom 10 (associativity_of_join) R->L }
% 294.30/37.66 join(join(X, X), Y)
% 294.30/37.66 = { by axiom 3 (idempotence_of_join) }
% 294.30/37.66 join(X, Y)
% 294.30/37.66
% 294.30/37.66 Lemma 21: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 294.30/37.66 Proof:
% 294.30/37.66 join(X, join(Y, Z))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.66 join(join(Y, Z), X)
% 294.30/37.66 = { by axiom 10 (associativity_of_join) }
% 294.30/37.66 join(Y, join(Z, X))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 join(Y, join(X, Z))
% 294.30/37.66
% 294.30/37.66 Lemma 22: meet(X, join(meet(X, Y), complement(join(X, Y)))) = meet(X, join(Y, complement(join(X, Y)))).
% 294.30/37.66 Proof:
% 294.30/37.66 meet(X, join(meet(X, Y), complement(join(X, Y))))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.66 meet(X, join(meet(Y, X), complement(join(X, Y))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.66 meet(X, join(complement(join(X, Y)), meet(Y, X)))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.66 meet(X, join(complement(join(X, Y)), meet(X, Y)))
% 294.30/37.66 = { by lemma 17 R->L }
% 294.30/37.66 meet(X, meet(join(complement(join(X, Y)), meet(X, Y)), join(Y, join(complement(join(X, Y)), meet(X, Y)))))
% 294.30/37.66 = { by lemma 21 R->L }
% 294.30/37.66 meet(X, meet(join(complement(join(X, Y)), meet(X, Y)), join(complement(join(X, Y)), join(Y, meet(X, Y)))))
% 294.30/37.66 = { by lemma 19 }
% 294.30/37.66 meet(X, meet(join(complement(join(X, Y)), meet(X, Y)), join(complement(join(X, Y)), Y)))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 meet(X, meet(join(complement(join(X, Y)), meet(X, Y)), join(Y, complement(join(X, Y)))))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.66 meet(X, meet(join(Y, complement(join(X, Y))), join(complement(join(X, Y)), meet(X, Y))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 meet(X, meet(join(Y, complement(join(X, Y))), join(meet(X, Y), complement(join(X, Y)))))
% 294.30/37.66 = { by axiom 8 (associativity_of_meet) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(meet(X, Y), complement(join(X, Y))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(meet(X, Y), complement(join(Y, X))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, Y)))
% 294.30/37.66 = { by lemma 13 R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, zero))))
% 294.30/37.66 = { by axiom 2 (complement_meet) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, meet(join(Y, X), complement(join(Y, X)))))))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, meet(complement(join(Y, X)), join(Y, X))))))
% 294.30/37.66 = { by axiom 11 (equation_H55) R->L }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(complement(join(Y, X)), Y))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, complement(join(Y, X))))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(Y, X)), meet(X, join(Y, complement(join(X, Y))))))
% 294.30/37.66 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.66 meet(meet(X, join(Y, complement(join(X, Y)))), join(complement(join(X, Y)), meet(X, join(Y, complement(join(X, Y))))))
% 294.30/37.66 = { by lemma 17 }
% 294.30/37.66 meet(X, join(Y, complement(join(X, Y))))
% 294.30/37.66
% 294.30/37.66 Lemma 23: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 294.30/37.66 Proof:
% 294.30/37.66 meet(X, meet(Y, join(X, Z)))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.66 meet(X, meet(join(X, Z), Y))
% 294.30/37.66 = { by axiom 8 (associativity_of_meet) R->L }
% 294.30/37.66 meet(meet(X, join(X, Z)), Y)
% 294.30/37.66 = { by axiom 7 (absorption1) }
% 294.30/37.66 meet(X, Y)
% 294.30/37.66
% 294.30/37.66 Lemma 24: meet(X, complement(join(X, Y))) = zero.
% 294.30/37.66 Proof:
% 294.30/37.66 meet(X, complement(join(X, Y)))
% 294.30/37.66 = { by lemma 23 R->L }
% 294.30/37.66 meet(X, meet(complement(join(X, Y)), join(X, Y)))
% 294.30/37.66 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.66 meet(X, meet(join(X, Y), complement(join(X, Y))))
% 294.30/37.66 = { by axiom 2 (complement_meet) }
% 294.30/37.66 meet(X, zero)
% 294.30/37.66 = { by lemma 15 }
% 294.30/37.66 zero
% 294.30/37.66
% 294.30/37.66 Lemma 25: join(X, join(Y, complement(join(X, Y)))) = one.
% 294.30/37.66 Proof:
% 294.30/37.66 join(X, join(Y, complement(join(X, Y))))
% 294.30/37.66 = { by axiom 10 (associativity_of_join) R->L }
% 294.30/37.66 join(join(X, Y), complement(join(X, Y)))
% 294.30/37.66 = { by axiom 5 (complement_join) }
% 294.30/37.66 one
% 294.30/37.66
% 294.30/37.66 Lemma 26: join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))) = complement(X).
% 294.30/37.66 Proof:
% 294.30/37.66 join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))
% 294.30/37.66 = { by axiom 12 (meet_join_complement) R->L }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(complement(join(Y, X)), complement(join(X, complement(join(X, Y)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(complement(join(Y, X)), complement(join(X, complement(join(Y, X)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 22 R->L }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(meet(X, complement(join(Y, X))), complement(join(X, complement(join(Y, X)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(meet(X, complement(join(X, Y))), complement(join(X, complement(join(Y, X)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 24 }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, join(zero, complement(join(X, complement(join(Y, X)))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 14 }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(meet(X, complement(join(X, complement(join(Y, X))))), zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 24 }
% 294.30/37.67 ifeq(join(X, join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), one, ifeq(zero, zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 25 }
% 294.30/37.67 ifeq(one, one, ifeq(zero, zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.67 ifeq(zero, zero, complement(X), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.67 complement(X)
% 294.30/37.67
% 294.30/37.67 Lemma 27: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 294.30/37.67 Proof:
% 294.30/37.67 meet(complement(X), complement(join(X, Y)))
% 294.30/37.67 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.67 meet(complement(join(X, Y)), complement(X))
% 294.30/37.67 = { by lemma 26 R->L }
% 294.30/37.67 meet(complement(join(X, Y)), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by axiom 7 (absorption1) }
% 294.30/37.67 complement(join(X, Y))
% 294.30/37.67
% 294.30/37.67 Lemma 28: meet(complement(complement(X)), complement(complement(join(X, Y)))) = complement(complement(X)).
% 294.30/37.67 Proof:
% 294.30/37.67 meet(complement(complement(X)), complement(complement(join(X, Y))))
% 294.30/37.67 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.67 meet(complement(complement(join(X, Y))), complement(complement(X)))
% 294.30/37.67 = { by lemma 26 R->L }
% 294.30/37.67 meet(complement(complement(join(X, Y))), complement(join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))))
% 294.30/37.67 = { by lemma 27 }
% 294.30/37.67 complement(join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))))
% 294.30/37.67 = { by lemma 26 }
% 294.30/37.67 complement(complement(X))
% 294.30/37.67
% 294.30/37.67 Lemma 29: join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))) = complement(Y).
% 294.30/37.67 Proof:
% 294.30/37.67 join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
% 294.30/37.67 = { by axiom 12 (meet_join_complement) R->L }
% 294.30/37.67 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.67 = { by lemma 22 R->L }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(Y, meet(X, complement(Y))), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(Y, meet(complement(Y), X)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 8 (associativity_of_meet) R->L }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(meet(Y, complement(Y)), X), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 2 (complement_meet) }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(zero, X), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(X, zero), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by lemma 15 }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(zero, complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by lemma 14 }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, complement(join(Y, meet(X, complement(Y))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by lemma 24 }
% 294.30/37.68 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by lemma 25 }
% 294.30/37.68 ifeq(one, one, ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.68 ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 294.30/37.68 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.68 complement(Y)
% 294.30/37.68
% 294.30/37.68 Lemma 30: join(meet(X, Y), complement(join(complement(Y), meet(X, Y)))) = Y.
% 294.30/37.68 Proof:
% 294.30/37.68 join(meet(X, Y), complement(join(complement(Y), meet(X, Y))))
% 294.30/37.68 = { by lemma 16 R->L }
% 294.30/37.68 join(meet(X, Y), complement(join(complement(Y), meet(X, complement(complement(Y))))))
% 294.30/37.68 = { by lemma 16 R->L }
% 294.30/37.68 join(meet(X, complement(complement(Y))), complement(join(complement(Y), meet(X, complement(complement(Y))))))
% 294.30/37.68 = { by lemma 29 }
% 294.30/37.68 complement(complement(Y))
% 294.30/37.68 = { by lemma 16 }
% 294.30/37.68 Y
% 294.30/37.68
% 294.30/37.68 Lemma 31: join(complement(X), complement(join(complement(X), meet(X, Y)))) = complement(meet(X, Y)).
% 294.30/37.68 Proof:
% 294.30/37.68 join(complement(X), complement(join(complement(X), meet(X, Y))))
% 294.30/37.68 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.68 join(complement(X), complement(join(complement(X), meet(Y, X))))
% 294.30/37.68 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.68 join(complement(X), complement(join(meet(Y, X), complement(X))))
% 294.30/37.68 = { by lemma 30 R->L }
% 294.30/37.68 join(complement(X), complement(join(meet(Y, X), complement(join(meet(Y, X), complement(join(complement(X), meet(Y, X))))))))
% 294.30/37.68 = { by lemma 30 R->L }
% 294.30/37.68 join(complement(join(meet(Y, X), complement(join(complement(X), meet(Y, X))))), complement(join(meet(Y, X), complement(join(meet(Y, X), complement(join(complement(X), meet(Y, X))))))))
% 294.30/37.68 = { by lemma 26 }
% 294.30/37.68 complement(meet(Y, X))
% 294.30/37.68 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.68 complement(meet(X, Y))
% 294.30/37.68
% 294.30/37.68 Lemma 32: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 294.30/37.68 Proof:
% 294.30/37.68 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.69 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.69 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 294.30/37.69 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(X, Y))
% 294.30/37.69 = { by axiom 7 (absorption1) R->L }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(meet(X, Y), join(meet(X, Y), Z)))
% 294.30/37.69 = { by axiom 8 (associativity_of_meet) }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(X, meet(Y, join(meet(X, Y), Z))))
% 294.30/37.69 = { by lemma 18 R->L }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(X, Y), Z))))
% 294.30/37.69 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(X, Y)))))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.69 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 294.30/37.69 = { by lemma 19 }
% 294.30/37.69 meet(X, join(Z, meet(Y, X)))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.69 meet(X, join(Z, meet(X, Y)))
% 294.30/37.69
% 294.30/37.69 Lemma 33: join(X, meet(Y, join(X, meet(Y, Z)))) = join(X, meet(Y, Z)).
% 294.30/37.69 Proof:
% 294.30/37.69 join(X, meet(Y, join(X, meet(Y, Z))))
% 294.30/37.69 = { by lemma 32 R->L }
% 294.30/37.69 join(X, join(meet(Y, Z), meet(Y, join(X, meet(Y, Z)))))
% 294.30/37.69 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.69 join(X, join(meet(Y, Z), meet(Y, join(meet(Y, Z), X))))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.69 join(X, join(meet(Y, Z), meet(join(meet(Y, Z), X), Y)))
% 294.30/37.69 = { by lemma 21 R->L }
% 294.30/37.69 join(meet(Y, Z), join(X, meet(join(meet(Y, Z), X), Y)))
% 294.30/37.69 = { by axiom 10 (associativity_of_join) R->L }
% 294.30/37.69 join(join(meet(Y, Z), X), meet(join(meet(Y, Z), X), Y))
% 294.30/37.69 = { by axiom 9 (absorption2) }
% 294.30/37.69 join(meet(Y, Z), X)
% 294.30/37.69 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.69 join(X, meet(Y, Z))
% 294.30/37.69
% 294.30/37.69 Lemma 34: meet(X, join(complement(X), meet(X, Y))) = meet(X, Y).
% 294.30/37.69 Proof:
% 294.30/37.69 meet(X, join(complement(X), meet(X, Y)))
% 294.30/37.69 = { by lemma 16 R->L }
% 294.30/37.69 complement(complement(meet(X, join(complement(X), meet(X, Y)))))
% 294.30/37.69 = { by lemma 31 R->L }
% 294.30/37.69 complement(join(complement(X), complement(join(complement(X), meet(X, join(complement(X), meet(X, Y)))))))
% 294.30/37.69 = { by lemma 33 }
% 294.30/37.69 complement(join(complement(X), complement(join(complement(X), meet(X, Y)))))
% 294.30/37.69 = { by lemma 31 }
% 294.30/37.69 complement(complement(meet(X, Y)))
% 294.30/37.69 = { by lemma 16 }
% 294.30/37.69 meet(X, Y)
% 294.30/37.69
% 294.30/37.69 Lemma 35: join(complement(X), meet(X, Y)) = join(Y, complement(join(X, Y))).
% 294.30/37.69 Proof:
% 294.30/37.69 join(complement(X), meet(X, Y))
% 294.30/37.69 = { by lemma 16 R->L }
% 294.30/37.69 complement(complement(join(complement(X), meet(X, Y))))
% 294.30/37.69 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.69 complement(complement(join(complement(X), meet(Y, X))))
% 294.30/37.69 = { by axiom 12 (meet_join_complement) R->L }
% 294.30/37.69 complement(ifeq(join(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y))), complement(join(complement(X), meet(Y, X)))), one, ifeq(meet(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y))), complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.69 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.69 complement(ifeq(join(complement(join(complement(X), meet(Y, X))), join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), one, ifeq(meet(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y))), complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.69 = { by lemma 25 }
% 294.30/37.70 complement(ifeq(one, one, ifeq(meet(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y))), complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.70 complement(ifeq(meet(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y))), complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X))))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.70 complement(ifeq(meet(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X)))))), complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by lemma 17 R->L }
% 294.30/37.70 complement(ifeq(meet(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X)))))), meet(complement(join(complement(X), meet(Y, X))), join(Y, complement(join(complement(X), meet(Y, X)))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by lemma 18 R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X)))))), join(Y, complement(join(complement(X), meet(Y, X)))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(join(Y, complement(join(complement(X), meet(Y, X)))), join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X)))))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(join(Y, complement(join(complement(X), meet(Y, X)))), join(complement(join(Y, complement(join(complement(X), meet(Y, X))))), Y))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by lemma 16 R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(join(Y, complement(join(complement(X), meet(Y, X)))), join(complement(join(Y, complement(join(complement(X), meet(Y, X))))), complement(complement(Y))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by lemma 16 R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), join(complement(join(Y, complement(join(complement(X), meet(Y, X))))), complement(complement(Y))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.70 = { by lemma 28 R->L }
% 294.30/37.70 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), join(complement(join(Y, complement(join(complement(X), meet(Y, X))))), meet(complement(complement(Y)), complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), join(complement(join(Y, complement(join(complement(X), meet(Y, X))))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), complement(complement(Y)))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by lemma 16 R->L }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), join(complement(complement(complement(join(Y, complement(join(complement(X), meet(Y, X))))))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), complement(complement(Y)))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by lemma 34 }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))), complement(complement(Y)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), meet(complement(complement(Y)), complement(complement(join(Y, complement(join(complement(X), meet(Y, X)))))))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by lemma 28 }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), complement(complement(Y))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by lemma 16 }
% 294.30/37.71 complement(ifeq(meet(complement(join(complement(X), meet(Y, X))), Y), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.71 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(complement(join(complement(X), meet(Y, X))), Y)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.71 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, X))))))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by lemma 16 R->L }
% 294.30/37.71 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(Y, complement(join(complement(X), meet(Y, complement(complement(X))))))))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.71 = { by axiom 9 (absorption2) R->L }
% 294.30/37.72 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(join(Y, meet(Y, complement(complement(X)))), complement(join(complement(X), meet(Y, complement(complement(X))))))))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by axiom 10 (associativity_of_join) }
% 294.30/37.72 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(Y, join(meet(Y, complement(complement(X))), complement(join(complement(X), meet(Y, complement(complement(X)))))))))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 29 }
% 294.30/37.72 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(Y, complement(complement(X)))))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 16 }
% 294.30/37.72 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, X)))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 16 R->L }
% 294.30/37.72 complement(ifeq(meet(Y, complement(join(complement(X), meet(Y, complement(complement(X)))))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 27 R->L }
% 294.30/37.72 complement(ifeq(meet(Y, meet(complement(complement(X)), complement(join(complement(X), meet(Y, complement(complement(X))))))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.72 complement(ifeq(meet(Y, meet(complement(complement(X)), complement(join(complement(X), meet(complement(complement(X)), Y))))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.72 complement(ifeq(meet(Y, meet(complement(complement(X)), complement(join(meet(complement(complement(X)), Y), complement(X))))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 18 }
% 294.30/37.72 complement(ifeq(meet(complement(complement(X)), meet(Y, complement(join(meet(complement(complement(X)), Y), complement(X))))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by axiom 8 (associativity_of_meet) R->L }
% 294.30/37.72 complement(ifeq(meet(meet(complement(complement(X)), Y), complement(join(meet(complement(complement(X)), Y), complement(X)))), zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by lemma 24 }
% 294.30/37.72 complement(ifeq(zero, zero, complement(join(Y, complement(join(Y, X)))), complement(join(complement(X), meet(Y, X)))))
% 294.30/37.72 = { by axiom 6 (ifeq_axiom) }
% 294.30/37.72 complement(complement(join(Y, complement(join(Y, X)))))
% 294.30/37.72 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.72 complement(complement(join(Y, complement(join(X, Y)))))
% 294.30/37.72 = { by lemma 16 }
% 294.30/37.72 join(Y, complement(join(X, Y)))
% 294.30/37.72
% 294.30/37.72 Lemma 36: meet(X, join(complement(X), Y)) = meet(X, Y).
% 294.30/37.72 Proof:
% 294.30/37.72 meet(X, join(complement(X), Y))
% 294.30/37.72 = { by lemma 26 R->L }
% 294.30/37.72 meet(X, join(join(complement(join(X, Y)), complement(join(X, complement(join(X, Y))))), Y))
% 294.30/37.72 = { by lemma 20 R->L }
% 294.30/37.72 meet(X, join(join(complement(join(X, Y)), join(complement(join(X, Y)), complement(join(X, complement(join(X, Y)))))), Y))
% 294.30/37.72 = { by lemma 26 }
% 294.30/37.72 meet(X, join(join(complement(join(X, Y)), complement(X)), Y))
% 294.30/37.72 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.72 meet(X, join(join(complement(X), complement(join(X, Y))), Y))
% 294.30/37.72 = { by axiom 10 (associativity_of_join) }
% 294.30/37.72 meet(X, join(complement(X), join(complement(join(X, Y)), Y)))
% 294.30/37.72 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.72 meet(X, join(complement(X), join(Y, complement(join(X, Y)))))
% 294.30/37.72 = { by lemma 35 R->L }
% 294.30/37.72 meet(X, join(complement(X), join(complement(X), meet(X, Y))))
% 294.30/37.72 = { by lemma 20 }
% 294.30/37.72 meet(X, join(complement(X), meet(X, Y)))
% 294.30/37.72 = { by lemma 34 }
% 294.30/37.72 meet(X, Y)
% 294.30/37.72
% 294.30/37.72 Lemma 37: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 294.30/37.72 Proof:
% 294.30/37.72 meet(X, join(Y, meet(X, Z)))
% 294.30/37.72 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.72 meet(X, join(Y, meet(Z, X)))
% 294.30/37.72 = { by lemma 16 R->L }
% 294.30/37.72 meet(X, join(Y, meet(Z, complement(complement(X)))))
% 294.30/37.72 = { by lemma 36 R->L }
% 294.30/37.72 meet(X, join(complement(X), join(Y, meet(Z, complement(complement(X))))))
% 294.30/37.72 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.72 meet(X, join(complement(X), join(Y, meet(complement(complement(X)), Z))))
% 294.30/37.72 = { by lemma 16 R->L }
% 294.30/37.72 meet(X, join(complement(complement(complement(X))), join(Y, meet(complement(complement(X)), Z))))
% 294.30/37.72 = { by lemma 21 R->L }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), meet(complement(complement(X)), Z))))
% 294.30/37.73 = { by axiom 1 (commutativity_of_meet) R->L }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), meet(Z, complement(complement(X))))))
% 294.30/37.73 = { by lemma 33 R->L }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), meet(Z, join(complement(complement(complement(X))), meet(Z, complement(complement(X))))))))
% 294.30/37.73 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), meet(Z, join(complement(complement(complement(X))), meet(complement(complement(X)), Z))))))
% 294.30/37.73 = { by lemma 35 }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), meet(Z, join(Z, complement(join(complement(complement(X)), Z)))))))
% 294.30/37.73 = { by axiom 7 (absorption1) }
% 294.30/37.73 meet(X, join(Y, join(complement(complement(complement(X))), Z)))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 meet(X, join(Y, join(Z, complement(complement(complement(X))))))
% 294.30/37.73 = { by lemma 16 }
% 294.30/37.73 meet(X, join(Y, join(Z, complement(X))))
% 294.30/37.73 = { by lemma 21 }
% 294.30/37.73 meet(X, join(Z, join(Y, complement(X))))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 meet(X, join(Z, join(complement(X), Y)))
% 294.30/37.73 = { by lemma 21 }
% 294.30/37.73 meet(X, join(complement(X), join(Z, Y)))
% 294.30/37.73 = { by lemma 36 }
% 294.30/37.73 meet(X, join(Z, Y))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 meet(X, join(Y, Z))
% 294.30/37.73
% 294.30/37.73 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 294.30/37.73 Proof:
% 294.30/37.73 meet(a, join(b, c))
% 294.30/37.73 = { by lemma 37 R->L }
% 294.30/37.73 meet(a, join(b, meet(a, c)))
% 294.30/37.73 = { by lemma 19 R->L }
% 294.30/37.73 join(meet(a, join(b, meet(a, c))), meet(b, meet(a, join(b, meet(a, c)))))
% 294.30/37.73 = { by lemma 23 }
% 294.30/37.73 join(meet(a, join(b, meet(a, c))), meet(b, a))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 join(meet(b, a), meet(a, join(b, meet(a, c))))
% 294.30/37.73 = { by axiom 1 (commutativity_of_meet) }
% 294.30/37.73 join(meet(a, b), meet(a, join(b, meet(a, c))))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.73 join(meet(a, b), meet(a, join(meet(a, c), b)))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) R->L }
% 294.30/37.73 join(meet(a, join(meet(a, c), b)), meet(a, b))
% 294.30/37.73 = { by lemma 37 R->L }
% 294.30/37.73 join(meet(a, join(meet(a, c), meet(a, b))), meet(a, b))
% 294.30/37.73 = { by lemma 32 R->L }
% 294.30/37.73 join(join(meet(a, b), meet(a, join(meet(a, c), meet(a, b)))), meet(a, b))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 join(join(meet(a, b), meet(a, join(meet(a, b), meet(a, c)))), meet(a, b))
% 294.30/37.73 = { by axiom 10 (associativity_of_join) }
% 294.30/37.73 join(meet(a, b), join(meet(a, join(meet(a, b), meet(a, c))), meet(a, b)))
% 294.30/37.73 = { by axiom 4 (commutativity_of_join) }
% 294.30/37.73 join(meet(a, b), join(meet(a, b), meet(a, join(meet(a, b), meet(a, c)))))
% 294.30/37.73 = { by lemma 33 }
% 294.30/37.73 join(meet(a, b), join(meet(a, b), meet(a, c)))
% 294.30/37.73 = { by lemma 20 }
% 294.30/37.73 join(meet(a, b), meet(a, c))
% 294.30/37.73 % SZS output end Proof
% 294.30/37.73
% 294.30/37.73 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------