%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT211-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n009.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 11:44:33 AM UTC 2026
% Result : Unsatisfiable 188.72s 24.25s
% Output : Proof 190.34s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LAT211-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.36 % Computer : n009.cluster.edu
% 0.09/0.36 % Model : x86_64 x86_64
% 0.09/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.36 % Memory : 8046.5625MB
% 0.09/0.36 % OS : Linux 6.8.0-71-generic
% 0.09/0.36 % CPULimit : 300
% 0.09/0.36 % WCLimit : 300
% 0.09/0.36 % DateTime : Sun Sep 27 14:06:14 UTC 2026
% 0.09/0.36 % CPUTime :
% 0.09/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 188.72/24.25 Command-line arguments: --no-flatten-goal
% 188.72/24.25
% 188.72/24.25 % SZS status Unsatisfiable
% 188.72/24.25
% 190.34/24.42 % SZS output start Proof
% 190.34/24.42 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 190.34/24.42 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 190.34/24.42 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 190.34/24.42 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 190.34/24.42 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 190.34/24.42 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 190.34/24.42 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 190.34/24.42 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 190.34/24.42 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 190.34/24.42 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 190.34/24.42 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 190.34/24.42 Axiom 12 (equation_H69): meet(X, join(Y, Z)) = join(meet(X, join(Z, meet(X, Y))), meet(X, join(Y, meet(X, Z)))).
% 190.34/24.42
% 190.34/24.42 Lemma 13: meet(X, one) = X.
% 190.34/24.42 Proof:
% 190.34/24.42 meet(X, one)
% 190.34/24.42 = { by axiom 3 (complement_join) R->L }
% 190.34/24.42 meet(X, join(X, complement(X)))
% 190.34/24.42 = { by axiom 9 (absorption1) }
% 190.34/24.42 X
% 190.34/24.42
% 190.34/24.42 Lemma 14: meet(one, X) = X.
% 190.34/24.42 Proof:
% 190.34/24.42 meet(one, X)
% 190.34/24.42 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.42 meet(X, one)
% 190.34/24.42 = { by lemma 13 }
% 190.34/24.42 X
% 190.34/24.42
% 190.34/24.42 Lemma 15: complement(one) = zero.
% 190.34/24.42 Proof:
% 190.34/24.42 complement(one)
% 190.34/24.42 = { by lemma 14 R->L }
% 190.34/24.42 meet(one, complement(one))
% 190.34/24.42 = { by axiom 5 (complement_meet) }
% 190.34/24.42 zero
% 190.34/24.42
% 190.34/24.42 Lemma 16: join(X, zero) = X.
% 190.34/24.42 Proof:
% 190.34/24.42 join(X, zero)
% 190.34/24.42 = { by axiom 5 (complement_meet) R->L }
% 190.34/24.42 join(X, meet(X, complement(X)))
% 190.34/24.42 = { by axiom 7 (absorption2) }
% 190.34/24.42 X
% 190.34/24.42
% 190.34/24.42 Lemma 17: meet(X, zero) = zero.
% 190.34/24.42 Proof:
% 190.34/24.42 meet(X, zero)
% 190.34/24.42 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.42 meet(zero, X)
% 190.34/24.42 = { by lemma 16 R->L }
% 190.34/24.42 join(meet(zero, X), zero)
% 190.34/24.42 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.42 join(zero, meet(zero, X))
% 190.34/24.42 = { by axiom 7 (absorption2) }
% 190.34/24.42 zero
% 190.34/24.42
% 190.34/24.42 Lemma 18: join(X, one) = one.
% 190.34/24.42 Proof:
% 190.34/24.42 join(X, one)
% 190.34/24.42 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.42 join(one, X)
% 190.34/24.42 = { by lemma 14 R->L }
% 190.34/24.42 join(one, meet(one, X))
% 190.34/24.42 = { by axiom 7 (absorption2) }
% 190.34/24.42 one
% 190.34/24.42
% 190.34/24.42 Lemma 19: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 190.34/24.42 Proof:
% 190.34/24.42 meet(X, meet(Y, join(X, Z)))
% 190.34/24.42 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.42 meet(X, meet(join(X, Z), Y))
% 190.34/24.42 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.42 meet(meet(X, join(X, Z)), Y)
% 190.34/24.42 = { by axiom 9 (absorption1) }
% 190.34/24.43 meet(X, Y)
% 190.34/24.43
% 190.34/24.43 Lemma 20: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 190.34/24.43 Proof:
% 190.34/24.43 meet(X, meet(join(X, Y), Z))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 meet(X, meet(Z, join(X, Y)))
% 190.34/24.43 = { by lemma 19 }
% 190.34/24.43 meet(X, Z)
% 190.34/24.43
% 190.34/24.43 Lemma 21: meet(X, join(Y, X)) = X.
% 190.34/24.43 Proof:
% 190.34/24.43 meet(X, join(Y, X))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.43 meet(X, join(X, Y))
% 190.34/24.43 = { by axiom 9 (absorption1) }
% 190.34/24.43 X
% 190.34/24.43
% 190.34/24.43 Lemma 22: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, join(Y, Z))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.43 join(join(Y, Z), X)
% 190.34/24.43 = { by axiom 8 (associativity_of_join) }
% 190.34/24.43 join(Y, join(Z, X))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.43 join(Y, join(X, Z))
% 190.34/24.43
% 190.34/24.43 Lemma 23: join(X, meet(Y, X)) = X.
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, meet(Y, X))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 join(X, meet(X, Y))
% 190.34/24.43 = { by axiom 7 (absorption2) }
% 190.34/24.43 X
% 190.34/24.43
% 190.34/24.43 Lemma 24: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, join(Y, meet(X, Z)))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.43 join(X, join(meet(X, Z), Y))
% 190.34/24.43 = { by axiom 8 (associativity_of_join) R->L }
% 190.34/24.43 join(join(X, meet(X, Z)), Y)
% 190.34/24.43 = { by axiom 7 (absorption2) }
% 190.34/24.43 join(X, Y)
% 190.34/24.43
% 190.34/24.43 Lemma 25: join(X, join(Y, join(Z, meet(W, join(X, Y))))) = join(X, join(Y, Z)).
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, join(Y, join(Z, meet(W, join(X, Y)))))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 join(X, join(Y, join(Z, meet(join(X, Y), W))))
% 190.34/24.43 = { by axiom 8 (associativity_of_join) R->L }
% 190.34/24.43 join(join(X, Y), join(Z, meet(join(X, Y), W)))
% 190.34/24.43 = { by lemma 24 }
% 190.34/24.43 join(join(X, Y), Z)
% 190.34/24.43 = { by axiom 8 (associativity_of_join) }
% 190.34/24.43 join(X, join(Y, Z))
% 190.34/24.43
% 190.34/24.43 Lemma 26: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 190.34/24.43 Proof:
% 190.34/24.43 meet(Y, meet(X, Z))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 meet(meet(X, Z), Y)
% 190.34/24.43 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.43 meet(X, meet(Z, Y))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.43 meet(X, meet(Y, Z))
% 190.34/24.43
% 190.34/24.43 Lemma 27: join(meet(X, Y), meet(X, join(Z, Y))) = meet(X, join(Z, Y)).
% 190.34/24.43 Proof:
% 190.34/24.43 join(meet(X, Y), meet(X, join(Z, Y)))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 join(meet(Y, X), meet(X, join(Z, Y)))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.43 join(meet(X, join(Z, Y)), meet(Y, X))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 join(meet(X, join(Z, Y)), meet(X, Y))
% 190.34/24.43 = { by lemma 21 R->L }
% 190.34/24.43 join(meet(X, join(Z, Y)), meet(X, meet(Y, join(Z, Y))))
% 190.34/24.43 = { by lemma 26 R->L }
% 190.34/24.43 join(meet(X, join(Z, Y)), meet(Y, meet(X, join(Z, Y))))
% 190.34/24.43 = { by lemma 23 }
% 190.34/24.43 meet(X, join(Z, Y))
% 190.34/24.43
% 190.34/24.43 Lemma 28: join(X, meet(Y, join(Z, meet(X, Y)))) = join(X, meet(Y, join(X, Z))).
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, meet(Y, join(Z, meet(X, Y))))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 join(X, meet(Y, join(Z, meet(Y, X))))
% 190.34/24.43 = { by lemma 23 R->L }
% 190.34/24.43 join(X, join(meet(Y, join(Z, meet(Y, X))), meet(Z, meet(Y, join(Z, meet(Y, X))))))
% 190.34/24.43 = { by lemma 19 }
% 190.34/24.43 join(X, join(meet(Y, join(Z, meet(Y, X))), meet(Z, Y)))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.43 join(X, join(meet(Z, Y), meet(Y, join(Z, meet(Y, X)))))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.43 join(X, join(meet(Y, Z), meet(Y, join(Z, meet(Y, X)))))
% 190.34/24.43 = { by lemma 25 R->L }
% 190.34/24.43 join(X, join(meet(Y, Z), join(meet(Y, join(Z, meet(Y, X))), meet(Y, join(X, meet(Y, Z))))))
% 190.34/24.43 = { by axiom 12 (equation_H69) R->L }
% 190.34/24.43 join(X, join(meet(Y, Z), meet(Y, join(X, Z))))
% 190.34/24.43 = { by lemma 27 }
% 190.34/24.43 join(X, meet(Y, join(X, Z)))
% 190.34/24.43
% 190.34/24.43 Lemma 29: join(complement(X), meet(X, join(Y, complement(X)))) = join(complement(X), meet(X, Y)).
% 190.34/24.43 Proof:
% 190.34/24.43 join(complement(X), meet(X, join(Y, complement(X))))
% 190.34/24.43 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.43 join(complement(X), meet(X, join(complement(X), Y)))
% 190.34/24.43 = { by lemma 28 R->L }
% 190.34/24.43 join(complement(X), meet(X, join(Y, meet(complement(X), X))))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.43 join(complement(X), meet(X, join(Y, meet(X, complement(X)))))
% 190.34/24.43 = { by axiom 5 (complement_meet) }
% 190.34/24.43 join(complement(X), meet(X, join(Y, zero)))
% 190.34/24.43 = { by lemma 16 }
% 190.34/24.43 join(complement(X), meet(X, Y))
% 190.34/24.43
% 190.34/24.43 Lemma 30: join(X, meet(complement(X), join(X, Y))) = join(X, meet(Y, complement(X))).
% 190.34/24.43 Proof:
% 190.34/24.43 join(X, meet(complement(X), join(X, Y)))
% 190.34/24.43 = { by lemma 28 R->L }
% 190.34/24.43 join(X, meet(complement(X), join(Y, meet(X, complement(X)))))
% 190.34/24.43 = { by axiom 5 (complement_meet) }
% 190.34/24.43 join(X, meet(complement(X), join(Y, zero)))
% 190.34/24.43 = { by lemma 16 }
% 190.34/24.43 join(X, meet(complement(X), Y))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.43 join(X, meet(Y, complement(X)))
% 190.34/24.43
% 190.34/24.43 Lemma 31: meet(X, meet(Y, complement(X))) = zero.
% 190.34/24.43 Proof:
% 190.34/24.43 meet(X, meet(Y, complement(X)))
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 meet(X, meet(complement(X), Y))
% 190.34/24.43 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.43 meet(meet(X, complement(X)), Y)
% 190.34/24.43 = { by axiom 5 (complement_meet) }
% 190.34/24.43 meet(zero, Y)
% 190.34/24.43 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.43 meet(Y, zero)
% 190.34/24.43 = { by lemma 17 }
% 190.34/24.43 zero
% 190.34/24.43
% 190.34/24.43 Lemma 32: meet(X, complement(complement(X))) = complement(complement(X)).
% 190.34/24.43 Proof:
% 190.34/24.43 meet(X, complement(complement(X)))
% 190.34/24.43 = { by axiom 11 (meet_join_complement) R->L }
% 190.34/24.43 ifeq(join(complement(X), meet(X, complement(complement(X)))), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by lemma 30 R->L }
% 190.34/24.44 ifeq(join(complement(X), meet(complement(complement(X)), join(complement(X), X))), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.44 ifeq(join(complement(X), meet(complement(complement(X)), join(X, complement(X)))), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by axiom 3 (complement_join) }
% 190.34/24.44 ifeq(join(complement(X), meet(complement(complement(X)), one)), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by lemma 13 }
% 190.34/24.44 ifeq(join(complement(X), complement(complement(X))), one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by axiom 3 (complement_join) }
% 190.34/24.44 ifeq(one, one, ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X)))), meet(X, complement(complement(X))))
% 190.34/24.44 = { by axiom 6 (ifeq_axiom) }
% 190.34/24.44 ifeq(meet(complement(X), meet(X, complement(complement(X)))), zero, complement(complement(X)), meet(X, complement(complement(X))))
% 190.34/24.44 = { by lemma 31 }
% 190.34/24.44 ifeq(zero, zero, complement(complement(X)), meet(X, complement(complement(X))))
% 190.34/24.44 = { by axiom 6 (ifeq_axiom) }
% 190.34/24.44 complement(complement(X))
% 190.34/24.44
% 190.34/24.44 Lemma 33: complement(complement(X)) = X.
% 190.34/24.44 Proof:
% 190.34/24.44 complement(complement(X))
% 190.34/24.44 = { by axiom 1 (idempotence_of_join) R->L }
% 190.34/24.44 complement(complement(join(X, X)))
% 190.34/24.44 = { by axiom 6 (ifeq_axiom) R->L }
% 190.34/24.44 ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 6 (ifeq_axiom) R->L }
% 190.34/24.44 ifeq(one, one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by lemma 18 R->L }
% 190.34/24.44 ifeq(join(X, one), one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.44 ifeq(join(one, X), one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 3 (complement_join) R->L }
% 190.34/24.44 ifeq(join(join(complement(join(X, X)), complement(complement(join(X, X)))), X), one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 8 (associativity_of_join) }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(complement(complement(join(X, X))), X)), one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(zero, zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 5 (complement_meet) R->L }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), complement(complement(join(X, X)))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by lemma 16 R->L }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(complement(complement(join(X, X))), zero)), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by lemma 17 R->L }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(complement(complement(join(X, X))), meet(X, zero))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by axiom 5 (complement_meet) R->L }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(complement(complement(join(X, X))), meet(X, meet(join(X, X), complement(join(X, X)))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.44 = { by lemma 20 }
% 190.34/24.44 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(complement(complement(join(X, X))), meet(X, complement(join(X, X))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(complement(complement(join(X, X))), meet(complement(join(X, X)), X))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by lemma 21 R->L }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), meet(join(complement(complement(join(X, X))), meet(complement(join(X, X)), X)), join(X, join(complement(complement(join(X, X))), meet(complement(join(X, X)), X))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by lemma 22 R->L }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), meet(join(complement(complement(join(X, X))), meet(complement(join(X, X)), X)), join(complement(complement(join(X, X))), join(X, meet(complement(join(X, X)), X))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by lemma 23 }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), meet(join(complement(complement(join(X, X))), meet(complement(join(X, X)), X)), join(complement(complement(join(X, X))), X))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), meet(join(complement(complement(join(X, X))), meet(complement(join(X, X)), X)), join(X, complement(complement(join(X, X)))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), meet(join(X, complement(complement(join(X, X)))), join(complement(complement(join(X, X))), meet(complement(join(X, X)), X)))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(meet(complement(join(X, X)), join(X, complement(complement(join(X, X))))), join(complement(complement(join(X, X))), meet(complement(join(X, X)), X))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by lemma 29 R->L }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(meet(complement(join(X, X)), join(X, complement(complement(join(X, X))))), join(complement(complement(join(X, X))), meet(complement(join(X, X)), join(X, complement(complement(join(X, X))))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by lemma 21 }
% 190.34/24.45 ifeq(join(complement(join(X, X)), join(X, complement(complement(join(X, X))))), one, ifeq(meet(complement(join(X, X)), join(X, complement(complement(join(X, X))))), zero, complement(complement(join(X, X))), join(X, complement(complement(join(X, X))))), join(X, complement(complement(join(X, X)))))
% 190.34/24.45 = { by axiom 11 (meet_join_complement) }
% 190.34/24.45 join(X, complement(complement(join(X, X))))
% 190.34/24.45 = { by axiom 1 (idempotence_of_join) }
% 190.34/24.45 join(X, complement(complement(X)))
% 190.34/24.45 = { by lemma 32 R->L }
% 190.34/24.45 join(X, meet(X, complement(complement(X))))
% 190.34/24.45 = { by axiom 7 (absorption2) }
% 190.34/24.45 X
% 190.34/24.45
% 190.34/24.45 Lemma 34: meet(X, meet(Y, complement(complement(X)))) = meet(Y, complement(complement(X))).
% 190.34/24.45 Proof:
% 190.34/24.45 meet(X, meet(Y, complement(complement(X))))
% 190.34/24.45 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.45 meet(X, meet(complement(complement(X)), Y))
% 190.34/24.45 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.46 meet(meet(X, complement(complement(X))), Y)
% 190.34/24.46 = { by lemma 32 }
% 190.34/24.46 meet(complement(complement(X)), Y)
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.46 meet(Y, complement(complement(X)))
% 190.34/24.46
% 190.34/24.46 Lemma 35: meet(complement(X), complement(meet(X, Y))) = complement(X).
% 190.34/24.46 Proof:
% 190.34/24.46 meet(complement(X), complement(meet(X, Y)))
% 190.34/24.46 = { by axiom 11 (meet_join_complement) R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(X), complement(meet(X, Y)))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(X), complement(meet(Y, X)))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(meet(Y, X)), complement(X))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by lemma 30 R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(X), join(X, complement(meet(Y, X))))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(X), join(X, complement(meet(X, Y))))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by lemma 24 R->L }
% 190.34/24.46 ifeq(join(X, meet(complement(X), join(X, join(complement(meet(X, Y)), meet(X, Y))))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.46 ifeq(join(X, meet(complement(X), join(X, join(meet(X, Y), complement(meet(X, Y)))))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 3 (complement_join) }
% 190.34/24.46 ifeq(join(X, meet(complement(X), join(X, one))), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by lemma 18 }
% 190.34/24.46 ifeq(join(X, meet(complement(X), one)), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by lemma 13 }
% 190.34/24.46 ifeq(join(X, complement(X)), one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 3 (complement_join) }
% 190.34/24.46 ifeq(one, one, ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 6 (ifeq_axiom) }
% 190.34/24.46 ifeq(meet(X, meet(complement(X), complement(meet(X, Y)))), zero, complement(X), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.46 ifeq(meet(X, meet(complement(meet(X, Y)), complement(X))), zero, complement(X), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by lemma 31 }
% 190.34/24.46 ifeq(zero, zero, complement(X), meet(complement(X), complement(meet(X, Y))))
% 190.34/24.46 = { by axiom 6 (ifeq_axiom) }
% 190.34/24.46 complement(X)
% 190.34/24.46
% 190.34/24.46 Lemma 36: join(X, join(Y, complement(join(X, Y)))) = one.
% 190.34/24.46 Proof:
% 190.34/24.46 join(X, join(Y, complement(join(X, Y))))
% 190.34/24.46 = { by axiom 8 (associativity_of_join) R->L }
% 190.34/24.46 join(join(X, Y), complement(join(X, Y)))
% 190.34/24.46 = { by axiom 3 (complement_join) }
% 190.34/24.46 one
% 190.34/24.46
% 190.34/24.46 Lemma 37: join(meet(X, complement(Y)), complement(meet(complement(Y), join(X, Y)))) = one.
% 190.34/24.46 Proof:
% 190.34/24.46 join(meet(X, complement(Y)), complement(meet(complement(Y), join(X, Y))))
% 190.34/24.46 = { by lemma 23 R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(complement(meet(complement(Y), join(X, Y))), meet(complement(complement(Y)), complement(meet(complement(Y), join(X, Y))))))
% 190.34/24.46 = { by lemma 35 }
% 190.34/24.46 join(meet(X, complement(Y)), join(complement(meet(complement(Y), join(X, Y))), complement(complement(Y))))
% 190.34/24.46 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.46 join(meet(X, complement(Y)), join(complement(complement(Y)), complement(meet(complement(Y), join(X, Y)))))
% 190.34/24.46 = { by lemma 33 }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(meet(complement(Y), join(X, Y)))))
% 190.34/24.46 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(meet(complement(Y), join(Y, X)))))
% 190.34/24.46 = { by lemma 27 R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(complement(Y), X), meet(complement(Y), join(Y, X))))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), join(Y, X))))))
% 190.34/24.46 = { by lemma 21 R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(meet(complement(Y), join(Y, X)), join(Y, meet(complement(Y), join(Y, X))))))))
% 190.34/24.46 = { by lemma 30 }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(meet(complement(Y), join(Y, X)), join(Y, meet(X, complement(Y))))))))
% 190.34/24.46 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), meet(join(Y, X), join(Y, meet(X, complement(Y)))))))))
% 190.34/24.46 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), meet(join(X, Y), join(Y, meet(X, complement(Y)))))))))
% 190.34/24.46 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), meet(join(Y, meet(X, complement(Y))), join(X, Y)))))))
% 190.34/24.46 = { by lemma 24 R->L }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), meet(join(Y, meet(X, complement(Y))), join(X, join(Y, meet(X, complement(Y))))))))))
% 190.34/24.46 = { by lemma 21 }
% 190.34/24.46 join(meet(X, complement(Y)), join(Y, complement(join(meet(X, complement(Y)), meet(complement(Y), join(Y, meet(X, complement(Y))))))))
% 190.34/24.46 = { by lemma 22 R->L }
% 190.34/24.46 join(Y, join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), meet(complement(Y), join(Y, meet(X, complement(Y))))))))
% 190.34/24.46 = { by lemma 25 R->L }
% 190.34/24.46 join(Y, join(meet(X, complement(Y)), join(complement(join(meet(X, complement(Y)), meet(complement(Y), join(Y, meet(X, complement(Y)))))), meet(complement(Y), join(Y, meet(X, complement(Y)))))))
% 190.34/24.46 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.46 join(Y, join(meet(X, complement(Y)), join(meet(complement(Y), join(Y, meet(X, complement(Y)))), complement(join(meet(X, complement(Y)), meet(complement(Y), join(Y, meet(X, complement(Y)))))))))
% 190.34/24.46 = { by lemma 36 }
% 190.34/24.46 join(Y, one)
% 190.34/24.46 = { by lemma 18 }
% 190.34/24.47 one
% 190.34/24.47
% 190.34/24.47 Lemma 38: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 190.34/24.47 Proof:
% 190.34/24.47 meet(complement(X), join(X, Y))
% 190.34/24.47 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.47 meet(complement(X), join(Y, X))
% 190.34/24.47 = { by lemma 33 R->L }
% 190.34/24.47 complement(complement(meet(complement(X), join(Y, X))))
% 190.34/24.47 = { by lemma 16 R->L }
% 190.34/24.47 complement(join(complement(meet(complement(X), join(Y, X))), zero))
% 190.34/24.47 = { by lemma 15 R->L }
% 190.34/24.47 complement(join(complement(meet(complement(X), join(Y, X))), complement(one)))
% 190.34/24.47 = { by lemma 37 R->L }
% 190.34/24.47 complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))))
% 190.34/24.47 = { by axiom 6 (ifeq_axiom) R->L }
% 190.34/24.47 ifeq(zero, zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 17 R->L }
% 190.34/24.47 ifeq(meet(Y, zero), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 5 (complement_meet) R->L }
% 190.34/24.47 ifeq(meet(Y, meet(meet(complement(X), join(Y, X)), complement(meet(complement(X), join(Y, X))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.47 ifeq(meet(Y, meet(complement(X), meet(join(Y, X), complement(meet(complement(X), join(Y, X)))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 26 R->L }
% 190.34/24.47 ifeq(meet(complement(X), meet(Y, meet(join(Y, X), complement(meet(complement(X), join(Y, X)))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 20 }
% 190.34/24.47 ifeq(meet(complement(X), meet(Y, complement(meet(complement(X), join(Y, X))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.47 ifeq(meet(meet(Y, complement(meet(complement(X), join(Y, X)))), complement(X)), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.47 ifeq(meet(Y, meet(complement(meet(complement(X), join(Y, X))), complement(X))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.47 ifeq(meet(Y, meet(complement(X), complement(meet(complement(X), join(Y, X))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 16 R->L }
% 190.34/24.47 ifeq(meet(Y, meet(complement(X), join(complement(meet(complement(X), join(Y, X))), zero))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 15 R->L }
% 190.34/24.47 ifeq(meet(Y, meet(complement(X), join(complement(meet(complement(X), join(Y, X))), complement(one)))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.47 ifeq(meet(meet(Y, complement(X)), join(complement(meet(complement(X), join(Y, X))), complement(one))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by lemma 37 R->L }
% 190.34/24.47 ifeq(meet(meet(Y, complement(X)), join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.47 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.47 ifeq(meet(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))), meet(Y, complement(X))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X)))
% 190.34/24.48 = { by axiom 6 (ifeq_axiom) R->L }
% 190.34/24.48 ifeq(one, one, ifeq(meet(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))), meet(Y, complement(X))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X))), meet(Y, complement(X)))
% 190.34/24.48 = { by lemma 36 R->L }
% 190.34/24.48 ifeq(join(meet(Y, complement(X)), join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), one, ifeq(meet(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))), meet(Y, complement(X))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X))), meet(Y, complement(X)))
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.48 ifeq(join(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))), meet(Y, complement(X))), one, ifeq(meet(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X)))))), meet(Y, complement(X))), zero, complement(join(complement(meet(complement(X), join(Y, X))), complement(join(meet(Y, complement(X)), complement(meet(complement(X), join(Y, X))))))), meet(Y, complement(X))), meet(Y, complement(X)))
% 190.34/24.48 = { by axiom 11 (meet_join_complement) }
% 190.34/24.48 meet(Y, complement(X))
% 190.34/24.48
% 190.34/24.48 Lemma 39: meet(X, join(complement(X), Y)) = meet(X, Y).
% 190.34/24.48 Proof:
% 190.34/24.48 meet(X, join(complement(X), Y))
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.48 meet(X, join(Y, complement(X)))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.48 meet(join(Y, complement(X)), X)
% 190.34/24.48 = { by lemma 33 R->L }
% 190.34/24.48 meet(join(Y, complement(X)), complement(complement(X)))
% 190.34/24.48 = { by lemma 34 R->L }
% 190.34/24.48 meet(X, meet(join(Y, complement(X)), complement(complement(X))))
% 190.34/24.48 = { by axiom 10 (associativity_of_meet) R->L }
% 190.34/24.48 meet(meet(X, join(Y, complement(X))), complement(complement(X)))
% 190.34/24.48 = { by lemma 38 R->L }
% 190.34/24.48 meet(complement(complement(X)), join(complement(X), meet(X, join(Y, complement(X)))))
% 190.34/24.48 = { by lemma 29 }
% 190.34/24.48 meet(complement(complement(X)), join(complement(X), meet(X, Y)))
% 190.34/24.48 = { by lemma 38 }
% 190.34/24.48 meet(meet(X, Y), complement(complement(X)))
% 190.34/24.48 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.48 meet(X, meet(Y, complement(complement(X))))
% 190.34/24.48 = { by lemma 34 }
% 190.34/24.48 meet(Y, complement(complement(X)))
% 190.34/24.48 = { by lemma 33 }
% 190.34/24.48 meet(Y, X)
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.48 meet(X, Y)
% 190.34/24.48
% 190.34/24.48 Lemma 40: meet(X, join(Y, join(complement(X), Z))) = meet(X, join(Y, Z)).
% 190.34/24.48 Proof:
% 190.34/24.48 meet(X, join(Y, join(complement(X), Z)))
% 190.34/24.48 = { by lemma 22 R->L }
% 190.34/24.48 meet(X, join(complement(X), join(Y, Z)))
% 190.34/24.48 = { by lemma 39 }
% 190.34/24.48 meet(X, join(Y, Z))
% 190.34/24.48
% 190.34/24.48 Lemma 41: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 190.34/24.48 Proof:
% 190.34/24.48 join(X, join(Y, meet(Z, join(X, Y))))
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.48 join(X, join(Y, meet(Z, join(Y, X))))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.48 join(X, join(Y, meet(join(Y, X), Z)))
% 190.34/24.48 = { by lemma 22 R->L }
% 190.34/24.48 join(Y, join(X, meet(join(Y, X), Z)))
% 190.34/24.48 = { by axiom 8 (associativity_of_join) R->L }
% 190.34/24.48 join(join(Y, X), meet(join(Y, X), Z))
% 190.34/24.48 = { by axiom 7 (absorption2) }
% 190.34/24.48 join(Y, X)
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.48 join(X, Y)
% 190.34/24.48
% 190.34/24.48 Lemma 42: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 190.34/24.48 Proof:
% 190.34/24.48 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.48 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.48 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(X, Y))
% 190.34/24.48 = { by axiom 9 (absorption1) R->L }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(meet(X, Y), join(meet(X, Y), Z)))
% 190.34/24.48 = { by axiom 10 (associativity_of_meet) }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(X, meet(Y, join(meet(X, Y), Z))))
% 190.34/24.48 = { by lemma 26 R->L }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(meet(X, Y), Z))))
% 190.34/24.48 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(X, Y)))))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.48 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 190.34/24.48 = { by lemma 23 }
% 190.34/24.48 meet(X, join(Z, meet(Y, X)))
% 190.34/24.48 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.48 meet(X, join(Z, meet(X, Y)))
% 190.34/24.49
% 190.34/24.49 Lemma 43: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 190.34/24.49 Proof:
% 190.34/24.49 meet(complement(X), complement(join(X, Y)))
% 190.34/24.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.49 meet(complement(join(X, Y)), complement(X))
% 190.34/24.49 = { by axiom 9 (absorption1) R->L }
% 190.34/24.49 meet(complement(join(X, Y)), complement(meet(X, join(X, Y))))
% 190.34/24.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.49 meet(complement(join(X, Y)), complement(meet(join(X, Y), X)))
% 190.34/24.49 = { by lemma 35 }
% 190.34/24.49 complement(join(X, Y))
% 190.34/24.49
% 190.34/24.49 Lemma 44: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 190.34/24.49 Proof:
% 190.34/24.49 meet(X, join(Y, meet(X, Z)))
% 190.34/24.49 = { by axiom 4 (commutativity_of_meet) R->L }
% 190.34/24.49 meet(X, join(Y, meet(Z, X)))
% 190.34/24.49 = { by lemma 33 R->L }
% 190.34/24.49 meet(X, join(Y, meet(Z, complement(complement(X)))))
% 190.34/24.49 = { by lemma 40 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, complement(complement(X))))))
% 190.34/24.49 = { by lemma 41 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), join(meet(Z, complement(complement(X))), meet(Z, join(complement(X), meet(Z, complement(complement(X)))))))))
% 190.34/24.49 = { by lemma 42 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, complement(complement(X))))))))
% 190.34/24.49 = { by lemma 13 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, meet(complement(complement(X)), one)))))))
% 190.34/24.49 = { by lemma 36 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, meet(complement(complement(X)), join(complement(X), join(complement(Z), complement(join(complement(X), complement(Z))))))))))))
% 190.34/24.49 = { by lemma 38 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, meet(join(complement(Z), complement(join(complement(X), complement(Z)))), complement(complement(X)))))))))
% 190.34/24.49 = { by axiom 4 (commutativity_of_meet) }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, meet(complement(complement(X)), join(complement(Z), complement(join(complement(X), complement(Z)))))))))))
% 190.34/24.49 = { by lemma 26 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(complement(complement(X)), meet(Z, join(complement(Z), complement(join(complement(X), complement(Z)))))))))))
% 190.34/24.49 = { by lemma 39 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(complement(complement(X)), meet(Z, complement(join(complement(X), complement(Z))))))))))
% 190.34/24.49 = { by lemma 26 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, meet(complement(complement(X)), complement(join(complement(X), complement(Z))))))))))
% 190.34/24.49 = { by lemma 43 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, complement(join(complement(X), complement(Z)))))))))
% 190.34/24.49 = { by axiom 2 (commutativity_of_join) R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(Z, complement(join(complement(Z), complement(X)))))))))
% 190.34/24.49 = { by lemma 33 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), meet(complement(complement(Z)), complement(join(complement(Z), complement(X)))))))))
% 190.34/24.49 = { by lemma 43 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(X), complement(join(complement(Z), complement(X))))))))
% 190.34/24.49 = { by lemma 39 R->L }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, join(complement(Z), join(complement(X), complement(join(complement(Z), complement(X)))))))))
% 190.34/24.49 = { by lemma 36 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), meet(Z, one))))
% 190.34/24.49 = { by lemma 13 }
% 190.34/24.49 meet(X, join(Y, join(complement(X), Z)))
% 190.34/24.49 = { by lemma 40 }
% 190.34/24.49 meet(X, join(Y, Z))
% 190.34/24.49
% 190.34/24.49 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 190.34/24.49 Proof:
% 190.34/24.49 meet(a, join(b, c))
% 190.34/24.49 = { by lemma 44 R->L }
% 190.34/24.49 meet(a, join(b, meet(a, c)))
% 190.34/24.49 = { by lemma 42 R->L }
% 190.34/24.49 join(meet(a, c), meet(a, join(b, meet(a, c))))
% 190.34/24.49 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.49 join(meet(a, c), meet(a, join(meet(a, c), b)))
% 190.34/24.49 = { by lemma 44 R->L }
% 190.34/24.49 join(meet(a, c), meet(a, join(meet(a, c), meet(a, b))))
% 190.34/24.49 = { by lemma 42 R->L }
% 190.34/24.49 join(meet(a, c), join(meet(a, b), meet(a, join(meet(a, c), meet(a, b)))))
% 190.34/24.49 = { by lemma 41 }
% 190.34/24.49 join(meet(a, c), meet(a, b))
% 190.34/24.49 = { by axiom 2 (commutativity_of_join) }
% 190.34/24.49 join(meet(a, b), meet(a, c))
% 190.34/24.49 % SZS output end Proof
% 190.34/24.49
% 190.34/24.49 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------