%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT214-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n007.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 185.97s 24.00s
% Output : Proof 188.28s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LAT214-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.37 % Computer : n007.cluster.edu
% 0.12/0.37 % Model : x86_64 x86_64
% 0.12/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.37 % Memory : 8046.5625MB
% 0.12/0.37 % OS : Linux 6.8.0-71-generic
% 0.12/0.37 % CPULimit : 300
% 0.12/0.37 % WCLimit : 300
% 0.12/0.37 % DateTime : Sun Sep 27 14:04:54 UTC 2026
% 0.12/0.38 % CPUTime :
% 0.12/0.38 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 185.97/24.00 Command-line arguments: --no-flatten-goal
% 185.97/24.00
% 185.97/24.00 % SZS status Unsatisfiable
% 185.97/24.00
% 187.51/24.16 % SZS output start Proof
% 187.51/24.16 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 187.51/24.16 Axiom 2 (complement_join): join(X, complement(X)) = one.
% 187.51/24.16 Axiom 3 (idempotence_of_meet): meet(X, X) = X.
% 187.51/24.16 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 187.51/24.16 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 187.51/24.16 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 187.51/24.16 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 187.51/24.16 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 187.51/24.16 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 187.51/24.16 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 187.51/24.16 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 187.51/24.17 Axiom 12 (equation_H79): meet(X, join(Y, meet(Z, join(X, W)))) = meet(X, join(meet(X, join(Y, meet(X, Z))), meet(Z, W))).
% 187.51/24.17
% 187.51/24.17 Lemma 13: meet(X, one) = X.
% 187.51/24.17 Proof:
% 187.51/24.17 meet(X, one)
% 187.51/24.17 = { by axiom 2 (complement_join) R->L }
% 187.51/24.17 meet(X, join(X, complement(X)))
% 187.51/24.17 = { by axiom 9 (absorption1) }
% 187.51/24.17 X
% 187.51/24.17
% 187.51/24.17 Lemma 14: meet(one, X) = X.
% 187.51/24.17 Proof:
% 187.51/24.17 meet(one, X)
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.17 meet(X, one)
% 187.51/24.17 = { by lemma 13 }
% 187.51/24.17 X
% 187.51/24.17
% 187.51/24.17 Lemma 15: complement(one) = zero.
% 187.51/24.17 Proof:
% 187.51/24.17 complement(one)
% 187.51/24.17 = { by lemma 14 R->L }
% 187.51/24.17 meet(one, complement(one))
% 187.51/24.17 = { by axiom 5 (complement_meet) }
% 187.51/24.17 zero
% 187.51/24.17
% 187.51/24.17 Lemma 16: join(X, zero) = X.
% 187.51/24.17 Proof:
% 187.51/24.17 join(X, zero)
% 187.51/24.17 = { by axiom 5 (complement_meet) R->L }
% 187.51/24.17 join(X, meet(X, complement(X)))
% 187.51/24.17 = { by axiom 7 (absorption2) }
% 187.51/24.17 X
% 187.51/24.17
% 187.51/24.17 Lemma 17: meet(X, zero) = zero.
% 187.51/24.17 Proof:
% 187.51/24.17 meet(X, zero)
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.17 meet(zero, X)
% 187.51/24.17 = { by lemma 16 R->L }
% 187.51/24.17 join(meet(zero, X), zero)
% 187.51/24.17 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.17 join(zero, meet(zero, X))
% 187.51/24.17 = { by axiom 7 (absorption2) }
% 187.51/24.17 zero
% 187.51/24.17
% 187.51/24.17 Lemma 18: join(X, one) = one.
% 187.51/24.17 Proof:
% 187.51/24.17 join(X, one)
% 187.51/24.17 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.17 join(one, X)
% 187.51/24.17 = { by lemma 14 R->L }
% 187.51/24.17 join(one, meet(one, X))
% 187.51/24.17 = { by axiom 7 (absorption2) }
% 187.51/24.17 one
% 187.51/24.17
% 187.51/24.17 Lemma 19: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 187.51/24.17 Proof:
% 187.51/24.17 meet(X, meet(Y, Z))
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.17 meet(meet(Y, Z), X)
% 187.51/24.17 = { by axiom 10 (associativity_of_meet) }
% 187.51/24.17 meet(Y, meet(Z, X))
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) }
% 187.51/24.17 meet(Y, meet(X, Z))
% 187.51/24.17
% 187.51/24.17 Lemma 20: meet(X, join(Y, X)) = X.
% 187.51/24.17 Proof:
% 187.51/24.17 meet(X, join(Y, X))
% 187.51/24.17 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.17 meet(X, join(X, Y))
% 187.51/24.17 = { by axiom 9 (absorption1) }
% 187.51/24.17 X
% 187.51/24.17
% 187.51/24.17 Lemma 21: join(X, meet(Y, X)) = X.
% 187.51/24.17 Proof:
% 187.51/24.17 join(X, meet(Y, X))
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.17 join(X, meet(X, Y))
% 187.51/24.17 = { by axiom 7 (absorption2) }
% 187.51/24.17 X
% 187.51/24.17
% 187.51/24.17 Lemma 22: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 187.51/24.17 Proof:
% 187.51/24.17 join(X, join(Y, meet(X, Z)))
% 187.51/24.17 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.17 join(X, join(meet(X, Z), Y))
% 187.51/24.17 = { by axiom 8 (associativity_of_join) R->L }
% 187.51/24.17 join(join(X, meet(X, Z)), Y)
% 187.51/24.17 = { by axiom 7 (absorption2) }
% 187.51/24.17 join(X, Y)
% 187.51/24.17
% 187.51/24.17 Lemma 23: meet(X, join(complement(X), meet(X, Y))) = meet(X, join(Y, complement(X))).
% 187.51/24.17 Proof:
% 187.51/24.17 meet(X, join(complement(X), meet(X, Y)))
% 187.51/24.17 = { by axiom 7 (absorption2) R->L }
% 187.51/24.17 meet(X, join(join(complement(X), meet(X, Y)), meet(join(complement(X), meet(X, Y)), join(X, Y))))
% 187.51/24.17 = { by axiom 12 (equation_H79) }
% 187.51/24.17 meet(X, join(meet(X, join(join(complement(X), meet(X, Y)), meet(X, join(complement(X), meet(X, Y))))), meet(join(complement(X), meet(X, Y)), Y)))
% 187.51/24.17 = { by lemma 21 }
% 187.51/24.17 meet(X, join(meet(X, join(complement(X), meet(X, Y))), meet(join(complement(X), meet(X, Y)), Y)))
% 187.51/24.17 = { by axiom 4 (commutativity_of_meet) }
% 187.51/24.17 meet(X, join(meet(X, join(complement(X), meet(X, Y))), meet(Y, join(complement(X), meet(X, Y)))))
% 187.51/24.17 = { by axiom 12 (equation_H79) R->L }
% 187.51/24.17 meet(X, join(complement(X), meet(Y, join(X, join(complement(X), meet(X, Y))))))
% 187.51/24.17 = { by lemma 22 }
% 187.51/24.17 meet(X, join(complement(X), meet(Y, join(X, complement(X)))))
% 187.51/24.17 = { by axiom 2 (complement_join) }
% 187.51/24.17 meet(X, join(complement(X), meet(Y, one)))
% 187.51/24.18 = { by lemma 13 }
% 187.51/24.18 meet(X, join(complement(X), Y))
% 187.51/24.18 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.18 meet(X, join(Y, complement(X)))
% 187.51/24.18
% 187.51/24.18 Lemma 24: meet(X, join(complement(X), meet(Y, X))) = meet(X, join(Y, complement(X))).
% 187.51/24.18 Proof:
% 187.51/24.18 meet(X, join(complement(X), meet(Y, X)))
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(X, join(complement(X), meet(X, Y)))
% 187.51/24.18 = { by lemma 23 }
% 187.51/24.18 meet(X, join(Y, complement(X)))
% 187.51/24.18
% 187.51/24.18 Lemma 25: join(X, join(Y, complement(X))) = one.
% 187.51/24.18 Proof:
% 187.51/24.18 join(X, join(Y, complement(X)))
% 187.51/24.18 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.18 join(X, join(complement(X), Y))
% 187.51/24.18 = { by axiom 8 (associativity_of_join) R->L }
% 187.51/24.18 join(join(X, complement(X)), Y)
% 187.51/24.18 = { by axiom 2 (complement_join) }
% 187.51/24.18 join(one, Y)
% 187.51/24.18 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.18 join(Y, one)
% 187.51/24.18 = { by lemma 18 }
% 187.51/24.18 one
% 187.51/24.18
% 187.51/24.18 Lemma 26: join(X, complement(complement(X))) = complement(complement(X)).
% 187.51/24.18 Proof:
% 187.51/24.18 join(X, complement(complement(X)))
% 187.51/24.18 = { by axiom 11 (meet_join_complement) R->L }
% 187.51/24.18 ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(X, complement(complement(X)))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by lemma 24 R->L }
% 187.51/24.18 ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(complement(complement(X)), meet(X, complement(X)))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by axiom 5 (complement_meet) }
% 187.51/24.18 ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), join(complement(complement(X)), zero)), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by lemma 16 }
% 187.51/24.18 ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(meet(complement(X), complement(complement(X))), zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by axiom 5 (complement_meet) }
% 187.51/24.18 ifeq(join(complement(X), join(X, complement(complement(X)))), one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by lemma 25 }
% 187.51/24.18 ifeq(one, one, ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X)))), join(X, complement(complement(X))))
% 187.51/24.18 = { by axiom 6 (ifeq_axiom) }
% 187.51/24.18 ifeq(zero, zero, complement(complement(X)), join(X, complement(complement(X))))
% 187.51/24.18 = { by axiom 6 (ifeq_axiom) }
% 187.51/24.18 complement(complement(X))
% 187.51/24.18
% 187.51/24.18 Lemma 27: meet(X, meet(Y, complement(X))) = zero.
% 187.51/24.18 Proof:
% 187.51/24.18 meet(X, meet(Y, complement(X)))
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(X, meet(complement(X), Y))
% 187.51/24.18 = { by axiom 10 (associativity_of_meet) R->L }
% 187.51/24.18 meet(meet(X, complement(X)), Y)
% 187.51/24.18 = { by axiom 5 (complement_meet) }
% 187.51/24.18 meet(zero, Y)
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(Y, zero)
% 187.51/24.18 = { by lemma 17 }
% 187.51/24.18 zero
% 187.51/24.18
% 187.51/24.18 Lemma 28: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 187.51/24.18 Proof:
% 187.51/24.18 meet(X, meet(Y, join(Z, meet(X, Y))))
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(X, meet(Y, join(Z, meet(Y, X))))
% 187.51/24.18 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.18 meet(X, meet(Y, join(meet(Y, X), Z)))
% 187.51/24.18 = { by lemma 19 R->L }
% 187.51/24.18 meet(Y, meet(X, join(meet(Y, X), Z)))
% 187.51/24.18 = { by axiom 10 (associativity_of_meet) R->L }
% 187.51/24.18 meet(meet(Y, X), join(meet(Y, X), Z))
% 187.51/24.18 = { by axiom 9 (absorption1) }
% 187.51/24.18 meet(Y, X)
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) }
% 187.51/24.18 meet(X, Y)
% 187.51/24.18
% 187.51/24.18 Lemma 29: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 187.51/24.18 Proof:
% 187.51/24.18 meet(X, meet(Y, join(X, Z)))
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(X, meet(join(X, Z), Y))
% 187.51/24.18 = { by axiom 10 (associativity_of_meet) R->L }
% 187.51/24.18 meet(meet(X, join(X, Z)), Y)
% 187.51/24.18 = { by axiom 9 (absorption1) }
% 187.51/24.18 meet(X, Y)
% 187.51/24.18
% 187.51/24.18 Lemma 30: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 187.51/24.18 Proof:
% 187.51/24.18 meet(X, meet(join(X, Y), Z))
% 187.51/24.18 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.18 meet(X, meet(Z, join(X, Y)))
% 187.51/24.18 = { by lemma 29 }
% 187.51/24.18 meet(X, Z)
% 187.51/24.18
% 187.51/24.18 Lemma 31: join(X, join(Y, complement(join(X, Y)))) = one.
% 187.51/24.18 Proof:
% 187.51/24.18 join(X, join(Y, complement(join(X, Y))))
% 187.51/24.18 = { by axiom 8 (associativity_of_join) R->L }
% 187.51/24.18 join(join(X, Y), complement(join(X, Y)))
% 187.51/24.18 = { by axiom 2 (complement_join) }
% 187.51/24.18 one
% 187.51/24.18
% 187.51/24.19 Lemma 32: join(meet(X, Y), complement(meet(X, join(Y, complement(X))))) = one.
% 187.51/24.19 Proof:
% 187.51/24.19 join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))
% 187.51/24.19 = { by axiom 4 (commutativity_of_meet) R->L }
% 187.51/24.19 join(meet(Y, X), complement(meet(X, join(Y, complement(X)))))
% 187.51/24.19 = { by axiom 6 (ifeq_axiom) R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(zero, zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by axiom 6 (ifeq_axiom) R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(one, one, ifeq(zero, zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by lemma 25 R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(join(meet(X, join(Y, complement(X))), X)), complement(meet(X, join(Y, complement(X)))))), one, ifeq(zero, zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.19 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(zero, zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by axiom 5 (complement_meet) R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), complement(meet(X, join(Y, complement(X))))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by lemma 16 R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), zero)), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.19 = { by lemma 17 R->L }
% 187.51/24.19 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), meet(meet(X, join(Y, complement(X))), zero))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.20 = { by axiom 5 (complement_meet) R->L }
% 187.51/24.20 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), meet(meet(X, join(Y, complement(X))), meet(join(meet(X, join(Y, complement(X))), X), complement(join(meet(X, join(Y, complement(X))), X)))))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.20 = { by lemma 30 }
% 187.51/24.20 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), meet(meet(X, join(Y, complement(X))), complement(join(meet(X, join(Y, complement(X))), X))))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.20 = { by lemma 23 }
% 187.51/24.20 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(join(meet(X, join(Y, complement(X))), X)), complement(meet(X, join(Y, complement(X)))))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(meet(Y, X), ifeq(join(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), one, ifeq(meet(meet(X, join(Y, complement(X))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), zero, complement(meet(X, join(Y, complement(X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X)))))
% 187.51/24.20 = { by axiom 11 (meet_join_complement) }
% 187.51/24.20 join(meet(Y, X), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, join(Y, complement(X))), X))))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(meet(Y, X), join(complement(meet(X, join(Y, complement(X)))), complement(join(X, meet(X, join(Y, complement(X)))))))
% 187.51/24.20 = { by axiom 7 (absorption2) }
% 187.51/24.20 join(meet(Y, X), join(complement(meet(X, join(Y, complement(X)))), complement(X)))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(meet(Y, X), join(complement(X), complement(meet(X, join(Y, complement(X))))))
% 187.51/24.20 = { by lemma 21 R->L }
% 187.51/24.20 join(meet(Y, X), join(complement(X), complement(join(meet(X, join(Y, complement(X))), meet(Y, meet(X, join(Y, complement(X))))))))
% 187.51/24.20 = { by lemma 29 }
% 187.51/24.20 join(meet(Y, X), join(complement(X), complement(join(meet(X, join(Y, complement(X))), meet(Y, X)))))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(meet(Y, X), join(complement(X), complement(join(meet(Y, X), meet(X, join(Y, complement(X)))))))
% 187.51/24.20 = { by lemma 24 R->L }
% 187.51/24.20 join(meet(Y, X), join(complement(X), complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X)))))))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) R->L }
% 187.51/24.20 join(join(complement(X), complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X)))))), meet(Y, X))
% 187.51/24.20 = { by axiom 8 (associativity_of_join) }
% 187.51/24.20 join(complement(X), join(complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))), meet(Y, X)))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(complement(X), join(meet(Y, X), complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X)))))))
% 187.51/24.20 = { by axiom 8 (associativity_of_join) R->L }
% 187.51/24.20 join(join(complement(X), meet(Y, X)), complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))))
% 187.51/24.20 = { by lemma 22 R->L }
% 187.51/24.20 join(join(complement(X), meet(Y, X)), join(complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))), meet(join(complement(X), meet(Y, X)), X)))
% 187.51/24.20 = { by axiom 8 (associativity_of_join) }
% 187.51/24.20 join(complement(X), join(meet(Y, X), join(complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))), meet(join(complement(X), meet(Y, X)), X))))
% 187.51/24.20 = { by axiom 4 (commutativity_of_meet) }
% 187.51/24.20 join(complement(X), join(meet(Y, X), join(complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))), meet(X, join(complement(X), meet(Y, X))))))
% 187.51/24.20 = { by axiom 1 (commutativity_of_join) }
% 187.51/24.20 join(complement(X), join(meet(Y, X), join(meet(X, join(complement(X), meet(Y, X))), complement(join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))))))
% 187.51/24.20 = { by lemma 31 }
% 187.51/24.20 join(complement(X), one)
% 187.51/24.20 = { by lemma 18 }
% 188.28/24.21 one
% 188.28/24.21
% 188.28/24.21 Lemma 33: meet(X, join(Y, complement(X))) = meet(X, Y).
% 188.28/24.21 Proof:
% 188.28/24.21 meet(X, join(Y, complement(X)))
% 188.28/24.21 = { by axiom 9 (absorption1) R->L }
% 188.28/24.21 meet(meet(X, join(Y, complement(X))), join(meet(X, join(Y, complement(X))), complement(complement(meet(X, join(Y, complement(X)))))))
% 188.28/24.21 = { by lemma 26 }
% 188.28/24.21 meet(meet(X, join(Y, complement(X))), complement(complement(meet(X, join(Y, complement(X))))))
% 188.28/24.21 = { by axiom 3 (idempotence_of_meet) R->L }
% 188.28/24.21 meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))
% 188.28/24.21 = { by axiom 11 (meet_join_complement) R->L }
% 188.28/24.21 ifeq(join(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.21 = { by lemma 26 R->L }
% 188.28/24.21 ifeq(join(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), join(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.21 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.21 ifeq(join(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.21 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.21 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.22 = { by lemma 28 R->L }
% 188.28/24.22 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), complement(meet(meet(X, join(Y, complement(X))), meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 188.28/24.22 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), complement(meet(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(X, join(Y, complement(X)))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.22 = { by lemma 22 R->L }
% 188.28/24.22 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), join(complement(meet(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(X, join(Y, complement(X))))), meet(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(X, join(Y, complement(X)))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.22 = { by axiom 1 (commutativity_of_join) }
% 188.28/24.22 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), join(meet(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(X, join(Y, complement(X)))), complement(meet(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(X, join(Y, complement(X))))))), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by axiom 2 (complement_join) }
% 188.28/24.23 ifeq(join(meet(meet(X, join(Y, complement(X))), join(complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), one), one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by lemma 18 }
% 188.28/24.23 ifeq(one, one, ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by axiom 6 (ifeq_axiom) }
% 188.28/24.23 ifeq(meet(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))))), zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by lemma 27 }
% 188.28/24.23 ifeq(zero, zero, complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X)))))), meet(meet(X, join(Y, complement(X))), complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by axiom 6 (ifeq_axiom) }
% 188.28/24.23 complement(complement(meet(meet(X, join(Y, complement(X))), meet(X, join(Y, complement(X))))))
% 188.28/24.23 = { by axiom 3 (idempotence_of_meet) }
% 188.28/24.23 complement(complement(meet(X, join(Y, complement(X)))))
% 188.28/24.23 = { by lemma 16 R->L }
% 188.28/24.23 complement(join(complement(meet(X, join(Y, complement(X)))), zero))
% 188.28/24.23 = { by lemma 15 R->L }
% 188.28/24.23 complement(join(complement(meet(X, join(Y, complement(X)))), complement(one)))
% 188.28/24.23 = { by lemma 32 R->L }
% 188.28/24.23 complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))))
% 188.28/24.23 = { by axiom 6 (ifeq_axiom) R->L }
% 188.28/24.23 ifeq(zero, zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.23 = { by lemma 17 R->L }
% 188.28/24.23 ifeq(meet(Y, zero), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.23 = { by axiom 5 (complement_meet) R->L }
% 188.28/24.23 ifeq(meet(Y, meet(meet(X, join(Y, complement(X))), complement(meet(X, join(Y, complement(X)))))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.23 = { by axiom 10 (associativity_of_meet) }
% 188.28/24.23 ifeq(meet(Y, meet(X, meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.23 = { by lemma 19 }
% 188.28/24.23 ifeq(meet(X, meet(Y, meet(join(Y, complement(X)), complement(meet(X, join(Y, complement(X))))))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by lemma 30 }
% 188.28/24.24 ifeq(meet(X, meet(Y, complement(meet(X, join(Y, complement(X)))))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by lemma 16 R->L }
% 188.28/24.24 ifeq(meet(X, meet(Y, join(complement(meet(X, join(Y, complement(X)))), zero))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by lemma 15 R->L }
% 188.28/24.24 ifeq(meet(X, meet(Y, join(complement(meet(X, join(Y, complement(X)))), complement(one)))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by axiom 10 (associativity_of_meet) R->L }
% 188.28/24.24 ifeq(meet(meet(X, Y), join(complement(meet(X, join(Y, complement(X)))), complement(one))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by lemma 32 R->L }
% 188.28/24.24 ifeq(meet(meet(X, Y), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by axiom 4 (commutativity_of_meet) R->L }
% 188.28/24.24 ifeq(meet(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))), meet(X, Y)), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y))
% 188.28/24.24 = { by axiom 6 (ifeq_axiom) R->L }
% 188.28/24.24 ifeq(one, one, ifeq(meet(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))), meet(X, Y)), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y)), meet(X, Y))
% 188.28/24.24 = { by lemma 31 R->L }
% 188.28/24.24 ifeq(join(meet(X, Y), join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), one, ifeq(meet(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))), meet(X, Y)), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y)), meet(X, Y))
% 188.28/24.24 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.24 ifeq(join(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))), meet(X, Y)), one, ifeq(meet(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X))))))), meet(X, Y)), zero, complement(join(complement(meet(X, join(Y, complement(X)))), complement(join(meet(X, Y), complement(meet(X, join(Y, complement(X)))))))), meet(X, Y)), meet(X, Y))
% 188.28/24.24 = { by axiom 11 (meet_join_complement) }
% 188.28/24.24 meet(X, Y)
% 188.28/24.24
% 188.28/24.24 Lemma 34: meet(X, join(meet(Y, complement(X)), meet(X, Z))) = meet(X, join(Z, meet(Y, complement(X)))).
% 188.28/24.24 Proof:
% 188.28/24.24 meet(X, join(meet(Y, complement(X)), meet(X, Z)))
% 188.28/24.24 = { by lemma 16 R->L }
% 188.28/24.24 meet(X, join(meet(Y, complement(X)), meet(X, join(Z, zero))))
% 188.28/24.24 = { by lemma 27 R->L }
% 188.28/24.24 meet(X, join(meet(Y, complement(X)), meet(X, join(Z, meet(X, meet(Y, complement(X)))))))
% 188.28/24.24 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.24 meet(X, join(meet(X, join(Z, meet(X, meet(Y, complement(X))))), meet(Y, complement(X))))
% 188.28/24.24 = { by axiom 3 (idempotence_of_meet) R->L }
% 188.28/24.24 meet(X, join(meet(X, join(Z, meet(X, meet(Y, complement(X))))), meet(meet(Y, complement(X)), meet(Y, complement(X)))))
% 188.28/24.24 = { by axiom 12 (equation_H79) R->L }
% 188.28/24.24 meet(X, join(Z, meet(meet(Y, complement(X)), join(X, meet(Y, complement(X))))))
% 188.28/24.24 = { by lemma 20 }
% 188.28/24.24 meet(X, join(Z, meet(Y, complement(X))))
% 188.28/24.24
% 188.28/24.24 Lemma 35: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
% 188.28/24.24 Proof:
% 188.28/24.24 meet(X, join(Y, meet(X, Z)))
% 188.28/24.24 = { by axiom 3 (idempotence_of_meet) R->L }
% 188.28/24.24 meet(meet(X, X), join(Y, meet(X, Z)))
% 188.28/24.24 = { by axiom 10 (associativity_of_meet) }
% 188.28/24.24 meet(X, meet(X, join(Y, meet(X, Z))))
% 188.28/24.24 = { by lemma 33 R->L }
% 188.28/24.24 meet(X, join(meet(X, join(Y, meet(X, Z))), complement(X)))
% 188.28/24.24 = { by lemma 21 R->L }
% 188.28/24.24 meet(X, join(meet(X, join(Y, meet(X, Z))), join(complement(X), meet(Z, complement(X)))))
% 188.28/24.24 = { by axiom 8 (associativity_of_join) R->L }
% 188.28/24.25 meet(X, join(join(meet(X, join(Y, meet(X, Z))), complement(X)), meet(Z, complement(X))))
% 188.28/24.25 = { by lemma 34 R->L }
% 188.28/24.25 meet(X, join(meet(Z, complement(X)), meet(X, join(meet(X, join(Y, meet(X, Z))), complement(X)))))
% 188.28/24.25 = { by lemma 33 }
% 188.28/24.25 meet(X, join(meet(Z, complement(X)), meet(X, meet(X, join(Y, meet(X, Z))))))
% 188.28/24.25 = { by lemma 34 }
% 188.28/24.25 meet(X, join(meet(X, join(Y, meet(X, Z))), meet(Z, complement(X))))
% 188.28/24.25 = { by axiom 12 (equation_H79) R->L }
% 188.28/24.25 meet(X, join(Y, meet(Z, join(X, complement(X)))))
% 188.28/24.25 = { by axiom 2 (complement_join) }
% 188.28/24.25 meet(X, join(Y, meet(Z, one)))
% 188.28/24.25 = { by lemma 13 }
% 188.28/24.25 meet(X, join(Y, Z))
% 188.28/24.25
% 188.28/24.25 Lemma 36: join(meet(X, Y), meet(X, join(meet(X, Y), Z))) = meet(X, join(Z, meet(X, Y))).
% 188.28/24.25 Proof:
% 188.28/24.25 join(meet(X, Y), meet(X, join(meet(X, Y), Z)))
% 188.28/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.25 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 188.28/24.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 188.28/24.25 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 188.28/24.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 188.28/24.25 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 188.28/24.25 = { by axiom 1 (commutativity_of_join) R->L }
% 188.28/24.25 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 188.28/24.25 = { by lemma 28 R->L }
% 188.28/24.25 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 188.28/24.25 = { by lemma 21 }
% 188.28/24.25 meet(X, join(Z, meet(Y, X)))
% 188.28/24.25 = { by axiom 4 (commutativity_of_meet) }
% 188.28/24.25 meet(X, join(Z, meet(X, Y)))
% 188.28/24.25
% 188.28/24.25 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 188.28/24.25 Proof:
% 188.28/24.25 meet(a, join(b, c))
% 188.28/24.25 = { by lemma 35 R->L }
% 188.28/24.25 meet(a, join(b, meet(a, c)))
% 188.28/24.25 = { by lemma 36 R->L }
% 188.28/24.25 join(meet(a, c), meet(a, join(meet(a, c), b)))
% 188.28/24.25 = { by lemma 35 R->L }
% 188.28/24.25 join(meet(a, c), meet(a, join(meet(a, c), meet(a, b))))
% 188.28/24.25 = { by lemma 36 }
% 188.28/24.25 meet(a, join(meet(a, b), meet(a, c)))
% 188.28/24.25 = { by axiom 7 (absorption2) R->L }
% 188.28/24.25 meet(join(a, meet(a, b)), join(meet(a, b), meet(a, c)))
% 188.28/24.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 188.28/24.25 meet(join(meet(a, b), meet(a, c)), join(a, meet(a, b)))
% 188.28/24.25 = { by lemma 22 R->L }
% 188.28/24.25 meet(join(meet(a, b), meet(a, c)), join(a, join(meet(a, b), meet(a, c))))
% 188.28/24.25 = { by lemma 20 }
% 188.28/24.25 join(meet(a, b), meet(a, c))
% 188.28/24.25 % SZS output end Proof
% 188.28/24.25
% 188.28/24.25 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------