%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT187-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n002.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:44:30 AM UTC 2026
% Result : Unsatisfiable 130.51s 17.06s
% Output : Proof 132.12s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : LAT187-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.17/0.43 % Computer : n002.cluster.edu
% 0.17/0.43 % Model : x86_64 x86_64
% 0.17/0.43 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.43 % Memory : 8046.5625MB
% 0.17/0.43 % OS : Linux 6.8.0-71-generic
% 0.17/0.43 % CPULimit : 300
% 0.17/0.44 % WCLimit : 300
% 0.17/0.44 % DateTime : Sun Sep 27 14:08:06 UTC 2026
% 0.17/0.44 % CPUTime :
% 0.17/0.44 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 130.51/17.06 Command-line arguments: --no-flatten-goal
% 130.51/17.06
% 130.51/17.06 % SZS status Unsatisfiable
% 130.51/17.06
% 132.12/17.21 % SZS output start Proof
% 132.12/17.21 Axiom 1 (idempotence_of_join): join(X, X) = X.
% 132.12/17.21 Axiom 2 (commutativity_of_join): join(X, Y) = join(Y, X).
% 132.12/17.21 Axiom 3 (complement_join): join(X, complement(X)) = one.
% 132.12/17.21 Axiom 4 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 132.12/17.21 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 132.12/17.21 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 132.12/17.21 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 132.12/17.21 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 132.12/17.21 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 132.12/17.21 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 132.12/17.21 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 132.12/17.21 Axiom 12 (equation_H15): meet(X, join(meet(X, Y), meet(X, Z))) = meet(X, join(meet(X, Y), join(meet(X, Z), meet(Z, join(X, Y))))).
% 132.12/17.21
% 132.12/17.21 Lemma 13: meet(X, one) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, one)
% 132.12/17.21 = { by axiom 3 (complement_join) R->L }
% 132.12/17.21 meet(X, join(X, complement(X)))
% 132.12/17.21 = { by axiom 9 (absorption1) }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 14: join(X, zero) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, zero)
% 132.12/17.21 = { by axiom 5 (complement_meet) R->L }
% 132.12/17.21 join(X, meet(X, complement(X)))
% 132.12/17.21 = { by axiom 7 (absorption2) }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 15: meet(X, zero) = zero.
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, zero)
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(zero, X)
% 132.12/17.21 = { by lemma 14 R->L }
% 132.12/17.21 join(meet(zero, X), zero)
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.21 join(zero, meet(zero, X))
% 132.12/17.21 = { by axiom 7 (absorption2) }
% 132.12/17.21 zero
% 132.12/17.21
% 132.12/17.21 Lemma 16: meet(one, X) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 meet(one, X)
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(X, one)
% 132.12/17.21 = { by lemma 13 }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 17: join(X, one) = one.
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, one)
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 join(one, X)
% 132.12/17.21 = { by lemma 16 R->L }
% 132.12/17.21 join(one, meet(one, X))
% 132.12/17.21 = { by axiom 7 (absorption2) }
% 132.12/17.21 one
% 132.12/17.21
% 132.12/17.21 Lemma 18: ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 132.12/17.21 = { by axiom 11 (meet_join_complement) }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 19: complement(complement(X)) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 complement(complement(X))
% 132.12/17.21 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.21 ifeq(zero, zero, complement(complement(X)), X)
% 132.12/17.21 = { by axiom 5 (complement_meet) R->L }
% 132.12/17.21 ifeq(meet(X, complement(X)), zero, complement(complement(X)), X)
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 ifeq(meet(complement(X), X), zero, complement(complement(X)), X)
% 132.12/17.21 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.21 ifeq(one, one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 132.12/17.21 = { by axiom 3 (complement_join) R->L }
% 132.12/17.21 ifeq(join(X, complement(X)), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 132.12/17.21 = { by lemma 18 }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 20: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, meet(Y, Z))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(meet(Y, Z), X)
% 132.12/17.21 = { by axiom 10 (associativity_of_meet) }
% 132.12/17.21 meet(Y, meet(Z, X))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.21 meet(Y, meet(X, Z))
% 132.12/17.21
% 132.12/17.21 Lemma 21: meet(X, join(Y, X)) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, join(Y, X))
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 meet(X, join(X, Y))
% 132.12/17.21 = { by axiom 9 (absorption1) }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 22: join(X, meet(Y, X)) = X.
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, meet(Y, X))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 join(X, meet(X, Y))
% 132.12/17.21 = { by axiom 7 (absorption2) }
% 132.12/17.21 X
% 132.12/17.21
% 132.12/17.21 Lemma 23: join(X, join(X, Y)) = join(X, Y).
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, join(X, Y))
% 132.12/17.21 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.21 join(join(X, X), Y)
% 132.12/17.21 = { by axiom 1 (idempotence_of_join) }
% 132.12/17.21 join(X, Y)
% 132.12/17.21
% 132.12/17.21 Lemma 24: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 132.12/17.21 Proof:
% 132.12/17.21 join(Y, join(X, Z))
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 join(join(X, Z), Y)
% 132.12/17.21 = { by axiom 8 (associativity_of_join) }
% 132.12/17.21 join(X, join(Z, Y))
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.21 join(X, join(Y, Z))
% 132.12/17.21
% 132.12/17.21 Lemma 25: meet(X, meet(Y, complement(X))) = zero.
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, meet(Y, complement(X)))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(X, meet(complement(X), Y))
% 132.12/17.21 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.21 meet(meet(X, complement(X)), Y)
% 132.12/17.21 = { by axiom 5 (complement_meet) }
% 132.12/17.21 meet(zero, Y)
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(Y, zero)
% 132.12/17.21 = { by lemma 15 }
% 132.12/17.21 zero
% 132.12/17.21
% 132.12/17.21 Lemma 26: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, join(Y, meet(X, Z)))
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 join(X, join(meet(X, Z), Y))
% 132.12/17.21 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.21 join(join(X, meet(X, Z)), Y)
% 132.12/17.21 = { by axiom 7 (absorption2) }
% 132.12/17.21 join(X, Y)
% 132.12/17.21
% 132.12/17.21 Lemma 27: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 132.12/17.21 Proof:
% 132.12/17.21 join(X, join(meet(X, Y), Z))
% 132.12/17.21 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.21 join(X, join(Z, meet(X, Y)))
% 132.12/17.21 = { by lemma 26 }
% 132.12/17.21 join(X, Z)
% 132.12/17.21
% 132.12/17.21 Lemma 28: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, meet(Y, join(X, Z)))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(X, meet(join(X, Z), Y))
% 132.12/17.21 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.21 meet(meet(X, join(X, Z)), Y)
% 132.12/17.21 = { by axiom 9 (absorption1) }
% 132.12/17.21 meet(X, Y)
% 132.12/17.21
% 132.12/17.21 Lemma 29: meet(join(X, Y), join(Y, meet(X, Z))) = join(Y, meet(X, Z)).
% 132.12/17.21 Proof:
% 132.12/17.21 meet(join(X, Y), join(Y, meet(X, Z)))
% 132.12/17.21 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.21 meet(join(Y, meet(X, Z)), join(X, Y))
% 132.12/17.21 = { by lemma 26 R->L }
% 132.12/17.21 meet(join(Y, meet(X, Z)), join(X, join(Y, meet(X, Z))))
% 132.12/17.21 = { by lemma 21 }
% 132.12/17.21 join(Y, meet(X, Z))
% 132.12/17.21
% 132.12/17.21 Lemma 30: meet(X, join(meet(X, Y), meet(X, Z))) = join(meet(X, Y), meet(X, Z)).
% 132.12/17.21 Proof:
% 132.12/17.21 meet(X, join(meet(X, Y), meet(X, Z)))
% 132.12/17.21 = { by axiom 7 (absorption2) R->L }
% 132.12/17.21 meet(join(X, meet(X, Y)), join(meet(X, Y), meet(X, Z)))
% 132.12/17.21 = { by lemma 29 }
% 132.12/17.21 join(meet(X, Y), meet(X, Z))
% 132.12/17.21
% 132.12/17.21 Lemma 31: meet(X, join(meet(X, Y), meet(Z, join(X, Y)))) = join(meet(X, Y), meet(X, Z)).
% 132.12/17.21 Proof:
% 132.12/17.22 meet(X, join(meet(X, Y), meet(Z, join(X, Y))))
% 132.12/17.22 = { by lemma 22 R->L }
% 132.12/17.22 meet(X, join(meet(X, Y), join(meet(Z, join(X, Y)), meet(X, meet(Z, join(X, Y))))))
% 132.12/17.22 = { by lemma 28 }
% 132.12/17.22 meet(X, join(meet(X, Y), join(meet(Z, join(X, Y)), meet(X, Z))))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.22 meet(X, join(meet(X, Y), join(meet(X, Z), meet(Z, join(X, Y)))))
% 132.12/17.22 = { by axiom 12 (equation_H15) R->L }
% 132.12/17.22 meet(X, join(meet(X, Y), meet(X, Z)))
% 132.12/17.22 = { by lemma 30 }
% 132.12/17.22 join(meet(X, Y), meet(X, Z))
% 132.12/17.22
% 132.12/17.22 Lemma 32: join(X, join(Y, complement(X))) = one.
% 132.12/17.22 Proof:
% 132.12/17.22 join(X, join(Y, complement(X)))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.22 join(X, join(complement(X), Y))
% 132.12/17.22 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.22 join(join(X, complement(X)), Y)
% 132.12/17.22 = { by axiom 3 (complement_join) }
% 132.12/17.22 join(one, Y)
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.22 join(Y, one)
% 132.12/17.22 = { by lemma 17 }
% 132.12/17.22 one
% 132.12/17.22
% 132.12/17.22 Lemma 33: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 132.12/17.22 Proof:
% 132.12/17.22 meet(X, meet(Y, complement(meet(X, Y))))
% 132.12/17.22 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.22 meet(meet(X, Y), complement(meet(X, Y)))
% 132.12/17.22 = { by axiom 5 (complement_meet) }
% 132.12/17.22 zero
% 132.12/17.22
% 132.12/17.22 Lemma 34: meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))) = complement(Y).
% 132.12/17.22 Proof:
% 132.12/17.22 meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))
% 132.12/17.22 = { by axiom 11 (meet_join_complement) R->L }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 27 R->L }
% 132.12/17.22 ifeq(join(Y, join(meet(Y, join(X, complement(Y))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.22 ifeq(join(Y, join(meet(join(X, complement(Y)), Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 31 R->L }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), join(join(X, complement(Y)), Y))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), join(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 32 }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), meet(complement(meet(Y, join(X, complement(Y)))), one)))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 13 }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), join(meet(join(X, complement(Y)), Y), complement(meet(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), join(meet(Y, join(X, complement(Y))), complement(meet(Y, join(X, complement(Y))))))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 3 (complement_join) }
% 132.12/17.22 ifeq(join(Y, meet(join(X, complement(Y)), one)), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 13 }
% 132.12/17.22 ifeq(join(Y, join(X, complement(Y))), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 32 }
% 132.12/17.22 ifeq(one, one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 6 (ifeq_axiom) }
% 132.12/17.22 ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by lemma 33 }
% 132.12/17.22 ifeq(zero, zero, complement(Y), meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))
% 132.12/17.22 = { by axiom 6 (ifeq_axiom) }
% 132.12/17.22 complement(Y)
% 132.12/17.22
% 132.12/17.22 Lemma 35: meet(join(X, Y), complement(meet(complement(Y), join(X, Y)))) = Y.
% 132.12/17.22 Proof:
% 132.12/17.22 meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))
% 132.12/17.22 = { by lemma 19 R->L }
% 132.12/17.22 meet(join(X, Y), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 132.12/17.22 = { by lemma 19 R->L }
% 132.12/17.22 meet(join(X, complement(complement(Y))), complement(meet(complement(Y), join(X, complement(complement(Y))))))
% 132.12/17.22 = { by lemma 34 }
% 132.12/17.22 complement(complement(Y))
% 132.12/17.22 = { by lemma 19 }
% 132.12/17.22 Y
% 132.12/17.22
% 132.12/17.22 Lemma 36: join(X, join(Y, complement(meet(Z, join(X, Y))))) = one.
% 132.12/17.22 Proof:
% 132.12/17.22 join(X, join(Y, complement(meet(Z, join(X, Y)))))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.22 join(X, join(Y, complement(meet(Z, join(Y, X)))))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.22 join(X, join(Y, complement(meet(join(Y, X), Z))))
% 132.12/17.22 = { by lemma 24 }
% 132.12/17.22 join(Y, join(X, complement(meet(join(Y, X), Z))))
% 132.12/17.22 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.22 join(join(Y, X), complement(meet(join(Y, X), Z)))
% 132.12/17.22 = { by lemma 27 R->L }
% 132.12/17.22 join(join(Y, X), join(meet(join(Y, X), Z), complement(meet(join(Y, X), Z))))
% 132.12/17.22 = { by axiom 3 (complement_join) }
% 132.12/17.22 join(join(Y, X), one)
% 132.12/17.22 = { by lemma 17 }
% 132.12/17.22 one
% 132.12/17.22
% 132.12/17.22 Lemma 37: meet(join(X, complement(Y)), join(Z, complement(Y))) = join(complement(Y), meet(X, join(Z, complement(Y)))).
% 132.12/17.22 Proof:
% 132.12/17.22 meet(join(X, complement(Y)), join(Z, complement(Y)))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(X, complement(Y)))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(complement(Y), X))
% 132.12/17.22 = { by lemma 13 R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(complement(Y), meet(X, one)))
% 132.12/17.22 = { by lemma 36 R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(complement(Y), meet(X, join(Z, join(complement(Y), complement(meet(Y, join(Z, complement(Y)))))))))
% 132.12/17.22 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(complement(Y), meet(X, join(join(Z, complement(Y)), complement(meet(Y, join(Z, complement(Y))))))))
% 132.12/17.22 = { by lemma 34 R->L }
% 132.12/17.22 meet(join(Z, complement(Y)), join(meet(join(Z, complement(Y)), complement(meet(Y, join(Z, complement(Y))))), meet(X, join(join(Z, complement(Y)), complement(meet(Y, join(Z, complement(Y))))))))
% 132.12/17.22 = { by lemma 31 }
% 132.12/17.22 join(meet(join(Z, complement(Y)), complement(meet(Y, join(Z, complement(Y))))), meet(join(Z, complement(Y)), X))
% 132.12/17.22 = { by lemma 34 }
% 132.12/17.22 join(complement(Y), meet(join(Z, complement(Y)), X))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.22 join(complement(Y), meet(X, join(Z, complement(Y))))
% 132.12/17.22
% 132.12/17.22 Lemma 38: join(X, meet(Z, join(X, Y))) = join(X, meet(Y, join(X, Z))).
% 132.12/17.22 Proof:
% 132.12/17.22 join(X, meet(Z, join(X, Y)))
% 132.12/17.22 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.22 join(X, meet(Z, join(Y, X)))
% 132.12/17.22 = { by lemma 19 R->L }
% 132.12/17.22 join(X, meet(Z, join(Y, complement(complement(X)))))
% 132.12/17.22 = { by lemma 19 R->L }
% 132.12/17.22 join(complement(complement(X)), meet(Z, join(Y, complement(complement(X)))))
% 132.12/17.22 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(complement(complement(X)), meet(join(Y, complement(complement(X))), Z))
% 132.12/17.23 = { by lemma 34 R->L }
% 132.12/17.23 join(meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), meet(join(Y, complement(complement(X))), Z))
% 132.12/17.23 = { by lemma 31 R->L }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), meet(Z, join(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), meet(join(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), Z)))
% 132.12/17.23 = { by lemma 34 }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(complement(complement(X)), meet(join(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), Z)))
% 132.12/17.23 = { by axiom 8 (associativity_of_join) }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(complement(complement(X)), meet(join(Y, join(complement(complement(X)), complement(meet(complement(X), join(Y, complement(complement(X))))))), Z)))
% 132.12/17.23 = { by lemma 36 }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(complement(complement(X)), meet(one, Z)))
% 132.12/17.23 = { by lemma 16 }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(complement(complement(X)), Z))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.23 meet(join(Y, complement(complement(X))), join(Z, complement(complement(X))))
% 132.12/17.23 = { by lemma 37 }
% 132.12/17.23 join(complement(complement(X)), meet(Y, join(Z, complement(complement(X)))))
% 132.12/17.23 = { by lemma 19 }
% 132.12/17.23 join(complement(complement(X)), meet(Y, join(Z, X)))
% 132.12/17.23 = { by lemma 19 }
% 132.12/17.23 join(X, meet(Y, join(Z, X)))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.23 join(X, meet(Y, join(X, Z)))
% 132.12/17.23
% 132.12/17.23 Lemma 39: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 132.12/17.23 Proof:
% 132.12/17.23 join(X, join(Y, meet(Z, join(X, Y))))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.23 join(X, join(Y, meet(Z, join(Y, X))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(X, join(Y, meet(join(Y, X), Z)))
% 132.12/17.23 = { by lemma 24 }
% 132.12/17.23 join(Y, join(X, meet(join(Y, X), Z)))
% 132.12/17.23 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.23 join(join(Y, X), meet(join(Y, X), Z))
% 132.12/17.23 = { by axiom 7 (absorption2) }
% 132.12/17.23 join(Y, X)
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.23 join(X, Y)
% 132.12/17.23
% 132.12/17.23 Lemma 40: meet(join(X, Y), join(X, meet(Z, join(X, Y)))) = join(X, meet(Z, join(X, Y))).
% 132.12/17.23 Proof:
% 132.12/17.23 meet(join(X, Y), join(X, meet(Z, join(X, Y))))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.23 meet(join(Y, X), join(X, meet(Z, join(X, Y))))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.23 meet(join(Y, X), join(X, meet(Z, join(Y, X))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 meet(join(X, meet(Z, join(Y, X))), join(Y, X))
% 132.12/17.23 = { by lemma 39 R->L }
% 132.12/17.23 meet(join(X, meet(Z, join(Y, X))), join(Y, join(X, meet(Z, join(Y, X)))))
% 132.12/17.23 = { by lemma 21 }
% 132.12/17.23 join(X, meet(Z, join(Y, X)))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.23 join(X, meet(Z, join(X, Y)))
% 132.12/17.23
% 132.12/17.23 Lemma 41: join(X, meet(join(X, Y), Z)) = join(X, meet(Y, join(X, Z))).
% 132.12/17.23 Proof:
% 132.12/17.23 join(X, meet(join(X, Y), Z))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(X, meet(Z, join(X, Y)))
% 132.12/17.23 = { by lemma 38 R->L }
% 132.12/17.23 join(X, meet(Y, join(X, Z)))
% 132.12/17.23
% 132.12/17.23 Lemma 42: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 132.12/17.23 Proof:
% 132.12/17.23 meet(X, meet(Y, join(Z, meet(X, Y))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 meet(X, meet(Y, join(Z, meet(Y, X))))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.23 meet(X, meet(Y, join(meet(Y, X), Z)))
% 132.12/17.23 = { by lemma 20 R->L }
% 132.12/17.23 meet(Y, meet(X, join(meet(Y, X), Z)))
% 132.12/17.23 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.23 meet(meet(Y, X), join(meet(Y, X), Z))
% 132.12/17.23 = { by axiom 9 (absorption1) }
% 132.12/17.23 meet(Y, X)
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.23 meet(X, Y)
% 132.12/17.23
% 132.12/17.23 Lemma 43: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 132.12/17.23 Proof:
% 132.12/17.23 meet(X, meet(join(X, Y), Z))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 meet(X, meet(Z, join(X, Y)))
% 132.12/17.23 = { by lemma 28 }
% 132.12/17.23 meet(X, Z)
% 132.12/17.23
% 132.12/17.23 Lemma 44: join(X, meet(Y, join(X, complement(join(X, Y))))) = X.
% 132.12/17.23 Proof:
% 132.12/17.23 join(X, meet(Y, join(X, complement(join(X, Y)))))
% 132.12/17.23 = { by lemma 38 R->L }
% 132.12/17.23 join(X, meet(complement(join(X, Y)), join(X, Y)))
% 132.12/17.23 = { by lemma 23 R->L }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), join(X, Y)))
% 132.12/17.23 = { by lemma 23 R->L }
% 132.12/17.23 join(X, join(X, meet(complement(join(X, join(X, Y))), join(X, Y))))
% 132.12/17.23 = { by lemma 40 R->L }
% 132.12/17.23 join(X, meet(join(X, Y), join(X, meet(complement(join(X, join(X, Y))), join(X, Y)))))
% 132.12/17.23 = { by lemma 41 R->L }
% 132.12/17.23 join(X, meet(join(X, join(X, Y)), meet(complement(join(X, join(X, Y))), join(X, Y))))
% 132.12/17.23 = { by lemma 20 R->L }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), meet(join(X, join(X, Y)), join(X, Y))))
% 132.12/17.23 = { by lemma 39 R->L }
% 132.12/17.23 join(X, join(meet(complement(join(X, join(X, Y))), meet(join(X, join(X, Y)), join(X, Y))), meet(complement(join(X, join(X, Y))), join(X, meet(complement(join(X, join(X, Y))), meet(join(X, join(X, Y)), join(X, Y)))))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(X, join(meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))), meet(complement(join(X, join(X, Y))), join(X, meet(complement(join(X, join(X, Y))), meet(join(X, join(X, Y)), join(X, Y)))))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(X, join(meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))), meet(complement(join(X, join(X, Y))), join(X, meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y))))))))
% 132.12/17.23 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.23 join(X, join(meet(complement(join(X, join(X, Y))), join(X, meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))))), meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y))))))
% 132.12/17.23 = { by lemma 42 R->L }
% 132.12/17.23 join(X, join(meet(complement(join(X, join(X, Y))), join(X, meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))))), meet(meet(join(X, join(X, Y)), join(X, Y)), meet(complement(join(X, join(X, Y))), join(X, meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))))))))
% 132.12/17.23 = { by lemma 22 }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), join(X, meet(meet(join(X, join(X, Y)), join(X, Y)), complement(join(X, join(X, Y)))))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), join(X, meet(complement(join(X, join(X, Y))), meet(join(X, join(X, Y)), join(X, Y))))))
% 132.12/17.23 = { by lemma 20 }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), join(X, meet(join(X, join(X, Y)), meet(complement(join(X, join(X, Y))), join(X, Y))))))
% 132.12/17.23 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.23 join(X, meet(complement(join(X, join(X, Y))), join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(X, join(X, Y))))))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 join(X, meet(complement(join(join(X, Y), X)), join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(X, join(X, Y))))))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 join(X, meet(complement(join(join(X, Y), X)), join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X)))))
% 132.12/17.24 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.24 join(X, meet(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), complement(join(join(X, Y), X))))
% 132.12/17.24 = { by lemma 39 R->L }
% 132.12/17.24 join(X, meet(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), complement(join(join(X, Y), join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X)))))))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 join(X, meet(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), complement(join(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), join(X, Y)))))
% 132.12/17.24 = { by lemma 43 R->L }
% 132.12/17.24 join(X, meet(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), meet(join(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), join(X, Y)), complement(join(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), join(X, Y))))))
% 132.12/17.24 = { by axiom 5 (complement_meet) }
% 132.12/17.24 join(X, meet(join(X, meet(meet(complement(join(X, join(X, Y))), join(X, Y)), join(join(X, Y), X))), zero))
% 132.12/17.24 = { by lemma 15 }
% 132.12/17.24 join(X, zero)
% 132.12/17.24 = { by lemma 14 }
% 132.12/17.24 X
% 132.12/17.24
% 132.12/17.24 Lemma 45: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 132.12/17.24 Proof:
% 132.12/17.24 meet(X, join(Y, meet(X, join(Y, Z))))
% 132.12/17.24 = { by lemma 40 R->L }
% 132.12/17.24 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 132.12/17.24 = { by lemma 42 }
% 132.12/17.24 meet(X, join(Y, Z))
% 132.12/17.24
% 132.12/17.24 Lemma 46: ifeq(meet(X, Y), zero, complement(join(X, complement(join(X, Y)))), Y) = Y.
% 132.12/17.24 Proof:
% 132.12/17.24 ifeq(meet(X, Y), zero, complement(join(X, complement(join(X, Y)))), Y)
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 ifeq(meet(X, Y), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.24 ifeq(meet(Y, X), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by lemma 44 R->L }
% 132.12/17.24 ifeq(meet(Y, join(X, meet(Y, join(X, complement(join(X, Y)))))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by lemma 45 }
% 132.12/17.24 ifeq(meet(Y, join(X, complement(join(X, Y)))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.24 ifeq(meet(Y, join(X, complement(join(Y, X)))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.24 ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y)
% 132.12/17.24 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.24 ifeq(one, one, ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y), Y)
% 132.12/17.24 = { by axiom 3 (complement_join) R->L }
% 132.12/17.24 ifeq(join(join(Y, X), complement(join(Y, X))), one, ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y), Y)
% 132.12/17.24 = { by axiom 8 (associativity_of_join) }
% 132.12/17.24 ifeq(join(Y, join(X, complement(join(Y, X)))), one, ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y), Y)
% 132.12/17.24 = { by lemma 18 }
% 132.12/17.24 Y
% 132.12/17.24
% 132.12/17.24 Lemma 47: complement(join(X, complement(join(X, meet(Y, complement(X)))))) = meet(Y, complement(X)).
% 132.12/17.24 Proof:
% 132.12/17.24 complement(join(X, complement(join(X, meet(Y, complement(X))))))
% 132.12/17.24 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.24 ifeq(zero, zero, complement(join(X, complement(join(X, meet(Y, complement(X)))))), meet(Y, complement(X)))
% 132.12/17.24 = { by lemma 25 R->L }
% 132.12/17.24 ifeq(meet(X, meet(Y, complement(X))), zero, complement(join(X, complement(join(X, meet(Y, complement(X)))))), meet(Y, complement(X)))
% 132.12/17.24 = { by lemma 46 }
% 132.12/17.24 meet(Y, complement(X))
% 132.12/17.24
% 132.12/17.24 Lemma 48: join(X, complement(join(X, meet(Y, complement(X))))) = complement(meet(Y, complement(X))).
% 132.12/17.24 Proof:
% 132.12/17.24 join(X, complement(join(X, meet(Y, complement(X)))))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 complement(complement(join(X, complement(join(X, meet(Y, complement(X)))))))
% 132.12/17.24 = { by lemma 47 }
% 132.12/17.24 complement(meet(Y, complement(X)))
% 132.12/17.24
% 132.12/17.24 Lemma 49: meet(join(X, Y), join(X, Z)) = join(X, meet(Y, join(X, Z))).
% 132.12/17.24 Proof:
% 132.12/17.24 meet(join(X, Y), join(X, Z))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 meet(join(X, Y), join(Z, X))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 meet(join(X, Y), join(Z, complement(complement(X))))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 meet(join(complement(complement(X)), Y), join(Z, complement(complement(X))))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 meet(join(Y, complement(complement(X))), join(Z, complement(complement(X))))
% 132.12/17.24 = { by lemma 37 }
% 132.12/17.24 join(complement(complement(X)), meet(Y, join(Z, complement(complement(X)))))
% 132.12/17.24 = { by lemma 19 }
% 132.12/17.24 join(X, meet(Y, join(Z, complement(complement(X)))))
% 132.12/17.24 = { by lemma 19 }
% 132.12/17.24 join(X, meet(Y, join(Z, X)))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.24 join(X, meet(Y, join(X, Z)))
% 132.12/17.24
% 132.12/17.24 Lemma 50: complement(join(X, meet(Y, complement(X)))) = meet(complement(X), complement(meet(Y, complement(X)))).
% 132.12/17.24 Proof:
% 132.12/17.24 complement(join(X, meet(Y, complement(X))))
% 132.12/17.24 = { by lemma 34 R->L }
% 132.12/17.24 meet(join(X, complement(join(X, meet(Y, complement(X))))), complement(meet(join(X, meet(Y, complement(X))), join(X, complement(join(X, meet(Y, complement(X))))))))
% 132.12/17.24 = { by lemma 48 }
% 132.12/17.24 meet(complement(meet(Y, complement(X))), complement(meet(join(X, meet(Y, complement(X))), join(X, complement(join(X, meet(Y, complement(X))))))))
% 132.12/17.24 = { by lemma 49 }
% 132.12/17.24 meet(complement(meet(Y, complement(X))), complement(join(X, meet(meet(Y, complement(X)), join(X, complement(join(X, meet(Y, complement(X)))))))))
% 132.12/17.24 = { by lemma 44 }
% 132.12/17.24 meet(complement(meet(Y, complement(X))), complement(X))
% 132.12/17.24 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.24 meet(complement(X), complement(meet(Y, complement(X))))
% 132.12/17.24
% 132.12/17.24 Lemma 51: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 132.12/17.24 Proof:
% 132.12/17.24 join(X, meet(Y, complement(meet(X, Y))))
% 132.12/17.24 = { by lemma 27 R->L }
% 132.12/17.24 join(X, join(meet(X, Y), meet(Y, complement(meet(X, Y)))))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 join(X, join(meet(X, Y), meet(complement(complement(Y)), complement(meet(X, Y)))))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 join(X, join(meet(X, Y), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y)))))))
% 132.12/17.24 = { by lemma 19 R->L }
% 132.12/17.24 join(X, join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y)))))))
% 132.12/17.24 = { by axiom 11 (meet_join_complement) R->L }
% 132.12/17.24 join(X, ifeq(join(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.24 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.24 join(X, ifeq(join(complement(Y), join(meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))), meet(X, complement(complement(Y))))), one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.24 = { by lemma 50 R->L }
% 132.12/17.24 join(X, ifeq(join(complement(Y), join(complement(join(complement(Y), meet(X, complement(complement(Y))))), meet(X, complement(complement(Y))))), one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.24 = { by axiom 8 (associativity_of_join) R->L }
% 132.12/17.24 join(X, ifeq(join(join(complement(Y), complement(join(complement(Y), meet(X, complement(complement(Y)))))), meet(X, complement(complement(Y)))), one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by lemma 47 R->L }
% 132.12/17.25 join(X, ifeq(join(join(complement(Y), complement(join(complement(Y), meet(X, complement(complement(Y)))))), complement(join(complement(Y), complement(join(complement(Y), meet(X, complement(complement(Y)))))))), one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 3 (complement_join) }
% 132.12/17.25 join(X, ifeq(one, one, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 6 (ifeq_axiom) }
% 132.12/17.25 join(X, ifeq(meet(complement(Y), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 join(X, ifeq(meet(complement(Y), join(meet(complement(complement(Y)), X), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.25 join(X, ifeq(meet(complement(Y), join(meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))), meet(complement(complement(Y)), X))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by lemma 31 R->L }
% 132.12/17.25 join(X, ifeq(meet(complement(Y), meet(complement(complement(Y)), join(meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))), meet(X, join(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 join(X, ifeq(meet(complement(Y), meet(join(meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))), meet(X, join(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))), complement(complement(Y)))), zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by lemma 25 }
% 132.12/17.25 join(X, ifeq(zero, zero, complement(complement(Y)), join(meet(X, complement(complement(Y))), meet(complement(complement(Y)), complement(meet(X, complement(complement(Y))))))))
% 132.12/17.25 = { by axiom 6 (ifeq_axiom) }
% 132.12/17.25 join(X, complement(complement(Y)))
% 132.12/17.25 = { by lemma 19 }
% 132.12/17.25 join(X, Y)
% 132.12/17.25
% 132.12/17.25 Lemma 52: meet(complement(Y), join(Y, X)) = meet(X, complement(meet(Y, X))).
% 132.12/17.25 Proof:
% 132.12/17.25 meet(complement(Y), join(Y, X))
% 132.12/17.25 = { by lemma 19 R->L }
% 132.12/17.25 complement(complement(meet(complement(Y), join(Y, X))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.25 complement(complement(meet(complement(Y), join(X, Y))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 complement(complement(meet(join(X, Y), complement(Y))))
% 132.12/17.25 = { by lemma 35 R->L }
% 132.12/17.25 complement(complement(meet(join(X, Y), complement(meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 complement(complement(meet(join(X, Y), complement(meet(complement(meet(complement(Y), join(X, Y))), join(X, Y))))))
% 132.12/17.25 = { by lemma 19 R->L }
% 132.12/17.25 complement(complement(meet(join(X, Y), complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))))
% 132.12/17.25 = { by lemma 19 R->L }
% 132.12/17.25 complement(complement(meet(complement(complement(join(X, Y))), complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 complement(complement(meet(complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))), complement(complement(join(X, Y))))))
% 132.12/17.25 = { by lemma 48 R->L }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(join(complement(join(X, Y)), meet(complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))), complement(complement(join(X, Y))))))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(join(complement(join(X, Y)), meet(complement(complement(join(X, Y))), complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(join(meet(complement(complement(join(X, Y))), complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y)))))), complement(join(X, Y))))))
% 132.12/17.25 = { by lemma 50 R->L }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(join(complement(join(complement(join(X, Y)), meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y)))))), complement(join(X, Y))))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(join(complement(join(X, Y)), complement(join(complement(join(X, Y)), meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))))))
% 132.12/17.25 = { by lemma 48 }
% 132.12/17.25 complement(join(complement(join(X, Y)), complement(complement(meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))))
% 132.12/17.25 = { by lemma 19 }
% 132.12/17.25 complement(join(complement(join(X, Y)), meet(complement(meet(complement(Y), join(X, Y))), complement(complement(join(X, Y))))))
% 132.12/17.25 = { by lemma 19 }
% 132.12/17.25 complement(join(complement(join(X, Y)), meet(complement(meet(complement(Y), join(X, Y))), join(X, Y))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.25 complement(join(complement(join(X, Y)), meet(join(X, Y), complement(meet(complement(Y), join(X, Y))))))
% 132.12/17.25 = { by lemma 35 }
% 132.12/17.25 complement(join(complement(join(X, Y)), Y))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.25 complement(join(Y, complement(join(X, Y))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.25 complement(join(Y, complement(join(Y, X))))
% 132.12/17.25 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.25 ifeq(zero, zero, complement(join(Y, complement(join(Y, X)))), meet(X, complement(meet(Y, X))))
% 132.12/17.25 = { by lemma 33 R->L }
% 132.12/17.25 ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(join(Y, complement(join(Y, X)))), meet(X, complement(meet(Y, X))))
% 132.12/17.25 = { by lemma 51 R->L }
% 132.12/17.25 ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(join(Y, complement(join(Y, meet(X, complement(meet(Y, X))))))), meet(X, complement(meet(Y, X))))
% 132.12/17.25 = { by lemma 46 }
% 132.12/17.25 meet(X, complement(meet(Y, X)))
% 132.12/17.25
% 132.12/17.25 Lemma 53: ifeq(join(X, meet(complement(X), Y)), one, complement(X), meet(complement(X), Y)) = meet(Y, complement(X)).
% 132.12/17.25 Proof:
% 132.12/17.25 ifeq(join(X, meet(complement(X), Y)), one, complement(X), meet(complement(X), Y))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 ifeq(join(X, meet(complement(X), Y)), one, complement(X), meet(Y, complement(X)))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 ifeq(join(X, meet(Y, complement(X))), one, complement(X), meet(Y, complement(X)))
% 132.12/17.25 = { by axiom 6 (ifeq_axiom) R->L }
% 132.12/17.25 ifeq(join(X, meet(Y, complement(X))), one, ifeq(zero, zero, complement(X), meet(Y, complement(X))), meet(Y, complement(X)))
% 132.12/17.25 = { by lemma 25 R->L }
% 132.12/17.25 ifeq(join(X, meet(Y, complement(X))), one, ifeq(meet(X, meet(Y, complement(X))), zero, complement(X), meet(Y, complement(X))), meet(Y, complement(X)))
% 132.12/17.25 = { by axiom 11 (meet_join_complement) }
% 132.12/17.25 meet(Y, complement(X))
% 132.12/17.25
% 132.12/17.25 Lemma 54: meet(X, join(Y, meet(join(Y, Z), X))) = meet(X, join(Y, Z)).
% 132.12/17.25 Proof:
% 132.12/17.25 meet(X, join(Y, meet(join(Y, Z), X)))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 meet(X, join(Y, meet(X, join(Y, Z))))
% 132.12/17.25 = { by lemma 45 }
% 132.12/17.25 meet(X, join(Y, Z))
% 132.12/17.25
% 132.12/17.25 Lemma 55: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 132.12/17.25 Proof:
% 132.12/17.25 meet(X, complement(meet(Y, X)))
% 132.12/17.25 = { by lemma 52 R->L }
% 132.12/17.25 meet(complement(Y), join(Y, X))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.25 meet(complement(Y), join(X, Y))
% 132.12/17.25 = { by lemma 19 R->L }
% 132.12/17.25 meet(complement(Y), join(X, complement(complement(Y))))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 meet(join(X, complement(complement(Y))), complement(Y))
% 132.12/17.25 = { by lemma 53 R->L }
% 132.12/17.25 ifeq(join(Y, meet(complement(Y), join(X, complement(complement(Y))))), one, complement(Y), meet(complement(Y), join(X, complement(complement(Y)))))
% 132.12/17.25 = { by lemma 54 R->L }
% 132.12/17.25 ifeq(join(Y, meet(complement(Y), join(X, complement(complement(Y))))), one, complement(Y), meet(complement(Y), join(X, meet(join(X, complement(complement(Y))), complement(Y)))))
% 132.12/17.25 = { by lemma 54 R->L }
% 132.12/17.25 ifeq(join(Y, meet(complement(Y), join(X, meet(join(X, complement(complement(Y))), complement(Y))))), one, complement(Y), meet(complement(Y), join(X, meet(join(X, complement(complement(Y))), complement(Y)))))
% 132.12/17.25 = { by lemma 53 }
% 132.12/17.25 meet(join(X, meet(join(X, complement(complement(Y))), complement(Y))), complement(Y))
% 132.12/17.25 = { by lemma 41 }
% 132.12/17.25 meet(join(X, meet(complement(complement(Y)), join(X, complement(Y)))), complement(Y))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.25 meet(complement(Y), join(X, meet(complement(complement(Y)), join(X, complement(Y)))))
% 132.12/17.25 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.25 meet(complement(Y), join(X, meet(complement(complement(Y)), join(complement(Y), X))))
% 132.12/17.25 = { by lemma 52 }
% 132.12/17.25 meet(complement(Y), join(X, meet(X, complement(meet(complement(Y), X)))))
% 132.12/17.25 = { by axiom 7 (absorption2) }
% 132.12/17.25 meet(complement(Y), X)
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.25 meet(X, complement(Y))
% 132.12/17.25
% 132.12/17.25 Lemma 56: join(X, meet(complement(X), Y)) = join(X, Y).
% 132.12/17.25 Proof:
% 132.12/17.25 join(X, meet(complement(X), Y))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 join(X, meet(Y, complement(X)))
% 132.12/17.25 = { by lemma 55 R->L }
% 132.12/17.25 join(X, meet(Y, complement(meet(X, Y))))
% 132.12/17.25 = { by lemma 51 }
% 132.12/17.25 join(X, Y)
% 132.12/17.25
% 132.12/17.25 Lemma 57: join(meet(X, join(Y, Z)), meet(Y, meet(W, X))) = meet(X, join(Y, Z)).
% 132.12/17.25 Proof:
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(Y, meet(W, X)))
% 132.12/17.25 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(meet(Y, W), X))
% 132.12/17.25 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(X, meet(Y, W)))
% 132.12/17.25 = { by lemma 43 R->L }
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(X, meet(Y, meet(join(Y, Z), W))))
% 132.12/17.25 = { by lemma 20 R->L }
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(X, meet(join(Y, Z), meet(Y, W))))
% 132.12/17.25 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.25 join(meet(X, join(Y, Z)), meet(meet(X, join(Y, Z)), meet(Y, W)))
% 132.12/17.25 = { by axiom 7 (absorption2) }
% 132.12/17.26 meet(X, join(Y, Z))
% 132.12/17.26
% 132.12/17.26 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 132.12/17.26 Proof:
% 132.12/17.26 meet(a, join(b, c))
% 132.12/17.26 = { by lemma 57 R->L }
% 132.12/17.26 join(meet(a, join(b, c)), meet(b, meet(X, a)))
% 132.12/17.26 = { by axiom 9 (absorption1) R->L }
% 132.12/17.26 meet(join(meet(a, join(b, c)), meet(b, meet(X, a))), join(join(meet(a, join(b, c)), meet(b, meet(X, a))), join(b, meet(a, join(b, c)))))
% 132.12/17.26 = { by axiom 8 (associativity_of_join) }
% 132.12/17.26 meet(join(meet(a, join(b, c)), meet(b, meet(X, a))), join(meet(a, join(b, c)), join(meet(b, meet(X, a)), join(b, meet(a, join(b, c))))))
% 132.12/17.26 = { by lemma 24 R->L }
% 132.12/17.26 meet(join(meet(a, join(b, c)), meet(b, meet(X, a))), join(meet(a, join(b, c)), join(b, join(meet(b, meet(X, a)), meet(a, join(b, c))))))
% 132.12/17.26 = { by lemma 27 }
% 132.12/17.26 meet(join(meet(a, join(b, c)), meet(b, meet(X, a))), join(meet(a, join(b, c)), join(b, meet(a, join(b, c)))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 meet(join(meet(a, join(b, c)), join(b, meet(a, join(b, c)))), join(meet(a, join(b, c)), meet(b, meet(X, a))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.26 meet(join(join(b, meet(a, join(b, c))), meet(a, join(b, c))), join(meet(a, join(b, c)), meet(b, meet(X, a))))
% 132.12/17.26 = { by axiom 8 (associativity_of_join) }
% 132.12/17.26 meet(join(b, join(meet(a, join(b, c)), meet(a, join(b, c)))), join(meet(a, join(b, c)), meet(b, meet(X, a))))
% 132.12/17.26 = { by lemma 57 }
% 132.12/17.26 meet(join(b, join(meet(a, join(b, c)), meet(a, join(b, c)))), meet(a, join(b, c)))
% 132.12/17.26 = { by lemma 20 }
% 132.12/17.26 meet(a, meet(join(b, join(meet(a, join(b, c)), meet(a, join(b, c)))), join(b, c)))
% 132.12/17.26 = { by lemma 49 }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(b, c))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(c, b))))
% 132.12/17.26 = { by lemma 19 R->L }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(c, complement(complement(b))))))
% 132.12/17.26 = { by lemma 56 R->L }
% 132.12/17.26 meet(a, join(b, meet(complement(b), meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(c, complement(complement(b)))))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.26 meet(a, join(b, meet(complement(b), meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(complement(complement(b)), c)))))
% 132.12/17.26 = { by lemma 19 R->L }
% 132.12/17.26 meet(a, join(b, meet(complement(complement(complement(b))), meet(join(meet(a, join(b, c)), meet(a, join(b, c))), join(complement(complement(b)), c)))))
% 132.12/17.26 = { by lemma 20 R->L }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), meet(complement(complement(complement(b))), join(complement(complement(b)), c)))))
% 132.12/17.26 = { by lemma 52 }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), meet(c, complement(meet(complement(complement(b)), c))))))
% 132.12/17.26 = { by lemma 55 }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), meet(c, complement(complement(complement(b)))))))
% 132.12/17.26 = { by lemma 19 }
% 132.12/17.26 meet(a, join(b, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), meet(c, complement(b)))))
% 132.12/17.26 = { by lemma 20 }
% 132.12/17.26 meet(a, join(b, meet(c, meet(join(meet(a, join(b, c)), meet(a, join(b, c))), complement(b)))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 meet(a, join(b, meet(c, meet(complement(b), join(meet(a, join(b, c)), meet(a, join(b, c)))))))
% 132.12/17.26 = { by lemma 20 }
% 132.12/17.26 meet(a, join(b, meet(complement(b), meet(c, join(meet(a, join(b, c)), meet(a, join(b, c)))))))
% 132.12/17.26 = { by lemma 56 }
% 132.12/17.26 meet(a, join(b, meet(c, join(meet(a, join(b, c)), meet(a, join(b, c))))))
% 132.12/17.26 = { by axiom 1 (idempotence_of_join) }
% 132.12/17.26 meet(a, join(b, meet(c, meet(a, join(b, c)))))
% 132.12/17.26 = { by lemma 20 R->L }
% 132.12/17.26 meet(a, join(b, meet(a, meet(c, join(b, c)))))
% 132.12/17.26 = { by lemma 21 }
% 132.12/17.26 meet(a, join(b, meet(a, c)))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), a)
% 132.12/17.26 = { by lemma 42 R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(a, join(meet(a, c), meet(join(b, meet(a, c)), a))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(a, join(meet(join(b, meet(a, c)), a), meet(a, c))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(a, join(meet(join(b, meet(a, c)), a), meet(c, a))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(a, join(meet(a, join(b, meet(a, c))), meet(c, a))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(a, join(meet(c, a), meet(a, join(b, meet(a, c))))))
% 132.12/17.26 = { by lemma 22 R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), meet(join(a, meet(c, a)), join(meet(c, a), meet(a, join(b, meet(a, c))))))
% 132.12/17.26 = { by lemma 29 }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(c, a), meet(a, join(b, meet(a, c)))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(a, join(b, meet(a, c))), meet(c, a)))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(join(b, meet(a, c)), a), meet(c, a)))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(join(b, meet(a, c)), a), meet(a, c)))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(a, join(b, meet(a, c))), meet(a, c)))
% 132.12/17.26 = { by lemma 19 R->L }
% 132.12/17.26 meet(join(b, meet(a, c)), join(meet(a, join(b, meet(a, c))), complement(complement(meet(a, c)))))
% 132.12/17.26 = { by lemma 19 R->L }
% 132.12/17.26 meet(join(b, complement(complement(meet(a, c)))), join(meet(a, join(b, meet(a, c))), complement(complement(meet(a, c)))))
% 132.12/17.26 = { by lemma 37 }
% 132.12/17.26 join(complement(complement(meet(a, c))), meet(b, join(meet(a, join(b, meet(a, c))), complement(complement(meet(a, c))))))
% 132.12/17.26 = { by lemma 19 }
% 132.12/17.26 join(meet(a, c), meet(b, join(meet(a, join(b, meet(a, c))), complement(complement(meet(a, c))))))
% 132.12/17.26 = { by lemma 19 }
% 132.12/17.26 join(meet(a, c), meet(b, join(meet(a, join(b, meet(a, c))), meet(a, c))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.26 join(meet(a, c), meet(b, join(meet(a, c), meet(a, join(b, meet(a, c))))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.26 join(meet(a, c), meet(b, join(meet(a, c), meet(a, join(meet(a, c), b)))))
% 132.12/17.26 = { by lemma 30 R->L }
% 132.12/17.26 join(meet(a, c), meet(b, meet(a, join(meet(a, c), meet(a, join(meet(a, c), b))))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 join(meet(a, c), meet(b, meet(join(meet(a, c), meet(a, join(meet(a, c), b))), a)))
% 132.12/17.26 = { by axiom 10 (associativity_of_meet) R->L }
% 132.12/17.26 join(meet(a, c), meet(meet(b, join(meet(a, c), meet(a, join(meet(a, c), b)))), a))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 join(meet(a, c), meet(a, meet(b, join(meet(a, c), meet(a, join(meet(a, c), b))))))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) R->L }
% 132.12/17.26 join(meet(a, c), meet(a, meet(b, join(meet(a, c), meet(a, join(b, meet(a, c)))))))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) R->L }
% 132.12/17.26 join(meet(a, c), meet(a, meet(b, join(meet(a, c), meet(join(b, meet(a, c)), a)))))
% 132.12/17.26 = { by lemma 20 R->L }
% 132.12/17.26 join(meet(a, c), meet(b, meet(a, join(meet(a, c), meet(join(b, meet(a, c)), a)))))
% 132.12/17.26 = { by lemma 43 R->L }
% 132.12/17.26 join(meet(a, c), meet(b, meet(join(b, meet(a, c)), meet(a, join(meet(a, c), meet(join(b, meet(a, c)), a))))))
% 132.12/17.26 = { by lemma 42 }
% 132.12/17.26 join(meet(a, c), meet(b, meet(join(b, meet(a, c)), a)))
% 132.12/17.26 = { by lemma 43 }
% 132.12/17.26 join(meet(a, c), meet(b, a))
% 132.12/17.26 = { by axiom 4 (commutativity_of_meet) }
% 132.12/17.26 join(meet(a, c), meet(a, b))
% 132.12/17.26 = { by axiom 2 (commutativity_of_join) }
% 132.12/17.26 join(meet(a, b), meet(a, c))
% 132.12/17.26 % SZS output end Proof
% 132.12/17.26
% 132.12/17.26 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------