%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LAT207-1 : TPTP v9.3.1. Released v3.1.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:44:32 AM UTC 2026
% Result : Unsatisfiable 147.23s 19.11s
% Output : Proof 150.39s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.05 % Problem : LAT207-1 : TPTP v9.3.1. Released v3.1.0.
% 0.00/0.07 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.19/0.45 % Computer : n003.cluster.edu
% 0.19/0.45 % Model : x86_64 x86_64
% 0.19/0.45 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.19/0.45 % Memory : 8046.5625MB
% 0.19/0.45 % OS : Linux 6.8.0-71-generic
% 0.19/0.45 % CPULimit : 300
% 0.19/0.45 % WCLimit : 300
% 0.19/0.45 % DateTime : Sun Sep 27 14:07:55 UTC 2026
% 0.19/0.46 % CPUTime :
% 0.19/0.46 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 147.23/19.11 Command-line arguments: --no-flatten-goal
% 147.23/19.11
% 147.23/19.11 % SZS status Unsatisfiable
% 147.23/19.11
% 149.40/19.43 % SZS output start Proof
% 149.40/19.43 Axiom 1 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 149.40/19.43 Axiom 2 (complement_meet): meet(X, complement(X)) = zero.
% 149.40/19.43 Axiom 3 (idempotence_of_join): join(X, X) = X.
% 149.40/19.43 Axiom 4 (commutativity_of_join): join(X, Y) = join(Y, X).
% 149.40/19.43 Axiom 5 (complement_join): join(X, complement(X)) = one.
% 149.40/19.43 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 149.40/19.43 Axiom 7 (absorption1): meet(X, join(X, Y)) = X.
% 149.40/19.43 Axiom 8 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 149.40/19.43 Axiom 9 (absorption2): join(X, meet(X, Y)) = X.
% 149.40/19.43 Axiom 10 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 149.40/19.44 Axiom 11 (equation_H61): meet(join(X, Y), join(X, Z)) = join(X, meet(join(X, Y), join(meet(X, Y), Z))).
% 149.40/19.44 Axiom 12 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 149.40/19.44
% 149.40/19.44 Lemma 13: join(X, zero) = X.
% 149.40/19.44 Proof:
% 149.40/19.44 join(X, zero)
% 149.40/19.44 = { by axiom 2 (complement_meet) R->L }
% 149.40/19.44 join(X, meet(X, complement(X)))
% 149.40/19.44 = { by axiom 9 (absorption2) }
% 149.40/19.44 X
% 149.40/19.44
% 149.40/19.44 Lemma 14: join(zero, X) = X.
% 149.40/19.44 Proof:
% 149.40/19.44 join(zero, X)
% 149.40/19.44 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.44 join(X, zero)
% 149.40/19.44 = { by lemma 13 }
% 149.40/19.44 X
% 149.40/19.44
% 149.40/19.44 Lemma 15: meet(X, zero) = zero.
% 149.40/19.44 Proof:
% 149.40/19.44 meet(X, zero)
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.44 meet(zero, X)
% 149.40/19.44 = { by lemma 14 R->L }
% 149.40/19.44 meet(zero, join(zero, X))
% 149.40/19.44 = { by axiom 7 (absorption1) }
% 149.40/19.44 zero
% 149.40/19.44
% 149.40/19.44 Lemma 16: meet(one, X) = X.
% 149.40/19.44 Proof:
% 149.40/19.44 meet(one, X)
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.44 meet(X, one)
% 149.40/19.44 = { by axiom 5 (complement_join) R->L }
% 149.40/19.44 meet(X, join(X, complement(X)))
% 149.40/19.44 = { by axiom 7 (absorption1) }
% 149.40/19.44 X
% 149.40/19.44
% 149.40/19.44 Lemma 17: join(X, one) = one.
% 149.40/19.44 Proof:
% 149.40/19.44 join(X, one)
% 149.40/19.44 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.44 join(one, X)
% 149.40/19.44 = { by lemma 16 R->L }
% 149.40/19.44 meet(one, join(one, X))
% 149.40/19.44 = { by axiom 7 (absorption1) }
% 149.40/19.44 one
% 149.40/19.44
% 149.40/19.44 Lemma 18: ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(X), Y), Y) = Y.
% 149.40/19.44 Proof:
% 149.40/19.44 ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(X), Y), Y)
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.44 ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y)
% 149.40/19.44 = { by axiom 12 (meet_join_complement) }
% 149.40/19.44 Y
% 149.40/19.44
% 149.40/19.44 Lemma 19: complement(complement(X)) = X.
% 149.40/19.44 Proof:
% 149.40/19.44 complement(complement(X))
% 149.40/19.44 = { by axiom 6 (ifeq_axiom) R->L }
% 149.40/19.44 ifeq(one, one, complement(complement(X)), X)
% 149.40/19.44 = { by axiom 5 (complement_join) R->L }
% 149.40/19.44 ifeq(join(X, complement(X)), one, complement(complement(X)), X)
% 149.40/19.44 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.44 ifeq(join(complement(X), X), one, complement(complement(X)), X)
% 149.40/19.44 = { by axiom 6 (ifeq_axiom) R->L }
% 149.40/19.44 ifeq(join(complement(X), X), one, ifeq(zero, zero, complement(complement(X)), X), X)
% 149.40/19.44 = { by axiom 2 (complement_meet) R->L }
% 149.40/19.44 ifeq(join(complement(X), X), one, ifeq(meet(X, complement(X)), zero, complement(complement(X)), X), X)
% 149.40/19.44 = { by lemma 18 }
% 149.40/19.44 X
% 149.40/19.44
% 149.40/19.44 Lemma 20: meet(X, meet(Y, complement(X))) = zero.
% 149.40/19.44 Proof:
% 149.40/19.44 meet(X, meet(Y, complement(X)))
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.44 meet(X, meet(complement(X), Y))
% 149.40/19.44 = { by axiom 8 (associativity_of_meet) R->L }
% 149.40/19.44 meet(meet(X, complement(X)), Y)
% 149.40/19.44 = { by axiom 2 (complement_meet) }
% 149.40/19.44 meet(zero, Y)
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.44 meet(Y, zero)
% 149.40/19.44 = { by lemma 15 }
% 149.40/19.44 zero
% 149.40/19.44
% 149.40/19.44 Lemma 21: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 149.40/19.44 Proof:
% 149.40/19.44 meet(X, meet(join(X, Y), Z))
% 149.40/19.44 = { by axiom 8 (associativity_of_meet) R->L }
% 149.40/19.44 meet(meet(X, join(X, Y)), Z)
% 149.40/19.44 = { by axiom 7 (absorption1) }
% 149.40/19.44 meet(X, Z)
% 149.40/19.44
% 149.40/19.44 Lemma 22: meet(join(X, meet(Y, complement(X))), join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X)))))) = meet(Y, complement(X)).
% 149.40/19.44 Proof:
% 149.40/19.44 meet(join(X, meet(Y, complement(X))), join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X))))))
% 149.40/19.44 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.44 meet(join(meet(Y, complement(X)), X), join(meet(Y, complement(X)), complement(join(X, meet(Y, complement(X))))))
% 149.40/19.44 = { by axiom 11 (equation_H61) }
% 149.40/19.44 join(meet(Y, complement(X)), meet(join(meet(Y, complement(X)), X), join(meet(meet(Y, complement(X)), X), complement(join(X, meet(Y, complement(X)))))))
% 149.40/19.44 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.44 join(meet(Y, complement(X)), meet(join(meet(Y, complement(X)), X), join(meet(X, meet(Y, complement(X))), complement(join(X, meet(Y, complement(X)))))))
% 149.40/19.44 = { by lemma 20 }
% 149.40/19.44 join(meet(Y, complement(X)), meet(join(meet(Y, complement(X)), X), join(zero, complement(join(X, meet(Y, complement(X)))))))
% 149.40/19.44 = { by lemma 14 }
% 149.40/19.44 join(meet(Y, complement(X)), meet(join(meet(Y, complement(X)), X), complement(join(X, meet(Y, complement(X))))))
% 149.40/19.44 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.44 join(meet(Y, complement(X)), meet(join(X, meet(Y, complement(X))), complement(join(X, meet(Y, complement(X))))))
% 149.40/19.44 = { by axiom 2 (complement_meet) }
% 149.40/19.44 join(meet(Y, complement(X)), zero)
% 149.40/19.44 = { by lemma 13 }
% 149.40/19.44 meet(Y, complement(X))
% 149.40/19.44
% 149.40/19.44 Lemma 23: join(X, join(Y, complement(join(X, Y)))) = one.
% 149.40/19.44 Proof:
% 149.40/19.44 join(X, join(Y, complement(join(X, Y))))
% 149.40/19.44 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.44 join(join(X, Y), complement(join(X, Y)))
% 149.40/19.44 = { by axiom 5 (complement_join) }
% 149.40/19.44 one
% 149.40/19.44
% 149.40/19.44 Lemma 24: join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))) = complement(Y).
% 149.40/19.44 Proof:
% 149.40/19.45 join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))
% 149.40/19.45 = { by axiom 12 (meet_join_complement) R->L }
% 149.40/19.45 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by lemma 21 R->L }
% 149.40/19.45 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, meet(join(Y, meet(X, complement(Y))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by lemma 22 }
% 149.40/19.45 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(meet(Y, meet(X, complement(Y))), zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by lemma 20 }
% 149.40/19.45 ifeq(join(Y, join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), one, ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by lemma 23 }
% 149.40/19.45 ifeq(one, one, ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y)))))), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by axiom 6 (ifeq_axiom) }
% 149.40/19.45 ifeq(zero, zero, complement(Y), join(meet(X, complement(Y)), complement(join(Y, meet(X, complement(Y))))))
% 149.40/19.45 = { by axiom 6 (ifeq_axiom) }
% 149.40/19.45 complement(Y)
% 149.40/19.45
% 149.40/19.45 Lemma 25: meet(join(X, Y), join(X, complement(join(X, Y)))) = X.
% 149.40/19.45 Proof:
% 149.40/19.45 meet(join(X, Y), join(X, complement(join(X, Y))))
% 149.40/19.45 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.45 meet(join(X, Y), join(complement(join(X, Y)), X))
% 149.40/19.45 = { by axiom 7 (absorption1) R->L }
% 149.40/19.45 meet(join(X, Y), join(complement(join(X, Y)), meet(X, join(X, Y))))
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.45 meet(join(complement(join(X, Y)), meet(X, join(X, Y))), join(X, Y))
% 149.40/19.45 = { by lemma 19 R->L }
% 149.40/19.45 meet(join(complement(join(X, Y)), meet(X, join(X, Y))), complement(complement(join(X, Y))))
% 149.40/19.45 = { by lemma 24 R->L }
% 149.40/19.45 meet(join(complement(join(X, Y)), meet(X, join(X, Y))), join(meet(X, complement(complement(join(X, Y)))), complement(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))))))
% 149.40/19.45 = { by lemma 19 R->L }
% 149.40/19.45 meet(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))), join(meet(X, complement(complement(join(X, Y)))), complement(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))))))
% 149.40/19.45 = { by lemma 22 }
% 149.40/19.45 meet(X, complement(complement(join(X, Y))))
% 149.40/19.45 = { by lemma 19 }
% 149.40/19.45 meet(X, join(X, Y))
% 149.40/19.45 = { by axiom 7 (absorption1) }
% 149.40/19.45 X
% 149.40/19.45
% 149.40/19.45 Lemma 26: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 149.40/19.45 Proof:
% 149.40/19.45 meet(Y, meet(X, Z))
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.45 meet(meet(X, Z), Y)
% 149.40/19.45 = { by axiom 8 (associativity_of_meet) }
% 149.40/19.45 meet(X, meet(Z, Y))
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.45 meet(X, meet(Y, Z))
% 149.40/19.45
% 149.40/19.45 Lemma 27: meet(X, join(Y, X)) = X.
% 149.40/19.45 Proof:
% 149.40/19.45 meet(X, join(Y, X))
% 149.40/19.45 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.45 meet(X, join(X, Y))
% 149.40/19.45 = { by axiom 7 (absorption1) }
% 149.40/19.45 X
% 149.40/19.45
% 149.40/19.45 Lemma 28: ifeq(meet(X, Y), zero, complement(join(X, complement(join(X, Y)))), Y) = Y.
% 149.40/19.45 Proof:
% 149.40/19.45 ifeq(meet(X, Y), zero, complement(join(X, complement(join(X, Y)))), Y)
% 149.40/19.45 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.45 ifeq(meet(X, Y), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.45 ifeq(meet(Y, X), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by lemma 25 R->L }
% 149.40/19.45 ifeq(meet(Y, meet(join(X, Y), join(X, complement(join(X, Y))))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.45 ifeq(meet(Y, meet(join(X, complement(join(X, Y))), join(X, Y))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by lemma 26 }
% 149.40/19.45 ifeq(meet(join(X, complement(join(X, Y))), meet(Y, join(X, Y))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by lemma 27 }
% 149.40/19.45 ifeq(meet(join(X, complement(join(X, Y))), Y), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.45 ifeq(meet(Y, join(X, complement(join(X, Y)))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.45 ifeq(meet(Y, join(X, complement(join(Y, X)))), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.45 ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y)
% 149.40/19.45 = { by axiom 6 (ifeq_axiom) R->L }
% 149.40/19.45 ifeq(one, one, ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y), Y)
% 149.40/19.45 = { by lemma 23 R->L }
% 149.40/19.45 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)
% 149.40/19.45 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.45 ifeq(join(join(X, complement(join(Y, X))), Y), one, ifeq(meet(join(X, complement(join(Y, X))), Y), zero, complement(join(X, complement(join(Y, X)))), Y), Y)
% 149.40/19.45 = { by axiom 12 (meet_join_complement) }
% 149.40/19.45 Y
% 149.40/19.45
% 149.40/19.45 Lemma 29: join(X, complement(join(X, meet(Y, complement(X))))) = complement(meet(Y, complement(X))).
% 149.40/19.45 Proof:
% 149.40/19.46 join(X, complement(join(X, meet(Y, complement(X)))))
% 149.40/19.46 = { by lemma 19 R->L }
% 149.40/19.46 complement(complement(join(X, complement(join(X, meet(Y, complement(X)))))))
% 149.40/19.46 = { by axiom 6 (ifeq_axiom) R->L }
% 149.40/19.46 complement(ifeq(zero, zero, complement(join(X, complement(join(X, meet(Y, complement(X)))))), meet(Y, complement(X))))
% 149.40/19.46 = { by lemma 20 R->L }
% 149.40/19.46 complement(ifeq(meet(X, meet(Y, complement(X))), zero, complement(join(X, complement(join(X, meet(Y, complement(X)))))), meet(Y, complement(X))))
% 149.40/19.46 = { by lemma 28 }
% 149.40/19.46 complement(meet(Y, complement(X)))
% 149.40/19.46
% 149.40/19.46 Lemma 30: join(complement(X), complement(join(complement(X), meet(Y, X)))) = complement(meet(X, Y)).
% 149.40/19.46 Proof:
% 149.40/19.46 join(complement(X), complement(join(complement(X), meet(Y, X))))
% 149.40/19.46 = { by lemma 19 R->L }
% 149.40/19.46 join(complement(X), complement(join(complement(X), meet(Y, complement(complement(X))))))
% 149.40/19.46 = { by lemma 29 }
% 149.40/19.46 complement(meet(Y, complement(complement(X))))
% 149.40/19.46 = { by lemma 19 }
% 149.40/19.46 complement(meet(Y, X))
% 149.40/19.46 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.46 complement(meet(X, Y))
% 149.40/19.46
% 149.40/19.46 Lemma 31: join(complement(X), complement(join(complement(X), meet(X, Y)))) = complement(meet(X, Y)).
% 149.40/19.46 Proof:
% 149.40/19.46 join(complement(X), complement(join(complement(X), meet(X, Y))))
% 149.40/19.46 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.46 join(complement(X), complement(join(complement(X), meet(Y, X))))
% 149.40/19.46 = { by lemma 30 }
% 149.40/19.46 complement(meet(X, Y))
% 149.40/19.46
% 149.40/19.46 Lemma 32: join(X, join(X, Y)) = join(X, Y).
% 149.40/19.46 Proof:
% 149.40/19.46 join(X, join(X, Y))
% 149.40/19.46 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.46 join(join(X, X), Y)
% 149.40/19.46 = { by axiom 3 (idempotence_of_join) }
% 149.40/19.46 join(X, Y)
% 149.40/19.46
% 149.40/19.46 Lemma 33: join(X, meet(join(X, Y), join(X, Z))) = meet(join(X, Y), join(X, Z)).
% 149.40/19.46 Proof:
% 149.40/19.46 join(X, meet(join(X, Y), join(X, Z)))
% 149.40/19.46 = { by axiom 11 (equation_H61) }
% 149.40/19.46 join(X, join(X, meet(join(X, Y), join(meet(X, Y), Z))))
% 149.40/19.46 = { by lemma 32 }
% 149.40/19.46 join(X, meet(join(X, Y), join(meet(X, Y), Z)))
% 149.40/19.46 = { by axiom 11 (equation_H61) R->L }
% 149.40/19.46 meet(join(X, Y), join(X, Z))
% 149.40/19.46
% 149.40/19.46 Lemma 34: join(X, join(Y, meet(X, Z))) = join(X, Y).
% 149.40/19.46 Proof:
% 149.40/19.46 join(X, join(Y, meet(X, Z)))
% 149.40/19.46 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.46 join(X, join(meet(X, Z), Y))
% 149.40/19.46 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.46 join(join(X, meet(X, Z)), Y)
% 149.40/19.46 = { by axiom 9 (absorption2) }
% 149.40/19.46 join(X, Y)
% 149.40/19.46
% 149.40/19.46 Lemma 35: join(X, meet(join(X, Y), join(Y, Z))) = join(X, Y).
% 149.40/19.46 Proof:
% 149.40/19.46 join(X, meet(join(X, Y), join(Y, Z)))
% 149.40/19.46 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.46 join(X, meet(join(Y, X), join(Y, Z)))
% 149.40/19.46 = { by lemma 33 R->L }
% 149.40/19.46 join(X, join(Y, meet(join(Y, X), join(Y, Z))))
% 149.40/19.46 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.46 join(join(X, Y), meet(join(Y, X), join(Y, Z)))
% 149.40/19.46 = { by lemma 33 R->L }
% 149.40/19.46 join(join(X, Y), join(Y, meet(join(Y, X), join(Y, Z))))
% 149.40/19.46 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.46 join(join(X, Y), join(Y, meet(join(X, Y), join(Y, Z))))
% 149.40/19.46 = { by lemma 34 }
% 149.40/19.46 join(join(X, Y), Y)
% 149.40/19.46 = { by axiom 10 (associativity_of_join) }
% 149.40/19.46 join(X, join(Y, Y))
% 149.40/19.46 = { by axiom 3 (idempotence_of_join) }
% 149.40/19.46 join(X, Y)
% 149.40/19.46
% 149.40/19.46 Lemma 36: meet(complement(X), join(X, complement(join(X, Y)))) = complement(join(X, Y)).
% 149.40/19.47 Proof:
% 149.40/19.47 meet(complement(X), join(X, complement(join(X, Y))))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 meet(complement(X), join(complement(join(X, Y)), X))
% 149.40/19.47 = { by axiom 7 (absorption1) R->L }
% 149.40/19.47 meet(complement(X), join(complement(join(X, Y)), meet(X, join(X, Y))))
% 149.40/19.47 = { by axiom 7 (absorption1) R->L }
% 149.40/19.47 meet(complement(meet(X, join(X, Y))), join(complement(join(X, Y)), meet(X, join(X, Y))))
% 149.40/19.47 = { by lemma 19 R->L }
% 149.40/19.47 meet(complement(meet(X, join(X, Y))), join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))))
% 149.40/19.47 = { by lemma 19 R->L }
% 149.40/19.47 meet(complement(meet(X, complement(complement(join(X, Y))))), join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 meet(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))), complement(meet(X, complement(complement(join(X, Y))))))
% 149.40/19.47 = { by lemma 29 R->L }
% 149.40/19.47 meet(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))), join(complement(join(X, Y)), complement(join(complement(join(X, Y)), meet(X, complement(complement(join(X, Y))))))))
% 149.40/19.47 = { by lemma 25 }
% 149.40/19.47 complement(join(X, Y))
% 149.40/19.47
% 149.40/19.47 Lemma 37: complement(join(complement(X), meet(X, Y))) = meet(X, complement(meet(X, Y))).
% 149.40/19.47 Proof:
% 149.40/19.47 complement(join(complement(X), meet(X, Y)))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 complement(join(complement(X), meet(Y, X)))
% 149.40/19.47 = { by lemma 36 R->L }
% 149.40/19.47 meet(complement(complement(X)), join(complement(X), complement(join(complement(X), meet(Y, X)))))
% 149.40/19.47 = { by lemma 30 }
% 149.40/19.47 meet(complement(complement(X)), complement(meet(X, Y)))
% 149.40/19.47 = { by lemma 19 }
% 149.40/19.47 meet(X, complement(meet(X, Y)))
% 149.40/19.47
% 149.40/19.47 Lemma 38: complement(join(X, complement(join(X, Y)))) = meet(complement(X), join(X, Y)).
% 149.40/19.47 Proof:
% 149.40/19.47 complement(join(X, complement(join(X, Y))))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 complement(join(complement(join(X, Y)), X))
% 149.40/19.47 = { by lemma 35 R->L }
% 149.40/19.47 complement(join(complement(join(X, Y)), meet(join(complement(join(X, Y)), X), join(X, Y))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.47 complement(join(complement(join(X, Y)), meet(join(X, Y), join(complement(join(X, Y)), X))))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.47 complement(join(complement(join(X, Y)), meet(join(X, Y), join(X, complement(join(X, Y))))))
% 149.40/19.47 = { by lemma 37 }
% 149.40/19.47 meet(join(X, Y), complement(meet(join(X, Y), join(X, complement(join(X, Y))))))
% 149.40/19.47 = { by lemma 25 }
% 149.40/19.47 meet(join(X, Y), complement(X))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.47 meet(complement(X), join(X, Y))
% 149.40/19.47
% 149.40/19.47 Lemma 39: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 149.40/19.47 Proof:
% 149.40/19.47 meet(X, meet(Y, complement(meet(X, Y))))
% 149.40/19.47 = { by axiom 8 (associativity_of_meet) R->L }
% 149.40/19.47 meet(meet(X, Y), complement(meet(X, Y)))
% 149.40/19.47 = { by axiom 2 (complement_meet) }
% 149.40/19.47 zero
% 149.40/19.47
% 149.40/19.47 Lemma 40: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 149.40/19.47 Proof:
% 149.40/19.47 join(X, join(meet(X, Y), Z))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 join(X, join(Z, meet(X, Y)))
% 149.40/19.47 = { by lemma 34 }
% 149.40/19.47 join(X, Z)
% 149.40/19.47
% 149.40/19.47 Lemma 41: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 149.40/19.47 Proof:
% 149.40/19.47 join(X, meet(Y, complement(meet(X, Y))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 join(X, meet(Y, complement(meet(Y, X))))
% 149.40/19.47 = { by lemma 37 R->L }
% 149.40/19.47 join(X, complement(join(complement(Y), meet(Y, X))))
% 149.40/19.47 = { by lemma 19 R->L }
% 149.40/19.47 join(X, complement(join(complement(Y), meet(complement(complement(Y)), X))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 join(X, complement(join(complement(Y), meet(X, complement(complement(Y))))))
% 149.40/19.47 = { by lemma 40 R->L }
% 149.40/19.47 join(X, join(meet(X, complement(complement(Y))), complement(join(complement(Y), meet(X, complement(complement(Y)))))))
% 149.40/19.47 = { by lemma 24 }
% 149.40/19.47 join(X, complement(complement(Y)))
% 149.40/19.47 = { by lemma 19 }
% 149.40/19.47 join(X, Y)
% 149.40/19.47
% 149.40/19.47 Lemma 42: meet(complement(Y), join(Y, X)) = meet(X, complement(meet(Y, X))).
% 149.40/19.47 Proof:
% 149.40/19.47 meet(complement(Y), join(Y, X))
% 149.40/19.47 = { by lemma 38 R->L }
% 149.40/19.47 complement(join(Y, complement(join(Y, X))))
% 149.40/19.47 = { by axiom 6 (ifeq_axiom) R->L }
% 149.40/19.47 ifeq(zero, zero, complement(join(Y, complement(join(Y, X)))), meet(X, complement(meet(Y, X))))
% 149.40/19.47 = { by lemma 39 R->L }
% 149.40/19.47 ifeq(meet(Y, meet(X, complement(meet(Y, X)))), zero, complement(join(Y, complement(join(Y, X)))), meet(X, complement(meet(Y, X))))
% 149.40/19.47 = { by lemma 41 R->L }
% 149.40/19.47 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))))
% 149.40/19.47 = { by lemma 28 }
% 149.40/19.47 meet(X, complement(meet(Y, X)))
% 149.40/19.47
% 149.40/19.47 Lemma 43: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 149.40/19.47 Proof:
% 149.40/19.47 meet(X, meet(Y, join(Z, meet(X, Y))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 meet(X, meet(Y, join(Z, meet(Y, X))))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 meet(X, meet(Y, join(meet(Y, X), Z)))
% 149.40/19.47 = { by lemma 26 }
% 149.40/19.47 meet(Y, meet(X, join(meet(Y, X), Z)))
% 149.40/19.47 = { by axiom 8 (associativity_of_meet) R->L }
% 149.40/19.47 meet(meet(Y, X), join(meet(Y, X), Z))
% 149.40/19.47 = { by axiom 7 (absorption1) }
% 149.40/19.47 meet(Y, X)
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.47 meet(X, Y)
% 149.40/19.47
% 149.40/19.47 Lemma 44: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 149.40/19.47 Proof:
% 149.40/19.47 join(X, join(Y, Z))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 join(join(Y, Z), X)
% 149.40/19.47 = { by axiom 10 (associativity_of_join) }
% 149.40/19.47 join(Y, join(Z, X))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.47 join(Y, join(X, Z))
% 149.40/19.47
% 149.40/19.47 Lemma 45: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 149.40/19.47 Proof:
% 149.40/19.47 join(X, join(Y, meet(Z, join(X, Y))))
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.47 join(X, join(Y, meet(Z, join(Y, X))))
% 149.40/19.47 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.47 join(X, join(Y, meet(join(Y, X), Z)))
% 149.40/19.47 = { by lemma 44 R->L }
% 149.40/19.47 join(Y, join(X, meet(join(Y, X), Z)))
% 149.40/19.47 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.47 join(join(Y, X), meet(join(Y, X), Z))
% 149.40/19.47 = { by axiom 9 (absorption2) }
% 149.40/19.47 join(Y, X)
% 149.40/19.47 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.47 join(X, Y)
% 149.40/19.47
% 149.40/19.47 Lemma 46: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 149.40/19.48 Proof:
% 149.40/19.48 meet(X, complement(meet(Y, X)))
% 149.40/19.48 = { by lemma 42 R->L }
% 149.40/19.48 meet(complement(Y), join(Y, X))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 meet(complement(Y), join(X, Y))
% 149.40/19.48 = { by lemma 43 R->L }
% 149.40/19.48 meet(complement(Y), meet(join(X, Y), join(X, meet(complement(Y), join(X, Y)))))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 meet(complement(Y), meet(join(Y, X), join(X, meet(complement(Y), join(X, Y)))))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 meet(complement(Y), meet(join(Y, X), join(X, meet(complement(Y), join(Y, X)))))
% 149.40/19.48 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.48 meet(complement(Y), meet(join(X, meet(complement(Y), join(Y, X))), join(Y, X)))
% 149.40/19.48 = { by lemma 45 R->L }
% 149.40/19.48 meet(complement(Y), meet(join(X, meet(complement(Y), join(Y, X))), join(Y, join(X, meet(complement(Y), join(Y, X))))))
% 149.40/19.48 = { by lemma 27 }
% 149.40/19.48 meet(complement(Y), join(X, meet(complement(Y), join(Y, X))))
% 149.40/19.48 = { by lemma 42 }
% 149.40/19.48 meet(complement(Y), join(X, meet(X, complement(meet(Y, X)))))
% 149.40/19.48 = { by axiom 9 (absorption2) }
% 149.40/19.48 meet(complement(Y), X)
% 149.40/19.48 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.48 meet(X, complement(Y))
% 149.40/19.48
% 149.40/19.48 Lemma 47: join(X, join(Y, complement(X))) = one.
% 149.40/19.48 Proof:
% 149.40/19.48 join(X, join(Y, complement(X)))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 join(X, join(complement(X), Y))
% 149.40/19.48 = { by axiom 10 (associativity_of_join) R->L }
% 149.40/19.48 join(join(X, complement(X)), Y)
% 149.40/19.48 = { by axiom 5 (complement_join) }
% 149.40/19.48 join(one, Y)
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 join(Y, one)
% 149.40/19.48 = { by lemma 17 }
% 149.40/19.48 one
% 149.40/19.48
% 149.40/19.48 Lemma 48: join(X, meet(Y, X)) = X.
% 149.40/19.48 Proof:
% 149.40/19.48 join(X, meet(Y, X))
% 149.40/19.48 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.48 join(X, meet(X, Y))
% 149.40/19.48 = { by axiom 9 (absorption2) }
% 149.40/19.48 X
% 149.40/19.48
% 149.40/19.48 Lemma 49: join(X, join(Y, meet(Z, X))) = join(X, Y).
% 149.40/19.48 Proof:
% 149.40/19.48 join(X, join(Y, meet(Z, X)))
% 149.40/19.48 = { by lemma 44 R->L }
% 149.40/19.48 join(Y, join(X, meet(Z, X)))
% 149.40/19.48 = { by lemma 48 }
% 149.40/19.48 join(Y, X)
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.48 join(X, Y)
% 149.40/19.48
% 149.40/19.48 Lemma 50: join(meet(X, complement(Y)), meet(Z, complement(Y))) = meet(complement(Y), join(X, meet(Z, complement(Y)))).
% 149.40/19.48 Proof:
% 149.40/19.48 join(meet(X, complement(Y)), meet(Z, complement(Y)))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(X, complement(Y)))
% 149.40/19.48 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), X))
% 149.40/19.48 = { by lemma 14 R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), join(zero, X)))
% 149.40/19.48 = { by lemma 15 R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), join(meet(meet(Z, complement(Y)), zero), X)))
% 149.40/19.48 = { by axiom 2 (complement_meet) R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), join(meet(meet(Z, complement(Y)), meet(join(meet(Z, complement(Y)), Y), complement(join(meet(Z, complement(Y)), Y)))), X)))
% 149.40/19.48 = { by lemma 21 }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), join(meet(meet(Z, complement(Y)), complement(join(meet(Z, complement(Y)), Y))), X)))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(complement(Y), join(meet(meet(Z, complement(Y)), complement(join(Y, meet(Z, complement(Y))))), X)))
% 149.40/19.48 = { by lemma 24 R->L }
% 149.40/19.48 join(meet(Z, complement(Y)), meet(join(meet(Z, complement(Y)), complement(join(Y, meet(Z, complement(Y))))), join(meet(meet(Z, complement(Y)), complement(join(Y, meet(Z, complement(Y))))), X)))
% 149.40/19.48 = { by axiom 11 (equation_H61) R->L }
% 149.40/19.48 meet(join(meet(Z, complement(Y)), complement(join(Y, meet(Z, complement(Y))))), join(meet(Z, complement(Y)), X))
% 149.40/19.48 = { by lemma 24 }
% 149.40/19.48 meet(complement(Y), join(meet(Z, complement(Y)), X))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.48 meet(complement(Y), join(X, meet(Z, complement(Y))))
% 149.40/19.48
% 149.40/19.48 Lemma 51: join(X, meet(Y, join(X, Z))) = meet(join(Y, X), join(X, Z)).
% 149.40/19.48 Proof:
% 149.40/19.48 join(X, meet(Y, join(X, Z)))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.48 join(meet(Y, join(X, Z)), X)
% 149.40/19.48 = { by lemma 35 R->L }
% 149.40/19.48 join(meet(Y, join(X, Z)), meet(join(meet(Y, join(X, Z)), X), join(X, Z)))
% 149.40/19.48 = { by axiom 4 (commutativity_of_join) }
% 149.40/19.48 join(meet(Y, join(X, Z)), meet(join(X, meet(Y, join(X, Z))), join(X, Z)))
% 149.40/19.48 = { by lemma 19 R->L }
% 149.40/19.48 join(meet(Y, join(X, Z)), meet(join(X, meet(Y, join(X, Z))), complement(complement(join(X, Z)))))
% 149.40/19.48 = { by lemma 19 R->L }
% 149.40/19.49 join(meet(Y, complement(complement(join(X, Z)))), meet(join(X, meet(Y, join(X, Z))), complement(complement(join(X, Z)))))
% 149.40/19.49 = { by lemma 50 }
% 149.40/19.49 meet(complement(complement(join(X, Z))), join(Y, meet(join(X, meet(Y, join(X, Z))), complement(complement(join(X, Z))))))
% 149.40/19.49 = { by lemma 19 }
% 149.40/19.49 meet(join(X, Z), join(Y, meet(join(X, meet(Y, join(X, Z))), complement(complement(join(X, Z))))))
% 149.40/19.49 = { by lemma 19 }
% 149.40/19.49 meet(join(X, Z), join(Y, meet(join(X, meet(Y, join(X, Z))), join(X, Z))))
% 149.40/19.49 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.49 meet(join(X, Z), join(Y, meet(join(X, Z), join(X, meet(Y, join(X, Z))))))
% 149.40/19.49 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.49 meet(join(X, Z), join(Y, meet(join(X, Z), join(meet(Y, join(X, Z)), X))))
% 149.40/19.49 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.49 meet(join(X, Z), join(Y, meet(join(meet(Y, join(X, Z)), X), join(X, Z))))
% 149.40/19.49 = { by lemma 40 R->L }
% 149.40/19.49 meet(join(X, Z), join(Y, join(meet(Y, join(X, Z)), meet(join(meet(Y, join(X, Z)), X), join(X, Z)))))
% 149.40/19.49 = { by lemma 35 }
% 149.40/19.49 meet(join(X, Z), join(Y, join(meet(Y, join(X, Z)), X)))
% 149.40/19.49 = { by lemma 40 }
% 149.40/19.49 meet(join(X, Z), join(Y, X))
% 149.40/19.49 = { by axiom 1 (commutativity_of_meet) }
% 149.40/19.49 meet(join(Y, X), join(X, Z))
% 149.40/19.49
% 149.40/19.49 Lemma 52: meet(complement(X), complement(join(X, Y))) = complement(join(X, Y)).
% 149.40/19.49 Proof:
% 149.40/19.49 meet(complement(X), complement(join(X, Y)))
% 149.40/19.49 = { by axiom 4 (commutativity_of_join) R->L }
% 149.40/19.49 meet(complement(X), complement(join(Y, X)))
% 149.40/19.49 = { by axiom 1 (commutativity_of_meet) R->L }
% 149.40/19.49 meet(complement(join(Y, X)), complement(X))
% 149.40/19.49 = { by lemma 19 R->L }
% 149.40/19.49 meet(complement(join(Y, X)), complement(complement(complement(X))))
% 149.40/19.49 = { by axiom 3 (idempotence_of_join) R->L }
% 149.40/19.49 meet(complement(join(Y, X)), join(complement(complement(complement(X))), complement(complement(complement(X)))))
% 149.40/19.49 = { by lemma 24 R->L }
% 149.40/19.49 meet(complement(join(Y, X)), join(complement(complement(complement(X))), join(meet(join(complement(complement(X)), Y), complement(complement(complement(X)))), complement(join(complement(complement(X)), meet(join(complement(complement(X)), Y), complement(complement(complement(X)))))))))
% 150.27/19.49 = { by axiom 4 (commutativity_of_join) R->L }
% 150.27/19.49 meet(complement(join(Y, X)), join(complement(complement(complement(X))), join(complement(join(complement(complement(X)), meet(join(complement(complement(X)), Y), complement(complement(complement(X)))))), meet(join(complement(complement(X)), Y), complement(complement(complement(X)))))))
% 150.27/19.49 = { by lemma 49 }
% 150.27/19.49 meet(complement(join(Y, X)), join(complement(complement(complement(X))), complement(join(complement(complement(X)), meet(join(complement(complement(X)), Y), complement(complement(complement(X))))))))
% 150.27/19.49 = { by lemma 19 }
% 150.27/19.49 meet(complement(join(Y, X)), join(complement(X), complement(join(complement(complement(X)), meet(join(complement(complement(X)), Y), complement(complement(complement(X))))))))
% 150.27/19.49 = { by lemma 19 }
% 150.27/19.49 meet(complement(join(Y, X)), join(complement(X), complement(join(complement(complement(X)), meet(join(complement(complement(X)), Y), complement(X))))))
% 150.27/19.49 = { by axiom 1 (commutativity_of_meet) }
% 150.27/19.49 meet(complement(join(Y, X)), join(complement(X), complement(join(complement(complement(X)), meet(complement(X), join(complement(complement(X)), Y))))))
% 150.27/19.49 = { by lemma 51 }
% 150.27/19.50 meet(complement(join(Y, X)), join(complement(X), complement(meet(join(complement(X), complement(complement(X))), join(complement(complement(X)), Y)))))
% 150.27/19.50 = { by axiom 5 (complement_join) }
% 150.27/19.50 meet(complement(join(Y, X)), join(complement(X), complement(meet(one, join(complement(complement(X)), Y)))))
% 150.27/19.50 = { by lemma 16 }
% 150.27/19.50 meet(complement(join(Y, X)), join(complement(X), complement(join(complement(complement(X)), Y))))
% 150.27/19.50 = { by axiom 4 (commutativity_of_join) }
% 150.27/19.50 meet(complement(join(Y, X)), join(complement(X), complement(join(Y, complement(complement(X))))))
% 150.27/19.50 = { by lemma 19 }
% 150.27/19.50 meet(complement(join(Y, X)), join(complement(X), complement(join(Y, X))))
% 150.27/19.50 = { by lemma 27 }
% 150.27/19.50 complement(join(Y, X))
% 150.27/19.50 = { by axiom 4 (commutativity_of_join) }
% 150.27/19.50 complement(join(X, Y))
% 150.27/19.50
% 150.27/19.50 Lemma 53: join(X, meet(Y, complement(X))) = join(X, Y).
% 150.27/19.50 Proof:
% 150.27/19.50 join(X, meet(Y, complement(X)))
% 150.27/19.50 = { by lemma 46 R->L }
% 150.27/19.50 join(X, meet(Y, complement(meet(X, Y))))
% 150.27/19.50 = { by lemma 41 }
% 150.27/19.50 join(X, Y)
% 150.27/19.50
% 150.27/19.50 Lemma 54: complement(join(X, Y)) = meet(complement(X), complement(Y)).
% 150.27/19.50 Proof:
% 150.27/19.50 complement(join(X, Y))
% 150.27/19.50 = { by axiom 4 (commutativity_of_join) R->L }
% 150.27/19.50 complement(join(Y, X))
% 150.27/19.50 = { by lemma 52 R->L }
% 150.27/19.50 meet(complement(Y), complement(join(Y, X)))
% 150.27/19.50 = { by lemma 53 R->L }
% 150.27/19.50 meet(complement(Y), complement(join(Y, meet(X, complement(Y)))))
% 150.27/19.50 = { by lemma 52 }
% 150.27/19.50 complement(join(Y, meet(X, complement(Y))))
% 150.27/19.50 = { by lemma 36 R->L }
% 150.27/19.50 meet(complement(Y), join(Y, complement(join(Y, meet(X, complement(Y))))))
% 150.27/19.50 = { by lemma 29 }
% 150.27/19.50 meet(complement(Y), complement(meet(X, complement(Y))))
% 150.27/19.50 = { by lemma 46 }
% 150.27/19.50 meet(complement(Y), complement(X))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) }
% 150.27/19.50 meet(complement(X), complement(Y))
% 150.27/19.50
% 150.27/19.50 Lemma 55: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 150.27/19.50 Proof:
% 150.27/19.50 complement(meet(X, Y))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.27/19.50 complement(meet(Y, X))
% 150.27/19.50 = { by lemma 31 R->L }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(Y, X))))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(X, Y))))
% 150.27/19.50 = { by lemma 19 R->L }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(X, complement(complement(Y))))))
% 150.27/19.50 = { by lemma 46 R->L }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(X, complement(meet(complement(Y), X))))))
% 150.27/19.50 = { by lemma 42 R->L }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(complement(complement(Y)), join(complement(Y), X)))))
% 150.27/19.50 = { by lemma 19 }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(Y, join(complement(Y), X)))))
% 150.27/19.50 = { by axiom 4 (commutativity_of_join) }
% 150.27/19.50 join(complement(Y), complement(join(complement(Y), meet(Y, join(X, complement(Y))))))
% 150.27/19.50 = { by lemma 31 }
% 150.27/19.50 complement(meet(Y, join(X, complement(Y))))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.27/19.50 complement(meet(join(X, complement(Y)), Y))
% 150.27/19.50 = { by lemma 30 R->L }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(join(complement(join(X, complement(Y))), meet(Y, join(X, complement(Y))))))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(join(complement(join(X, complement(Y))), meet(join(X, complement(Y)), Y))))
% 150.27/19.50 = { by lemma 19 R->L }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(complement(complement(join(complement(join(X, complement(Y))), meet(join(X, complement(Y)), Y))))))
% 150.27/19.50 = { by lemma 37 }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(complement(meet(join(X, complement(Y)), complement(meet(join(X, complement(Y)), Y))))))
% 150.27/19.50 = { by axiom 1 (commutativity_of_meet) }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))))))
% 150.27/19.50 = { by axiom 6 (ifeq_axiom) R->L }
% 150.27/19.50 join(complement(join(X, complement(Y))), complement(ifeq(one, one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.50 = { by lemma 47 R->L }
% 150.39/19.50 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, join(meet(join(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))), join(meet(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))), X)), complement(Y))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.50 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.50 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, join(complement(Y), meet(join(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))), join(meet(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))), X)))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.50 = { by axiom 11 (equation_H61) R->L }
% 150.39/19.50 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))), join(complement(Y), X))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(complement(Y), X), join(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(X, complement(Y)), join(complement(Y), complement(join(complement(Y), meet(join(X, complement(Y)), complement(complement(Y)))))))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by lemma 29 }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(join(X, complement(Y)), complement(complement(Y)))))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by lemma 19 }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(join(X, complement(Y)), Y)))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), Y), one, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y)))
% 150.39/19.51 = { by axiom 6 (ifeq_axiom) R->L }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), Y), one, ifeq(zero, zero, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y), Y)))
% 150.39/19.51 = { by lemma 39 R->L }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(ifeq(join(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y))))), Y), one, ifeq(meet(Y, meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), zero, complement(meet(join(X, complement(Y)), complement(meet(Y, join(X, complement(Y)))))), Y), Y)))
% 150.39/19.51 = { by lemma 18 }
% 150.39/19.51 join(complement(join(X, complement(Y))), complement(Y))
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.51 join(complement(Y), complement(join(X, complement(Y))))
% 150.39/19.51 = { by lemma 54 }
% 150.39/19.51 join(complement(Y), meet(complement(X), complement(complement(Y))))
% 150.39/19.51 = { by lemma 53 }
% 150.39/19.51 join(complement(Y), complement(X))
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.51 join(complement(X), complement(Y))
% 150.39/19.51
% 150.39/19.51 Lemma 56: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 150.39/19.51 Proof:
% 150.39/19.51 join(complement(X), meet(Y, X))
% 150.39/19.51 = { by lemma 19 R->L }
% 150.39/19.51 join(complement(X), complement(complement(meet(Y, X))))
% 150.39/19.51 = { by lemma 55 R->L }
% 150.39/19.51 complement(meet(X, complement(meet(Y, X))))
% 150.39/19.51 = { by lemma 42 R->L }
% 150.39/19.51 complement(meet(complement(Y), join(Y, X)))
% 150.39/19.51 = { by lemma 38 R->L }
% 150.39/19.51 complement(complement(join(Y, complement(join(Y, X)))))
% 150.39/19.51 = { by lemma 19 }
% 150.39/19.51 join(Y, complement(join(Y, X)))
% 150.39/19.51 = { by lemma 54 }
% 150.39/19.51 join(Y, meet(complement(Y), complement(X)))
% 150.39/19.51 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.51 join(Y, meet(complement(X), complement(Y)))
% 150.39/19.51 = { by lemma 53 }
% 150.39/19.51 join(Y, complement(X))
% 150.39/19.51
% 150.39/19.51 Lemma 57: ifeq(join(X, complement(Y)), one, join(X, Y), X) = X.
% 150.39/19.51 Proof:
% 150.39/19.51 ifeq(join(X, complement(Y)), one, join(X, Y), X)
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.51 ifeq(join(complement(Y), X), one, join(X, Y), X)
% 150.39/19.51 = { by lemma 53 R->L }
% 150.39/19.51 ifeq(join(complement(Y), X), one, join(X, meet(Y, complement(X))), X)
% 150.39/19.51 = { by lemma 19 R->L }
% 150.39/19.51 ifeq(join(complement(Y), complement(complement(X))), one, join(X, meet(Y, complement(X))), X)
% 150.39/19.51 = { by lemma 55 R->L }
% 150.39/19.51 ifeq(complement(meet(Y, complement(X))), one, join(X, meet(Y, complement(X))), X)
% 150.39/19.51 = { by lemma 29 R->L }
% 150.39/19.51 ifeq(join(X, complement(join(X, meet(Y, complement(X))))), one, join(X, meet(Y, complement(X))), X)
% 150.39/19.51 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.51 ifeq(join(complement(join(X, meet(Y, complement(X)))), X), one, join(X, meet(Y, complement(X))), X)
% 150.39/19.51 = { by axiom 7 (absorption1) R->L }
% 150.39/19.51 ifeq(join(complement(join(X, meet(Y, complement(X)))), X), one, join(X, meet(Y, complement(X))), meet(X, join(X, meet(Y, complement(X)))))
% 150.39/19.51 = { by axiom 7 (absorption1) R->L }
% 150.39/19.51 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, join(X, meet(Y, complement(X))))), one, join(X, meet(Y, complement(X))), meet(X, join(X, meet(Y, complement(X)))))
% 150.39/19.51 = { by lemma 19 R->L }
% 150.39/19.51 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, join(X, meet(Y, complement(X))))), one, complement(complement(join(X, meet(Y, complement(X))))), meet(X, join(X, meet(Y, complement(X)))))
% 150.39/19.51 = { by lemma 19 R->L }
% 150.39/19.51 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, join(X, meet(Y, complement(X))))), one, complement(complement(join(X, meet(Y, complement(X))))), meet(X, complement(complement(join(X, meet(Y, complement(X)))))))
% 150.39/19.51 = { by lemma 19 R->L }
% 150.39/19.52 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), one, complement(complement(join(X, meet(Y, complement(X))))), meet(X, complement(complement(join(X, meet(Y, complement(X)))))))
% 150.39/19.52 = { by axiom 6 (ifeq_axiom) R->L }
% 150.39/19.52 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), one, ifeq(zero, zero, complement(complement(join(X, meet(Y, complement(X))))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), meet(X, complement(complement(join(X, meet(Y, complement(X)))))))
% 150.39/19.52 = { by lemma 20 R->L }
% 150.39/19.52 ifeq(join(complement(join(X, meet(Y, complement(X)))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), one, ifeq(meet(complement(join(X, meet(Y, complement(X)))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), zero, complement(complement(join(X, meet(Y, complement(X))))), meet(X, complement(complement(join(X, meet(Y, complement(X))))))), meet(X, complement(complement(join(X, meet(Y, complement(X)))))))
% 150.39/19.52 = { by axiom 12 (meet_join_complement) }
% 150.39/19.52 meet(X, complement(complement(join(X, meet(Y, complement(X))))))
% 150.39/19.52 = { by lemma 19 }
% 150.39/19.52 meet(X, join(X, meet(Y, complement(X))))
% 150.39/19.52 = { by axiom 7 (absorption1) }
% 150.39/19.52 X
% 150.39/19.52
% 150.39/19.52 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 150.39/19.52 Proof:
% 150.39/19.52 meet(a, join(b, c))
% 150.39/19.52 = { by lemma 27 R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(b, meet(a, join(b, c))))
% 150.39/19.52 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(b, meet(join(b, c), a)))
% 150.39/19.52 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(meet(join(b, c), a), b))
% 150.39/19.52 = { by lemma 34 R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(meet(join(b, c), a), join(b, meet(meet(join(b, c), a), complement(meet(b, complement(c)))))))
% 150.39/19.52 = { by lemma 19 R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(meet(join(b, c), a), join(b, meet(complement(complement(meet(join(b, c), a))), complement(meet(b, complement(c)))))))
% 150.39/19.52 = { by lemma 54 R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(meet(join(b, c), a), join(b, complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.52 = { by lemma 56 R->L }
% 150.39/19.52 meet(meet(a, join(b, c)), join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.52 = { by axiom 6 (ifeq_axiom) R->L }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(one, one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by lemma 23 R->L }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), meet(b, join(complement(meet(join(b, c), a)), meet(complement(c), b))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by lemma 48 R->L }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(complement(c), b))), meet(complement(c), meet(b, join(complement(meet(join(b, c), a)), meet(complement(c), b))))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by lemma 43 }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(complement(c), b))), meet(complement(c), b)))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.53 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), join(meet(complement(c), b), meet(b, join(complement(meet(join(b, c), a)), meet(complement(c), b)))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.53 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), join(meet(complement(c), b), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), join(meet(b, complement(c)), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by lemma 45 }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))), complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), one, join(meet(join(b, c), a), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(complement(meet(join(b, c), a)), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))), one, join(join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))), meet(join(b, c), a)), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.54 meet(meet(a, join(b, c)), ifeq(join(join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))), complement(meet(join(b, c), a))), one, join(join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))), meet(join(b, c), a)), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c)))))))
% 150.39/19.54 = { by lemma 57 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(complement(join(complement(meet(join(b, c), a)), meet(b, complement(c)))), meet(b, join(complement(meet(join(b, c), a)), meet(b, complement(c))))))
% 150.39/19.54 = { by lemma 56 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, complement(join(complement(meet(join(b, c), a)), meet(b, complement(c))))))
% 150.39/19.54 = { by lemma 54 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(complement(complement(meet(join(b, c), a))), complement(meet(b, complement(c))))))
% 150.39/19.54 = { by lemma 19 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(meet(join(b, c), a), complement(meet(b, complement(c))))))
% 150.39/19.54 = { by lemma 55 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(meet(join(b, c), a), join(complement(b), complement(complement(c))))))
% 150.39/19.54 = { by lemma 19 }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(meet(join(b, c), a), join(complement(b), c))))
% 150.39/19.54 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(join(complement(b), c), meet(join(b, c), a))))
% 150.39/19.54 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.54 meet(meet(a, join(b, c)), join(b, meet(join(c, complement(b)), meet(join(b, c), a))))
% 150.39/19.55 = { by lemma 27 R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(b, meet(join(c, complement(b)), meet(join(b, c), a))), join(join(c, complement(b)), join(b, meet(join(c, complement(b)), meet(join(b, c), a))))))
% 150.39/19.55 = { by lemma 34 }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(b, meet(join(c, complement(b)), meet(join(b, c), a))), join(join(c, complement(b)), b)))
% 150.39/19.55 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(join(c, complement(b)), b), join(b, meet(join(c, complement(b)), meet(join(b, c), a)))))
% 150.39/19.55 = { by axiom 4 (commutativity_of_join) }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(b, join(c, complement(b))), join(b, meet(join(c, complement(b)), meet(join(b, c), a)))))
% 150.39/19.55 = { by lemma 26 R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(b, join(c, complement(b))), join(b, meet(join(b, c), meet(join(c, complement(b)), a)))))
% 150.39/19.55 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(join(b, join(c, complement(b))), join(b, meet(join(b, c), meet(a, join(c, complement(b)))))))
% 150.39/19.55 = { by lemma 47 }
% 150.39/19.55 meet(meet(a, join(b, c)), meet(one, join(b, meet(join(b, c), meet(a, join(c, complement(b)))))))
% 150.39/19.55 = { by lemma 16 }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(join(b, c), meet(a, join(c, complement(b))))))
% 150.39/19.55 = { by lemma 26 R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, meet(join(b, c), join(c, complement(b))))))
% 150.39/19.55 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, meet(join(c, b), join(c, complement(b))))))
% 150.39/19.55 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, meet(join(c, b), join(complement(b), c)))))
% 150.39/19.55 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, meet(join(complement(b), c), join(c, b)))))
% 150.39/19.55 = { by lemma 51 R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, join(c, meet(complement(b), join(c, b))))))
% 150.39/19.55 = { by axiom 6 (ifeq_axiom) R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(one, one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by lemma 17 R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(join(c, b), one), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by axiom 5 (complement_join) R->L }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(join(c, b), join(meet(join(c, b), complement(b)), complement(meet(join(c, b), complement(b))))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by lemma 40 }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(join(c, b), complement(meet(join(c, b), complement(b)))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(join(c, b), complement(meet(complement(b), join(c, b)))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by axiom 10 (associativity_of_join) }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, join(b, complement(meet(complement(b), join(c, b))))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.55 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, join(b, complement(meet(join(c, b), complement(b))))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.55 = { by lemma 29 R->L }
% 150.39/19.56 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, join(b, join(b, complement(join(b, meet(join(c, b), complement(b))))))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.56 = { by lemma 32 }
% 150.39/19.56 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, join(b, complement(join(b, meet(join(c, b), complement(b)))))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.56 = { by lemma 29 }
% 150.39/19.56 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, complement(meet(join(c, b), complement(b)))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.56 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.56 meet(meet(a, join(b, c)), join(b, meet(a, ifeq(join(c, complement(meet(complement(b), join(c, b)))), one, join(c, meet(complement(b), join(c, b))), c))))
% 150.39/19.56 = { by lemma 57 }
% 150.39/19.56 meet(meet(a, join(b, c)), join(b, meet(a, c)))
% 150.39/19.56 = { by axiom 8 (associativity_of_meet) }
% 150.39/19.56 meet(a, meet(join(b, c), join(b, meet(a, c))))
% 150.39/19.56 = { by axiom 4 (commutativity_of_join) R->L }
% 150.39/19.56 meet(a, meet(join(c, b), join(b, meet(a, c))))
% 150.39/19.56 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.39/19.56 meet(a, meet(join(b, meet(a, c)), join(c, b)))
% 150.39/19.56 = { by lemma 49 R->L }
% 150.39/19.56 meet(a, meet(join(b, meet(a, c)), join(c, join(b, meet(a, c)))))
% 150.39/19.56 = { by lemma 27 }
% 150.39/19.56 meet(a, join(b, meet(a, c)))
% 150.39/19.56 = { by axiom 1 (commutativity_of_meet) R->L }
% 150.39/19.56 meet(a, join(b, meet(c, a)))
% 150.39/19.56 = { by lemma 19 R->L }
% 150.39/19.56 meet(a, join(b, meet(c, complement(complement(a)))))
% 150.39/19.56 = { by lemma 19 R->L }
% 150.39/19.56 meet(complement(complement(a)), join(b, meet(c, complement(complement(a)))))
% 150.39/19.56 = { by lemma 50 R->L }
% 150.39/19.56 join(meet(b, complement(complement(a))), meet(c, complement(complement(a))))
% 150.39/19.56 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.56 join(meet(complement(complement(a)), b), meet(c, complement(complement(a))))
% 150.39/19.56 = { by lemma 19 }
% 150.39/19.56 join(meet(a, b), meet(c, complement(complement(a))))
% 150.39/19.56 = { by lemma 19 }
% 150.39/19.56 join(meet(a, b), meet(c, a))
% 150.39/19.56 = { by axiom 1 (commutativity_of_meet) }
% 150.39/19.56 join(meet(a, b), meet(a, c))
% 150.39/19.56 % SZS output end Proof
% 150.39/19.56
% 150.39/19.56 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------