TSTP Solution File: LAT190-10 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LAT190-10 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n009.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 06:27:42 EDT 2023
% Result : Unsatisfiable 81.41s 10.80s
% Output : Proof 83.03s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : LAT190-10 : TPTP v8.1.2. Released v7.5.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Thu Aug 24 06:07:21 EDT 2023
% 0.13/0.34 % CPUTime :
% 81.41/10.80 Command-line arguments: --no-flatten-goal
% 81.41/10.80
% 81.41/10.80 % SZS status Unsatisfiable
% 81.41/10.80
% 82.64/10.90 % SZS output start Proof
% 82.64/10.90 Axiom 1 (commutativity_of_join): join(X, Y) = join(Y, X).
% 82.64/10.90 Axiom 2 (idempotence_of_meet): meet(X, X) = X.
% 82.64/10.90 Axiom 3 (commutativity_of_meet): meet(X, Y) = meet(Y, X).
% 82.64/10.90 Axiom 4 (complement_join): join(X, complement(X)) = one.
% 82.64/10.90 Axiom 5 (complement_meet): meet(X, complement(X)) = zero.
% 82.64/10.90 Axiom 6 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 82.64/10.90 Axiom 7 (absorption2): join(X, meet(X, Y)) = X.
% 82.64/10.90 Axiom 8 (associativity_of_join): join(join(X, Y), Z) = join(X, join(Y, Z)).
% 82.64/10.90 Axiom 9 (absorption1): meet(X, join(X, Y)) = X.
% 82.64/10.90 Axiom 10 (associativity_of_meet): meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 82.64/10.90 Axiom 11 (meet_join_complement): ifeq(join(X, Y), one, ifeq(meet(X, Y), zero, complement(X), Y), Y) = Y.
% 82.64/10.90 Axiom 12 (equation_H18): join(meet(X, Y), meet(X, Z)) = meet(X, join(meet(X, Y), join(meet(X, Z), meet(Y, join(X, Z))))).
% 82.64/10.90
% 82.64/10.90 Lemma 13: meet(X, one) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 meet(X, one)
% 82.64/10.90 = { by axiom 4 (complement_join) R->L }
% 82.64/10.90 meet(X, join(X, complement(X)))
% 82.64/10.90 = { by axiom 9 (absorption1) }
% 82.64/10.90 X
% 82.64/10.90
% 82.64/10.90 Lemma 14: meet(one, X) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 meet(one, X)
% 82.64/10.90 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.90 meet(X, one)
% 82.64/10.90 = { by lemma 13 }
% 82.64/10.90 X
% 82.64/10.90
% 82.64/10.90 Lemma 15: join(X, one) = one.
% 82.64/10.90 Proof:
% 82.64/10.90 join(X, one)
% 82.64/10.90 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.90 join(one, X)
% 82.64/10.90 = { by lemma 14 R->L }
% 82.64/10.90 join(one, meet(one, X))
% 82.64/10.90 = { by axiom 7 (absorption2) }
% 82.64/10.90 one
% 82.64/10.90
% 82.64/10.90 Lemma 16: join(X, zero) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 join(X, zero)
% 82.64/10.90 = { by axiom 5 (complement_meet) R->L }
% 82.64/10.90 join(X, meet(X, complement(X)))
% 82.64/10.90 = { by axiom 7 (absorption2) }
% 82.64/10.90 X
% 82.64/10.90
% 82.64/10.90 Lemma 17: ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 ifeq(join(X, Y), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 82.64/10.90 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.90 ifeq(join(Y, X), one, ifeq(meet(Y, X), zero, complement(Y), X), X)
% 82.64/10.90 = { by axiom 11 (meet_join_complement) }
% 82.64/10.90 X
% 82.64/10.90
% 82.64/10.90 Lemma 18: complement(complement(X)) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 complement(complement(X))
% 82.64/10.90 = { by axiom 6 (ifeq_axiom) R->L }
% 82.64/10.90 ifeq(zero, zero, complement(complement(X)), X)
% 82.64/10.90 = { by axiom 5 (complement_meet) R->L }
% 82.64/10.90 ifeq(meet(X, complement(X)), zero, complement(complement(X)), X)
% 82.64/10.90 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.90 ifeq(meet(complement(X), X), zero, complement(complement(X)), X)
% 82.64/10.90 = { by axiom 6 (ifeq_axiom) R->L }
% 82.64/10.90 ifeq(one, one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 82.64/10.90 = { by axiom 4 (complement_join) R->L }
% 82.64/10.90 ifeq(join(X, complement(X)), one, ifeq(meet(complement(X), X), zero, complement(complement(X)), X), X)
% 82.64/10.90 = { by lemma 17 }
% 82.64/10.90 X
% 82.64/10.90
% 82.64/10.90 Lemma 19: meet(X, meet(X, Y)) = meet(X, Y).
% 82.64/10.90 Proof:
% 82.64/10.90 meet(X, meet(X, Y))
% 82.64/10.90 = { by axiom 10 (associativity_of_meet) R->L }
% 82.64/10.90 meet(meet(X, X), Y)
% 82.64/10.90 = { by axiom 2 (idempotence_of_meet) }
% 82.64/10.90 meet(X, Y)
% 82.64/10.90
% 82.64/10.90 Lemma 20: meet(X, meet(Y, X)) = meet(X, Y).
% 82.64/10.90 Proof:
% 82.64/10.90 meet(X, meet(Y, X))
% 82.64/10.90 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.90 meet(X, meet(X, Y))
% 82.64/10.90 = { by lemma 19 }
% 82.64/10.90 meet(X, Y)
% 82.64/10.90
% 82.64/10.90 Lemma 21: meet(X, join(Y, X)) = X.
% 82.64/10.90 Proof:
% 82.64/10.90 meet(X, join(Y, X))
% 82.64/10.90 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.90 meet(X, join(X, Y))
% 82.64/10.90 = { by axiom 9 (absorption1) }
% 82.64/10.91 X
% 82.64/10.91
% 82.64/10.91 Lemma 22: join(X, meet(Y, X)) = X.
% 82.64/10.91 Proof:
% 82.64/10.91 join(X, meet(Y, X))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.91 join(X, meet(X, Y))
% 82.64/10.91 = { by axiom 7 (absorption2) }
% 82.64/10.91 X
% 82.64/10.91
% 82.64/10.91 Lemma 23: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 82.64/10.91 Proof:
% 82.64/10.91 join(Y, join(X, Z))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 join(join(X, Z), Y)
% 82.64/10.91 = { by axiom 8 (associativity_of_join) }
% 82.64/10.91 join(X, join(Z, Y))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 join(X, join(Y, Z))
% 82.64/10.91
% 82.64/10.91 Lemma 24: meet(X, meet(Y, join(X, Z))) = meet(X, Y).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, meet(Y, join(X, Z)))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.91 meet(X, meet(join(X, Z), Y))
% 82.64/10.91 = { by axiom 10 (associativity_of_meet) R->L }
% 82.64/10.91 meet(meet(X, join(X, Z)), Y)
% 82.64/10.91 = { by axiom 9 (absorption1) }
% 82.64/10.91 meet(X, Y)
% 82.64/10.91
% 82.64/10.91 Lemma 25: meet(X, meet(join(X, Y), Z)) = meet(X, Z).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, meet(join(X, Y), Z))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.91 meet(X, meet(Z, join(X, Y)))
% 82.64/10.91 = { by lemma 24 }
% 82.64/10.91 meet(X, Z)
% 82.64/10.91
% 82.64/10.91 Lemma 26: join(X, join(Y, meet(Z, join(X, Y)))) = join(X, Y).
% 82.64/10.91 Proof:
% 82.64/10.91 join(X, join(Y, meet(Z, join(X, Y))))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.91 join(X, join(Y, meet(join(X, Y), Z)))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) R->L }
% 82.64/10.91 join(join(X, Y), meet(join(X, Y), Z))
% 82.64/10.91 = { by axiom 7 (absorption2) }
% 82.64/10.91 join(X, Y)
% 82.64/10.91
% 82.64/10.91 Lemma 27: join(X, join(meet(X, Y), Z)) = join(X, Z).
% 82.64/10.91 Proof:
% 82.64/10.91 join(X, join(meet(X, Y), Z))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) R->L }
% 82.64/10.91 join(join(X, meet(X, Y)), Z)
% 82.64/10.91 = { by axiom 7 (absorption2) }
% 82.64/10.91 join(X, Z)
% 82.64/10.91
% 82.64/10.91 Lemma 28: join(meet(X, Y), meet(Y, join(X, Z))) = meet(Y, join(X, Z)).
% 82.64/10.91 Proof:
% 82.64/10.91 join(meet(X, Y), meet(Y, join(X, Z)))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 join(meet(Y, join(X, Z)), meet(X, Y))
% 82.64/10.91 = { by lemma 24 R->L }
% 82.64/10.91 join(meet(Y, join(X, Z)), meet(X, meet(Y, join(X, Z))))
% 82.64/10.91 = { by lemma 22 }
% 82.64/10.91 meet(Y, join(X, Z))
% 82.64/10.91
% 82.64/10.91 Lemma 29: meet(X, join(meet(X, Y), meet(Z, join(X, Y)))) = join(meet(X, Z), meet(X, Y)).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, join(meet(X, Y), meet(Z, join(X, Y))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 meet(X, join(meet(Z, join(X, Y)), meet(X, Y)))
% 82.64/10.91 = { by lemma 28 R->L }
% 82.64/10.91 meet(X, join(join(meet(X, Z), meet(Z, join(X, Y))), meet(X, Y)))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) }
% 82.64/10.91 meet(X, join(meet(X, Z), join(meet(Z, join(X, Y)), meet(X, Y))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 meet(X, join(meet(X, Z), join(meet(X, Y), meet(Z, join(X, Y)))))
% 82.64/10.91 = { by axiom 12 (equation_H18) R->L }
% 82.64/10.91 join(meet(X, Z), meet(X, Y))
% 82.64/10.91
% 82.64/10.91 Lemma 30: join(X, join(Y, complement(X))) = one.
% 82.64/10.91 Proof:
% 82.64/10.91 join(X, join(Y, complement(X)))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 join(X, join(complement(X), Y))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) R->L }
% 82.64/10.91 join(join(X, complement(X)), Y)
% 82.64/10.91 = { by axiom 4 (complement_join) }
% 82.64/10.91 join(one, Y)
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 join(Y, one)
% 82.64/10.91 = { by lemma 15 }
% 82.64/10.91 one
% 82.64/10.91
% 82.64/10.91 Lemma 31: meet(X, meet(Y, complement(meet(X, Y)))) = zero.
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, meet(Y, complement(meet(X, Y))))
% 82.64/10.91 = { by axiom 10 (associativity_of_meet) R->L }
% 82.64/10.91 meet(meet(X, Y), complement(meet(X, Y)))
% 82.64/10.91 = { by axiom 5 (complement_meet) }
% 82.64/10.91 zero
% 82.64/10.91
% 82.64/10.91 Lemma 32: meet(join(X, Y), complement(meet(complement(X), join(X, Y)))) = X.
% 82.64/10.91 Proof:
% 82.64/10.91 meet(join(X, Y), complement(meet(complement(X), join(X, Y))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 meet(join(Y, X), complement(meet(complement(X), join(X, Y))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 meet(join(Y, X), complement(meet(complement(X), join(Y, X))))
% 82.64/10.91 = { by lemma 18 R->L }
% 82.64/10.91 meet(join(Y, X), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 82.64/10.91 = { by lemma 18 R->L }
% 82.64/10.91 meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))
% 82.64/10.91 = { by axiom 11 (meet_join_complement) R->L }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 27 R->L }
% 82.64/10.91 ifeq(join(complement(X), join(meet(complement(X), join(Y, complement(complement(X)))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 ifeq(join(complement(X), join(meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), meet(complement(X), join(Y, complement(complement(X)))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.91 ifeq(join(complement(X), join(meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))), meet(join(Y, complement(complement(X))), complement(X)))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 29 R->L }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(X)), meet(complement(meet(complement(X), join(Y, complement(complement(X))))), join(join(Y, complement(complement(X))), complement(X)))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(X)), meet(complement(meet(complement(X), join(Y, complement(complement(X))))), join(complement(X), join(Y, complement(complement(X)))))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 30 }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(X)), meet(complement(meet(complement(X), join(Y, complement(complement(X))))), one)))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 13 }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), join(meet(join(Y, complement(complement(X))), complement(X)), complement(meet(complement(X), join(Y, complement(complement(X)))))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), join(meet(complement(X), join(Y, complement(complement(X)))), complement(meet(complement(X), join(Y, complement(complement(X)))))))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 4 (complement_join) }
% 82.64/10.91 ifeq(join(complement(X), meet(join(Y, complement(complement(X))), one)), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 13 }
% 82.64/10.91 ifeq(join(complement(X), join(Y, complement(complement(X)))), one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 30 }
% 82.64/10.91 ifeq(one, one, ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 6 (ifeq_axiom) }
% 82.64/10.91 ifeq(meet(complement(X), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X))))))), zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by lemma 31 }
% 82.64/10.91 ifeq(zero, zero, complement(complement(X)), meet(join(Y, complement(complement(X))), complement(meet(complement(X), join(Y, complement(complement(X)))))))
% 82.64/10.91 = { by axiom 6 (ifeq_axiom) }
% 82.64/10.91 complement(complement(X))
% 82.64/10.91 = { by lemma 18 }
% 82.64/10.91 X
% 82.64/10.91
% 82.64/10.91 Lemma 33: meet(join(X, Y), join(X, Z)) = join(X, meet(Z, join(X, Y))).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(join(X, Y), join(X, Z))
% 82.64/10.91 = { by lemma 13 R->L }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, one)))
% 82.64/10.91 = { by lemma 15 R->L }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), one))))
% 82.64/10.91 = { by axiom 4 (complement_join) R->L }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), join(meet(join(X, Y), complement(X)), complement(meet(join(X, Y), complement(X))))))))
% 82.64/10.91 = { by lemma 27 }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(join(X, Y), complement(X)))))))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(join(X, Y), complement(X))))))))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(X, join(Y, complement(meet(complement(X), join(X, Y))))))))
% 82.64/10.91 = { by axiom 8 (associativity_of_join) R->L }
% 82.64/10.91 meet(join(X, Y), join(X, meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 82.64/10.91 = { by lemma 32 R->L }
% 82.64/10.91 meet(join(X, Y), join(meet(join(X, Y), complement(meet(complement(X), join(X, Y)))), meet(Z, join(join(X, Y), complement(meet(complement(X), join(X, Y)))))))
% 82.64/10.91 = { by lemma 29 }
% 82.64/10.91 join(meet(join(X, Y), Z), meet(join(X, Y), complement(meet(complement(X), join(X, Y)))))
% 82.64/10.91 = { by lemma 32 }
% 82.64/10.91 join(meet(join(X, Y), Z), X)
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 join(X, meet(join(X, Y), Z))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.91 join(X, meet(Z, join(X, Y)))
% 82.64/10.91
% 82.64/10.91 Lemma 34: meet(X, meet(Y, join(Z, meet(X, Y)))) = meet(X, Y).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, meet(Y, join(Z, meet(X, Y))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 meet(X, meet(Y, join(meet(X, Y), Z)))
% 82.64/10.91 = { by axiom 10 (associativity_of_meet) R->L }
% 82.64/10.91 meet(meet(X, Y), join(meet(X, Y), Z))
% 82.64/10.91 = { by axiom 9 (absorption1) }
% 82.64/10.91 meet(X, Y)
% 82.64/10.91
% 82.64/10.91 Lemma 35: meet(X, join(Y, meet(X, join(Y, Z)))) = meet(X, join(Y, Z)).
% 82.64/10.91 Proof:
% 82.64/10.91 meet(X, join(Y, meet(X, join(Y, Z))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.91 meet(X, join(Y, meet(X, join(Z, Y))))
% 82.64/10.91 = { by lemma 21 R->L }
% 82.64/10.91 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, join(Y, meet(X, join(Z, Y))))))
% 82.64/10.91 = { by lemma 26 }
% 82.64/10.91 meet(X, meet(join(Y, meet(X, join(Z, Y))), join(Z, Y)))
% 82.64/10.91 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.91 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Z, Y)))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 meet(X, meet(join(Z, Y), join(Y, meet(X, join(Y, Z)))))
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.91 meet(X, meet(join(Y, Z), join(Y, meet(X, join(Y, Z)))))
% 82.64/10.91 = { by lemma 34 }
% 82.64/10.91 meet(X, join(Y, Z))
% 82.64/10.91
% 82.64/10.91 Lemma 36: join(X, join(Y, meet(Z, X))) = join(X, Y).
% 82.64/10.91 Proof:
% 82.64/10.91 join(X, join(Y, meet(Z, X)))
% 82.64/10.91 = { by lemma 23 }
% 82.64/10.91 join(Y, join(X, meet(Z, X)))
% 82.64/10.91 = { by lemma 22 }
% 82.64/10.91 join(Y, X)
% 82.64/10.91 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.92 join(X, Y)
% 82.64/10.92
% 82.64/10.92 Lemma 37: join(meet(X, Y), meet(X, join(Z, meet(X, Y)))) = meet(X, join(Z, meet(X, Y))).
% 82.64/10.92 Proof:
% 82.64/10.92 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.92 join(meet(Y, X), meet(X, join(Z, meet(X, Y))))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.92 join(meet(Y, X), meet(X, join(Z, meet(Y, X))))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.92 join(meet(X, join(Z, meet(Y, X))), meet(Y, X))
% 82.64/10.92 = { by lemma 34 R->L }
% 82.64/10.92 join(meet(X, join(Z, meet(Y, X))), meet(Y, meet(X, join(Z, meet(Y, X)))))
% 82.64/10.92 = { by lemma 22 }
% 82.64/10.92 meet(X, join(Z, meet(Y, X)))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.92 meet(X, join(Z, meet(X, Y)))
% 82.64/10.92
% 82.64/10.92 Lemma 38: meet(join(X, Y), join(Y, Z)) = join(Y, meet(Z, join(X, Y))).
% 82.64/10.92 Proof:
% 82.64/10.92 meet(join(X, Y), join(Y, Z))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.92 meet(join(Y, X), join(Y, Z))
% 82.64/10.92 = { by lemma 33 }
% 82.64/10.92 join(Y, meet(Z, join(Y, X)))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.92 join(Y, meet(Z, join(X, Y)))
% 82.64/10.92
% 82.64/10.92 Lemma 39: meet(X, join(Y, meet(X, Z))) = join(meet(X, Z), meet(X, Y)).
% 82.64/10.92 Proof:
% 82.64/10.92 meet(X, join(Y, meet(X, Z)))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.92 meet(X, join(meet(X, Z), Y))
% 82.64/10.92 = { by axiom 7 (absorption2) R->L }
% 82.64/10.92 meet(join(X, meet(X, Z)), join(meet(X, Z), Y))
% 82.64/10.92 = { by lemma 38 }
% 82.64/10.92 join(meet(X, Z), meet(Y, join(X, meet(X, Z))))
% 82.64/10.92 = { by axiom 7 (absorption2) }
% 82.64/10.92 join(meet(X, Z), meet(Y, X))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.92 join(meet(X, Z), meet(X, Y))
% 82.64/10.92
% 82.64/10.92 Lemma 40: join(meet(X, Y), meet(X, join(meet(X, Y), Z))) = join(meet(X, Y), meet(X, Z)).
% 82.64/10.92 Proof:
% 82.64/10.92 join(meet(X, Y), meet(X, join(meet(X, Y), Z)))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.92 join(meet(X, Y), meet(X, join(Z, meet(X, Y))))
% 82.64/10.92 = { by lemma 37 }
% 82.64/10.92 meet(X, join(Z, meet(X, Y)))
% 82.64/10.92 = { by lemma 39 }
% 82.64/10.92 join(meet(X, Y), meet(X, Z))
% 82.64/10.92
% 82.64/10.92 Lemma 41: join(meet(X, Y), meet(X, complement(meet(X, Y)))) = X.
% 82.64/10.92 Proof:
% 82.64/10.92 join(meet(X, Y), meet(X, complement(meet(X, Y))))
% 82.64/10.92 = { by lemma 40 R->L }
% 82.64/10.92 join(meet(X, Y), meet(X, join(meet(X, Y), complement(meet(X, Y)))))
% 82.64/10.92 = { by axiom 4 (complement_join) }
% 82.64/10.92 join(meet(X, Y), meet(X, one))
% 82.64/10.92 = { by lemma 13 }
% 82.64/10.92 join(meet(X, Y), X)
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.92 join(X, meet(X, Y))
% 82.64/10.92 = { by axiom 7 (absorption2) }
% 82.64/10.92 X
% 82.64/10.92
% 82.64/10.92 Lemma 42: join(X, meet(Y, complement(meet(X, Y)))) = join(X, Y).
% 82.64/10.92 Proof:
% 82.64/10.92 join(X, meet(Y, complement(meet(X, Y))))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 82.64/10.92 join(X, meet(Y, complement(meet(Y, X))))
% 82.64/10.92 = { by lemma 36 R->L }
% 82.64/10.92 join(X, join(meet(Y, complement(meet(Y, X))), meet(Y, X)))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) }
% 82.64/10.92 join(X, join(meet(Y, X), meet(Y, complement(meet(Y, X)))))
% 82.64/10.92 = { by lemma 41 }
% 82.64/10.92 join(X, Y)
% 82.64/10.92
% 82.64/10.92 Lemma 43: complement(meet(Y, complement(meet(Y, X)))) = join(X, complement(join(Y, X))).
% 82.64/10.92 Proof:
% 82.64/10.92 complement(meet(Y, complement(meet(Y, X))))
% 82.64/10.92 = { by lemma 25 R->L }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, meet(join(Y, X), X)))))
% 82.64/10.92 = { by lemma 16 R->L }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), zero)))))
% 82.64/10.92 = { by axiom 7 (absorption2) R->L }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), join(zero, meet(zero, join(Y, meet(X, join(X, Y))))))))))
% 82.64/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), join(meet(zero, join(Y, meet(X, join(X, Y)))), zero))))))
% 82.64/10.92 = { by lemma 16 }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(zero, join(Y, meet(X, join(X, Y)))))))))
% 82.64/10.92 = { by axiom 3 (commutativity_of_meet) }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(join(Y, meet(X, join(X, Y))), zero))))))
% 82.64/10.92 = { by axiom 5 (complement_meet) R->L }
% 82.64/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(join(Y, meet(X, join(X, Y))), meet(join(join(Y, meet(X, join(X, Y))), X), complement(join(join(Y, meet(X, join(X, Y))), X)))))))))
% 82.64/10.92 = { by lemma 25 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(join(Y, meet(X, join(X, Y))), complement(join(join(Y, meet(X, join(X, Y))), X))))))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(join(Y, meet(X, join(X, Y))), complement(join(X, join(Y, meet(X, join(X, Y)))))))))))
% 83.03/10.92 = { by lemma 26 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(join(Y, meet(X, join(X, Y))), complement(join(X, Y))))))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(complement(join(X, Y)), join(Y, meet(X, join(X, Y)))))))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(complement(join(X, Y)), join(Y, meet(X, join(Y, X)))))))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(complement(join(Y, X)), join(Y, meet(X, join(Y, X)))))))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(complement(join(Y, X)), join(Y, meet(join(Y, X), X))))))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(complement(join(Y, X)), join(meet(join(Y, X), X), Y)))))))
% 83.03/10.92 = { by lemma 33 R->L }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, meet(join(meet(join(Y, X), X), Y), join(meet(join(Y, X), X), complement(join(Y, X))))))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, meet(join(meet(join(Y, X), X), complement(join(Y, X))), join(meet(join(Y, X), X), Y))))))
% 83.03/10.92 = { by lemma 33 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), meet(Y, join(meet(join(Y, X), X), complement(join(Y, X)))))))))
% 83.03/10.92 = { by lemma 35 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(meet(join(Y, X), X), complement(join(Y, X)))))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(complement(join(Y, X)), meet(join(Y, X), X))))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(complement(join(Y, X)), meet(X, join(Y, X)))))))
% 83.03/10.92 = { by lemma 21 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(complement(join(Y, X)), X)))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, join(X, complement(join(Y, X)))))))
% 83.03/10.92 = { by axiom 6 (ifeq_axiom) R->L }
% 83.03/10.92 ifeq(one, one, complement(meet(Y, complement(meet(Y, join(X, complement(join(Y, X))))))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 4 (complement_join) R->L }
% 83.03/10.92 ifeq(join(join(Y, X), complement(join(Y, X))), one, complement(meet(Y, complement(meet(Y, join(X, complement(join(Y, X))))))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 8 (associativity_of_join) }
% 83.03/10.92 ifeq(join(Y, join(X, complement(join(Y, X)))), one, complement(meet(Y, complement(meet(Y, join(X, complement(join(Y, X))))))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.92 ifeq(join(Y, join(X, complement(join(Y, X)))), one, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.92 ifeq(join(join(X, complement(join(Y, X))), Y), one, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 6 (ifeq_axiom) R->L }
% 83.03/10.92 ifeq(join(join(X, complement(join(Y, X))), Y), one, ifeq(zero, zero, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by lemma 42 R->L }
% 83.03/10.92 ifeq(join(join(X, complement(join(Y, X))), meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), one, ifeq(zero, zero, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by lemma 31 R->L }
% 83.03/10.92 ifeq(join(join(X, complement(join(Y, X))), meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), one, ifeq(meet(join(X, complement(join(Y, X))), meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), zero, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.92 ifeq(join(join(X, complement(join(Y, X))), meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), one, ifeq(meet(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y))), join(X, complement(join(Y, X)))), zero, complement(meet(Y, complement(meet(join(X, complement(join(Y, X))), Y)))), join(X, complement(join(Y, X)))), join(X, complement(join(Y, X))))
% 83.03/10.92 = { by lemma 17 }
% 83.03/10.92 join(X, complement(join(Y, X)))
% 83.03/10.92
% 83.03/10.92 Lemma 44: join(complement(Y), meet(Y, X)) = join(X, complement(join(Y, X))).
% 83.03/10.92 Proof:
% 83.03/10.92 join(complement(Y), meet(Y, X))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.92 join(complement(Y), meet(X, Y))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.92 join(meet(X, Y), complement(Y))
% 83.03/10.92 = { by lemma 22 R->L }
% 83.03/10.92 join(meet(X, Y), complement(join(Y, meet(X, Y))))
% 83.03/10.92 = { by lemma 43 R->L }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, meet(X, Y)))))
% 83.03/10.92 = { by lemma 20 }
% 83.03/10.92 complement(meet(Y, complement(meet(Y, X))))
% 83.03/10.92 = { by lemma 43 }
% 83.03/10.92 join(X, complement(join(Y, X)))
% 83.03/10.92
% 83.03/10.92 Lemma 45: join(X, complement(join(Y, X))) = join(X, complement(Y)).
% 83.03/10.92 Proof:
% 83.03/10.92 join(X, complement(join(Y, X)))
% 83.03/10.92 = { by lemma 44 R->L }
% 83.03/10.92 join(complement(Y), meet(Y, X))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.92 join(complement(Y), meet(X, Y))
% 83.03/10.92 = { by lemma 26 R->L }
% 83.03/10.92 join(complement(Y), join(meet(X, Y), meet(X, join(complement(Y), meet(X, Y)))))
% 83.03/10.92 = { by lemma 37 }
% 83.03/10.92 join(complement(Y), meet(X, join(complement(Y), meet(X, Y))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.92 join(complement(Y), meet(X, join(complement(Y), meet(Y, X))))
% 83.03/10.92 = { by lemma 44 }
% 83.03/10.92 join(complement(Y), meet(X, join(X, complement(join(Y, X)))))
% 83.03/10.92 = { by axiom 9 (absorption1) }
% 83.03/10.92 join(complement(Y), X)
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.92 join(X, complement(Y))
% 83.03/10.92
% 83.03/10.92 Lemma 46: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 83.03/10.92 Proof:
% 83.03/10.92 join(complement(X), meet(X, Y))
% 83.03/10.92 = { by lemma 44 }
% 83.03/10.92 join(Y, complement(join(X, Y)))
% 83.03/10.92 = { by lemma 45 }
% 83.03/10.92 join(Y, complement(X))
% 83.03/10.92
% 83.03/10.92 Lemma 47: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 83.03/10.92 Proof:
% 83.03/10.92 join(X, complement(join(X, Y)))
% 83.03/10.92 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.92 join(X, complement(join(Y, X)))
% 83.03/10.92 = { by lemma 26 R->L }
% 83.03/10.92 join(X, join(complement(join(Y, X)), meet(join(complement(join(Y, X)), join(Y, X)), join(X, complement(join(Y, X))))))
% 83.03/10.92 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.92 join(X, join(complement(join(Y, X)), meet(join(X, complement(join(Y, X))), join(complement(join(Y, X)), join(Y, X)))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 join(X, join(complement(join(Y, X)), meet(join(X, complement(join(Y, X))), join(join(Y, X), complement(join(Y, X))))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 join(X, join(complement(join(Y, X)), meet(join(complement(join(Y, X)), X), join(join(Y, X), complement(join(Y, X))))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 join(X, join(complement(join(Y, X)), meet(join(join(Y, X), complement(join(Y, X))), join(complement(join(Y, X)), X))))
% 83.03/10.93 = { by lemma 21 R->L }
% 83.03/10.93 join(X, join(meet(complement(join(Y, X)), join(join(Y, X), complement(join(Y, X)))), meet(join(join(Y, X), complement(join(Y, X))), join(complement(join(Y, X)), X))))
% 83.03/10.93 = { by lemma 28 }
% 83.03/10.93 join(X, meet(join(join(Y, X), complement(join(Y, X))), join(complement(join(Y, X)), X)))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(X, meet(join(complement(join(Y, X)), X), join(join(Y, X), complement(join(Y, X)))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.93 join(X, meet(join(X, complement(join(Y, X))), join(join(Y, X), complement(join(Y, X)))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(X, meet(join(join(Y, X), complement(join(Y, X))), join(X, complement(join(Y, X)))))
% 83.03/10.93 = { by axiom 4 (complement_join) }
% 83.03/10.93 join(X, meet(one, join(X, complement(join(Y, X)))))
% 83.03/10.93 = { by lemma 14 }
% 83.03/10.93 join(X, join(X, complement(join(Y, X))))
% 83.03/10.93 = { by lemma 44 R->L }
% 83.03/10.93 join(X, join(complement(Y), meet(Y, X)))
% 83.03/10.93 = { by lemma 36 }
% 83.03/10.93 join(X, complement(Y))
% 83.03/10.93
% 83.03/10.93 Lemma 48: join(meet(X, Y), meet(Y, join(Z, meet(X, Y)))) = meet(Y, join(Z, meet(X, Y))).
% 83.03/10.93 Proof:
% 83.03/10.93 join(meet(X, Y), meet(Y, join(Z, meet(X, Y))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 join(meet(X, Y), meet(Y, join(Z, meet(Y, X))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 join(meet(Y, X), meet(Y, join(Z, meet(Y, X))))
% 83.03/10.93 = { by lemma 37 }
% 83.03/10.93 meet(Y, join(Z, meet(Y, X)))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 meet(Y, join(Z, meet(X, Y)))
% 83.03/10.93
% 83.03/10.93 Lemma 49: meet(join(X, Y), meet(Z, join(X, join(Y, W)))) = meet(Z, join(X, Y)).
% 83.03/10.93 Proof:
% 83.03/10.93 meet(join(X, Y), meet(Z, join(X, join(Y, W))))
% 83.03/10.93 = { by axiom 8 (associativity_of_join) R->L }
% 83.03/10.93 meet(join(X, Y), meet(Z, join(join(X, Y), W)))
% 83.03/10.93 = { by lemma 24 }
% 83.03/10.93 meet(join(X, Y), Z)
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 meet(Z, join(X, Y))
% 83.03/10.93
% 83.03/10.93 Lemma 50: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 83.03/10.93 Proof:
% 83.03/10.93 complement(join(X, complement(Y)))
% 83.03/10.93 = { by lemma 18 R->L }
% 83.03/10.93 complement(join(complement(complement(X)), complement(Y)))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 complement(join(complement(Y), complement(complement(X))))
% 83.03/10.93 = { by lemma 45 R->L }
% 83.03/10.93 complement(join(complement(Y), complement(join(complement(X), complement(Y)))))
% 83.03/10.93 = { by lemma 46 R->L }
% 83.03/10.93 complement(join(complement(Y), complement(join(complement(Y), meet(Y, complement(X))))))
% 83.03/10.93 = { by lemma 47 }
% 83.03/10.93 complement(join(complement(Y), complement(meet(Y, complement(X)))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 complement(join(complement(Y), complement(meet(complement(X), Y))))
% 83.03/10.93 = { by lemma 18 R->L }
% 83.03/10.93 complement(join(complement(Y), complement(meet(complement(X), complement(complement(Y))))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), complement(Y)))
% 83.03/10.93 = { by lemma 32 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(meet(complement(complement(Y)), join(complement(Y), meet(complement(X), complement(complement(Y)))))))))
% 83.03/10.93 = { by lemma 41 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(meet(complement(complement(Y)), join(complement(Y), meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))))))))))
% 83.03/10.93 = { by lemma 41 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), join(complement(Y), meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))))))))))
% 83.03/10.93 = { by lemma 48 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), join(complement(Y), meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))))))))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), complement(Y))))))))
% 83.03/10.93 = { by lemma 49 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), join(meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), meet(complement(X), join(meet(complement(complement(Y)), Z), join(meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))), W)))), complement(Y))))))))
% 83.03/10.93 = { by lemma 49 R->L }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), meet(complement(X), join(meet(complement(complement(Y)), Z), join(meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))), W)))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), join(meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), meet(complement(X), join(meet(complement(complement(Y)), Z), join(meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))), W)))), complement(Y))))))))
% 83.03/10.93 = { by lemma 40 }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), meet(complement(X), join(meet(complement(complement(Y)), Z), join(meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))), W)))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), complement(Y)))))))
% 83.03/10.93 = { by lemma 49 }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), meet(join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z)))), complement(Y)))))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))), meet(complement(Y), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))))))))
% 83.03/10.93 = { by lemma 41 }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), complement(complement(Y))), meet(complement(Y), join(meet(complement(complement(Y)), Z), meet(complement(complement(Y)), complement(meet(complement(complement(Y)), Z))))))))))
% 83.03/10.93 = { by lemma 41 }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), complement(complement(Y))), meet(complement(Y), complement(complement(Y))))))))
% 83.03/10.93 = { by axiom 5 (complement_meet) }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(join(meet(complement(X), complement(complement(Y))), zero)))))
% 83.03/10.93 = { by lemma 16 }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(join(complement(Y), meet(complement(X), complement(complement(Y)))), complement(meet(complement(X), complement(complement(Y)))))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 complement(join(complement(meet(complement(X), complement(complement(Y)))), meet(complement(meet(complement(X), complement(complement(Y)))), join(complement(Y), meet(complement(X), complement(complement(Y)))))))
% 83.03/10.93 = { by axiom 7 (absorption2) }
% 83.03/10.93 complement(complement(meet(complement(X), complement(complement(Y)))))
% 83.03/10.93 = { by lemma 18 }
% 83.03/10.93 complement(complement(meet(complement(X), Y)))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 complement(complement(meet(Y, complement(X))))
% 83.03/10.93 = { by lemma 18 }
% 83.03/10.93 meet(Y, complement(X))
% 83.03/10.93
% 83.03/10.93 Lemma 51: meet(X, join(Y, meet(Z, X))) = join(meet(X, Z), meet(X, Y)).
% 83.03/10.93 Proof:
% 83.03/10.93 meet(X, join(Y, meet(Z, X)))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 meet(X, join(Y, meet(X, Z)))
% 83.03/10.93 = { by lemma 39 }
% 83.03/10.93 join(meet(X, Z), meet(X, Y))
% 83.03/10.93
% 83.03/10.93 Lemma 52: meet(X, join(complement(X), Y)) = meet(X, Y).
% 83.03/10.93 Proof:
% 83.03/10.93 meet(X, join(complement(X), Y))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 meet(X, join(Y, complement(X)))
% 83.03/10.93 = { by lemma 28 R->L }
% 83.03/10.93 join(meet(Y, X), meet(X, join(Y, complement(X))))
% 83.03/10.93 = { by lemma 45 R->L }
% 83.03/10.93 join(meet(Y, X), meet(X, join(Y, complement(join(X, Y)))))
% 83.03/10.93 = { by lemma 44 R->L }
% 83.03/10.93 join(meet(Y, X), meet(X, join(complement(X), meet(X, Y))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(meet(Y, X), meet(X, join(complement(X), meet(Y, X))))
% 83.03/10.93 = { by lemma 48 }
% 83.03/10.93 meet(X, join(complement(X), meet(Y, X)))
% 83.03/10.93 = { by lemma 51 }
% 83.03/10.93 join(meet(X, Y), meet(X, complement(X)))
% 83.03/10.93 = { by axiom 5 (complement_meet) }
% 83.03/10.93 join(meet(X, Y), zero)
% 83.03/10.93 = { by lemma 16 }
% 83.03/10.93 meet(X, Y)
% 83.03/10.93
% 83.03/10.93 Goal 1 (prove_distributivity): meet(a, join(b, c)) = join(meet(a, b), meet(a, c)).
% 83.03/10.93 Proof:
% 83.03/10.93 meet(a, join(b, c))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 meet(a, join(c, b))
% 83.03/10.93 = { by lemma 28 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, b)))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(b, c)))
% 83.03/10.93 = { by lemma 52 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(complement(a), join(b, c))))
% 83.03/10.93 = { by axiom 8 (associativity_of_join) R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(join(complement(a), b), c)))
% 83.03/10.93 = { by lemma 42 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(join(complement(a), meet(b, complement(meet(complement(a), b)))), c)))
% 83.03/10.93 = { by axiom 8 (associativity_of_join) }
% 83.03/10.93 join(meet(c, a), meet(a, join(complement(a), join(meet(b, complement(meet(complement(a), b))), c))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.93 join(meet(c, a), meet(a, join(complement(a), join(c, meet(b, complement(meet(complement(a), b)))))))
% 83.03/10.93 = { by lemma 52 }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, complement(meet(complement(a), b))))))
% 83.03/10.93 = { by lemma 19 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, meet(b, complement(meet(complement(a), b)))))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, meet(b, complement(meet(b, complement(a))))))))
% 83.03/10.93 = { by lemma 18 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, complement(complement(meet(b, complement(meet(b, complement(a))))))))))
% 83.03/10.93 = { by lemma 43 }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, complement(join(complement(a), complement(join(b, complement(a)))))))))
% 83.03/10.93 = { by lemma 50 }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, meet(join(b, complement(a)), complement(complement(a)))))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, meet(complement(complement(a)), join(b, complement(a)))))))
% 83.03/10.93 = { by lemma 24 }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, complement(complement(a))))))
% 83.03/10.93 = { by lemma 18 }
% 83.03/10.93 join(meet(c, a), meet(a, join(c, meet(b, a))))
% 83.03/10.93 = { by lemma 51 }
% 83.03/10.93 join(meet(c, a), join(meet(a, b), meet(a, c)))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.93 join(meet(c, a), join(meet(a, c), meet(a, b)))
% 83.03/10.93 = { by axiom 12 (equation_H18) }
% 83.03/10.93 join(meet(c, a), meet(a, join(meet(a, c), join(meet(a, b), meet(c, join(a, b))))))
% 83.03/10.93 = { by lemma 19 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, meet(a, join(meet(a, c), join(meet(a, b), meet(c, join(a, b)))))))
% 83.03/10.93 = { by axiom 12 (equation_H18) R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(meet(a, c), meet(a, b))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(meet(c, a), meet(a, join(meet(c, a), meet(a, b))))
% 83.03/10.93 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.93 join(meet(c, a), meet(a, join(meet(c, a), meet(b, a))))
% 83.03/10.93 = { by lemma 18 R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(meet(c, a), complement(complement(meet(b, a))))))
% 83.03/10.93 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.93 join(meet(c, a), meet(a, join(complement(complement(meet(b, a))), meet(c, a))))
% 83.03/10.93 = { by lemma 38 R->L }
% 83.03/10.94 meet(join(complement(complement(meet(b, a))), meet(c, a)), join(meet(c, a), a))
% 83.03/10.94 = { by lemma 35 R->L }
% 83.03/10.94 meet(join(complement(complement(meet(b, a))), meet(c, a)), join(meet(c, a), meet(join(complement(complement(meet(b, a))), meet(c, a)), join(meet(c, a), a))))
% 83.03/10.94 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.94 meet(join(complement(complement(meet(b, a))), meet(c, a)), join(meet(c, a), meet(join(meet(c, a), a), join(complement(complement(meet(b, a))), meet(c, a)))))
% 83.03/10.94 = { by lemma 38 }
% 83.03/10.94 join(meet(c, a), meet(meet(join(meet(c, a), a), join(complement(complement(meet(b, a))), meet(c, a))), join(complement(complement(meet(b, a))), meet(c, a))))
% 83.03/10.94 = { by axiom 10 (associativity_of_meet) }
% 83.03/10.94 join(meet(c, a), meet(join(meet(c, a), a), meet(join(complement(complement(meet(b, a))), meet(c, a)), join(complement(complement(meet(b, a))), meet(c, a)))))
% 83.03/10.94 = { by lemma 33 }
% 83.03/10.94 join(meet(c, a), meet(join(meet(c, a), a), join(complement(complement(meet(b, a))), meet(meet(c, a), join(complement(complement(meet(b, a))), meet(c, a))))))
% 83.03/10.94 = { by lemma 21 }
% 83.03/10.94 join(meet(c, a), meet(join(meet(c, a), a), join(complement(complement(meet(b, a))), meet(c, a))))
% 83.03/10.94 = { by axiom 1 (commutativity_of_join) R->L }
% 83.03/10.94 join(meet(c, a), meet(join(meet(c, a), a), join(meet(c, a), complement(complement(meet(b, a))))))
% 83.03/10.94 = { by axiom 9 (absorption1) R->L }
% 83.03/10.94 join(meet(meet(c, a), join(meet(c, a), a)), meet(join(meet(c, a), a), join(meet(c, a), complement(complement(meet(b, a))))))
% 83.03/10.94 = { by lemma 28 }
% 83.03/10.94 meet(join(meet(c, a), a), join(meet(c, a), complement(complement(meet(b, a)))))
% 83.03/10.94 = { by lemma 33 }
% 83.03/10.94 join(meet(c, a), meet(complement(complement(meet(b, a))), join(meet(c, a), a)))
% 83.03/10.94 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.94 join(meet(c, a), meet(complement(complement(meet(b, a))), join(a, meet(c, a))))
% 83.03/10.94 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.94 join(meet(c, a), meet(join(a, meet(c, a)), complement(complement(meet(b, a)))))
% 83.03/10.94 = { by lemma 50 R->L }
% 83.03/10.94 join(meet(c, a), complement(join(complement(meet(b, a)), complement(join(a, meet(c, a))))))
% 83.03/10.94 = { by lemma 47 R->L }
% 83.03/10.94 join(meet(c, a), complement(join(meet(c, a), join(complement(meet(b, a)), complement(join(a, meet(c, a)))))))
% 83.03/10.94 = { by lemma 23 }
% 83.03/10.94 join(meet(c, a), complement(join(complement(meet(b, a)), join(meet(c, a), complement(join(a, meet(c, a)))))))
% 83.03/10.94 = { by lemma 44 R->L }
% 83.03/10.94 join(meet(c, a), complement(join(complement(meet(b, a)), join(complement(a), meet(a, meet(c, a))))))
% 83.03/10.94 = { by lemma 46 }
% 83.03/10.94 join(meet(c, a), complement(join(complement(meet(b, a)), join(meet(c, a), complement(a)))))
% 83.03/10.94 = { by lemma 23 R->L }
% 83.03/10.94 join(meet(c, a), complement(join(meet(c, a), join(complement(meet(b, a)), complement(a)))))
% 83.03/10.94 = { by lemma 47 }
% 83.03/10.94 join(meet(c, a), complement(join(complement(meet(b, a)), complement(a))))
% 83.03/10.94 = { by lemma 50 }
% 83.03/10.94 join(meet(c, a), meet(a, complement(complement(meet(b, a)))))
% 83.03/10.94 = { by lemma 18 }
% 83.03/10.94 join(meet(c, a), meet(a, meet(b, a)))
% 83.03/10.94 = { by lemma 20 }
% 83.03/10.94 join(meet(c, a), meet(a, b))
% 83.03/10.94 = { by axiom 1 (commutativity_of_join) }
% 83.03/10.94 join(meet(a, b), meet(c, a))
% 83.03/10.94 = { by axiom 3 (commutativity_of_meet) }
% 83.03/10.94 join(meet(b, a), meet(c, a))
% 83.03/10.94 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.94 join(meet(b, a), meet(a, c))
% 83.03/10.94 = { by axiom 3 (commutativity_of_meet) R->L }
% 83.03/10.94 join(meet(a, b), meet(a, c))
% 83.03/10.94 % SZS output end Proof
% 83.03/10.94
% 83.03/10.94 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------