%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL016+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n018.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 12:33:07 PM UTC 2026
% Result : Theorem 66.62s 8.82s
% Output : Proof 67.38s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : REL016+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.09/0.35 % Computer : n018.cluster.edu
% 0.09/0.35 % Model : x86_64 x86_64
% 0.09/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.35 % Memory : 8046.5625MB
% 0.09/0.35 % OS : Linux 6.8.0-71-generic
% 0.09/0.35 % CPULimit : 300
% 0.09/0.35 % WCLimit : 300
% 0.09/0.35 % DateTime : Sun Sep 27 22:53:38 UTC 2026
% 0.09/0.35 % CPUTime :
% 0.09/0.35 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 66.62/8.82 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 66.62/8.82
% 66.62/8.82 % SZS status Theorem
% 66.62/8.82
% 67.38/8.92 % SZS output start Proof
% 67.38/8.92 Axiom 1 (composition_identity): composition(X, one) = X.
% 67.38/8.92 Axiom 2 (def_zero): zero = meet(X, complement(X)).
% 67.38/8.92 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 67.38/8.92 Axiom 4 (def_top): top = join(X, complement(X)).
% 67.38/8.92 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 67.38/8.92 Axiom 6 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 67.38/8.92 Axiom 7 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 67.38/8.92 Axiom 8 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 67.38/8.92 Axiom 9 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 67.38/8.92 Axiom 10 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 67.38/8.92 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 67.38/8.92 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 67.38/8.92 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 67.38/8.92
% 67.38/8.92 Lemma 14: complement(top) = zero.
% 67.38/8.92 Proof:
% 67.38/8.92 complement(top)
% 67.38/8.92 = { by axiom 4 (def_top) }
% 67.38/8.92 complement(join(complement(X), complement(complement(X))))
% 67.38/8.92 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.92 meet(X, complement(X))
% 67.38/8.92 = { by axiom 2 (def_zero) R->L }
% 67.38/8.92 zero
% 67.38/8.92
% 67.38/8.92 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 67.38/8.92 Proof:
% 67.38/8.92 join(X, join(Y, complement(X)))
% 67.38/8.92 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.92 join(X, join(complement(X), Y))
% 67.38/8.92 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.92 join(join(X, complement(X)), Y)
% 67.38/8.92 = { by axiom 4 (def_top) R->L }
% 67.38/8.92 join(top, Y)
% 67.38/8.92 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.92 join(Y, top)
% 67.38/8.92
% 67.38/8.92 Lemma 16: join(X, join(complement(X), Y)) = join(Y, top).
% 67.38/8.92 Proof:
% 67.38/8.92 join(X, join(complement(X), Y))
% 67.38/8.92 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.92 join(X, join(Y, complement(X)))
% 67.38/8.92 = { by lemma 15 }
% 67.38/8.92 join(Y, top)
% 67.38/8.92
% 67.38/8.92 Lemma 17: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 67.38/8.92 Proof:
% 67.38/8.92 converse(composition(converse(X), Y))
% 67.38/8.92 = { by axiom 7 (converse_multiplicativity) }
% 67.38/8.92 composition(converse(Y), converse(converse(X)))
% 67.38/8.92 = { by axiom 5 (converse_idempotence) }
% 67.38/8.92 composition(converse(Y), X)
% 67.38/8.92
% 67.38/8.92 Lemma 18: composition(converse(one), X) = X.
% 67.38/8.92 Proof:
% 67.38/8.92 composition(converse(one), X)
% 67.38/8.92 = { by lemma 17 R->L }
% 67.38/8.92 converse(composition(converse(X), one))
% 67.38/8.92 = { by axiom 1 (composition_identity) }
% 67.38/8.92 converse(converse(X))
% 67.38/8.92 = { by axiom 5 (converse_idempotence) }
% 67.38/8.92 X
% 67.38/8.92
% 67.38/8.92 Lemma 19: join(complement(X), complement(X)) = complement(X).
% 67.38/8.92 Proof:
% 67.38/8.92 join(complement(X), complement(X))
% 67.38/8.92 = { by lemma 18 R->L }
% 67.38/8.92 join(complement(X), complement(composition(converse(one), X)))
% 67.38/8.92 = { by axiom 1 (composition_identity) R->L }
% 67.38/8.92 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 67.38/8.92 = { by axiom 8 (composition_associativity) R->L }
% 67.38/8.92 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 67.38/8.92 = { by lemma 18 }
% 67.38/8.92 join(complement(X), complement(composition(one, X)))
% 67.38/8.92 = { by axiom 5 (converse_idempotence) R->L }
% 67.38/8.92 join(complement(X), complement(composition(converse(converse(one)), X)))
% 67.38/8.92 = { by lemma 18 R->L }
% 67.38/8.92 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 67.38/8.92 = { by axiom 5 (converse_idempotence) R->L }
% 67.38/8.93 join(complement(X), composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))), complement(X))
% 67.38/8.93 = { by axiom 13 (converse_cancellativity) }
% 67.38/8.93 complement(X)
% 67.38/8.93
% 67.38/8.93 Lemma 20: join(X, join(Y, complement(join(X, Y)))) = top.
% 67.38/8.93 Proof:
% 67.38/8.93 join(X, join(Y, complement(join(X, Y))))
% 67.38/8.93 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.93 join(join(X, Y), complement(join(X, Y)))
% 67.38/8.93 = { by axiom 4 (def_top) R->L }
% 67.38/8.93 top
% 67.38/8.93
% 67.38/8.93 Lemma 21: join(X, top) = top.
% 67.38/8.93 Proof:
% 67.38/8.93 join(X, top)
% 67.38/8.93 = { by axiom 4 (def_top) }
% 67.38/8.93 join(X, join(complement(X), complement(complement(X))))
% 67.38/8.93 = { by lemma 16 }
% 67.38/8.93 join(complement(complement(X)), top)
% 67.38/8.93 = { by lemma 15 R->L }
% 67.38/8.93 join(complement(complement(X)), join(complement(complement(X)), complement(complement(complement(X)))))
% 67.38/8.93 = { by lemma 19 R->L }
% 67.38/8.93 join(complement(complement(X)), join(complement(complement(X)), complement(join(complement(complement(X)), complement(complement(X))))))
% 67.38/8.93 = { by lemma 20 }
% 67.38/8.93 top
% 67.38/8.93
% 67.38/8.93 Lemma 22: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 join(meet(X, Y), complement(join(complement(X), Y)))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.93 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 67.38/8.93 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 23: join(meet(X, Y), join(Z, complement(join(complement(X), Y)))) = join(X, Z).
% 67.38/8.93 Proof:
% 67.38/8.93 join(meet(X, Y), join(Z, complement(join(complement(X), Y))))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(meet(X, Y), join(complement(join(complement(X), Y)), Z))
% 67.38/8.93 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.93 join(join(meet(X, Y), complement(join(complement(X), Y))), Z)
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 join(X, Z)
% 67.38/8.93
% 67.38/8.93 Lemma 24: meet(Y, X) = meet(X, Y).
% 67.38/8.93 Proof:
% 67.38/8.93 meet(Y, X)
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.93 complement(join(complement(Y), complement(X)))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 complement(join(complement(X), complement(Y)))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.93 meet(X, Y)
% 67.38/8.93
% 67.38/8.93 Lemma 25: complement(join(meet(X, Y), complement(Z))) = meet(Z, join(complement(X), complement(Y))).
% 67.38/8.93 Proof:
% 67.38/8.93 complement(join(meet(X, Y), complement(Z)))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.93 complement(join(complement(join(complement(X), complement(Y))), complement(Z)))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.93 meet(join(complement(X), complement(Y)), Z)
% 67.38/8.93 = { by lemma 24 R->L }
% 67.38/8.93 meet(Z, join(complement(X), complement(Y)))
% 67.38/8.93
% 67.38/8.93 Lemma 26: meet(join(X, complement(Y)), join(complement(X), complement(Y))) = complement(Y).
% 67.38/8.93 Proof:
% 67.38/8.93 meet(join(X, complement(Y)), join(complement(X), complement(Y)))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 meet(join(X, complement(Y)), join(complement(Y), complement(X)))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 meet(join(complement(Y), X), join(complement(Y), complement(X)))
% 67.38/8.93 = { by lemma 25 R->L }
% 67.38/8.93 complement(join(meet(Y, X), complement(join(complement(Y), X))))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 complement(Y)
% 67.38/8.93
% 67.38/8.93 Lemma 27: join(meet(X, Y), meet(X, complement(Y))) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 join(meet(X, Y), meet(X, complement(Y)))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(meet(X, complement(Y)), meet(X, Y))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.93 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 28: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 67.38/8.93 Proof:
% 67.38/8.93 join(meet(X, Y), meet(Y, complement(X)))
% 67.38/8.93 = { by lemma 24 }
% 67.38/8.93 join(meet(Y, X), meet(Y, complement(X)))
% 67.38/8.93 = { by lemma 27 }
% 67.38/8.93 Y
% 67.38/8.93
% 67.38/8.93 Lemma 29: join(zero, complement(X)) = complement(X).
% 67.38/8.93 Proof:
% 67.38/8.93 join(zero, complement(X))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(complement(X), zero)
% 67.38/8.93 = { by lemma 14 R->L }
% 67.38/8.93 join(complement(X), complement(top))
% 67.38/8.93 = { by lemma 21 R->L }
% 67.38/8.93 join(complement(X), complement(join(meet(complement(complement(X)), complement(complement(X))), top)))
% 67.38/8.93 = { by lemma 16 R->L }
% 67.38/8.93 join(complement(X), complement(join(zero, join(complement(zero), meet(complement(complement(X)), complement(complement(X)))))))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(complement(X), complement(join(zero, join(meet(complement(complement(X)), complement(complement(X))), complement(zero)))))
% 67.38/8.93 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.93 join(complement(X), complement(join(join(zero, meet(complement(complement(X)), complement(complement(X)))), complement(zero))))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.93 join(complement(X), complement(join(join(zero, complement(join(complement(complement(complement(X))), complement(complement(complement(X)))))), complement(zero))))
% 67.38/8.93 = { by axiom 2 (def_zero) }
% 67.38/8.93 join(complement(X), complement(join(join(meet(complement(complement(X)), complement(complement(complement(X)))), complement(join(complement(complement(complement(X))), complement(complement(complement(X)))))), complement(zero))))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 join(complement(X), complement(join(complement(complement(X)), complement(zero))))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.93 join(complement(X), meet(complement(X), zero))
% 67.38/8.93 = { by lemma 14 R->L }
% 67.38/8.93 join(complement(X), meet(complement(X), complement(top)))
% 67.38/8.93 = { by lemma 26 R->L }
% 67.38/8.93 join(meet(join(X, complement(X)), join(complement(X), complement(X))), meet(complement(X), complement(top)))
% 67.38/8.93 = { by axiom 4 (def_top) R->L }
% 67.38/8.93 join(meet(top, join(complement(X), complement(X))), meet(complement(X), complement(top)))
% 67.38/8.93 = { by lemma 19 }
% 67.38/8.93 join(meet(top, complement(X)), meet(complement(X), complement(top)))
% 67.38/8.93 = { by lemma 28 }
% 67.38/8.93 complement(X)
% 67.38/8.93
% 67.38/8.93 Lemma 30: join(X, zero) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 join(X, zero)
% 67.38/8.93 = { by lemma 23 R->L }
% 67.38/8.93 join(meet(X, Y), join(zero, complement(join(complement(X), Y))))
% 67.38/8.93 = { by lemma 29 }
% 67.38/8.93 join(meet(X, Y), complement(join(complement(X), Y)))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 31: complement(complement(X)) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 complement(complement(X))
% 67.38/8.93 = { by lemma 29 R->L }
% 67.38/8.93 join(zero, complement(complement(X)))
% 67.38/8.93 = { by axiom 2 (def_zero) }
% 67.38/8.93 join(meet(X, complement(X)), complement(complement(X)))
% 67.38/8.93 = { by lemma 19 R->L }
% 67.38/8.93 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 32: complement(join(zero, complement(X))) = meet(X, top).
% 67.38/8.93 Proof:
% 67.38/8.93 complement(join(zero, complement(X)))
% 67.38/8.93 = { by lemma 14 R->L }
% 67.38/8.93 complement(join(complement(top), complement(X)))
% 67.38/8.93 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.93 meet(top, X)
% 67.38/8.93 = { by lemma 24 R->L }
% 67.38/8.93 meet(X, top)
% 67.38/8.93
% 67.38/8.93 Lemma 33: meet(X, top) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 meet(X, top)
% 67.38/8.93 = { by lemma 32 R->L }
% 67.38/8.93 complement(join(zero, complement(X)))
% 67.38/8.93 = { by lemma 29 R->L }
% 67.38/8.93 join(zero, complement(join(zero, complement(X))))
% 67.38/8.93 = { by lemma 32 }
% 67.38/8.93 join(zero, meet(X, top))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(meet(X, top), zero)
% 67.38/8.93 = { by lemma 14 R->L }
% 67.38/8.93 join(meet(X, top), complement(top))
% 67.38/8.93 = { by lemma 21 R->L }
% 67.38/8.93 join(meet(X, top), complement(join(complement(X), top)))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 34: meet(top, X) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 meet(top, X)
% 67.38/8.93 = { by lemma 24 }
% 67.38/8.93 meet(X, top)
% 67.38/8.93 = { by lemma 33 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 35: join(X, complement(join(Y, complement(X)))) = X.
% 67.38/8.93 Proof:
% 67.38/8.93 join(X, complement(join(Y, complement(X))))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(X, complement(join(complement(X), Y)))
% 67.38/8.93 = { by lemma 23 R->L }
% 67.38/8.93 join(meet(X, Y), join(complement(join(complement(X), Y)), complement(join(complement(X), Y))))
% 67.38/8.93 = { by lemma 19 }
% 67.38/8.93 join(meet(X, Y), complement(join(complement(X), Y)))
% 67.38/8.93 = { by lemma 22 }
% 67.38/8.93 X
% 67.38/8.93
% 67.38/8.93 Lemma 36: join(X, join(X, Y)) = join(X, Y).
% 67.38/8.93 Proof:
% 67.38/8.93 join(X, join(X, Y))
% 67.38/8.93 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.93 join(X, join(Y, X))
% 67.38/8.93 = { by lemma 31 R->L }
% 67.38/8.93 join(X, join(Y, complement(complement(X))))
% 67.38/8.93 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.93 join(join(X, Y), complement(complement(X)))
% 67.38/8.93 = { by lemma 26 R->L }
% 67.38/8.93 join(join(X, Y), complement(meet(join(join(Y, complement(complement(X))), complement(X)), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(join(X, Y), complement(meet(join(complement(X), join(Y, complement(complement(X)))), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 join(join(X, Y), complement(meet(join(complement(X), join(complement(complement(X)), Y)), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.94 join(join(X, Y), complement(meet(join(join(complement(X), complement(complement(X))), Y), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by axiom 4 (def_top) R->L }
% 67.38/8.94 join(join(X, Y), complement(meet(join(top, Y), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(join(X, Y), complement(meet(join(Y, top), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by lemma 21 }
% 67.38/8.94 join(join(X, Y), complement(meet(top, join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 67.38/8.94 = { by lemma 34 }
% 67.38/8.94 join(join(X, Y), complement(join(complement(join(Y, complement(complement(X)))), complement(X))))
% 67.38/8.94 = { by lemma 31 }
% 67.38/8.94 join(join(X, Y), complement(join(complement(join(Y, X)), complement(X))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(join(X, Y), complement(join(complement(X), complement(join(Y, X)))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(join(X, Y), complement(join(complement(X), complement(join(X, Y)))))
% 67.38/8.94 = { by lemma 35 }
% 67.38/8.94 join(X, Y)
% 67.38/8.94
% 67.38/8.94 Lemma 37: join(Y, join(Z, X)) = join(X, join(Y, Z)).
% 67.38/8.94 Proof:
% 67.38/8.94 join(Y, join(Z, X))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 join(join(Z, X), Y)
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(join(X, Z), Y)
% 67.38/8.94 = { by axiom 10 (maddux2_join_associativity) R->L }
% 67.38/8.94 join(X, join(Z, Y))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(X, join(Y, Z))
% 67.38/8.94
% 67.38/8.94 Lemma 38: join(Z, join(Y, X)) = join(X, join(Y, Z)).
% 67.38/8.94 Proof:
% 67.38/8.94 join(Z, join(Y, X))
% 67.38/8.94 = { by lemma 37 }
% 67.38/8.94 join(X, join(Z, Y))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 join(X, join(Y, Z))
% 67.38/8.94
% 67.38/8.94 Lemma 39: meet(X, join(Y, complement(join(Y, complement(X))))) = X.
% 67.38/8.94 Proof:
% 67.38/8.94 meet(X, join(Y, complement(join(Y, complement(X)))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 meet(X, join(Y, complement(join(complement(X), Y))))
% 67.38/8.94 = { by lemma 30 R->L }
% 67.38/8.94 join(meet(X, join(Y, complement(join(complement(X), Y)))), zero)
% 67.38/8.94 = { by lemma 14 R->L }
% 67.38/8.94 join(meet(X, join(Y, complement(join(complement(X), Y)))), complement(top))
% 67.38/8.94 = { by lemma 20 R->L }
% 67.38/8.94 join(meet(X, join(Y, complement(join(complement(X), Y)))), complement(join(complement(X), join(Y, complement(join(complement(X), Y))))))
% 67.38/8.94 = { by lemma 22 }
% 67.38/8.94 X
% 67.38/8.94
% 67.38/8.94 Lemma 40: join(join(X, Y), Y) = join(X, Y).
% 67.38/8.94 Proof:
% 67.38/8.94 join(join(X, Y), Y)
% 67.38/8.94 = { by lemma 39 R->L }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), complement(join(join(X, Y), complement(Y))))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), complement(join(complement(Y), join(X, Y))))))
% 67.38/8.94 = { by lemma 38 }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), complement(join(Y, join(X, complement(Y)))))))
% 67.38/8.94 = { by lemma 15 }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), complement(join(X, top)))))
% 67.38/8.94 = { by lemma 21 }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), complement(top))))
% 67.38/8.94 = { by lemma 14 }
% 67.38/8.94 join(join(X, Y), meet(Y, join(join(X, Y), zero)))
% 67.38/8.94 = { by lemma 30 }
% 67.38/8.94 join(join(X, Y), meet(Y, join(X, Y)))
% 67.38/8.94 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.94 join(join(X, Y), complement(join(complement(Y), complement(join(X, Y)))))
% 67.38/8.94 = { by lemma 35 }
% 67.38/8.94 join(X, Y)
% 67.38/8.94
% 67.38/8.94 Lemma 41: meet(meet(X, complement(Y)), join(Z, X)) = meet(X, complement(Y)).
% 67.38/8.94 Proof:
% 67.38/8.94 meet(meet(X, complement(Y)), join(Z, X))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(X, Z))
% 67.38/8.94 = { by lemma 22 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), complement(join(complement(X), complement(Y)))), Z))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), complement(join(complement(Y), complement(X)))), Z))
% 67.38/8.94 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), meet(Y, X)), Z))
% 67.38/8.94 = { by lemma 24 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), meet(X, Y)), Z))
% 67.38/8.94 = { by axiom 10 (maddux2_join_associativity) R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(meet(X, complement(Y)), join(meet(X, Y), Z)))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.94 meet(meet(X, complement(Y)), join(meet(X, complement(Y)), join(Z, meet(X, Y))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), meet(X, complement(Y))))
% 67.38/8.94 = { by lemma 30 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), zero)))
% 67.38/8.94 = { by lemma 14 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), complement(top))))
% 67.38/8.94 = { by lemma 21 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), complement(join(join(Z, meet(X, Y)), top)))))
% 67.38/8.94 = { by lemma 31 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(complement(complement(meet(X, complement(Y)))), complement(join(join(Z, meet(X, Y)), top)))))
% 67.38/8.94 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(join(Z, meet(X, Y)), complement(complement(meet(X, complement(Y))))), complement(join(join(Z, meet(X, Y)), top))))
% 67.38/8.94 = { by lemma 15 R->L }
% 67.38/8.94 meet(meet(X, complement(Y)), join(join(join(Z, meet(X, Y)), complement(complement(meet(X, complement(Y))))), complement(join(complement(meet(X, complement(Y))), join(join(Z, meet(X, Y)), complement(complement(meet(X, complement(Y)))))))))
% 67.38/8.94 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 meet(meet(X, complement(Y)), join(join(join(Z, meet(X, Y)), complement(complement(meet(X, complement(Y))))), complement(join(join(join(Z, meet(X, Y)), complement(complement(meet(X, complement(Y))))), complement(meet(X, complement(Y)))))))
% 67.38/8.95 = { by lemma 39 }
% 67.38/8.95 meet(X, complement(Y))
% 67.38/8.95
% 67.38/8.95 Lemma 42: join(meet(X, complement(Y)), Y) = join(X, Y).
% 67.38/8.95 Proof:
% 67.38/8.95 join(meet(X, complement(Y)), Y)
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(Y, meet(X, complement(Y)))
% 67.38/8.95 = { by lemma 41 R->L }
% 67.38/8.95 join(Y, meet(meet(X, complement(Y)), join(Y, X)))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(Y, meet(meet(X, complement(Y)), join(X, Y)))
% 67.38/8.95 = { by lemma 31 R->L }
% 67.38/8.95 join(complement(complement(Y)), meet(meet(X, complement(Y)), join(X, Y)))
% 67.38/8.95 = { by lemma 31 R->L }
% 67.38/8.95 join(complement(complement(Y)), meet(meet(X, complement(Y)), join(X, complement(complement(Y)))))
% 67.38/8.95 = { by lemma 24 R->L }
% 67.38/8.95 join(complement(complement(Y)), meet(meet(complement(Y), X), join(X, complement(complement(Y)))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(complement(complement(Y)), meet(meet(complement(Y), X), join(complement(complement(Y)), X)))
% 67.38/8.95 = { by lemma 24 }
% 67.38/8.95 join(complement(complement(Y)), meet(join(complement(complement(Y)), X), meet(complement(Y), X)))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(complement(Y)))
% 67.38/8.95 = { by lemma 22 R->L }
% 67.38/8.95 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(join(complement(complement(Y)), X)))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(join(complement(join(complement(complement(Y)), X)), meet(complement(Y), X))))
% 67.38/8.95 = { by lemma 22 }
% 67.38/8.95 join(complement(complement(Y)), X)
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.95 join(X, complement(complement(Y)))
% 67.38/8.95 = { by lemma 31 }
% 67.38/8.95 join(X, Y)
% 67.38/8.95
% 67.38/8.95 Lemma 43: join(complement(X), meet(Y, complement(X))) = complement(X).
% 67.38/8.95 Proof:
% 67.38/8.95 join(complement(X), meet(Y, complement(X)))
% 67.38/8.95 = { by lemma 33 R->L }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), top)
% 67.38/8.95 = { by lemma 21 R->L }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), join(complement(join(complement(complement(X)), Y)), top))
% 67.38/8.95 = { by lemma 16 R->L }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), join(meet(complement(X), Y), join(complement(meet(complement(X), Y)), complement(join(complement(complement(X)), Y)))))
% 67.38/8.95 = { by lemma 23 }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), join(complement(X), complement(meet(complement(X), Y))))
% 67.38/8.95 = { by lemma 24 }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), join(complement(X), complement(meet(Y, complement(X)))))
% 67.38/8.95 = { by lemma 25 R->L }
% 67.38/8.95 complement(join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X))))))
% 67.38/8.95 = { by lemma 22 }
% 67.38/8.95 complement(X)
% 67.38/8.95
% 67.38/8.95 Lemma 44: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 67.38/8.95 Proof:
% 67.38/8.95 join(composition(X, Y), composition(X, Z))
% 67.38/8.95 = { by axiom 5 (converse_idempotence) R->L }
% 67.38/8.95 join(composition(X, Y), composition(converse(converse(X)), Z))
% 67.38/8.95 = { by axiom 5 (converse_idempotence) R->L }
% 67.38/8.95 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 67.38/8.95 = { by lemma 17 R->L }
% 67.38/8.95 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 67.38/8.95 = { by axiom 9 (converse_additivity) R->L }
% 67.38/8.95 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 67.38/8.95 = { by axiom 7 (converse_multiplicativity) }
% 67.38/8.95 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 67.38/8.95 = { by axiom 11 (composition_distributivity) R->L }
% 67.38/8.95 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.95 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 67.38/8.95 = { by axiom 7 (converse_multiplicativity) }
% 67.38/8.95 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 67.38/8.95 = { by axiom 9 (converse_additivity) }
% 67.38/8.95 composition(converse(converse(X)), join(converse(converse(Y)), converse(converse(Z))))
% 67.38/8.95 = { by axiom 5 (converse_idempotence) }
% 67.38/8.95 composition(converse(converse(X)), join(Y, converse(converse(Z))))
% 67.38/8.95 = { by axiom 5 (converse_idempotence) }
% 67.38/8.95 composition(X, join(Y, converse(converse(Z))))
% 67.38/8.95 = { by axiom 5 (converse_idempotence) }
% 67.38/8.95 composition(X, join(Y, Z))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.95 composition(X, join(Z, Y))
% 67.38/8.95
% 67.38/8.95 Lemma 45: join(X, complement(join(Y, complement(join(X, Y))))) = X.
% 67.38/8.95 Proof:
% 67.38/8.95 join(X, complement(join(Y, complement(join(X, Y)))))
% 67.38/8.95 = { by lemma 31 R->L }
% 67.38/8.95 join(complement(complement(X)), complement(join(Y, complement(join(X, Y)))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(complement(join(Y, complement(join(X, Y)))), complement(complement(X)))
% 67.38/8.95 = { by lemma 34 R->L }
% 67.38/8.95 meet(top, join(complement(join(Y, complement(join(X, Y)))), complement(complement(X))))
% 67.38/8.95 = { by lemma 25 R->L }
% 67.38/8.95 complement(join(meet(join(Y, complement(join(X, Y))), complement(X)), complement(top)))
% 67.38/8.95 = { by lemma 14 }
% 67.38/8.95 complement(join(meet(join(Y, complement(join(X, Y))), complement(X)), zero))
% 67.38/8.95 = { by lemma 30 }
% 67.38/8.95 complement(meet(join(Y, complement(join(X, Y))), complement(X)))
% 67.38/8.95 = { by lemma 24 R->L }
% 67.38/8.95 complement(meet(complement(X), join(Y, complement(join(X, Y)))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 complement(meet(complement(X), join(Y, complement(join(Y, X)))))
% 67.38/8.95 = { by lemma 33 R->L }
% 67.38/8.95 complement(meet(complement(X), join(Y, complement(join(Y, meet(X, top))))))
% 67.38/8.95 = { by lemma 29 R->L }
% 67.38/8.95 complement(meet(join(zero, complement(X)), join(Y, complement(join(Y, meet(X, top))))))
% 67.38/8.95 = { by lemma 32 R->L }
% 67.38/8.95 complement(meet(join(zero, complement(X)), join(Y, complement(join(Y, complement(join(zero, complement(X))))))))
% 67.38/8.95 = { by lemma 39 }
% 67.38/8.95 complement(join(zero, complement(X)))
% 67.38/8.95 = { by lemma 29 }
% 67.38/8.95 complement(complement(X))
% 67.38/8.95 = { by lemma 31 }
% 67.38/8.95 X
% 67.38/8.95
% 67.38/8.95 Lemma 46: join(meet(X, Y), complement(join(X, complement(Y)))) = Y.
% 67.38/8.95 Proof:
% 67.38/8.95 join(meet(X, Y), complement(join(X, complement(Y))))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(meet(X, Y), complement(join(complement(Y), X)))
% 67.38/8.95 = { by lemma 24 R->L }
% 67.38/8.95 join(meet(Y, X), complement(join(complement(Y), X)))
% 67.38/8.95 = { by lemma 22 }
% 67.38/8.95 Y
% 67.38/8.95
% 67.38/8.95 Lemma 47: join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), Y))) = complement(Y).
% 67.38/8.95 Proof:
% 67.38/8.95 join(meet(X, complement(Y)), complement(join(meet(X, complement(Y)), Y)))
% 67.38/8.95 = { by lemma 42 }
% 67.38/8.95 join(meet(X, complement(Y)), complement(join(X, Y)))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(complement(join(X, Y)), meet(X, complement(Y)))
% 67.38/8.95 = { by lemma 31 R->L }
% 67.38/8.95 join(complement(join(X, complement(complement(Y)))), meet(X, complement(Y)))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 join(meet(X, complement(Y)), complement(join(X, complement(complement(Y)))))
% 67.38/8.95 = { by lemma 46 }
% 67.38/8.95 complement(Y)
% 67.38/8.95
% 67.38/8.95 Lemma 48: meet(complement(X), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X)) = join(meet(Y, complement(X)), meet(Z, complement(X))).
% 67.38/8.95 Proof:
% 67.38/8.95 meet(complement(X), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.95 = { by lemma 43 R->L }
% 67.38/8.95 meet(join(complement(X), meet(Y, complement(X))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 meet(join(meet(Y, complement(X)), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.95 = { by lemma 24 R->L }
% 67.38/8.95 meet(join(meet(complement(X), Y), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.95 = { by lemma 43 R->L }
% 67.38/8.95 meet(join(meet(complement(X), Y), join(complement(X), meet(Z, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.95 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.95 meet(join(meet(complement(X), Y), join(meet(Z, complement(X)), complement(X))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by lemma 28 R->L }
% 67.38/8.96 meet(join(meet(complement(X), Y), join(meet(Z, complement(X)), join(meet(Y, complement(X)), meet(complement(X), complement(Y))))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by lemma 38 R->L }
% 67.38/8.96 meet(join(meet(complement(X), Y), join(meet(complement(X), complement(Y)), join(meet(Y, complement(X)), meet(Z, complement(X))))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by axiom 10 (maddux2_join_associativity) }
% 67.38/8.96 meet(join(join(meet(complement(X), Y), meet(complement(X), complement(Y))), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by lemma 27 }
% 67.38/8.96 meet(join(complement(X), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.96 meet(join(join(meet(Y, complement(X)), meet(Z, complement(X))), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 meet(join(join(meet(Y, complement(X)), meet(Z, complement(X))), complement(X)), join(X, join(meet(Y, complement(X)), meet(Z, complement(X)))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 meet(join(complement(X), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(X, join(meet(Y, complement(X)), meet(Z, complement(X)))))
% 67.38/8.96 = { by lemma 33 R->L }
% 67.38/8.96 meet(join(complement(X), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(X, meet(join(meet(Y, complement(X)), meet(Z, complement(X))), top)))
% 67.38/8.96 = { by lemma 32 R->L }
% 67.38/8.96 meet(join(complement(X), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(X, complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))))
% 67.38/8.96 = { by lemma 33 R->L }
% 67.38/8.96 meet(join(complement(X), meet(join(meet(Y, complement(X)), meet(Z, complement(X))), top)), join(X, complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))))
% 67.38/8.96 = { by lemma 32 R->L }
% 67.38/8.96 meet(join(complement(X), complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))), join(X, complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))))
% 67.38/8.96 = { by lemma 24 }
% 67.38/8.96 meet(join(X, complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))), join(complement(X), complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))))
% 67.38/8.96 = { by lemma 26 }
% 67.38/8.96 complement(join(zero, complement(join(meet(Y, complement(X)), meet(Z, complement(X))))))
% 67.38/8.96 = { by lemma 32 }
% 67.38/8.96 meet(join(meet(Y, complement(X)), meet(Z, complement(X))), top)
% 67.38/8.96 = { by lemma 33 }
% 67.38/8.96 join(meet(Y, complement(X)), meet(Z, complement(X)))
% 67.38/8.96
% 67.38/8.96 Lemma 49: join(meet(composition(X, meet(Y, complement(Z))), complement(composition(X, Z))), composition(X, Z)) = join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, Z)).
% 67.38/8.96 Proof:
% 67.38/8.96 join(meet(composition(X, meet(Y, complement(Z))), complement(composition(X, Z))), composition(X, Z))
% 67.38/8.96 = { by lemma 42 }
% 67.38/8.96 join(composition(X, meet(Y, complement(Z))), composition(X, Z))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 join(composition(X, Z), composition(X, meet(Y, complement(Z))))
% 67.38/8.96 = { by lemma 44 }
% 67.38/8.96 composition(X, join(meet(Y, complement(Z)), Z))
% 67.38/8.96 = { by lemma 31 R->L }
% 67.38/8.96 composition(X, join(meet(Y, complement(Z)), complement(complement(Z))))
% 67.38/8.96 = { by lemma 41 R->L }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(complement(Z))))
% 67.38/8.96 = { by lemma 46 R->L }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Y, complement(complement(Z))))))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(complement(join(Y, complement(complement(Z)))), meet(Y, complement(Z))))))
% 67.38/8.96 = { by lemma 31 }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(complement(join(Y, Z)), meet(Y, complement(Z))))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Y, Z))))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.96 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Z, Y))))))
% 67.38/8.96 = { by lemma 46 }
% 67.38/8.96 composition(X, join(Z, Y))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 composition(X, join(Y, Z))
% 67.38/8.96 = { by lemma 44 R->L }
% 67.38/8.96 join(composition(X, Z), composition(X, Y))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) }
% 67.38/8.96 join(composition(X, Y), composition(X, Z))
% 67.38/8.96 = { by lemma 42 R->L }
% 67.38/8.96 join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, Z))
% 67.38/8.96
% 67.38/8.96 Goal 1 (goals): tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))) = tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(composition(x0, x1), complement(composition(x0, x2)))).
% 67.38/8.96 Proof:
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 67.38/8.96 = { by lemma 40 R->L }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))))
% 67.38/8.96 = { by lemma 48 R->L }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(complement(composition(x0, x2)), join(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), composition(x0, x2)))))
% 67.38/8.96 = { by lemma 24 }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))), composition(x0, x2)), complement(composition(x0, x2)))))
% 67.38/8.96 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(composition(x0, x2), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2))))), complement(composition(x0, x2)))))
% 67.38/8.96 = { by lemma 37 R->L }
% 67.38/8.96 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2))), complement(composition(x0, x2)))))
% 67.38/8.97 = { by lemma 49 R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), composition(x0, x2))), complement(composition(x0, x2)))))
% 67.38/8.97 = { by lemma 36 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), composition(x0, x2)), complement(composition(x0, x2)))))
% 67.38/8.97 = { by lemma 49 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), meet(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2)), complement(composition(x0, x2)))))
% 67.38/8.97 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2))), complement(complement(composition(x0, x2)))))))
% 67.38/8.97 = { by lemma 47 R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2))), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), complement(join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2)))))))))
% 67.38/8.97 = { by lemma 45 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 40 R->L }
% 67.38/8.97 tuple(join(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 48 R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0_2, x2_2)), join(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 24 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2)))), composition(x0_2, x2_2)), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(composition(x0_2, x2_2), join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))))), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 38 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(meet(composition(x0_2, x1_2), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 49 R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2))), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 36 }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2)), complement(composition(x0_2, x2_2)))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by axiom 6 (maddux4_definiton_of_meet) }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), complement(join(complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2))), complement(complement(composition(x0_2, x2_2)))))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 47 R->L }
% 67.38/8.97 tuple(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), complement(join(complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2))), complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), complement(join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), composition(x0_2, x2_2)))))))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 = { by lemma 45 }
% 67.38/8.97 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(composition(x0, x1), complement(composition(x0, x2))))
% 67.38/8.97 % SZS output end Proof
% 67.38/8.97
% 67.38/8.97 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------