%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL016+4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:07 PM UTC 2026
% Result : Theorem 45.73s 6.24s
% Output : Proof 46.56s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : REL016+4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.10/0.36 % Computer : n003.cluster.edu
% 0.10/0.36 % Model : x86_64 x86_64
% 0.10/0.36 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.10/0.36 % Memory : 8046.5625MB
% 0.10/0.36 % OS : Linux 6.8.0-71-generic
% 0.10/0.36 % CPULimit : 300
% 0.10/0.36 % WCLimit : 300
% 0.10/0.36 % DateTime : Sun Sep 27 22:53:26 UTC 2026
% 0.10/0.36 % CPUTime :
% 0.10/0.36 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 45.73/6.24 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
% 45.73/6.24
% 45.73/6.24 % SZS status Theorem
% 45.73/6.24
% 45.73/6.29 % SZS output start Proof
% 45.73/6.29 Axiom 1 (composition_identity): composition(X, one) = X.
% 45.73/6.29 Axiom 2 (def_zero): zero = meet(X, complement(X)).
% 45.73/6.29 Axiom 3 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 45.73/6.29 Axiom 4 (def_top): top = join(X, complement(X)).
% 45.73/6.29 Axiom 5 (converse_idempotence): converse(converse(X)) = X.
% 45.73/6.29 Axiom 6 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 45.73/6.29 Axiom 7 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 45.73/6.29 Axiom 8 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 45.73/6.29 Axiom 9 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 45.73/6.29 Axiom 10 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 45.73/6.29 Axiom 11 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 45.73/6.29 Axiom 12 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 45.73/6.29 Axiom 13 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 45.73/6.29 Axiom 14 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 45.73/6.29
% 45.73/6.29 Lemma 15: complement(top) = zero.
% 45.73/6.29 Proof:
% 45.73/6.29 complement(top)
% 45.73/6.29 = { by axiom 4 (def_top) }
% 45.73/6.29 complement(join(complement(X), complement(complement(X))))
% 45.73/6.29 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 45.73/6.29 meet(X, complement(X))
% 45.73/6.29 = { by axiom 2 (def_zero) R->L }
% 45.73/6.29 zero
% 45.73/6.29
% 45.73/6.29 Lemma 16: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 45.73/6.29 Proof:
% 45.73/6.29 join(X, join(Y, Z))
% 45.73/6.29 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.29 join(join(Y, Z), X)
% 45.73/6.29 = { by axiom 10 (maddux2_join_associativity) R->L }
% 45.73/6.29 join(Y, join(Z, X))
% 45.73/6.29 = { by axiom 3 (maddux1_join_commutativity) }
% 45.73/6.29 join(Y, join(X, Z))
% 45.73/6.29
% 45.73/6.29 Lemma 17: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 join(meet(X, Y), complement(join(complement(X), Y)))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 45.73/6.30 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 45.73/6.30 = { by axiom 12 (maddux3_a_kind_of_de_Morgan) R->L }
% 45.73/6.30 X
% 45.73/6.30
% 45.73/6.30 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 45.73/6.30 Proof:
% 45.73/6.30 join(X, join(Y, complement(X)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(X, join(complement(X), Y))
% 45.73/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 45.73/6.30 join(join(X, complement(X)), Y)
% 45.73/6.30 = { by axiom 4 (def_top) R->L }
% 45.73/6.30 join(top, Y)
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) }
% 45.73/6.30 join(Y, top)
% 45.73/6.30
% 45.73/6.30 Lemma 19: join(X, join(Y, complement(join(X, Y)))) = top.
% 45.73/6.30 Proof:
% 45.73/6.30 join(X, join(Y, complement(join(X, Y))))
% 45.73/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 45.73/6.30 join(join(X, Y), complement(join(X, Y)))
% 45.73/6.30 = { by axiom 4 (def_top) R->L }
% 45.73/6.30 top
% 45.73/6.30
% 45.73/6.30 Lemma 20: join(X, top) = top.
% 45.73/6.30 Proof:
% 45.73/6.30 join(X, top)
% 45.73/6.30 = { by axiom 4 (def_top) }
% 45.73/6.30 join(X, join(composition(Y, Z), complement(composition(Y, Z))))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(X, join(complement(composition(Y, Z)), composition(Y, Z)))
% 45.73/6.30 = { by lemma 16 R->L }
% 45.73/6.30 join(complement(composition(Y, Z)), join(X, composition(Y, Z)))
% 45.73/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 45.73/6.30 join(join(complement(composition(Y, Z)), X), composition(Y, Z))
% 45.73/6.30 = { by lemma 17 R->L }
% 45.73/6.30 join(join(complement(composition(Y, Z)), X), join(meet(composition(Y, Z), X), complement(join(complement(composition(Y, Z)), X))))
% 45.73/6.30 = { by lemma 18 }
% 45.73/6.30 join(meet(composition(Y, Z), X), top)
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) }
% 45.73/6.30 join(top, meet(composition(Y, Z), X))
% 45.73/6.30 = { by axiom 1 (composition_identity) R->L }
% 45.73/6.30 join(top, meet(composition(Y, Z), composition(X, one)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(meet(composition(Y, Z), composition(X, one)), top)
% 45.73/6.30 = { by axiom 4 (def_top) }
% 45.73/6.30 join(meet(composition(Y, Z), composition(X, one)), join(meet(composition(Y, meet(Z, composition(converse(Y), composition(X, one)))), composition(X, one)), complement(meet(composition(Y, meet(Z, composition(converse(Y), composition(X, one)))), composition(X, one)))))
% 45.73/6.30 = { by axiom 14 (modular_law_1) R->L }
% 45.73/6.30 join(meet(composition(Y, Z), composition(X, one)), join(meet(composition(Y, meet(Z, composition(converse(Y), composition(X, one)))), composition(X, one)), complement(join(meet(composition(Y, Z), composition(X, one)), meet(composition(Y, meet(Z, composition(converse(Y), composition(X, one)))), composition(X, one))))))
% 45.73/6.30 = { by lemma 19 }
% 45.73/6.30 top
% 45.73/6.30
% 45.73/6.30 Lemma 21: join(meet(X, Y), join(Z, complement(join(complement(X), Y)))) = join(X, Z).
% 45.73/6.30 Proof:
% 45.73/6.30 join(meet(X, Y), join(Z, complement(join(complement(X), Y))))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(meet(X, Y), join(complement(join(complement(X), Y)), Z))
% 45.73/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 45.73/6.30 join(join(meet(X, Y), complement(join(complement(X), Y))), Z)
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 join(X, Z)
% 45.73/6.30
% 45.73/6.30 Lemma 22: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 45.73/6.30 Proof:
% 45.73/6.30 converse(composition(converse(X), Y))
% 45.73/6.30 = { by axiom 7 (converse_multiplicativity) }
% 45.73/6.30 composition(converse(Y), converse(converse(X)))
% 45.73/6.30 = { by axiom 5 (converse_idempotence) }
% 45.73/6.30 composition(converse(Y), X)
% 45.73/6.30
% 45.73/6.30 Lemma 23: composition(converse(one), X) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 composition(converse(one), X)
% 45.73/6.30 = { by lemma 22 R->L }
% 45.73/6.30 converse(composition(converse(X), one))
% 45.73/6.30 = { by axiom 1 (composition_identity) }
% 45.73/6.30 converse(converse(X))
% 45.73/6.30 = { by axiom 5 (converse_idempotence) }
% 45.73/6.30 X
% 45.73/6.30
% 45.73/6.30 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 45.73/6.30 Proof:
% 45.73/6.30 join(complement(X), complement(X))
% 45.73/6.30 = { by lemma 23 R->L }
% 45.73/6.30 join(complement(X), complement(composition(converse(one), X)))
% 45.73/6.30 = { by axiom 1 (composition_identity) R->L }
% 45.73/6.30 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 45.73/6.30 = { by axiom 8 (composition_associativity) R->L }
% 45.73/6.30 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 45.73/6.30 = { by lemma 23 }
% 45.73/6.30 join(complement(X), complement(composition(one, X)))
% 45.73/6.30 = { by axiom 5 (converse_idempotence) R->L }
% 45.73/6.30 join(complement(X), complement(composition(converse(converse(one)), X)))
% 45.73/6.30 = { by lemma 23 R->L }
% 45.73/6.30 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 45.73/6.30 = { by axiom 5 (converse_idempotence) R->L }
% 45.73/6.30 join(complement(X), composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(composition(converse(converse(converse(one))), complement(composition(converse(converse(one)), X))), complement(X))
% 45.73/6.30 = { by axiom 13 (converse_cancellativity) }
% 45.73/6.30 complement(X)
% 45.73/6.30
% 45.73/6.30 Lemma 25: join(X, complement(meet(X, Y))) = top.
% 45.73/6.30 Proof:
% 45.73/6.30 join(X, complement(meet(X, Y)))
% 45.73/6.30 = { by lemma 21 R->L }
% 45.73/6.30 join(meet(X, Y), join(complement(meet(X, Y)), complement(join(complement(X), Y))))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(meet(X, Y), join(complement(join(complement(X), Y)), complement(meet(X, Y))))
% 45.73/6.30 = { by lemma 18 }
% 45.73/6.30 join(complement(join(complement(X), Y)), top)
% 45.73/6.30 = { by lemma 20 }
% 45.73/6.30 top
% 45.73/6.30
% 45.73/6.30 Lemma 26: meet(Y, X) = meet(X, Y).
% 45.73/6.30 Proof:
% 45.73/6.30 meet(Y, X)
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 45.73/6.30 complement(join(complement(Y), complement(X)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 complement(join(complement(X), complement(Y)))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 45.73/6.30 meet(X, Y)
% 45.73/6.30
% 45.73/6.30 Lemma 27: complement(join(zero, complement(X))) = meet(X, top).
% 45.73/6.30 Proof:
% 45.73/6.30 complement(join(zero, complement(X)))
% 45.73/6.30 = { by lemma 15 R->L }
% 45.73/6.30 complement(join(complement(top), complement(X)))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 45.73/6.30 meet(top, X)
% 45.73/6.30 = { by lemma 26 R->L }
% 45.73/6.30 meet(X, top)
% 45.73/6.30
% 45.73/6.30 Lemma 28: join(meet(X, Y), complement(join(complement(Y), X))) = Y.
% 45.73/6.30 Proof:
% 45.73/6.30 join(meet(X, Y), complement(join(complement(Y), X)))
% 45.73/6.30 = { by lemma 26 }
% 45.73/6.30 join(meet(Y, X), complement(join(complement(Y), X)))
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 Y
% 45.73/6.30
% 45.73/6.30 Lemma 29: complement(join(meet(X, Y), complement(Z))) = meet(Z, join(complement(X), complement(Y))).
% 45.73/6.30 Proof:
% 45.73/6.30 complement(join(meet(X, Y), complement(Z)))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 45.73/6.30 complement(join(complement(join(complement(X), complement(Y))), complement(Z)))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 45.73/6.30 meet(join(complement(X), complement(Y)), Z)
% 45.73/6.30 = { by lemma 26 R->L }
% 45.73/6.30 meet(Z, join(complement(X), complement(Y)))
% 45.73/6.30
% 45.73/6.30 Lemma 30: meet(join(X, complement(Y)), join(complement(X), complement(Y))) = complement(Y).
% 45.73/6.30 Proof:
% 45.73/6.30 meet(join(X, complement(Y)), join(complement(X), complement(Y)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 meet(join(X, complement(Y)), join(complement(Y), complement(X)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 meet(join(complement(Y), X), join(complement(Y), complement(X)))
% 45.73/6.30 = { by lemma 29 R->L }
% 45.73/6.30 complement(join(meet(Y, X), complement(join(complement(Y), X))))
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 complement(Y)
% 45.73/6.30
% 45.73/6.30 Lemma 31: join(meet(X, Y), meet(X, complement(Y))) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 join(meet(X, Y), meet(X, complement(Y)))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(meet(X, complement(Y)), meet(X, Y))
% 45.73/6.30 = { by axiom 6 (maddux4_definiton_of_meet) }
% 45.73/6.30 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 X
% 45.73/6.30
% 45.73/6.30 Lemma 32: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 45.73/6.30 Proof:
% 45.73/6.30 join(meet(X, Y), meet(Y, complement(X)))
% 45.73/6.30 = { by lemma 26 }
% 45.73/6.30 join(meet(Y, X), meet(Y, complement(X)))
% 45.73/6.30 = { by lemma 31 }
% 45.73/6.30 Y
% 45.73/6.30
% 45.73/6.30 Lemma 33: join(zero, complement(X)) = complement(X).
% 45.73/6.30 Proof:
% 45.73/6.30 join(zero, complement(X))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(complement(X), zero)
% 45.73/6.30 = { by lemma 15 R->L }
% 45.73/6.30 join(complement(X), complement(top))
% 45.73/6.30 = { by lemma 24 R->L }
% 45.73/6.30 join(complement(X), join(complement(top), complement(top)))
% 45.73/6.30 = { by lemma 15 }
% 45.73/6.30 join(complement(X), join(zero, complement(top)))
% 45.73/6.30 = { by lemma 20 R->L }
% 45.73/6.30 join(complement(X), join(zero, complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 15 R->L }
% 45.73/6.30 join(complement(X), join(complement(top), complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 25 R->L }
% 45.73/6.30 join(complement(X), join(complement(join(zero, complement(meet(zero, complement(X))))), complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 27 }
% 45.73/6.30 join(complement(X), join(meet(meet(zero, complement(X)), top), complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 26 R->L }
% 45.73/6.30 join(complement(X), join(meet(top, meet(zero, complement(X))), complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 26 R->L }
% 45.73/6.30 join(complement(X), join(meet(top, meet(complement(X), zero)), complement(join(complement(meet(complement(X), zero)), top))))
% 45.73/6.30 = { by lemma 28 }
% 45.73/6.30 join(complement(X), meet(complement(X), zero))
% 45.73/6.30 = { by lemma 15 R->L }
% 45.73/6.30 join(complement(X), meet(complement(X), complement(top)))
% 45.73/6.30 = { by lemma 30 R->L }
% 45.73/6.30 join(meet(join(X, complement(X)), join(complement(X), complement(X))), meet(complement(X), complement(top)))
% 45.73/6.30 = { by axiom 4 (def_top) R->L }
% 45.73/6.30 join(meet(top, join(complement(X), complement(X))), meet(complement(X), complement(top)))
% 45.73/6.30 = { by lemma 24 }
% 45.73/6.30 join(meet(top, complement(X)), meet(complement(X), complement(top)))
% 45.73/6.30 = { by lemma 32 }
% 45.73/6.30 complement(X)
% 45.73/6.30
% 45.73/6.30 Lemma 34: join(X, zero) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 join(X, zero)
% 45.73/6.30 = { by lemma 21 R->L }
% 45.73/6.30 join(meet(X, Y), join(zero, complement(join(complement(X), Y))))
% 45.73/6.30 = { by lemma 33 }
% 45.73/6.30 join(meet(X, Y), complement(join(complement(X), Y)))
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 X
% 45.73/6.30
% 45.73/6.30 Lemma 35: complement(complement(X)) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 complement(complement(X))
% 45.73/6.30 = { by lemma 24 R->L }
% 45.73/6.30 complement(join(complement(X), complement(X)))
% 45.73/6.30 = { by lemma 33 R->L }
% 45.73/6.30 join(zero, complement(join(complement(X), complement(X))))
% 45.73/6.30 = { by axiom 2 (def_zero) }
% 45.73/6.30 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 45.73/6.30 = { by lemma 17 }
% 45.73/6.30 X
% 45.73/6.30
% 45.73/6.30 Lemma 36: meet(X, top) = X.
% 45.73/6.30 Proof:
% 45.73/6.30 meet(X, top)
% 45.73/6.30 = { by lemma 27 R->L }
% 45.73/6.30 complement(join(zero, complement(X)))
% 45.73/6.30 = { by lemma 33 R->L }
% 45.73/6.30 join(zero, complement(join(zero, complement(X))))
% 45.73/6.30 = { by lemma 27 }
% 45.73/6.30 join(zero, meet(X, top))
% 45.73/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 45.73/6.30 join(meet(X, top), zero)
% 45.73/6.30 = { by lemma 15 R->L }
% 45.73/6.30 join(meet(X, top), complement(top))
% 45.73/6.30 = { by lemma 20 R->L }
% 46.56/6.30 join(meet(X, top), complement(join(complement(X), top)))
% 46.56/6.30 = { by lemma 17 }
% 46.56/6.30 X
% 46.56/6.30
% 46.56/6.30 Lemma 37: join(X, join(X, Y)) = join(X, Y).
% 46.56/6.30 Proof:
% 46.56/6.30 join(X, join(X, Y))
% 46.56/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.30 join(X, join(Y, X))
% 46.56/6.30 = { by lemma 35 R->L }
% 46.56/6.30 join(X, join(Y, complement(complement(X))))
% 46.56/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 46.56/6.30 join(join(X, Y), complement(complement(X)))
% 46.56/6.30 = { by lemma 30 R->L }
% 46.56/6.30 join(join(X, Y), complement(meet(join(join(Y, complement(complement(X))), complement(X)), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.30 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.30 join(join(X, Y), complement(meet(join(complement(X), join(Y, complement(complement(X)))), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.30 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.30 join(join(X, Y), complement(meet(join(complement(X), join(complement(complement(X)), Y)), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.30 = { by axiom 10 (maddux2_join_associativity) }
% 46.56/6.30 join(join(X, Y), complement(meet(join(join(complement(X), complement(complement(X))), Y), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.30 = { by axiom 4 (def_top) R->L }
% 46.56/6.30 join(join(X, Y), complement(meet(join(top, Y), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.30 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 join(join(X, Y), complement(meet(join(Y, top), join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.31 = { by lemma 20 }
% 46.56/6.31 join(join(X, Y), complement(meet(top, join(complement(join(Y, complement(complement(X)))), complement(X)))))
% 46.56/6.31 = { by lemma 26 }
% 46.56/6.31 join(join(X, Y), complement(meet(join(complement(join(Y, complement(complement(X)))), complement(X)), top)))
% 46.56/6.31 = { by lemma 36 }
% 46.56/6.31 join(join(X, Y), complement(join(complement(join(Y, complement(complement(X)))), complement(X))))
% 46.56/6.31 = { by lemma 35 }
% 46.56/6.31 join(join(X, Y), complement(join(complement(join(Y, X)), complement(X))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 join(join(X, Y), complement(join(complement(X), complement(join(Y, X)))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 join(join(X, Y), complement(join(complement(X), complement(join(X, Y)))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(join(X, Y), complement(join(complement(join(X, Y)), complement(X))))
% 46.56/6.31 = { by lemma 21 R->L }
% 46.56/6.31 join(meet(join(X, Y), complement(X)), join(complement(join(complement(join(X, Y)), complement(X))), complement(join(complement(join(X, Y)), complement(X)))))
% 46.56/6.31 = { by lemma 24 }
% 46.56/6.31 join(meet(join(X, Y), complement(X)), complement(join(complement(join(X, Y)), complement(X))))
% 46.56/6.31 = { by lemma 17 }
% 46.56/6.31 join(X, Y)
% 46.56/6.31
% 46.56/6.31 Lemma 38: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 46.56/6.31 Proof:
% 46.56/6.31 join(Z, join(X, Y))
% 46.56/6.31 = { by lemma 16 R->L }
% 46.56/6.31 join(X, join(Z, Y))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 join(X, join(Y, Z))
% 46.56/6.31
% 46.56/6.31 Lemma 39: meet(meet(X, complement(Y)), join(Z, X)) = meet(X, complement(Y)).
% 46.56/6.31 Proof:
% 46.56/6.31 meet(meet(X, complement(Y)), join(Z, X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(X, Z))
% 46.56/6.31 = { by lemma 17 R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), complement(join(complement(X), complement(Y)))), Z))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), complement(join(complement(Y), complement(X)))), Z))
% 46.56/6.31 = { by axiom 6 (maddux4_definiton_of_meet) R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), meet(Y, X)), Z))
% 46.56/6.31 = { by lemma 26 R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(meet(X, complement(Y)), meet(X, Y)), Z))
% 46.56/6.31 = { by axiom 10 (maddux2_join_associativity) R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(meet(X, complement(Y)), join(meet(X, Y), Z)))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 meet(meet(X, complement(Y)), join(meet(X, complement(Y)), join(Z, meet(X, Y))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), meet(X, complement(Y))))
% 46.56/6.31 = { by lemma 34 R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), zero)))
% 46.56/6.31 = { by lemma 15 R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), complement(top))))
% 46.56/6.31 = { by lemma 20 R->L }
% 46.56/6.31 meet(meet(X, complement(Y)), join(join(Z, meet(X, Y)), join(meet(X, complement(Y)), complement(join(join(Z, meet(X, Y)), top)))))
% 46.56/6.31 = { by lemma 35 R->L }
% 46.56/6.31 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)))))
% 46.56/6.31 = { by axiom 10 (maddux2_join_associativity) }
% 46.56/6.31 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))))
% 46.56/6.31 = { by lemma 18 R->L }
% 46.56/6.31 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)))))))))
% 46.56/6.31 = { by lemma 34 R->L }
% 46.56/6.31 join(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))))))))), zero)
% 46.56/6.31 = { by lemma 15 R->L }
% 46.56/6.31 join(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))))))))), complement(top))
% 46.56/6.31 = { by lemma 19 R->L }
% 46.56/6.31 join(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))))))))), complement(join(complement(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)))))))))))
% 46.56/6.31 = { by lemma 17 }
% 46.56/6.31 meet(X, complement(Y))
% 46.56/6.31
% 46.56/6.31 Lemma 40: join(meet(X, complement(Y)), Y) = join(X, Y).
% 46.56/6.31 Proof:
% 46.56/6.31 join(meet(X, complement(Y)), Y)
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(Y, meet(X, complement(Y)))
% 46.56/6.31 = { by lemma 39 R->L }
% 46.56/6.31 join(Y, meet(meet(X, complement(Y)), join(Y, X)))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(Y, meet(meet(X, complement(Y)), join(X, Y)))
% 46.56/6.31 = { by lemma 35 R->L }
% 46.56/6.31 join(complement(complement(Y)), meet(meet(X, complement(Y)), join(X, Y)))
% 46.56/6.31 = { by lemma 35 R->L }
% 46.56/6.31 join(complement(complement(Y)), meet(meet(X, complement(Y)), join(X, complement(complement(Y)))))
% 46.56/6.31 = { by lemma 26 R->L }
% 46.56/6.31 join(complement(complement(Y)), meet(meet(complement(Y), X), join(X, complement(complement(Y)))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(complement(complement(Y)), meet(meet(complement(Y), X), join(complement(complement(Y)), X)))
% 46.56/6.31 = { by lemma 26 }
% 46.56/6.31 join(complement(complement(Y)), meet(join(complement(complement(Y)), X), meet(complement(Y), X)))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(complement(Y)))
% 46.56/6.31 = { by lemma 17 R->L }
% 46.56/6.31 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(join(meet(complement(Y), X), complement(join(complement(complement(Y)), X)))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(meet(join(complement(complement(Y)), X), meet(complement(Y), X)), complement(join(complement(join(complement(complement(Y)), X)), meet(complement(Y), X))))
% 46.56/6.31 = { by lemma 17 }
% 46.56/6.31 join(complement(complement(Y)), X)
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 join(X, complement(complement(Y)))
% 46.56/6.31 = { by lemma 35 }
% 46.56/6.31 join(X, Y)
% 46.56/6.31
% 46.56/6.31 Lemma 41: join(complement(X), meet(Y, complement(X))) = complement(X).
% 46.56/6.31 Proof:
% 46.56/6.31 join(complement(X), meet(Y, complement(X)))
% 46.56/6.31 = { by lemma 36 R->L }
% 46.56/6.31 meet(join(complement(X), meet(Y, complement(X))), top)
% 46.56/6.31 = { by lemma 25 R->L }
% 46.56/6.31 meet(join(complement(X), meet(Y, complement(X))), join(complement(X), complement(meet(complement(X), Y))))
% 46.56/6.31 = { by lemma 26 }
% 46.56/6.31 meet(join(complement(X), meet(Y, complement(X))), join(complement(X), complement(meet(Y, complement(X)))))
% 46.56/6.31 = { by lemma 29 R->L }
% 46.56/6.31 complement(join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X))))))
% 46.56/6.31 = { by lemma 17 }
% 46.56/6.31 complement(X)
% 46.56/6.31
% 46.56/6.31 Lemma 42: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 46.56/6.31 Proof:
% 46.56/6.31 join(composition(X, Y), composition(X, Z))
% 46.56/6.31 = { by axiom 5 (converse_idempotence) R->L }
% 46.56/6.31 join(composition(X, Y), composition(converse(converse(X)), Z))
% 46.56/6.31 = { by axiom 5 (converse_idempotence) R->L }
% 46.56/6.31 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 46.56/6.31 = { by lemma 22 R->L }
% 46.56/6.31 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 46.56/6.31 = { by axiom 9 (converse_additivity) R->L }
% 46.56/6.31 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 46.56/6.31 = { by axiom 7 (converse_multiplicativity) }
% 46.56/6.31 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 46.56/6.31 = { by axiom 11 (composition_distributivity) R->L }
% 46.56/6.31 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 46.56/6.31 = { by axiom 7 (converse_multiplicativity) }
% 46.56/6.31 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 46.56/6.31 = { by axiom 9 (converse_additivity) }
% 46.56/6.31 composition(converse(converse(X)), join(converse(converse(Y)), converse(converse(Z))))
% 46.56/6.31 = { by axiom 5 (converse_idempotence) }
% 46.56/6.31 composition(converse(converse(X)), join(Y, converse(converse(Z))))
% 46.56/6.31 = { by axiom 5 (converse_idempotence) }
% 46.56/6.31 composition(X, join(Y, converse(converse(Z))))
% 46.56/6.31 = { by axiom 5 (converse_idempotence) }
% 46.56/6.31 composition(X, join(Y, Z))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 composition(X, join(Z, Y))
% 46.56/6.31
% 46.56/6.31 Lemma 43: meet(join(X, complement(Y)), join(X, Y)) = X.
% 46.56/6.31 Proof:
% 46.56/6.31 meet(join(X, complement(Y)), join(X, Y))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(join(X, complement(Y)), join(Y, X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(join(complement(Y), X), join(Y, X))
% 46.56/6.31 = { by lemma 36 R->L }
% 46.56/6.31 meet(join(complement(Y), X), join(Y, meet(X, top)))
% 46.56/6.31 = { by lemma 27 R->L }
% 46.56/6.31 meet(join(complement(Y), X), join(Y, complement(join(zero, complement(X)))))
% 46.56/6.31 = { by lemma 36 R->L }
% 46.56/6.31 meet(join(complement(Y), meet(X, top)), join(Y, complement(join(zero, complement(X)))))
% 46.56/6.31 = { by lemma 27 R->L }
% 46.56/6.31 meet(join(complement(Y), complement(join(zero, complement(X)))), join(Y, complement(join(zero, complement(X)))))
% 46.56/6.31 = { by lemma 26 }
% 46.56/6.31 meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), complement(join(zero, complement(X)))))
% 46.56/6.31 = { by lemma 30 }
% 46.56/6.31 complement(join(zero, complement(X)))
% 46.56/6.31 = { by lemma 27 }
% 46.56/6.31 meet(X, top)
% 46.56/6.31 = { by lemma 36 }
% 46.56/6.31 X
% 46.56/6.31
% 46.56/6.31 Lemma 44: meet(join(complement(X), Y), join(Y, X)) = Y.
% 46.56/6.31 Proof:
% 46.56/6.31 meet(join(complement(X), Y), join(Y, X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(join(Y, complement(X)), join(Y, X))
% 46.56/6.31 = { by lemma 43 }
% 46.56/6.31 Y
% 46.56/6.31
% 46.56/6.31 Lemma 45: join(meet(X, Y), complement(join(X, complement(Y)))) = Y.
% 46.56/6.31 Proof:
% 46.56/6.31 join(meet(X, Y), complement(join(X, complement(Y))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(meet(X, Y), complement(join(complement(Y), X)))
% 46.56/6.31 = { by lemma 28 }
% 46.56/6.31 Y
% 46.56/6.31
% 46.56/6.31 Lemma 46: meet(complement(X), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X)) = join(meet(Y, complement(X)), meet(Z, complement(X))).
% 46.56/6.31 Proof:
% 46.56/6.31 meet(complement(X), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 41 R->L }
% 46.56/6.31 meet(join(complement(X), meet(Z, complement(X))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(join(meet(Z, complement(X)), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 26 R->L }
% 46.56/6.31 meet(join(meet(complement(X), Z), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 41 R->L }
% 46.56/6.31 meet(join(meet(complement(X), Z), join(complement(X), meet(Y, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 meet(join(meet(complement(X), Z), join(meet(Y, complement(X)), complement(X))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 32 R->L }
% 46.56/6.31 meet(join(meet(complement(X), Z), join(meet(Y, complement(X)), join(meet(Z, complement(X)), meet(complement(X), complement(Z))))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 38 R->L }
% 46.56/6.31 meet(join(meet(complement(X), Z), join(meet(complement(X), complement(Z)), join(meet(Y, complement(X)), meet(Z, complement(X))))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by axiom 10 (maddux2_join_associativity) }
% 46.56/6.31 meet(join(join(meet(complement(X), Z), meet(complement(X), complement(Z))), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 31 }
% 46.56/6.31 meet(join(complement(X), join(meet(Y, complement(X)), meet(Z, complement(X)))), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 meet(join(join(meet(Y, complement(X)), meet(Z, complement(X))), complement(X)), join(join(meet(Y, complement(X)), meet(Z, complement(X))), X))
% 46.56/6.31 = { by lemma 43 }
% 46.56/6.31 join(meet(Y, complement(X)), meet(Z, complement(X)))
% 46.56/6.31
% 46.56/6.31 Lemma 47: 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)).
% 46.56/6.31 Proof:
% 46.56/6.31 join(meet(composition(X, meet(Y, complement(Z))), complement(composition(X, Z))), composition(X, Z))
% 46.56/6.31 = { by lemma 40 }
% 46.56/6.31 join(composition(X, meet(Y, complement(Z))), composition(X, Z))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.31 join(composition(X, Z), composition(X, meet(Y, complement(Z))))
% 46.56/6.31 = { by lemma 42 }
% 46.56/6.31 composition(X, join(meet(Y, complement(Z)), Z))
% 46.56/6.31 = { by lemma 35 R->L }
% 46.56/6.31 composition(X, join(meet(Y, complement(Z)), complement(complement(Z))))
% 46.56/6.31 = { by lemma 39 R->L }
% 46.56/6.31 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(complement(Z))))
% 46.56/6.31 = { by lemma 45 R->L }
% 46.56/6.31 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Y, complement(complement(Z))))))))
% 46.56/6.31 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.31 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(complement(join(Y, complement(complement(Z)))), meet(Y, complement(Z))))))
% 46.56/6.31 = { by lemma 35 }
% 46.56/6.32 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(complement(join(Y, Z)), meet(Y, complement(Z))))))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.32 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Y, Z))))))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.32 composition(X, join(meet(meet(Y, complement(Z)), join(Z, Y)), complement(join(meet(Y, complement(Z)), complement(join(Z, Y))))))
% 46.56/6.32 = { by lemma 45 }
% 46.56/6.32 composition(X, join(Z, Y))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.32 composition(X, join(Y, Z))
% 46.56/6.32 = { by lemma 42 R->L }
% 46.56/6.32 join(composition(X, Z), composition(X, Y))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.32 join(composition(X, Y), composition(X, Z))
% 46.56/6.32 = { by lemma 40 R->L }
% 46.56/6.32 join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, Z))
% 46.56/6.32
% 46.56/6.32 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)))).
% 46.56/6.32 Proof:
% 46.56/6.32 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)))))
% 46.56/6.32 = { by lemma 46 R->L }
% 46.56/6.32 tuple(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))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.32 tuple(meet(complement(composition(x0_2, x2_2)), 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)))))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.56/6.32 = { by lemma 38 R->L }
% 46.56/6.32 tuple(meet(complement(composition(x0_2, x2_2)), join(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), join(composition(x0_2, x2_2), meet(composition(x0_2, x1_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)))))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) }
% 46.56/6.32 tuple(meet(complement(composition(x0_2, x2_2)), 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)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.56/6.32 = { by lemma 47 R->L }
% 46.56/6.32 tuple(meet(complement(composition(x0_2, x2_2)), 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)))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.56/6.32 = { by lemma 37 }
% 46.56/6.32 tuple(meet(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))), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))
% 46.56/6.32 = { by lemma 41 R->L }
% 46.56/6.32 tuple(meet(join(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_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), 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)))))
% 46.56/6.32 = { by lemma 44 }
% 46.56/6.32 tuple(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)))))
% 46.56/6.32 = { by lemma 46 R->L }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), 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))))
% 46.56/6.32 = { by axiom 3 (maddux1_join_commutativity) R->L }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0, x2)), join(composition(x0, x2), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), meet(composition(x0, x1), complement(composition(x0, x2)))))))
% 46.56/6.32 = { by lemma 38 }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0, x2)), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2)))))
% 46.56/6.32 = { by lemma 47 R->L }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0, x2)), 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)))))
% 46.56/6.32 = { by lemma 37 }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0, x2)), join(meet(composition(x0, meet(x1, complement(x2))), complement(composition(x0, x2))), composition(x0, x2))))
% 46.56/6.32 = { by lemma 47 }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(complement(composition(x0, x2)), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2))))
% 46.56/6.32 = { by lemma 41 R->L }
% 46.56/6.32 tuple(meet(composition(x0_2, meet(x1_2, complement(x2_2))), complement(composition(x0_2, x2_2))), meet(join(complement(composition(x0, x2)), meet(composition(x0, x1), complement(composition(x0, x2)))), join(meet(composition(x0, x1), complement(composition(x0, x2))), composition(x0, x2))))
% 46.56/6.32 = { by lemma 44 }
% 46.56/6.32 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))))
% 46.56/6.32 % SZS output end Proof
% 46.56/6.32
% 46.56/6.32 RESULT: Theorem (the conjecture is true).
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