%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL030-4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n007.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 12:33:21 PM UTC 2026
% Result : Unsatisfiable 49.70s 6.78s
% Output : Proof 51.30s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL030-4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.12/0.37 % Computer : n007.cluster.edu
% 0.12/0.37 % Model : x86_64 x86_64
% 0.12/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.37 % Memory : 8046.5625MB
% 0.12/0.37 % OS : Linux 6.8.0-71-generic
% 0.12/0.37 % CPULimit : 300
% 0.12/0.37 % WCLimit : 300
% 0.12/0.37 % DateTime : Sun Sep 27 22:52:38 UTC 2026
% 0.12/0.37 % CPUTime :
% 0.12/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 49.70/6.78 Command-line arguments: --stitch /export/starexec/sandbox2/solver/bin/stitch --hint-skel-cost 0 --hint-skel-factor 0.5
% 49.70/6.78
% 49.70/6.78 % SZS status Unsatisfiable
% 49.70/6.78
% 51.30/6.94 % SZS output start Proof
% 51.30/6.94 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 51.30/6.94 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 51.30/6.94 Axiom 3 (def_top_12): top = join(X, complement(X)).
% 51.30/6.94 Axiom 4 (goals_17): join(sk1, one) = one.
% 51.30/6.94 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 51.30/6.94 Axiom 6 (composition_identity_6): composition(X, one) = X.
% 51.30/6.94 Axiom 7 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 51.30/6.94 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 51.30/6.94 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 51.30/6.94 Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 51.30/6.94 Axiom 11 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 51.30/6.94 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 51.30/6.94 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 51.30/6.94 Axiom 14 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 51.30/6.94
% 51.30/6.94 Lemma 15: complement(top) = zero.
% 51.30/6.94 Proof:
% 51.30/6.94 complement(top)
% 51.30/6.94 = { by axiom 3 (def_top_12) }
% 51.30/6.94 complement(join(complement(X), complement(complement(X))))
% 51.30/6.94 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.94 meet(X, complement(X))
% 51.30/6.94 = { by axiom 5 (def_zero_13) R->L }
% 51.30/6.94 zero
% 51.30/6.94
% 51.30/6.94 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 51.30/6.94 Proof:
% 51.30/6.94 join(meet(X, Y), complement(join(complement(X), Y)))
% 51.30/6.94 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 51.30/6.94 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 51.30/6.94 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 51.30/6.94 X
% 51.30/6.94
% 51.30/6.94 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 51.30/6.94 Proof:
% 51.30/6.94 join(meet(X, Y), meet(X, complement(Y)))
% 51.30/6.94 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.94 join(meet(X, complement(Y)), meet(X, Y))
% 51.30/6.94 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 51.30/6.94 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 51.30/6.94 = { by lemma 16 }
% 51.30/6.94 X
% 51.30/6.94
% 51.30/6.94 Lemma 18: meet(Y, X) = meet(X, Y).
% 51.30/6.94 Proof:
% 51.30/6.94 meet(Y, X)
% 51.30/6.94 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 51.30/6.94 complement(join(complement(Y), complement(X)))
% 51.30/6.94 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.94 complement(join(complement(X), complement(Y)))
% 51.30/6.94 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.95 meet(X, Y)
% 51.30/6.95
% 51.30/6.95 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 51.30/6.95 Proof:
% 51.30/6.95 complement(join(zero, complement(X)))
% 51.30/6.95 = { by lemma 15 R->L }
% 51.30/6.95 complement(join(complement(top), complement(X)))
% 51.30/6.95 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.95 meet(top, X)
% 51.30/6.95 = { by lemma 18 R->L }
% 51.30/6.95 meet(X, top)
% 51.30/6.95
% 51.30/6.95 Lemma 20: composition(converse(one), X) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 composition(converse(one), X)
% 51.30/6.95 = { by axiom 1 (converse_idempotence_8) R->L }
% 51.30/6.95 composition(converse(one), converse(converse(X)))
% 51.30/6.95 = { by axiom 10 (converse_multiplicativity_10) R->L }
% 51.30/6.95 converse(composition(converse(X), one))
% 51.30/6.95 = { by axiom 6 (composition_identity_6) }
% 51.30/6.95 converse(converse(X))
% 51.30/6.95 = { by axiom 1 (converse_idempotence_8) }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 21: composition(one, X) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 composition(one, X)
% 51.30/6.95 = { by lemma 20 R->L }
% 51.30/6.95 composition(converse(one), composition(one, X))
% 51.30/6.95 = { by axiom 11 (composition_associativity_5) }
% 51.30/6.95 composition(composition(converse(one), one), X)
% 51.30/6.95 = { by axiom 6 (composition_identity_6) }
% 51.30/6.95 composition(converse(one), X)
% 51.30/6.95 = { by lemma 20 }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 51.30/6.95 Proof:
% 51.30/6.95 join(complement(X), complement(X))
% 51.30/6.95 = { by lemma 20 R->L }
% 51.30/6.95 join(complement(X), composition(converse(one), complement(X)))
% 51.30/6.95 = { by lemma 21 R->L }
% 51.30/6.95 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.95 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 51.30/6.95 = { by axiom 14 (converse_cancellativity_11) }
% 51.30/6.95 complement(X)
% 51.30/6.95
% 51.30/6.95 Lemma 23: join(zero, complement(complement(X))) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 join(zero, complement(complement(X)))
% 51.30/6.95 = { by axiom 5 (def_zero_13) }
% 51.30/6.95 join(meet(X, complement(X)), complement(complement(X)))
% 51.30/6.95 = { by lemma 22 R->L }
% 51.30/6.95 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 51.30/6.95 = { by lemma 16 }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 24: meet(top, complement(X)) = complement(X).
% 51.30/6.95 Proof:
% 51.30/6.95 meet(top, complement(X))
% 51.30/6.95 = { by lemma 18 }
% 51.30/6.95 meet(complement(X), top)
% 51.30/6.95 = { by lemma 19 R->L }
% 51.30/6.95 complement(join(zero, complement(complement(X))))
% 51.30/6.95 = { by lemma 23 }
% 51.30/6.95 complement(X)
% 51.30/6.95
% 51.30/6.95 Lemma 25: complement(zero) = top.
% 51.30/6.95 Proof:
% 51.30/6.95 complement(zero)
% 51.30/6.95 = { by lemma 17 R->L }
% 51.30/6.95 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 51.30/6.95 = { by lemma 15 }
% 51.30/6.95 join(meet(complement(zero), top), meet(complement(zero), zero))
% 51.30/6.95 = { by lemma 18 R->L }
% 51.30/6.95 join(meet(complement(zero), top), meet(zero, complement(zero)))
% 51.30/6.95 = { by axiom 5 (def_zero_13) R->L }
% 51.30/6.95 join(meet(complement(zero), top), zero)
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.95 join(zero, meet(complement(zero), top))
% 51.30/6.95 = { by lemma 18 R->L }
% 51.30/6.95 join(zero, meet(top, complement(zero)))
% 51.30/6.95 = { by lemma 24 }
% 51.30/6.95 join(zero, complement(zero))
% 51.30/6.95 = { by axiom 3 (def_top_12) R->L }
% 51.30/6.95 top
% 51.30/6.95
% 51.30/6.95 Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
% 51.30/6.95 Proof:
% 51.30/6.95 join(X, join(Y, complement(X)))
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.95 join(X, join(complement(X), Y))
% 51.30/6.95 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.95 join(join(X, complement(X)), Y)
% 51.30/6.95 = { by axiom 3 (def_top_12) R->L }
% 51.30/6.95 join(top, Y)
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.95 join(Y, top)
% 51.30/6.95
% 51.30/6.95 Lemma 27: join(top, complement(X)) = top.
% 51.30/6.95 Proof:
% 51.30/6.95 join(top, complement(X))
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.95 join(complement(X), top)
% 51.30/6.95 = { by lemma 26 R->L }
% 51.30/6.95 join(X, join(complement(X), complement(X)))
% 51.30/6.95 = { by lemma 22 }
% 51.30/6.95 join(X, complement(X))
% 51.30/6.95 = { by axiom 3 (def_top_12) R->L }
% 51.30/6.95 top
% 51.30/6.95
% 51.30/6.95 Lemma 28: join(Y, top) = join(X, top).
% 51.30/6.95 Proof:
% 51.30/6.95 join(Y, top)
% 51.30/6.95 = { by lemma 27 R->L }
% 51.30/6.95 join(Y, join(top, complement(Y)))
% 51.30/6.95 = { by lemma 26 }
% 51.30/6.95 join(top, top)
% 51.30/6.95 = { by lemma 26 R->L }
% 51.30/6.95 join(X, join(top, complement(X)))
% 51.30/6.95 = { by lemma 27 }
% 51.30/6.95 join(X, top)
% 51.30/6.95
% 51.30/6.95 Lemma 29: join(X, top) = top.
% 51.30/6.95 Proof:
% 51.30/6.95 join(X, top)
% 51.30/6.95 = { by lemma 28 }
% 51.30/6.95 join(complement(top), top)
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.95 join(top, complement(top))
% 51.30/6.95 = { by axiom 3 (def_top_12) R->L }
% 51.30/6.95 top
% 51.30/6.95
% 51.30/6.95 Lemma 30: join(zero, meet(X, top)) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 join(zero, meet(X, top))
% 51.30/6.95 = { by lemma 25 R->L }
% 51.30/6.95 join(zero, meet(X, complement(zero)))
% 51.30/6.95 = { by lemma 15 R->L }
% 51.30/6.95 join(complement(top), meet(X, complement(zero)))
% 51.30/6.95 = { by lemma 27 R->L }
% 51.30/6.95 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 51.30/6.95 = { by lemma 25 R->L }
% 51.30/6.95 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 51.30/6.95 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.95 join(meet(zero, X), meet(X, complement(zero)))
% 51.30/6.95 = { by lemma 18 R->L }
% 51.30/6.95 join(meet(X, zero), meet(X, complement(zero)))
% 51.30/6.95 = { by lemma 17 }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 31: join(zero, complement(X)) = complement(X).
% 51.30/6.95 Proof:
% 51.30/6.95 join(zero, complement(X))
% 51.30/6.95 = { by lemma 24 R->L }
% 51.30/6.95 join(zero, meet(top, complement(X)))
% 51.30/6.95 = { by lemma 18 }
% 51.30/6.95 join(zero, meet(complement(X), top))
% 51.30/6.95 = { by lemma 30 }
% 51.30/6.95 complement(X)
% 51.30/6.95
% 51.30/6.95 Lemma 32: meet(X, top) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 meet(X, top)
% 51.30/6.95 = { by lemma 19 R->L }
% 51.30/6.95 complement(join(zero, complement(X)))
% 51.30/6.95 = { by lemma 31 R->L }
% 51.30/6.95 join(zero, complement(join(zero, complement(X))))
% 51.30/6.95 = { by lemma 19 }
% 51.30/6.95 join(zero, meet(X, top))
% 51.30/6.95 = { by lemma 30 }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 33: join(X, zero) = X.
% 51.30/6.95 Proof:
% 51.30/6.95 join(X, zero)
% 51.30/6.95 = { by lemma 15 R->L }
% 51.30/6.95 join(X, complement(top))
% 51.30/6.95 = { by lemma 29 R->L }
% 51.30/6.95 join(X, complement(join(complement(X), top)))
% 51.30/6.95 = { by lemma 32 R->L }
% 51.30/6.95 join(meet(X, top), complement(join(complement(X), top)))
% 51.30/6.95 = { by lemma 16 }
% 51.30/6.95 X
% 51.30/6.95
% 51.30/6.95 Lemma 34: join(one, sk1) = one.
% 51.30/6.95 Proof:
% 51.30/6.95 join(one, sk1)
% 51.30/6.95 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.95 join(sk1, one)
% 51.30/6.95 = { by axiom 4 (goals_17) }
% 51.30/6.95 one
% 51.30/6.95
% 51.30/6.95 Lemma 35: join(meet(complement(X), Y), meet(Y, X)) = Y.
% 51.30/6.95 Proof:
% 51.30/6.95 join(meet(complement(X), Y), meet(Y, X))
% 51.30/6.95 = { by lemma 18 }
% 51.30/6.96 join(meet(Y, complement(X)), meet(Y, X))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(meet(Y, X), meet(Y, complement(X)))
% 51.30/6.96 = { by lemma 17 }
% 51.30/6.96 Y
% 51.30/6.96
% 51.30/6.96 Lemma 36: complement(complement(X)) = X.
% 51.30/6.96 Proof:
% 51.30/6.96 complement(complement(X))
% 51.30/6.96 = { by lemma 35 R->L }
% 51.30/6.96 join(meet(complement(X), complement(complement(X))), meet(complement(complement(X)), X))
% 51.30/6.96 = { by axiom 5 (def_zero_13) R->L }
% 51.30/6.96 join(zero, meet(complement(complement(X)), X))
% 51.30/6.96 = { by lemma 18 R->L }
% 51.30/6.96 join(zero, meet(X, complement(complement(X))))
% 51.30/6.96 = { by axiom 5 (def_zero_13) }
% 51.30/6.96 join(meet(X, complement(X)), meet(X, complement(complement(X))))
% 51.30/6.96 = { by lemma 17 }
% 51.30/6.96 X
% 51.30/6.96
% 51.30/6.96 Lemma 37: meet(top, X) = X.
% 51.30/6.96 Proof:
% 51.30/6.96 meet(top, X)
% 51.30/6.96 = { by lemma 18 }
% 51.30/6.96 meet(X, top)
% 51.30/6.96 = { by lemma 32 }
% 51.30/6.96 X
% 51.30/6.96
% 51.30/6.96 Lemma 38: complement(join(meet(X, Y), complement(Z))) = meet(Z, join(complement(X), complement(Y))).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(join(meet(X, Y), complement(Z)))
% 51.30/6.96 = { by axiom 7 (maddux4_definiton_of_meet_4) }
% 51.30/6.96 complement(join(complement(join(complement(X), complement(Y))), complement(Z)))
% 51.30/6.96 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.96 meet(join(complement(X), complement(Y)), Z)
% 51.30/6.96 = { by lemma 18 R->L }
% 51.30/6.96 meet(Z, join(complement(X), complement(Y)))
% 51.30/6.96
% 51.30/6.96 Lemma 39: complement(meet(X, Y)) = join(complement(X), complement(Y)).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(meet(X, Y))
% 51.30/6.96 = { by lemma 18 }
% 51.30/6.96 complement(meet(Y, X))
% 51.30/6.96 = { by lemma 33 R->L }
% 51.30/6.96 complement(join(meet(Y, X), zero))
% 51.30/6.96 = { by lemma 15 R->L }
% 51.30/6.96 complement(join(meet(Y, X), complement(top)))
% 51.30/6.96 = { by lemma 38 }
% 51.30/6.96 meet(top, join(complement(Y), complement(X)))
% 51.30/6.96 = { by lemma 37 }
% 51.30/6.96 join(complement(Y), complement(X))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.96 join(complement(X), complement(Y))
% 51.30/6.96
% 51.30/6.96 Lemma 40: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 51.30/6.96 Proof:
% 51.30/6.96 join(X, join(Y, Z))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(join(Y, Z), X)
% 51.30/6.96 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 51.30/6.96 join(Y, join(Z, X))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.96 join(Y, join(X, Z))
% 51.30/6.96
% 51.30/6.96 Lemma 41: join(X, meet(Y, X)) = X.
% 51.30/6.96 Proof:
% 51.30/6.96 join(X, meet(Y, X))
% 51.30/6.96 = { by lemma 18 }
% 51.30/6.96 join(X, meet(X, Y))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(meet(X, Y), X)
% 51.30/6.96 = { by lemma 16 R->L }
% 51.30/6.96 join(meet(X, Y), join(meet(X, Y), complement(join(complement(X), Y))))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(meet(X, Y), join(complement(join(complement(X), Y)), meet(X, Y)))
% 51.30/6.96 = { by lemma 32 R->L }
% 51.30/6.96 join(meet(X, Y), join(complement(join(complement(X), Y)), meet(meet(X, Y), top)))
% 51.30/6.96 = { by lemma 19 R->L }
% 51.30/6.96 join(meet(X, Y), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 51.30/6.96 = { by lemma 32 R->L }
% 51.30/6.96 join(meet(meet(X, Y), top), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 51.30/6.96 = { by lemma 19 R->L }
% 51.30/6.96 join(complement(join(zero, complement(meet(X, Y)))), join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y))))))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(complement(join(zero, complement(meet(X, Y)))), join(complement(join(zero, complement(meet(X, Y)))), complement(join(complement(X), Y))))
% 51.30/6.96 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.96 join(join(complement(join(zero, complement(meet(X, Y)))), complement(join(zero, complement(meet(X, Y))))), complement(join(complement(X), Y)))
% 51.30/6.96 = { by lemma 22 }
% 51.30/6.96 join(complement(join(zero, complement(meet(X, Y)))), complement(join(complement(X), Y)))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.96 join(complement(join(complement(X), Y)), complement(join(zero, complement(meet(X, Y)))))
% 51.30/6.96 = { by lemma 19 }
% 51.30/6.96 join(complement(join(complement(X), Y)), meet(meet(X, Y), top))
% 51.30/6.96 = { by lemma 32 }
% 51.30/6.96 join(complement(join(complement(X), Y)), meet(X, Y))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.96 join(meet(X, Y), complement(join(complement(X), Y)))
% 51.30/6.96 = { by lemma 16 }
% 51.30/6.96 X
% 51.30/6.96
% 51.30/6.96 Lemma 42: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(join(X, complement(Y)))
% 51.30/6.96 = { by lemma 36 R->L }
% 51.30/6.96 complement(join(complement(complement(X)), complement(Y)))
% 51.30/6.96 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.96 meet(complement(X), Y)
% 51.30/6.96 = { by lemma 18 R->L }
% 51.30/6.96 meet(Y, complement(X))
% 51.30/6.96
% 51.30/6.96 Lemma 43: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(join(complement(X), Y))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 complement(join(Y, complement(X)))
% 51.30/6.96 = { by lemma 42 }
% 51.30/6.96 meet(X, complement(Y))
% 51.30/6.96
% 51.30/6.96 Lemma 44: complement(meet(Y, complement(X))) = join(X, complement(Y)).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(meet(Y, complement(X)))
% 51.30/6.96 = { by lemma 39 }
% 51.30/6.96 join(complement(Y), complement(complement(X)))
% 51.30/6.96 = { by lemma 36 }
% 51.30/6.96 join(complement(Y), X)
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.96 join(X, complement(Y))
% 51.30/6.96
% 51.30/6.96 Lemma 45: join(X, join(complement(X), Y)) = top.
% 51.30/6.96 Proof:
% 51.30/6.96 join(X, join(complement(X), Y))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 join(X, join(Y, complement(X)))
% 51.30/6.96 = { by lemma 26 }
% 51.30/6.96 join(Y, top)
% 51.30/6.96 = { by lemma 28 R->L }
% 51.30/6.96 join(Z, top)
% 51.30/6.96 = { by lemma 29 }
% 51.30/6.96 top
% 51.30/6.96
% 51.30/6.96 Lemma 46: meet(X, join(X, join(Y, Z))) = X.
% 51.30/6.96 Proof:
% 51.30/6.96 meet(X, join(X, join(Y, Z)))
% 51.30/6.96 = { by lemma 40 R->L }
% 51.30/6.96 meet(X, join(Y, join(X, Z)))
% 51.30/6.96 = { by lemma 33 R->L }
% 51.30/6.96 join(meet(X, join(Y, join(X, Z))), zero)
% 51.30/6.96 = { by lemma 15 R->L }
% 51.30/6.96 join(meet(X, join(Y, join(X, Z))), complement(top))
% 51.30/6.96 = { by lemma 36 R->L }
% 51.30/6.96 join(meet(X, join(Y, join(complement(complement(X)), Z))), complement(top))
% 51.30/6.96 = { by lemma 29 R->L }
% 51.30/6.96 join(meet(X, join(Y, join(complement(complement(X)), Z))), complement(join(Y, top)))
% 51.30/6.96 = { by lemma 45 R->L }
% 51.30/6.96 join(meet(X, join(Y, join(complement(complement(X)), Z))), complement(join(Y, join(complement(X), join(complement(complement(X)), Z)))))
% 51.30/6.96 = { by lemma 40 }
% 51.30/6.96 join(meet(X, join(Y, join(complement(complement(X)), Z))), complement(join(complement(X), join(Y, join(complement(complement(X)), Z)))))
% 51.30/6.96 = { by lemma 16 }
% 51.30/6.96 X
% 51.30/6.96
% 51.30/6.96 Lemma 47: meet(X, join(X, Y)) = X.
% 51.30/6.96 Proof:
% 51.30/6.96 meet(X, join(X, Y))
% 51.30/6.96 = { by lemma 16 R->L }
% 51.30/6.96 meet(X, join(X, join(meet(Y, Z), complement(join(complement(Y), Z)))))
% 51.30/6.96 = { by lemma 46 }
% 51.30/6.96 X
% 51.30/6.96
% 51.30/6.96 Lemma 48: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 51.30/6.96 Proof:
% 51.30/6.96 converse(join(X, converse(Y)))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 converse(join(converse(Y), X))
% 51.30/6.96 = { by axiom 8 (converse_additivity_9) }
% 51.30/6.96 join(converse(converse(Y)), converse(X))
% 51.30/6.96 = { by axiom 1 (converse_idempotence_8) }
% 51.30/6.96 join(Y, converse(X))
% 51.30/6.96
% 51.30/6.96 Lemma 49: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 51.30/6.96 Proof:
% 51.30/6.96 converse(join(converse(X), Y))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 converse(join(Y, converse(X)))
% 51.30/6.96 = { by lemma 48 }
% 51.30/6.96 join(X, converse(Y))
% 51.30/6.96
% 51.30/6.96 Lemma 50: join(X, converse(complement(converse(X)))) = top.
% 51.30/6.96 Proof:
% 51.30/6.96 join(X, converse(complement(converse(X))))
% 51.30/6.96 = { by lemma 49 R->L }
% 51.30/6.96 converse(join(converse(X), complement(converse(X))))
% 51.30/6.96 = { by axiom 3 (def_top_12) R->L }
% 51.30/6.96 converse(top)
% 51.30/6.96 = { by lemma 29 R->L }
% 51.30/6.96 converse(join(Y, top))
% 51.30/6.96 = { by axiom 8 (converse_additivity_9) }
% 51.30/6.96 join(converse(Y), converse(top))
% 51.30/6.96 = { by axiom 3 (def_top_12) }
% 51.30/6.96 join(converse(Y), converse(join(converse(complement(converse(Y))), complement(converse(complement(converse(Y)))))))
% 51.30/6.96 = { by lemma 49 }
% 51.30/6.96 join(converse(Y), join(complement(converse(Y)), converse(complement(converse(complement(converse(Y)))))))
% 51.30/6.96 = { by lemma 45 }
% 51.30/6.96 top
% 51.30/6.96
% 51.30/6.96 Lemma 51: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 51.30/6.96 Proof:
% 51.30/6.96 complement(join(complement(X), complement(Y)))
% 51.30/6.96 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.96 meet(X, Y)
% 51.30/6.96 = { by lemma 18 R->L }
% 51.30/6.96 meet(Y, X)
% 51.30/6.96
% 51.30/6.96 Lemma 52: meet(join(X, complement(Y)), join(complement(X), complement(Y))) = complement(Y).
% 51.30/6.96 Proof:
% 51.30/6.96 meet(join(X, complement(Y)), join(complement(X), complement(Y)))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 meet(join(X, complement(Y)), join(complement(Y), complement(X)))
% 51.30/6.96 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.96 meet(join(complement(Y), X), join(complement(Y), complement(X)))
% 51.30/6.96 = { by lemma 38 R->L }
% 51.30/6.96 complement(join(meet(Y, X), complement(join(complement(Y), X))))
% 51.30/6.96 = { by lemma 16 }
% 51.30/6.96 complement(Y)
% 51.30/6.96
% 51.30/6.96 Lemma 53: meet(X, join(complement(X), complement(Y))) = meet(X, complement(Y)).
% 51.30/6.96 Proof:
% 51.30/6.96 meet(X, join(complement(X), complement(Y)))
% 51.30/6.96 = { by lemma 41 R->L }
% 51.30/6.96 meet(X, join(complement(X), join(complement(Y), meet(X, complement(Y)))))
% 51.30/6.96 = { by lemma 43 R->L }
% 51.30/6.96 meet(X, join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
% 51.30/6.96 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.96 meet(X, join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
% 51.30/6.97 = { by lemma 39 R->L }
% 51.30/6.97 meet(X, join(complement(meet(X, Y)), complement(join(complement(X), Y))))
% 51.30/6.97 = { by lemma 16 R->L }
% 51.30/6.97 meet(join(meet(X, Y), complement(join(complement(X), Y))), join(complement(meet(X, Y)), complement(join(complement(X), Y))))
% 51.30/6.97 = { by lemma 52 }
% 51.30/6.97 complement(join(complement(X), Y))
% 51.30/6.97 = { by lemma 43 }
% 51.30/6.97 meet(X, complement(Y))
% 51.30/6.97
% 51.30/6.97 Lemma 54: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 51.30/6.97 Proof:
% 51.30/6.97 join(meet(X, Y), meet(Y, complement(X)))
% 51.30/6.97 = { by lemma 18 }
% 51.30/6.97 join(meet(Y, X), meet(Y, complement(X)))
% 51.30/6.97 = { by lemma 17 }
% 51.30/6.97 Y
% 51.30/6.97
% 51.30/6.97 Lemma 55: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 51.30/6.97 Proof:
% 51.30/6.97 join(meet(X, Y), complement(join(Y, complement(X))))
% 51.30/6.97 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.97 join(meet(X, Y), complement(join(complement(X), Y)))
% 51.30/6.97 = { by lemma 16 }
% 51.30/6.97 X
% 51.30/6.97
% 51.30/6.97 Lemma 56: join(composition(X, Y), composition(X, complement(Y))) = composition(X, top).
% 51.30/6.97 Proof:
% 51.30/6.97 join(composition(X, Y), composition(X, complement(Y)))
% 51.30/6.97 = { by axiom 1 (converse_idempotence_8) R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(converse(complement(Y)))))
% 51.30/6.97 = { by lemma 33 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), zero))))
% 51.30/6.97 = { by axiom 5 (def_zero_13) }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), complement(complement(converse(complement(Y)))))))))
% 51.30/6.97 = { by lemma 52 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), meet(join(converse(complement(converse(complement(complement(converse(complement(Y))))))), complement(complement(converse(complement(Y))))), join(complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))), complement(complement(converse(complement(Y)))))))))))
% 51.30/6.97 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), meet(join(complement(complement(converse(complement(Y)))), converse(complement(converse(complement(complement(converse(complement(Y)))))))), join(complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))), complement(complement(converse(complement(Y)))))))))))
% 51.30/6.97 = { by lemma 50 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), meet(top, join(complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))), complement(complement(converse(complement(Y)))))))))))
% 51.30/6.97 = { by lemma 37 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), join(complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))), complement(complement(converse(complement(Y))))))))))
% 51.30/6.97 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y))))))))))))))
% 51.30/6.97 = { by lemma 53 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), meet(complement(converse(complement(Y))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))))))
% 51.30/6.97 = { by lemma 47 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(meet(converse(complement(Y)), join(converse(complement(Y)), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))), meet(complement(converse(complement(Y))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))))))
% 51.30/6.97 = { by lemma 36 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(meet(converse(complement(Y)), join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))), meet(complement(converse(complement(Y))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))))))
% 51.30/6.97 = { by lemma 53 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(meet(converse(complement(Y)), join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))), meet(complement(converse(complement(Y))), join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y))))))))))))))
% 51.30/6.97 = { by lemma 18 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(meet(converse(complement(Y)), join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y)))))))))), meet(join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y))))))))), complement(converse(complement(Y))))))))
% 51.30/6.97 = { by lemma 54 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(complement(complement(converse(complement(Y)))), complement(converse(complement(converse(complement(complement(converse(complement(Y))))))))))))
% 51.30/6.97 = { by lemma 36 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), complement(converse(complement(converse(complement(complement(converse(complement(Y))))))))))))
% 51.30/6.97 = { by lemma 36 }
% 51.30/6.97 join(composition(X, Y), composition(X, converse(join(converse(complement(Y)), complement(converse(complement(converse(converse(complement(Y))))))))))
% 51.30/6.97 = { by lemma 49 }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(Y), converse(complement(converse(complement(converse(converse(complement(Y))))))))))
% 51.30/6.97 = { by axiom 1 (converse_idempotence_8) }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(Y), converse(complement(converse(complement(complement(Y))))))))
% 51.30/6.97 = { by lemma 55 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, join(meet(join(complement(Y), converse(complement(converse(complement(complement(Y)))))), join(complement(Y), complement(converse(complement(converse(complement(complement(Y)))))))), complement(join(join(complement(Y), complement(converse(complement(converse(complement(complement(Y))))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y))))))))))))
% 51.30/6.97 = { by lemma 38 R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(join(meet(Y, converse(complement(converse(complement(complement(Y)))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y))))))))), complement(join(join(complement(Y), complement(converse(complement(converse(complement(complement(Y))))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y))))))))))))
% 51.30/6.97 = { by lemma 16 }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(Y), complement(join(join(complement(Y), complement(converse(complement(converse(complement(complement(Y))))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y))))))))))))
% 51.30/6.97 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(Y), complement(join(complement(Y), join(complement(converse(complement(converse(complement(complement(Y)))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y)))))))))))))
% 51.30/6.97 = { by lemma 43 }
% 51.30/6.97 join(composition(X, Y), composition(X, join(complement(Y), meet(Y, complement(join(complement(converse(complement(converse(complement(complement(Y)))))), complement(join(complement(Y), converse(complement(converse(complement(complement(Y)))))))))))))
% 51.30/6.98 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/6.98 join(composition(X, Y), composition(X, join(complement(Y), meet(Y, meet(converse(complement(converse(complement(complement(Y))))), join(complement(Y), converse(complement(converse(complement(complement(Y)))))))))))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.98 join(composition(X, Y), composition(X, join(complement(Y), meet(Y, meet(converse(complement(converse(complement(complement(Y))))), join(converse(complement(converse(complement(complement(Y))))), complement(Y)))))))
% 51.30/6.98 = { by lemma 47 }
% 51.30/6.98 join(composition(X, Y), composition(X, join(complement(Y), meet(Y, converse(complement(converse(complement(complement(Y)))))))))
% 51.30/6.98 = { by lemma 18 R->L }
% 51.30/6.98 join(composition(X, Y), composition(X, join(complement(Y), meet(converse(complement(converse(complement(complement(Y))))), Y))))
% 51.30/6.98 = { by lemma 16 R->L }
% 51.30/6.98 join(composition(X, Y), composition(X, join(join(meet(complement(Y), converse(complement(converse(complement(complement(Y)))))), complement(join(complement(complement(Y)), converse(complement(converse(complement(complement(Y)))))))), meet(converse(complement(converse(complement(complement(Y))))), Y))))
% 51.30/6.98 = { by lemma 50 }
% 51.30/6.98 join(composition(X, Y), composition(X, join(join(meet(complement(Y), converse(complement(converse(complement(complement(Y)))))), complement(top)), meet(converse(complement(converse(complement(complement(Y))))), Y))))
% 51.30/6.98 = { by lemma 15 }
% 51.30/6.98 join(composition(X, Y), composition(X, join(join(meet(complement(Y), converse(complement(converse(complement(complement(Y)))))), zero), meet(converse(complement(converse(complement(complement(Y))))), Y))))
% 51.30/6.98 = { by lemma 33 }
% 51.30/6.98 join(composition(X, Y), composition(X, join(meet(complement(Y), converse(complement(converse(complement(complement(Y)))))), meet(converse(complement(converse(complement(complement(Y))))), Y))))
% 51.30/6.98 = { by lemma 35 }
% 51.30/6.98 join(composition(X, Y), composition(X, converse(complement(converse(complement(complement(Y)))))))
% 51.30/6.98 = { by lemma 36 }
% 51.30/6.98 join(composition(X, Y), composition(X, converse(complement(converse(Y)))))
% 51.30/6.98 = { by axiom 1 (converse_idempotence_8) R->L }
% 51.30/6.98 converse(converse(join(composition(X, Y), composition(X, converse(complement(converse(Y)))))))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.98 converse(converse(join(composition(X, converse(complement(converse(Y)))), composition(X, Y))))
% 51.30/6.98 = { by axiom 8 (converse_additivity_9) }
% 51.30/6.98 converse(join(converse(composition(X, converse(complement(converse(Y))))), converse(composition(X, Y))))
% 51.30/6.98 = { by axiom 10 (converse_multiplicativity_10) }
% 51.30/6.98 converse(join(composition(converse(converse(complement(converse(Y)))), converse(X)), converse(composition(X, Y))))
% 51.30/6.98 = { by axiom 1 (converse_idempotence_8) }
% 51.30/6.98 converse(join(composition(complement(converse(Y)), converse(X)), converse(composition(X, Y))))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.98 converse(join(converse(composition(X, Y)), composition(complement(converse(Y)), converse(X))))
% 51.30/6.98 = { by axiom 10 (converse_multiplicativity_10) }
% 51.30/6.98 converse(join(composition(converse(Y), converse(X)), composition(complement(converse(Y)), converse(X))))
% 51.30/6.98 = { by axiom 12 (composition_distributivity_7) R->L }
% 51.30/6.98 converse(composition(join(converse(Y), complement(converse(Y))), converse(X)))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.98 converse(composition(join(complement(converse(Y)), converse(Y)), converse(X)))
% 51.30/6.98 = { by lemma 48 R->L }
% 51.30/6.98 converse(composition(converse(join(Y, converse(complement(converse(Y))))), converse(X)))
% 51.30/6.98 = { by axiom 10 (converse_multiplicativity_10) R->L }
% 51.30/6.98 converse(converse(composition(X, join(Y, converse(complement(converse(Y)))))))
% 51.30/6.98 = { by axiom 1 (converse_idempotence_8) }
% 51.30/6.98 composition(X, join(Y, converse(complement(converse(Y)))))
% 51.30/6.98 = { by lemma 50 }
% 51.30/6.98 composition(X, top)
% 51.30/6.98
% 51.30/6.98 Lemma 57: complement(join(meet(X, complement(Y)), complement(Z))) = meet(Z, complement(meet(X, complement(Y)))).
% 51.30/6.98 Proof:
% 51.30/6.98 complement(join(meet(X, complement(Y)), complement(Z)))
% 51.30/6.98 = { by lemma 38 }
% 51.30/6.98 meet(Z, join(complement(X), complement(complement(Y))))
% 51.30/6.98 = { by lemma 36 }
% 51.30/6.98 meet(Z, join(complement(X), Y))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.98 meet(Z, join(Y, complement(X)))
% 51.30/6.98 = { by lemma 44 R->L }
% 51.30/6.98 meet(Z, complement(meet(X, complement(Y))))
% 51.30/6.98
% 51.30/6.98 Lemma 58: join(X, complement(meet(Y, complement(composition(sk1, X))))) = complement(meet(Y, complement(X))).
% 51.30/6.98 Proof:
% 51.30/6.98 join(X, complement(meet(Y, complement(composition(sk1, X)))))
% 51.30/6.98 = { by lemma 44 }
% 51.30/6.98 join(X, join(composition(sk1, X), complement(Y)))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.98 join(X, join(complement(Y), composition(sk1, X)))
% 51.30/6.98 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.98 join(join(X, complement(Y)), composition(sk1, X))
% 51.30/6.98 = { by lemma 44 R->L }
% 51.30/6.98 join(complement(meet(Y, complement(X))), composition(sk1, X))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.98 join(composition(sk1, X), complement(meet(Y, complement(X))))
% 51.30/6.98 = { by lemma 44 }
% 51.30/6.98 join(composition(sk1, X), join(X, complement(Y)))
% 51.30/6.98 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.98 join(join(composition(sk1, X), X), complement(Y))
% 51.30/6.98 = { by lemma 21 R->L }
% 51.30/6.98 join(join(composition(sk1, X), composition(one, X)), complement(Y))
% 51.30/6.98 = { by axiom 12 (composition_distributivity_7) R->L }
% 51.30/6.98 join(composition(join(sk1, one), X), complement(Y))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.98 join(composition(join(one, sk1), X), complement(Y))
% 51.30/6.98 = { by lemma 34 }
% 51.30/6.98 join(composition(one, X), complement(Y))
% 51.30/6.98 = { by lemma 21 }
% 51.30/6.98 join(X, complement(Y))
% 51.30/6.98 = { by lemma 44 R->L }
% 51.30/6.98 complement(meet(Y, complement(X)))
% 51.30/6.98
% 51.30/6.98 Lemma 59: complement(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y)))) = complement(meet(X, complement(composition(sk1, Y)))).
% 51.30/6.98 Proof:
% 51.30/6.98 complement(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))))
% 51.30/6.98 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.98 complement(join(meet(X, complement(Y)), meet(X, complement(composition(sk1, Y)))))
% 51.30/6.98 = { by lemma 32 R->L }
% 51.30/6.98 complement(join(meet(X, complement(Y)), meet(meet(X, complement(composition(sk1, Y))), top)))
% 51.30/6.98 = { by lemma 19 R->L }
% 51.30/6.98 complement(join(meet(X, complement(Y)), complement(join(zero, complement(meet(X, complement(composition(sk1, Y))))))))
% 51.30/6.98 = { by lemma 57 }
% 51.30/6.98 meet(join(zero, complement(meet(X, complement(composition(sk1, Y))))), complement(meet(X, complement(Y))))
% 51.30/6.98 = { by lemma 31 }
% 51.30/6.98 meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y))))
% 51.30/6.98 = { by lemma 23 R->L }
% 51.30/6.98 join(zero, complement(complement(meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))))
% 51.30/6.98 = { by lemma 36 }
% 51.30/6.98 join(zero, meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.98 = { by lemma 15 R->L }
% 51.30/6.98 join(complement(top), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.98 = { by lemma 29 R->L }
% 51.30/6.98 join(complement(join(Z, top)), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.98 = { by lemma 28 }
% 51.30/6.98 join(complement(join(Y, top)), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.98 = { by lemma 26 R->L }
% 51.30/6.98 join(complement(join(meet(X, complement(composition(sk1, Y))), join(Y, complement(meet(X, complement(composition(sk1, Y))))))), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.98 = { by lemma 58 }
% 51.30/6.99 join(complement(join(meet(X, complement(composition(sk1, Y))), complement(meet(X, complement(Y))))), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.99 = { by lemma 57 }
% 51.30/6.99 join(meet(meet(X, complement(Y)), complement(meet(X, complement(composition(sk1, Y))))), meet(complement(meet(X, complement(composition(sk1, Y)))), complement(meet(X, complement(Y)))))
% 51.30/6.99 = { by lemma 54 }
% 51.30/6.99 complement(meet(X, complement(composition(sk1, Y))))
% 51.30/6.99
% 51.30/6.99 Lemma 60: join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))) = meet(X, complement(composition(sk1, Y))).
% 51.30/6.99 Proof:
% 51.30/6.99 join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y)))
% 51.30/6.99 = { by lemma 32 R->L }
% 51.30/6.99 meet(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))), top)
% 51.30/6.99 = { by lemma 19 R->L }
% 51.30/6.99 complement(join(zero, complement(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))))))
% 51.30/6.99 = { by lemma 59 }
% 51.30/6.99 complement(join(zero, complement(meet(X, complement(composition(sk1, Y))))))
% 51.30/6.99 = { by lemma 19 }
% 51.30/6.99 meet(meet(X, complement(composition(sk1, Y))), top)
% 51.30/6.99 = { by lemma 32 }
% 51.30/6.99 meet(X, complement(composition(sk1, Y)))
% 51.30/6.99
% 51.30/6.99 Goal 1 (goals_18): tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))) = tuple(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))).
% 51.30/6.99 Proof:
% 51.30/6.99 tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 51.30/6.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/6.99 tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 55 R->L }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 59 }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(sk3), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 44 }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(sk3), join(composition(sk1, sk3), complement(composition(sk1, sk2)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 36 R->L }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), complement(complement(join(complement(sk3), composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 39 R->L }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(complement(meet(composition(sk1, sk2), complement(join(complement(sk3), composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/6.99 = { by lemma 53 R->L }
% 51.30/6.99 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(complement(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(join(complement(sk3), composition(sk1, sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 39 }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), complement(join(complement(composition(sk1, sk2)), complement(join(complement(sk3), composition(sk1, sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(complement(sk3), composition(sk1, sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(composition(sk1, sk3), complement(sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 36 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(complement(complement(composition(sk1, sk3))), complement(sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 39 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 36 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(complement(sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 21 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(composition(one, complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 34 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(composition(join(one, sk1), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by axiom 12 (composition_distributivity_7) }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(join(composition(one, complement(sk3)), composition(sk1, complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 21 }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(join(complement(sk3), composition(sk1, complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 43 }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), meet(sk3, complement(composition(sk1, complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 42 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), complement(join(composition(sk1, complement(sk3)), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 41 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), meet(X, complement(join(composition(sk1, complement(sk3)), complement(sk3))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 53 R->L }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), meet(X, join(complement(X), complement(join(composition(sk1, complement(sk3)), complement(sk3)))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.00 = { by lemma 18 }
% 51.30/7.00 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), meet(join(complement(X), complement(join(composition(sk1, complement(sk3)), complement(sk3)))), X)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 51 R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), complement(join(complement(X), complement(join(complement(X), complement(join(composition(sk1, complement(sk3)), complement(sk3)))))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 51 }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), complement(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(complement(composition(sk1, sk3)), join(complement(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(join(composition(sk1, complement(sk3)), complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 31 R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(join(zero, complement(composition(sk1, sk3))), join(complement(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(join(composition(sk1, complement(sk3)), complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 38 R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3))), complement(join(zero, complement(composition(sk1, sk3))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3))), complement(join(complement(composition(sk1, sk3)), zero))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 15 R->L }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3))), complement(join(complement(composition(sk1, sk3)), complement(top)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 51 }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3))), meet(top, composition(sk1, sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by lemma 37 }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3))), composition(sk1, sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/7.01 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), meet(join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)), join(composition(sk1, complement(sk3)), complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by lemma 18 R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by lemma 33 R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), zero)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by lemma 15 R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(top))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by axiom 3 (def_top_12) }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(join(join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), complement(X)), complement(join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), complement(X))))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by axiom 7 (maddux4_definiton_of_meet_4) R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(join(join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), complement(X)), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(meet(join(composition(sk1, complement(sk3)), complement(sk3)), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X))), complement(join(complement(join(composition(sk1, complement(sk3)), complement(sk3))), join(complement(X), meet(join(composition(sk1, complement(sk3)), complement(sk3)), X)))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by lemma 16 }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, sk3), join(composition(sk1, complement(sk3)), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.02 = { by lemma 40 R->L }
% 51.30/7.02 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(composition(sk1, complement(sk3)), join(composition(sk1, sk3), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by axiom 9 (maddux2_join_associativity_2) }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(complement(join(join(composition(sk1, complement(sk3)), composition(sk1, sk3)), complement(sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 42 }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(sk3, complement(join(composition(sk1, complement(sk3)), composition(sk1, sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(sk3, complement(join(composition(sk1, sk3), composition(sk1, complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 56 }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(meet(sk3, complement(composition(sk1, top)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 39 }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(complement(sk3), complement(complement(composition(sk1, top)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 36 }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(complement(sk3), composition(sk1, top)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 56 R->L }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(complement(sk3), join(composition(sk1, sk2), composition(sk1, complement(sk2)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 40 R->L }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), join(composition(sk1, sk2), join(complement(sk3), composition(sk1, complement(sk2)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 46 }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk2)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by axiom 2 (maddux1_join_commutativity_1) }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(composition(sk1, sk2), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by axiom 3 (def_top_12) R->L }
% 51.30/7.03 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 18 }
% 51.30/7.03 tuple(join(meet(complement(sk3), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 60 }
% 51.30/7.03 tuple(join(meet(complement(sk3), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 18 }
% 51.30/7.03 tuple(join(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(sk3)), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 42 R->L }
% 51.30/7.03 tuple(join(complement(join(sk3, complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 58 }
% 51.30/7.03 tuple(join(complement(complement(meet(composition(sk1, sk2), complement(sk3)))), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 36 }
% 51.30/7.03 tuple(join(meet(composition(sk1, sk2), complement(sk3)), complement(top)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 15 }
% 51.30/7.03 tuple(join(meet(composition(sk1, sk2), complement(sk3)), zero), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 33 }
% 51.30/7.03 tuple(meet(composition(sk1, sk2), complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 51.30/7.03 = { by lemma 60 }
% 51.30/7.03 tuple(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))
% 51.30/7.03 % SZS output end Proof
% 51.30/7.03
% 51.30/7.03 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------