%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL040-4 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n012.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 12:33:28 PM UTC 2026
% Result : Unsatisfiable 209.62s 26.80s
% Output : Proof 213.11s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : REL040-4 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.03 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.33 % Computer : n012.cluster.edu
% 0.08/0.33 % Model : x86_64 x86_64
% 0.08/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.33 % Memory : 8046.5625MB
% 0.08/0.33 % OS : Linux 6.8.0-71-generic
% 0.08/0.33 % CPULimit : 300
% 0.08/0.33 % WCLimit : 300
% 0.08/0.33 % DateTime : Sun Sep 27 22:56:04 UTC 2026
% 0.08/0.34 % CPUTime :
% 0.08/0.34 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 209.62/26.80 Command-line arguments: --stitch /export/starexec/sandbox2/solver/bin/stitch --hint-skel-cost 0 --hint-skel-factor 0.5
% 209.62/26.80
% 209.62/26.80 % SZS status Unsatisfiable
% 209.62/26.80
% 212.33/27.06 % SZS output start Proof
% 212.33/27.06 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 212.33/27.06 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 212.33/27.06 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 212.33/27.06 Axiom 4 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 212.33/27.06 Axiom 5 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 212.33/27.06 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 212.33/27.06 Axiom 7 (def_zero_13): zero = meet(X, complement(X)).
% 212.33/27.06 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 212.33/27.07 Axiom 9 (goals_17): join(composition(converse(sk1), sk1), one) = one.
% 212.33/27.07 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 212.33/27.07 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 212.33/27.07 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 212.33/27.07 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 212.33/27.07 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 212.33/27.07 Axiom 15 (modular_law_1_15): 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).
% 212.33/27.07 Axiom 16 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 212.33/27.07 Axiom 17 (dedekind_law_14): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 212.33/27.07
% 212.33/27.07 Lemma 18: complement(top) = zero.
% 212.33/27.07 Proof:
% 212.33/27.07 complement(top)
% 212.33/27.07 = { by axiom 6 (def_top_12) }
% 212.33/27.07 complement(join(complement(X), complement(complement(X))))
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 212.33/27.07 meet(X, complement(X))
% 212.33/27.07 = { by axiom 7 (def_zero_13) R->L }
% 212.33/27.07 zero
% 212.33/27.07
% 212.33/27.07 Lemma 19: meet(Y, X) = meet(X, Y).
% 212.33/27.07 Proof:
% 212.33/27.07 meet(Y, X)
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.07 complement(join(complement(Y), complement(X)))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 complement(join(complement(X), complement(Y)))
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 212.33/27.07 meet(X, Y)
% 212.33/27.07
% 212.33/27.07 Lemma 20: join(X, converse(complement(converse(X)))) = converse(top).
% 212.33/27.07 Proof:
% 212.33/27.07 join(X, converse(complement(converse(X))))
% 212.33/27.07 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.07 join(converse(converse(X)), converse(complement(converse(X))))
% 212.33/27.07 = { by axiom 4 (converse_additivity_9) R->L }
% 212.33/27.07 converse(join(converse(X), complement(converse(X))))
% 212.33/27.07 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.07 converse(top)
% 212.33/27.07
% 212.33/27.07 Lemma 21: join(X, join(Y, complement(X))) = join(Y, top).
% 212.33/27.07 Proof:
% 212.33/27.07 join(X, join(Y, complement(X)))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(X, join(complement(X), Y))
% 212.33/27.07 = { by axiom 8 (maddux2_join_associativity_2) }
% 212.33/27.07 join(join(X, complement(X)), Y)
% 212.33/27.07 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.07 join(top, Y)
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.07 join(Y, top)
% 212.33/27.07
% 212.33/27.07 Lemma 22: composition(converse(one), X) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 composition(converse(one), X)
% 212.33/27.07 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.07 composition(converse(one), converse(converse(X)))
% 212.33/27.07 = { by axiom 5 (converse_multiplicativity_10) R->L }
% 212.33/27.07 converse(composition(converse(X), one))
% 212.33/27.07 = { by axiom 3 (composition_identity_6) }
% 212.33/27.07 converse(converse(X))
% 212.33/27.07 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 23: converse(one) = one.
% 212.33/27.07 Proof:
% 212.33/27.07 converse(one)
% 212.33/27.07 = { by axiom 3 (composition_identity_6) R->L }
% 212.33/27.07 composition(converse(one), one)
% 212.33/27.07 = { by lemma 22 }
% 212.33/27.07 one
% 212.33/27.07
% 212.33/27.07 Lemma 24: composition(one, X) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 composition(one, X)
% 212.33/27.07 = { by lemma 23 R->L }
% 212.33/27.07 composition(converse(one), X)
% 212.33/27.07 = { by lemma 22 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 25: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 212.33/27.07 Proof:
% 212.33/27.07 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 212.33/27.07 = { by axiom 13 (converse_cancellativity_11) }
% 212.33/27.07 complement(X)
% 212.33/27.07
% 212.33/27.07 Lemma 26: join(complement(X), complement(X)) = complement(X).
% 212.33/27.07 Proof:
% 212.33/27.07 join(complement(X), complement(X))
% 212.33/27.07 = { by lemma 22 R->L }
% 212.33/27.07 join(complement(X), composition(converse(one), complement(X)))
% 212.33/27.07 = { by lemma 24 R->L }
% 212.33/27.07 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 212.33/27.07 = { by lemma 25 }
% 212.33/27.07 complement(X)
% 212.33/27.07
% 212.33/27.07 Lemma 27: join(top, complement(X)) = top.
% 212.33/27.07 Proof:
% 212.33/27.07 join(top, complement(X))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(complement(X), top)
% 212.33/27.07 = { by lemma 21 R->L }
% 212.33/27.07 join(X, join(complement(X), complement(X)))
% 212.33/27.07 = { by lemma 26 }
% 212.33/27.07 join(X, complement(X))
% 212.33/27.07 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.07 top
% 212.33/27.07
% 212.33/27.07 Lemma 28: join(Y, top) = join(X, top).
% 212.33/27.07 Proof:
% 212.33/27.07 join(Y, top)
% 212.33/27.07 = { by lemma 27 R->L }
% 212.33/27.07 join(Y, join(top, complement(Y)))
% 212.33/27.07 = { by lemma 21 }
% 212.33/27.07 join(top, top)
% 212.33/27.07 = { by lemma 21 R->L }
% 212.33/27.07 join(X, join(top, complement(X)))
% 212.33/27.07 = { by lemma 27 }
% 212.33/27.07 join(X, top)
% 212.33/27.07
% 212.33/27.07 Lemma 29: join(X, join(complement(X), Y)) = join(Z, top).
% 212.33/27.07 Proof:
% 212.33/27.07 join(X, join(complement(X), Y))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(X, join(Y, complement(X)))
% 212.33/27.07 = { by lemma 21 }
% 212.33/27.07 join(Y, top)
% 212.33/27.07 = { by lemma 28 R->L }
% 212.33/27.07 join(Z, top)
% 212.33/27.07
% 212.33/27.07 Lemma 30: join(X, top) = top.
% 212.33/27.07 Proof:
% 212.33/27.07 join(X, top)
% 212.33/27.07 = { by lemma 29 R->L }
% 212.33/27.07 join(Y, join(complement(Y), composition(converse(Z), complement(composition(Z, Y)))))
% 212.33/27.07 = { by lemma 25 }
% 212.33/27.07 join(Y, complement(Y))
% 212.33/27.07 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.07 top
% 212.33/27.07
% 212.33/27.07 Lemma 31: join(X, converse(top)) = top.
% 212.33/27.07 Proof:
% 212.33/27.07 join(X, converse(top))
% 212.33/27.07 = { by lemma 20 R->L }
% 212.33/27.07 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 212.33/27.07 = { by lemma 29 }
% 212.33/27.07 join(Y, top)
% 212.33/27.07 = { by lemma 30 }
% 212.33/27.07 top
% 212.33/27.07
% 212.33/27.07 Lemma 32: converse(top) = top.
% 212.33/27.07 Proof:
% 212.33/27.07 converse(top)
% 212.33/27.07 = { by lemma 31 R->L }
% 212.33/27.07 converse(join(X, converse(top)))
% 212.33/27.07 = { by axiom 4 (converse_additivity_9) }
% 212.33/27.07 join(converse(X), converse(converse(top)))
% 212.33/27.07 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.07 join(converse(X), top)
% 212.33/27.07 = { by lemma 30 }
% 212.33/27.07 top
% 212.33/27.07
% 212.33/27.07 Lemma 33: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 join(meet(X, Y), complement(join(complement(X), Y)))
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.07 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 212.33/27.07 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 34: join(zero, meet(X, top)) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 join(zero, meet(X, top))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(meet(X, top), zero)
% 212.33/27.07 = { by lemma 18 R->L }
% 212.33/27.07 join(meet(X, top), complement(top))
% 212.33/27.07 = { by lemma 32 R->L }
% 212.33/27.07 join(meet(X, converse(top)), complement(top))
% 212.33/27.07 = { by lemma 31 R->L }
% 212.33/27.07 join(meet(X, converse(top)), complement(join(complement(X), converse(top))))
% 212.33/27.07 = { by lemma 33 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 35: join(zero, meet(top, X)) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 join(zero, meet(top, X))
% 212.33/27.07 = { by lemma 19 }
% 212.33/27.07 join(zero, meet(X, top))
% 212.33/27.07 = { by lemma 34 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 36: complement(join(zero, complement(X))) = meet(X, top).
% 212.33/27.07 Proof:
% 212.33/27.07 complement(join(zero, complement(X)))
% 212.33/27.07 = { by lemma 18 R->L }
% 212.33/27.07 complement(join(complement(top), complement(X)))
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 212.33/27.07 meet(top, X)
% 212.33/27.07 = { by lemma 19 R->L }
% 212.33/27.07 meet(X, top)
% 212.33/27.07
% 212.33/27.07 Lemma 37: join(zero, complement(complement(X))) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 join(zero, complement(complement(X)))
% 212.33/27.07 = { by axiom 7 (def_zero_13) }
% 212.33/27.07 join(meet(X, complement(X)), complement(complement(X)))
% 212.33/27.07 = { by lemma 26 R->L }
% 212.33/27.07 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 212.33/27.07 = { by lemma 33 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 38: join(zero, complement(X)) = complement(X).
% 212.33/27.07 Proof:
% 212.33/27.07 join(zero, complement(X))
% 212.33/27.07 = { by lemma 37 R->L }
% 212.33/27.07 join(zero, complement(join(zero, complement(complement(X)))))
% 212.33/27.07 = { by lemma 36 }
% 212.33/27.07 join(zero, meet(complement(X), top))
% 212.33/27.07 = { by lemma 19 R->L }
% 212.33/27.07 join(zero, meet(top, complement(X)))
% 212.33/27.07 = { by lemma 35 }
% 212.33/27.07 complement(X)
% 212.33/27.07
% 212.33/27.07 Lemma 39: meet(X, top) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 meet(X, top)
% 212.33/27.07 = { by lemma 36 R->L }
% 212.33/27.07 complement(join(zero, complement(X)))
% 212.33/27.07 = { by lemma 38 R->L }
% 212.33/27.07 join(zero, complement(join(zero, complement(X))))
% 212.33/27.07 = { by lemma 36 }
% 212.33/27.07 join(zero, meet(X, top))
% 212.33/27.07 = { by lemma 34 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 40: meet(top, X) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 meet(top, X)
% 212.33/27.07 = { by lemma 19 }
% 212.33/27.07 meet(X, top)
% 212.33/27.07 = { by lemma 39 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 41: join(meet(X, Y), meet(X, complement(Y))) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 join(meet(X, Y), meet(X, complement(Y)))
% 212.33/27.07 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.07 join(meet(X, complement(Y)), meet(X, Y))
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.07 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 212.33/27.07 = { by lemma 33 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 42: complement(zero) = top.
% 212.33/27.07 Proof:
% 212.33/27.07 complement(zero)
% 212.33/27.07 = { by lemma 35 R->L }
% 212.33/27.07 join(zero, meet(top, complement(zero)))
% 212.33/27.07 = { by lemma 40 R->L }
% 212.33/27.07 join(meet(top, zero), meet(top, complement(zero)))
% 212.33/27.07 = { by lemma 41 }
% 212.33/27.07 top
% 212.33/27.07
% 212.33/27.07 Lemma 43: complement(complement(X)) = X.
% 212.33/27.07 Proof:
% 212.33/27.07 complement(complement(X))
% 212.33/27.07 = { by lemma 38 R->L }
% 212.33/27.07 join(zero, complement(complement(X)))
% 212.33/27.07 = { by lemma 37 }
% 212.33/27.07 X
% 212.33/27.07
% 212.33/27.07 Lemma 44: meet(X, X) = complement(complement(X)).
% 212.33/27.07 Proof:
% 212.33/27.07 meet(X, X)
% 212.33/27.07 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.08 complement(join(complement(X), complement(X)))
% 212.33/27.08 = { by lemma 26 }
% 212.33/27.08 complement(complement(X))
% 212.33/27.08
% 212.33/27.08 Lemma 45: join(zero, meet(X, X)) = X.
% 212.33/27.08 Proof:
% 212.33/27.08 join(zero, meet(X, X))
% 212.33/27.08 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.08 join(zero, complement(join(complement(X), complement(X))))
% 212.33/27.08 = { by axiom 7 (def_zero_13) }
% 212.33/27.08 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 212.33/27.08 = { by lemma 33 }
% 212.33/27.08 X
% 212.33/27.08
% 212.33/27.08 Lemma 46: join(converse(zero), X) = X.
% 212.33/27.08 Proof:
% 212.33/27.08 join(converse(zero), X)
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(X, converse(zero))
% 212.33/27.08 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.08 join(converse(converse(X)), converse(zero))
% 212.33/27.08 = { by lemma 43 R->L }
% 212.33/27.08 join(converse(complement(complement(converse(X)))), converse(zero))
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(converse(zero), converse(complement(complement(converse(X)))))
% 212.33/27.08 = { by lemma 44 R->L }
% 212.33/27.08 join(converse(zero), converse(meet(converse(X), converse(X))))
% 212.33/27.08 = { by axiom 4 (converse_additivity_9) R->L }
% 212.33/27.08 converse(join(zero, meet(converse(X), converse(X))))
% 212.33/27.08 = { by lemma 45 }
% 212.33/27.08 converse(converse(X))
% 212.33/27.08 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.08 X
% 212.33/27.08
% 212.33/27.08 Lemma 47: converse(zero) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 converse(zero)
% 212.33/27.08 = { by lemma 39 R->L }
% 212.33/27.08 meet(converse(zero), top)
% 212.33/27.08 = { by axiom 6 (def_top_12) }
% 212.33/27.08 meet(converse(zero), join(converse(zero), complement(converse(zero))))
% 212.33/27.08 = { by lemma 46 }
% 212.33/27.08 meet(converse(zero), complement(converse(zero)))
% 212.33/27.08 = { by axiom 7 (def_zero_13) R->L }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 48: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 212.33/27.08 Proof:
% 212.33/27.08 join(zero, join(X, complement(complement(Y))))
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(zero, join(complement(complement(Y)), X))
% 212.33/27.08 = { by lemma 44 R->L }
% 212.33/27.08 join(zero, join(meet(Y, Y), X))
% 212.33/27.08 = { by axiom 8 (maddux2_join_associativity_2) }
% 212.33/27.08 join(join(zero, meet(Y, Y)), X)
% 212.33/27.08 = { by lemma 45 }
% 212.33/27.08 join(Y, X)
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.08 join(X, Y)
% 212.33/27.08
% 212.33/27.08 Lemma 49: join(X, zero) = X.
% 212.33/27.08 Proof:
% 212.33/27.08 join(X, zero)
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(zero, X)
% 212.33/27.08 = { by lemma 48 R->L }
% 212.33/27.08 join(zero, join(zero, complement(complement(X))))
% 212.33/27.08 = { by lemma 38 }
% 212.33/27.08 join(zero, complement(complement(X)))
% 212.33/27.08 = { by lemma 37 }
% 212.33/27.08 X
% 212.33/27.08
% 212.33/27.08 Lemma 50: join(top, X) = top.
% 212.33/27.08 Proof:
% 212.33/27.08 join(top, X)
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(X, top)
% 212.33/27.08 = { by lemma 28 R->L }
% 212.33/27.08 join(Y, top)
% 212.33/27.08 = { by lemma 30 }
% 212.33/27.08 top
% 212.33/27.08
% 212.33/27.08 Lemma 51: join(zero, X) = X.
% 212.33/27.08 Proof:
% 212.33/27.08 join(zero, X)
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 join(X, zero)
% 212.33/27.08 = { by lemma 49 }
% 212.33/27.08 X
% 212.33/27.08
% 212.33/27.08 Lemma 52: meet(X, zero) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 meet(X, zero)
% 212.33/27.08 = { by lemma 19 }
% 212.33/27.08 meet(zero, X)
% 212.33/27.08 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.08 complement(join(complement(zero), complement(X)))
% 212.33/27.08 = { by lemma 42 }
% 212.33/27.08 complement(join(top, complement(X)))
% 212.33/27.08 = { by lemma 27 }
% 212.33/27.08 complement(top)
% 212.33/27.08 = { by lemma 18 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 53: meet(zero, X) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 meet(zero, X)
% 212.33/27.08 = { by lemma 19 }
% 212.33/27.08 meet(X, zero)
% 212.33/27.08 = { by lemma 52 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 54: composition(top, top) = top.
% 212.33/27.08 Proof:
% 212.33/27.08 composition(top, top)
% 212.33/27.08 = { by lemma 31 R->L }
% 212.33/27.08 composition(join(one, converse(top)), top)
% 212.33/27.08 = { by lemma 23 R->L }
% 212.33/27.08 composition(join(converse(one), converse(top)), top)
% 212.33/27.08 = { by axiom 12 (composition_distributivity_7) }
% 212.33/27.08 join(composition(converse(one), top), composition(converse(top), top))
% 212.33/27.08 = { by lemma 22 }
% 212.33/27.08 join(top, composition(converse(top), top))
% 212.33/27.08 = { by lemma 32 }
% 212.33/27.08 join(top, composition(top, top))
% 212.33/27.08 = { by lemma 50 }
% 212.33/27.08 top
% 212.33/27.08
% 212.33/27.08 Lemma 55: composition(top, zero) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 composition(top, zero)
% 212.33/27.08 = { by lemma 32 R->L }
% 212.33/27.08 composition(converse(top), zero)
% 212.33/27.08 = { by lemma 51 R->L }
% 212.33/27.08 join(zero, composition(converse(top), zero))
% 212.33/27.08 = { by lemma 18 R->L }
% 212.33/27.08 join(complement(top), composition(converse(top), zero))
% 212.33/27.08 = { by lemma 18 R->L }
% 212.33/27.08 join(complement(top), composition(converse(top), complement(top)))
% 212.33/27.08 = { by lemma 54 R->L }
% 212.33/27.08 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 212.33/27.08 = { by lemma 25 }
% 212.33/27.08 complement(top)
% 212.33/27.08 = { by lemma 18 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 56: composition(X, zero) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 composition(X, zero)
% 212.33/27.08 = { by lemma 51 R->L }
% 212.33/27.08 join(zero, composition(X, zero))
% 212.33/27.08 = { by lemma 55 R->L }
% 212.33/27.08 join(composition(top, zero), composition(X, zero))
% 212.33/27.08 = { by axiom 12 (composition_distributivity_7) R->L }
% 212.33/27.08 composition(join(top, X), zero)
% 212.33/27.08 = { by lemma 50 }
% 212.33/27.08 composition(top, zero)
% 212.33/27.08 = { by lemma 55 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 57: composition(converse(X), complement(composition(X, top))) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 composition(converse(X), complement(composition(X, top)))
% 212.33/27.08 = { by lemma 51 R->L }
% 212.33/27.08 join(zero, composition(converse(X), complement(composition(X, top))))
% 212.33/27.08 = { by lemma 18 R->L }
% 212.33/27.08 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 212.33/27.08 = { by lemma 25 }
% 212.33/27.08 complement(top)
% 212.33/27.08 = { by lemma 18 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 58: composition(zero, X) = zero.
% 212.33/27.08 Proof:
% 212.33/27.08 composition(zero, X)
% 212.33/27.08 = { by lemma 39 R->L }
% 212.33/27.08 meet(composition(zero, X), top)
% 212.33/27.08 = { by lemma 42 R->L }
% 212.33/27.08 meet(composition(zero, X), complement(zero))
% 212.33/27.08 = { by lemma 47 R->L }
% 212.33/27.08 meet(composition(converse(zero), X), complement(zero))
% 212.33/27.08 = { by lemma 18 R->L }
% 212.33/27.08 meet(composition(converse(complement(top)), X), complement(zero))
% 212.33/27.08 = { by lemma 54 R->L }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), X), complement(zero))
% 212.33/27.08 = { by lemma 47 R->L }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), X), complement(converse(zero)))
% 212.33/27.08 = { by lemma 57 R->L }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), X), complement(converse(composition(converse(top), complement(composition(top, top))))))
% 212.33/27.08 = { by axiom 5 (converse_multiplicativity_10) }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), converse(converse(top)))))
% 212.33/27.08 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top)))
% 212.33/27.08 = { by lemma 49 R->L }
% 212.33/27.08 join(meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top))), zero)
% 212.33/27.08 = { by lemma 53 R->L }
% 212.33/27.08 join(meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top))), meet(zero, complement(composition(converse(complement(composition(top, top))), top))))
% 212.33/27.08 = { by lemma 56 R->L }
% 212.33/27.08 join(meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top))), meet(composition(converse(complement(composition(top, top))), zero), complement(composition(converse(complement(composition(top, top))), top))))
% 212.33/27.08 = { by lemma 52 R->L }
% 212.33/27.08 join(meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top))), meet(composition(converse(complement(composition(top, top))), meet(X, zero)), complement(composition(converse(complement(composition(top, top))), top))))
% 212.33/27.08 = { by lemma 57 R->L }
% 212.33/27.08 join(meet(composition(converse(complement(composition(top, top))), X), complement(composition(converse(complement(composition(top, top))), top))), meet(composition(converse(complement(composition(top, top))), meet(X, composition(converse(converse(complement(composition(top, top)))), complement(composition(converse(complement(composition(top, top))), top))))), complement(composition(converse(complement(composition(top, top))), top))))
% 212.33/27.08 = { by axiom 15 (modular_law_1_15) }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), meet(X, composition(converse(converse(complement(composition(top, top)))), complement(composition(converse(complement(composition(top, top))), top))))), complement(composition(converse(complement(composition(top, top))), top)))
% 212.33/27.08 = { by lemma 57 }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), meet(X, zero)), complement(composition(converse(complement(composition(top, top))), top)))
% 212.33/27.08 = { by lemma 52 }
% 212.33/27.08 meet(composition(converse(complement(composition(top, top))), zero), complement(composition(converse(complement(composition(top, top))), top)))
% 212.33/27.08 = { by lemma 56 }
% 212.33/27.08 meet(zero, complement(composition(converse(complement(composition(top, top))), top)))
% 212.33/27.08 = { by lemma 53 }
% 212.33/27.08 zero
% 212.33/27.08
% 212.33/27.08 Lemma 59: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 212.33/27.08 Proof:
% 212.33/27.08 meet(X, join(complement(Y), complement(Z)))
% 212.33/27.08 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.08 meet(X, join(complement(Z), complement(Y)))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 meet(join(complement(Z), complement(Y)), X)
% 212.33/27.09 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.09 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 212.33/27.09 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 212.33/27.09 complement(join(meet(Z, Y), complement(X)))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.09 complement(join(complement(X), meet(Z, Y)))
% 212.33/27.09 = { by lemma 19 R->L }
% 212.33/27.09 complement(join(complement(X), meet(Y, Z)))
% 212.33/27.09
% 212.33/27.09 Lemma 60: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 212.33/27.09 Proof:
% 212.33/27.09 complement(join(X, complement(Y)))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.09 complement(join(complement(Y), X))
% 212.33/27.09 = { by lemma 40 R->L }
% 212.33/27.09 complement(join(complement(Y), meet(top, X)))
% 212.33/27.09 = { by lemma 59 R->L }
% 212.33/27.09 meet(Y, join(complement(top), complement(X)))
% 212.33/27.09 = { by lemma 18 }
% 212.33/27.09 meet(Y, join(zero, complement(X)))
% 212.33/27.09 = { by lemma 38 }
% 212.33/27.09 meet(Y, complement(X))
% 212.33/27.09
% 212.33/27.09 Lemma 61: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 212.33/27.09 Proof:
% 212.33/27.09 complement(meet(X, complement(Y)))
% 212.33/27.09 = { by lemma 51 R->L }
% 212.33/27.09 complement(join(zero, meet(X, complement(Y))))
% 212.33/27.09 = { by lemma 60 R->L }
% 212.33/27.09 complement(join(zero, complement(join(Y, complement(X)))))
% 212.33/27.09 = { by lemma 36 }
% 212.33/27.09 meet(join(Y, complement(X)), top)
% 212.33/27.09 = { by lemma 39 }
% 212.33/27.09 join(Y, complement(X))
% 212.33/27.09
% 212.33/27.09 Lemma 62: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 212.33/27.09 Proof:
% 212.33/27.09 complement(meet(complement(X), Y))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 complement(meet(Y, complement(X)))
% 212.33/27.09 = { by lemma 61 }
% 212.33/27.09 join(X, complement(Y))
% 212.33/27.09
% 212.33/27.09 Lemma 63: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 212.33/27.09 Proof:
% 212.33/27.09 complement(join(complement(X), Y))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.09 complement(join(Y, complement(X)))
% 212.33/27.09 = { by lemma 60 }
% 212.33/27.09 meet(X, complement(Y))
% 212.33/27.09
% 212.33/27.09 Lemma 64: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 212.33/27.09 Proof:
% 212.33/27.09 join(complement(X), complement(Y))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.09 join(complement(Y), complement(X))
% 212.33/27.09 = { by lemma 45 R->L }
% 212.33/27.09 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 212.33/27.09 = { by lemma 59 }
% 212.33/27.09 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 212.33/27.09 = { by lemma 38 }
% 212.33/27.09 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 212.33/27.09 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 212.33/27.09 complement(join(meet(Y, X), meet(Y, X)))
% 212.33/27.09 = { by lemma 43 R->L }
% 212.33/27.09 complement(join(meet(Y, X), complement(complement(meet(Y, X)))))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.09 complement(join(complement(complement(meet(Y, X))), meet(Y, X)))
% 212.33/27.09 = { by lemma 48 R->L }
% 212.33/27.09 complement(join(zero, join(complement(complement(meet(Y, X))), complement(complement(meet(Y, X))))))
% 212.33/27.09 = { by lemma 26 }
% 212.33/27.09 complement(join(zero, complement(complement(meet(Y, X)))))
% 212.33/27.09 = { by lemma 37 }
% 212.33/27.09 complement(meet(Y, X))
% 212.33/27.09 = { by lemma 19 R->L }
% 212.33/27.09 complement(meet(X, Y))
% 212.33/27.09
% 212.33/27.09 Lemma 65: join(X, complement(meet(X, Y))) = top.
% 212.33/27.09 Proof:
% 212.33/27.09 join(X, complement(meet(X, Y)))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 join(X, complement(meet(Y, X)))
% 212.33/27.09 = { by lemma 64 R->L }
% 212.33/27.09 join(X, join(complement(Y), complement(X)))
% 212.33/27.09 = { by lemma 21 }
% 212.33/27.09 join(complement(Y), top)
% 212.33/27.09 = { by lemma 30 }
% 212.33/27.09 top
% 212.33/27.09
% 212.33/27.09 Lemma 66: meet(X, meet(Y, complement(X))) = zero.
% 212.33/27.09 Proof:
% 212.33/27.09 meet(X, meet(Y, complement(X)))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 meet(X, meet(complement(X), Y))
% 212.33/27.09 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.09 complement(join(complement(X), complement(meet(complement(X), Y))))
% 212.33/27.09 = { by lemma 65 }
% 212.33/27.09 complement(top)
% 212.33/27.09 = { by lemma 18 }
% 212.33/27.09 zero
% 212.33/27.09
% 212.33/27.09 Lemma 67: meet(X, join(X, complement(Y))) = X.
% 212.33/27.09 Proof:
% 212.33/27.09 meet(X, join(X, complement(Y)))
% 212.33/27.09 = { by lemma 61 R->L }
% 212.33/27.09 meet(X, complement(meet(Y, complement(X))))
% 212.33/27.09 = { by lemma 63 R->L }
% 212.33/27.09 complement(join(complement(X), meet(Y, complement(X))))
% 212.33/27.09 = { by lemma 38 R->L }
% 212.33/27.09 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 212.33/27.09 = { by lemma 66 R->L }
% 212.33/27.09 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 212.33/27.09 = { by lemma 33 }
% 212.33/27.09 X
% 212.33/27.09
% 212.33/27.09 Lemma 68: join(X, meet(X, Y)) = X.
% 212.33/27.09 Proof:
% 212.33/27.09 join(X, meet(X, Y))
% 212.33/27.09 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.09 join(X, complement(join(complement(X), complement(Y))))
% 212.33/27.09 = { by lemma 62 R->L }
% 212.33/27.09 complement(meet(complement(X), join(complement(X), complement(Y))))
% 212.33/27.09 = { by lemma 67 }
% 212.33/27.09 complement(complement(X))
% 212.33/27.09 = { by lemma 43 }
% 212.33/27.09 X
% 212.33/27.09
% 212.33/27.09 Lemma 69: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 212.33/27.09 Proof:
% 212.33/27.09 join(Y, join(X, Z))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.09 join(join(X, Z), Y)
% 212.33/27.09 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 212.33/27.09 join(X, join(Z, Y))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.09 join(X, join(Y, Z))
% 212.33/27.09
% 212.33/27.09 Lemma 70: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 212.33/27.09 Proof:
% 212.33/27.09 join(Z, join(X, Y))
% 212.33/27.09 = { by lemma 69 }
% 212.33/27.09 join(X, join(Z, Y))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.09 join(X, join(Y, Z))
% 212.33/27.09
% 212.33/27.09 Lemma 71: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 212.33/27.09 Proof:
% 212.33/27.09 meet(complement(X), complement(Y))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 meet(complement(Y), complement(X))
% 212.33/27.09 = { by lemma 38 R->L }
% 212.33/27.09 meet(join(zero, complement(Y)), complement(X))
% 212.33/27.09 = { by lemma 60 R->L }
% 212.33/27.09 complement(join(X, complement(join(zero, complement(Y)))))
% 212.33/27.09 = { by lemma 36 }
% 212.33/27.09 complement(join(X, meet(Y, top)))
% 212.33/27.09 = { by lemma 39 }
% 212.33/27.09 complement(join(X, Y))
% 212.33/27.09
% 212.33/27.09 Lemma 72: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 212.33/27.09 Proof:
% 212.33/27.09 join(meet(X, Y), meet(Y, complement(X)))
% 212.33/27.09 = { by lemma 19 }
% 212.33/27.09 join(meet(Y, X), meet(Y, complement(X)))
% 212.33/27.09 = { by lemma 41 }
% 212.33/27.09 Y
% 212.33/27.09
% 212.33/27.09 Lemma 73: join(meet(X, Y), X) = X.
% 212.33/27.09 Proof:
% 212.33/27.09 join(meet(X, Y), X)
% 212.33/27.09 = { by lemma 39 R->L }
% 212.33/27.09 meet(join(meet(X, Y), X), top)
% 212.33/27.09 = { by lemma 36 R->L }
% 212.33/27.09 complement(join(zero, complement(join(meet(X, Y), X))))
% 212.33/27.09 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.10 complement(join(zero, complement(join(X, meet(X, Y)))))
% 212.33/27.10 = { by lemma 71 R->L }
% 212.33/27.10 complement(join(zero, meet(complement(X), complement(meet(X, Y)))))
% 212.33/27.10 = { by lemma 51 R->L }
% 212.33/27.10 complement(join(zero, join(zero, meet(complement(X), complement(meet(X, Y))))))
% 212.33/27.10 = { by lemma 18 R->L }
% 212.33/27.10 complement(join(zero, join(complement(top), meet(complement(X), complement(meet(X, Y))))))
% 212.33/27.10 = { by lemma 65 R->L }
% 212.33/27.10 complement(join(zero, join(complement(join(X, complement(meet(X, Y)))), meet(complement(X), complement(meet(X, Y))))))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 complement(join(zero, join(meet(meet(X, Y), complement(X)), meet(complement(X), complement(meet(X, Y))))))
% 212.33/27.10 = { by lemma 72 }
% 212.33/27.10 complement(join(zero, complement(X)))
% 212.33/27.10 = { by lemma 36 }
% 212.33/27.10 meet(X, top)
% 212.33/27.10 = { by lemma 39 }
% 212.33/27.10 X
% 212.33/27.10
% 212.33/27.10 Lemma 74: meet(X, join(X, Y)) = X.
% 212.33/27.10 Proof:
% 212.33/27.10 meet(X, join(X, Y))
% 212.33/27.10 = { by lemma 39 R->L }
% 212.33/27.10 meet(X, join(X, meet(Y, top)))
% 212.33/27.10 = { by lemma 36 R->L }
% 212.33/27.10 meet(X, join(X, complement(join(zero, complement(Y)))))
% 212.33/27.10 = { by lemma 67 }
% 212.33/27.10 X
% 212.33/27.10
% 212.33/27.10 Lemma 75: meet(complement(Z), meet(X, Y)) = meet(X, meet(Y, complement(Z))).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(complement(Z), meet(X, Y))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(meet(X, Y), complement(Z))
% 212.33/27.10 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.10 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 212.33/27.10 = { by lemma 71 }
% 212.33/27.10 complement(join(join(complement(X), complement(Y)), Z))
% 212.33/27.10 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 212.33/27.10 complement(join(complement(X), join(complement(Y), Z)))
% 212.33/27.10 = { by lemma 63 }
% 212.33/27.10 meet(X, complement(join(complement(Y), Z)))
% 212.33/27.10 = { by lemma 63 }
% 212.33/27.10 meet(X, meet(Y, complement(Z)))
% 212.33/27.10
% 212.33/27.10 Lemma 76: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(Y, meet(Z, X))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(Y, meet(X, Z))
% 212.33/27.10 = { by lemma 39 R->L }
% 212.33/27.10 meet(meet(Y, top), meet(X, Z))
% 212.33/27.10 = { by lemma 36 R->L }
% 212.33/27.10 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 212.33/27.10 = { by lemma 75 }
% 212.33/27.10 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 212.33/27.10 = { by lemma 36 }
% 212.33/27.10 meet(X, meet(Z, meet(Y, top)))
% 212.33/27.10 = { by lemma 39 }
% 212.33/27.10 meet(X, meet(Z, Y))
% 212.33/27.10 = { by lemma 19 R->L }
% 212.33/27.10 meet(X, meet(Y, Z))
% 212.33/27.10
% 212.33/27.10 Lemma 77: meet(X, complement(join(Y, X))) = zero.
% 212.33/27.10 Proof:
% 212.33/27.10 meet(X, complement(join(Y, X)))
% 212.33/27.10 = { by lemma 60 R->L }
% 212.33/27.10 complement(join(join(Y, X), complement(X)))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.10 complement(join(complement(X), join(Y, X)))
% 212.33/27.10 = { by lemma 70 }
% 212.33/27.10 complement(join(Y, join(X, complement(X))))
% 212.33/27.10 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.10 complement(join(Y, top))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 complement(join(top, Y))
% 212.33/27.10 = { by lemma 50 }
% 212.33/27.10 complement(top)
% 212.33/27.10 = { by lemma 18 }
% 212.33/27.10 zero
% 212.33/27.10
% 212.33/27.10 Lemma 78: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(complement(X), join(Y, complement(Z)))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(join(Y, complement(Z)), complement(X))
% 212.33/27.10 = { by lemma 60 R->L }
% 212.33/27.10 complement(join(X, complement(join(Y, complement(Z)))))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 complement(join(X, meet(Z, complement(Y))))
% 212.33/27.10
% 212.33/27.10 Lemma 79: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 212.33/27.10 Proof:
% 212.33/27.10 join(meet(X, Y), meet(complement(X), Y))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 join(meet(X, Y), meet(Y, complement(X)))
% 212.33/27.10 = { by lemma 72 }
% 212.33/27.10 Y
% 212.33/27.10
% 212.33/27.10 Lemma 80: meet(X, meet(complement(Y), join(X, Y))) = meet(complement(Y), join(X, Y)).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(X, meet(complement(Y), join(X, Y)))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(X, meet(join(X, Y), complement(Y)))
% 212.33/27.10 = { by lemma 60 R->L }
% 212.33/27.10 meet(X, complement(join(Y, complement(join(X, Y)))))
% 212.33/27.10 = { by lemma 71 R->L }
% 212.33/27.10 meet(X, complement(join(Y, meet(complement(X), complement(Y)))))
% 212.33/27.10 = { by lemma 49 R->L }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), zero)
% 212.33/27.10 = { by axiom 7 (def_zero_13) }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(join(Y, complement(complement(X))), complement(join(Y, complement(complement(X))))))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(join(Y, complement(complement(X))), meet(complement(X), complement(Y))))
% 212.33/27.10 = { by lemma 75 R->L }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(complement(Y), meet(join(Y, complement(complement(X))), complement(X))))
% 212.33/27.10 = { by lemma 19 R->L }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(complement(Y), meet(complement(X), join(Y, complement(complement(X))))))
% 212.33/27.10 = { by lemma 75 }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(complement(X), meet(join(Y, complement(complement(X))), complement(Y))))
% 212.33/27.10 = { by lemma 19 R->L }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(complement(X), meet(complement(Y), join(Y, complement(complement(X))))))
% 212.33/27.10 = { by lemma 78 }
% 212.33/27.10 join(meet(X, complement(join(Y, meet(complement(X), complement(Y))))), meet(complement(X), complement(join(Y, meet(complement(X), complement(Y))))))
% 212.33/27.10 = { by lemma 79 }
% 212.33/27.10 complement(join(Y, meet(complement(X), complement(Y))))
% 212.33/27.10 = { by lemma 71 }
% 212.33/27.10 complement(join(Y, complement(join(X, Y))))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 meet(join(X, Y), complement(Y))
% 212.33/27.10 = { by lemma 19 R->L }
% 212.33/27.10 meet(complement(Y), join(X, Y))
% 212.33/27.10
% 212.33/27.10 Lemma 81: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(X, complement(meet(X, Y)))
% 212.33/27.10 = { by lemma 63 R->L }
% 212.33/27.10 complement(join(complement(X), meet(X, Y)))
% 212.33/27.10 = { by lemma 59 R->L }
% 212.33/27.10 meet(X, join(complement(X), complement(Y)))
% 212.33/27.10 = { by lemma 61 R->L }
% 212.33/27.10 meet(X, complement(meet(Y, complement(complement(X)))))
% 212.33/27.10 = { by lemma 63 R->L }
% 212.33/27.10 complement(join(complement(X), meet(Y, complement(complement(X)))))
% 212.33/27.10 = { by lemma 78 R->L }
% 212.33/27.10 meet(complement(complement(X)), join(complement(X), complement(Y)))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 meet(complement(complement(X)), join(complement(Y), complement(X)))
% 212.33/27.10 = { by lemma 80 R->L }
% 212.33/27.10 meet(complement(Y), meet(complement(complement(X)), join(complement(Y), complement(X))))
% 212.33/27.10 = { by lemma 64 }
% 212.33/27.10 meet(complement(Y), meet(complement(complement(X)), complement(meet(Y, X))))
% 212.33/27.10 = { by lemma 71 }
% 212.33/27.10 meet(complement(Y), complement(join(complement(X), meet(Y, X))))
% 212.33/27.10 = { by lemma 71 }
% 212.33/27.10 complement(join(Y, join(complement(X), meet(Y, X))))
% 212.33/27.10 = { by lemma 69 }
% 212.33/27.10 complement(join(complement(X), join(Y, meet(Y, X))))
% 212.33/27.10 = { by lemma 63 }
% 212.33/27.10 meet(X, complement(join(Y, meet(Y, X))))
% 212.33/27.10 = { by lemma 68 }
% 212.33/27.10 meet(X, complement(Y))
% 212.33/27.10
% 212.33/27.10 Lemma 82: meet(complement(X), join(Y, X)) = meet(Y, complement(X)).
% 212.33/27.10 Proof:
% 212.33/27.10 meet(complement(X), join(Y, X))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.10 meet(complement(X), join(X, Y))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(join(X, Y), complement(X))
% 212.33/27.10 = { by lemma 60 R->L }
% 212.33/27.10 complement(join(X, complement(join(X, Y))))
% 212.33/27.10 = { by lemma 71 R->L }
% 212.33/27.10 meet(complement(X), complement(complement(join(X, Y))))
% 212.33/27.10 = { by lemma 71 R->L }
% 212.33/27.10 meet(complement(X), complement(meet(complement(X), complement(Y))))
% 212.33/27.10 = { by lemma 81 }
% 212.33/27.10 meet(complement(X), complement(complement(Y)))
% 212.33/27.10 = { by lemma 71 }
% 212.33/27.10 complement(join(X, complement(Y)))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 meet(Y, complement(X))
% 212.33/27.10
% 212.33/27.10 Lemma 83: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 212.33/27.10 Proof:
% 212.33/27.10 join(composition(X, Y), composition(X, Z))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.10 join(composition(X, Y), composition(X, converse(converse(Z))))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.10 join(composition(X, Y), composition(converse(converse(X)), converse(converse(Z))))
% 212.33/27.10 = { by axiom 5 (converse_multiplicativity_10) R->L }
% 212.33/27.10 join(composition(X, Y), converse(composition(converse(Z), converse(X))))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.10 join(converse(converse(composition(X, Y))), converse(composition(converse(Z), converse(X))))
% 212.33/27.10 = { by axiom 4 (converse_additivity_9) R->L }
% 212.33/27.10 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 212.33/27.10 = { by axiom 5 (converse_multiplicativity_10) }
% 212.33/27.10 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 212.33/27.10 = { by axiom 12 (composition_distributivity_7) R->L }
% 212.33/27.10 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 212.33/27.10 = { by axiom 5 (converse_multiplicativity_10) }
% 212.33/27.10 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.10 composition(X, converse(join(converse(Z), converse(Y))))
% 212.33/27.10 = { by axiom 4 (converse_additivity_9) }
% 212.33/27.10 composition(X, join(converse(converse(Z)), converse(converse(Y))))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.10 composition(X, join(converse(converse(Z)), Y))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 composition(X, join(Y, converse(converse(Z))))
% 212.33/27.10 = { by axiom 1 (converse_idempotence_8) }
% 212.33/27.10 composition(X, join(Y, Z))
% 212.33/27.10
% 212.33/27.10 Lemma 84: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 212.33/27.10 Proof:
% 212.33/27.10 join(composition(X, Z), composition(X, Y))
% 212.33/27.10 = { by lemma 83 }
% 212.33/27.10 composition(X, join(Z, Y))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 composition(X, join(Y, Z))
% 212.33/27.10
% 212.33/27.10 Lemma 85: complement(join(X, join(Y, complement(Z)))) = meet(Z, complement(join(X, Y))).
% 212.33/27.10 Proof:
% 212.33/27.10 complement(join(X, join(Y, complement(Z))))
% 212.33/27.10 = { by lemma 69 }
% 212.33/27.10 complement(join(Y, join(X, complement(Z))))
% 212.33/27.10 = { by axiom 8 (maddux2_join_associativity_2) }
% 212.33/27.10 complement(join(join(Y, X), complement(Z)))
% 212.33/27.10 = { by lemma 60 }
% 212.33/27.10 meet(Z, complement(join(Y, X)))
% 212.33/27.10 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.10 meet(Z, complement(join(X, Y)))
% 212.33/27.10
% 212.33/27.10 Lemma 86: meet(one, composition(converse(complement(X)), X)) = zero.
% 212.33/27.10 Proof:
% 212.33/27.10 meet(one, composition(converse(complement(X)), X))
% 212.33/27.10 = { by lemma 39 R->L }
% 212.33/27.10 meet(one, composition(converse(complement(X)), meet(X, top)))
% 212.33/27.10 = { by lemma 46 R->L }
% 212.33/27.10 meet(one, composition(join(converse(zero), converse(complement(X))), meet(X, top)))
% 212.33/27.10 = { by axiom 4 (converse_additivity_9) R->L }
% 212.33/27.10 meet(one, composition(converse(join(zero, complement(X))), meet(X, top)))
% 212.33/27.10 = { by lemma 36 R->L }
% 212.33/27.10 meet(one, composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))), one)
% 212.33/27.10 = { by lemma 43 R->L }
% 212.33/27.10 meet(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))), complement(complement(one)))
% 212.33/27.10 = { by lemma 25 R->L }
% 212.33/27.10 meet(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 212.33/27.10 = { by axiom 3 (composition_identity_6) }
% 212.33/27.10 meet(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 212.33/27.10 = { by lemma 63 }
% 212.33/27.10 meet(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))), meet(one, complement(composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 212.33/27.10 = { by lemma 66 }
% 212.33/27.10 zero
% 212.33/27.10
% 212.33/27.10 Lemma 87: join(X, join(Y, complement(join(X, Y)))) = top.
% 212.33/27.10 Proof:
% 212.33/27.10 join(X, join(Y, complement(join(X, Y))))
% 212.33/27.10 = { by axiom 8 (maddux2_join_associativity_2) }
% 212.33/27.10 join(join(X, Y), complement(join(X, Y)))
% 212.33/27.10 = { by axiom 6 (def_top_12) R->L }
% 212.33/27.10 top
% 212.33/27.10
% 212.33/27.10 Lemma 88: meet(complement(X), composition(converse(sk1), composition(sk1, X))) = zero.
% 212.33/27.10 Proof:
% 212.33/27.10 meet(complement(X), composition(converse(sk1), composition(sk1, X)))
% 212.33/27.10 = { by lemma 19 }
% 212.33/27.10 meet(composition(converse(sk1), composition(sk1, X)), complement(X))
% 212.33/27.10 = { by lemma 24 R->L }
% 212.33/27.10 meet(composition(converse(sk1), composition(sk1, X)), complement(composition(one, X)))
% 212.33/27.10 = { by axiom 9 (goals_17) R->L }
% 212.33/27.11 meet(composition(converse(sk1), composition(sk1, X)), complement(composition(join(composition(converse(sk1), sk1), one), X)))
% 212.33/27.11 = { by axiom 12 (composition_distributivity_7) }
% 212.33/27.11 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(composition(converse(sk1), sk1), X), composition(one, X))))
% 212.33/27.11 = { by axiom 10 (composition_associativity_5) R->L }
% 212.33/27.11 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(converse(sk1), composition(sk1, X)), composition(one, X))))
% 212.33/27.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 212.33/27.11 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(one, X), composition(converse(sk1), composition(sk1, X)))))
% 212.33/27.11 = { by lemma 24 }
% 212.33/27.11 meet(composition(converse(sk1), composition(sk1, X)), complement(join(X, composition(converse(sk1), composition(sk1, X)))))
% 212.33/27.11 = { by lemma 77 }
% 212.33/27.11 zero
% 212.33/27.11
% 212.33/27.11 Lemma 89: join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))) = meet(composition(X, Y), composition(X, Z)).
% 212.33/27.11 Proof:
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z)))
% 212.33/27.11 = { by lemma 74 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), join(composition(X, meet(Y, Z)), composition(X, Y))))
% 212.33/27.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), join(composition(X, Y), composition(X, meet(Y, Z)))))
% 212.33/27.11 = { by lemma 84 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), composition(X, join(meet(Y, Z), Y))))
% 212.33/27.11 = { by lemma 73 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), composition(X, Y)))
% 212.33/27.11 = { by lemma 19 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), composition(X, meet(Y, Z))))
% 212.33/27.11 = { by lemma 74 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), join(composition(X, meet(Y, Z)), composition(X, Z)))))
% 212.33/27.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), join(composition(X, Z), composition(X, meet(Y, Z))))))
% 212.33/27.11 = { by lemma 84 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, join(meet(Y, Z), Z)))))
% 212.33/27.11 = { by lemma 39 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, meet(join(meet(Y, Z), Z), top)))))
% 212.33/27.11 = { by lemma 36 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(join(meet(Y, Z), Z))))))))
% 212.33/27.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(join(Z, meet(Y, Z)))))))))
% 212.33/27.11 = { by lemma 71 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, meet(complement(Z), complement(meet(Y, Z)))))))))
% 212.33/27.11 = { by lemma 51 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(zero, meet(complement(Z), complement(meet(Y, Z))))))))))
% 212.33/27.11 = { by lemma 18 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(complement(top), meet(complement(Z), complement(meet(Y, Z))))))))))
% 212.33/27.11 = { by lemma 65 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(complement(join(Z, complement(meet(Z, Y)))), meet(complement(Z), complement(meet(Y, Z))))))))))
% 212.33/27.11 = { by lemma 19 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(complement(join(Z, complement(meet(Y, Z)))), meet(complement(Z), complement(meet(Y, Z))))))))))
% 212.33/27.11 = { by lemma 60 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(meet(meet(Y, Z), complement(Z)), meet(complement(Z), complement(meet(Y, Z))))))))))
% 212.33/27.11 = { by lemma 72 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(Z)))))))
% 212.33/27.11 = { by lemma 36 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, meet(Z, top)))))
% 212.33/27.11 = { by lemma 39 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, Z))))
% 212.33/27.11 = { by lemma 19 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, Z), composition(X, meet(Y, Z)))))
% 212.33/27.11 = { by lemma 76 }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), meet(composition(X, Y), composition(X, Z))))
% 212.33/27.11 = { by lemma 19 R->L }
% 212.33/27.11 join(meet(composition(X, Y), composition(X, Z)), meet(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))))
% 212.33/27.11 = { by lemma 68 }
% 212.33/27.12 meet(composition(X, Y), composition(X, Z))
% 212.33/27.12
% 212.33/27.12 Lemma 90: join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))) = meet(composition(sk1, meet(X, Y)), complement(composition(sk1, complement(Y)))).
% 212.33/27.12 Proof:
% 212.33/27.12 join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))
% 212.33/27.12 = { by lemma 39 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), top)
% 212.33/27.12 = { by lemma 42 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(zero))
% 212.33/27.12 = { by lemma 56 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), complement(converse(Y)))), zero)))
% 212.33/27.12 = { by lemma 88 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), complement(converse(Y)))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 39 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), complement(converse(meet(Y, top))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 36 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), complement(converse(complement(join(zero, complement(Y))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by axiom 1 (converse_idempotence_8) R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(converse(complement(converse(complement(join(zero, complement(Y))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 79 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(complement(join(zero, complement(Y))), converse(complement(converse(complement(join(zero, complement(Y))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 49 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), join(meet(complement(join(zero, complement(Y))), converse(complement(converse(complement(join(zero, complement(Y))))))), zero))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 53 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), join(meet(complement(join(zero, complement(Y))), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(Y)))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 58 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), join(meet(complement(join(zero, complement(Y))), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(composition(zero, complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y)))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 24 R->L }
% 212.33/27.12 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), join(meet(composition(one, complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(composition(zero, complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y)))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.12 = { by lemma 86 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), join(meet(composition(one, complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(Y)))))), converse(complement(join(zero, complement(Y)))))), complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y)))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by axiom 16 (modular_law_2_16) }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(Y)))))), converse(complement(join(zero, complement(Y)))))), complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 86 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(composition(zero, complement(join(zero, complement(Y)))), converse(complement(converse(complement(join(zero, complement(Y))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 58 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(Y))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 53 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y))))))), zero)))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 49 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(meet(join(zero, complement(Y)), converse(complement(converse(complement(join(zero, complement(Y)))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y)))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 38 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(zero, complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y))))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 18 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(complement(top), complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y))))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 32 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y))))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 20 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(complement(join(complement(join(zero, complement(Y))), converse(complement(converse(complement(join(zero, complement(Y)))))))), complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y))))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 63 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(meet(join(zero, complement(Y)), complement(converse(complement(converse(complement(join(zero, complement(Y)))))))), complement(join(complement(join(zero, complement(Y))), complement(converse(complement(converse(complement(join(zero, complement(Y))))))))))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 33 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(join(zero, complement(Y))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by axiom 4 (converse_additivity_9) }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), join(converse(zero), converse(complement(Y))))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by lemma 46 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(meet(sk1, composition(composition(sk1, Y), converse(complement(Y)))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y))))))
% 212.33/27.13 = { by axiom 17 (dedekind_law_14) R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(meet(composition(sk1, complement(Y)), composition(sk1, Y)), composition(meet(sk1, composition(composition(sk1, Y), converse(complement(Y)))), meet(complement(Y), composition(converse(sk1), composition(sk1, Y)))))))
% 212.33/27.13 = { by lemma 88 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(meet(composition(sk1, complement(Y)), composition(sk1, Y)), composition(meet(sk1, composition(composition(sk1, Y), converse(complement(Y)))), zero))))
% 212.33/27.13 = { by lemma 56 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(meet(composition(sk1, complement(Y)), composition(sk1, Y)), zero)))
% 212.33/27.13 = { by lemma 49 }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(meet(composition(sk1, complement(Y)), composition(sk1, Y))))
% 212.33/27.13 = { by lemma 19 R->L }
% 212.33/27.13 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(meet(composition(sk1, Y), composition(sk1, complement(Y)))))
% 212.33/27.13 = { by lemma 19 }
% 212.33/27.13 meet(complement(meet(composition(sk1, Y), composition(sk1, complement(Y)))), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))
% 212.33/27.13 = { by lemma 89 }
% 212.33/27.13 meet(complement(meet(composition(sk1, Y), composition(sk1, complement(Y)))), meet(composition(sk1, X), composition(sk1, Y)))
% 212.33/27.13 = { by lemma 76 R->L }
% 212.33/27.13 meet(composition(sk1, X), meet(composition(sk1, Y), complement(meet(composition(sk1, Y), composition(sk1, complement(Y))))))
% 212.33/27.13 = { by lemma 81 }
% 212.33/27.13 meet(composition(sk1, X), meet(composition(sk1, Y), complement(composition(sk1, complement(Y)))))
% 212.33/27.13 = { by lemma 76 }
% 212.33/27.13 meet(complement(composition(sk1, complement(Y))), meet(composition(sk1, X), composition(sk1, Y)))
% 212.33/27.13 = { by lemma 89 R->L }
% 212.33/27.13 meet(complement(composition(sk1, complement(Y))), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))
% 212.33/27.14 = { by lemma 19 R->L }
% 212.33/27.14 meet(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(composition(sk1, complement(Y))))
% 212.33/27.14 = { by lemma 82 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), composition(sk1, complement(Y))))
% 212.33/27.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))
% 212.33/27.14 = { by lemma 72 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), composition(sk1, X))))))
% 212.33/27.14 = { by lemma 85 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(composition(sk1, X), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))))))
% 212.33/27.14 = { by lemma 41 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), meet(composition(sk1, X), complement(composition(sk1, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))))))
% 212.33/27.14 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), join(meet(composition(sk1, X), complement(composition(sk1, Y))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))))
% 212.33/27.14 = { by lemma 89 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), join(meet(composition(sk1, X), complement(composition(sk1, Y))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))))
% 212.33/27.14 = { by lemma 81 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), join(meet(composition(sk1, X), complement(meet(composition(sk1, X), composition(sk1, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))))
% 212.33/27.14 = { by lemma 89 R->L }
% 212.33/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), join(meet(composition(sk1, X), complement(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))))
% 213.11/27.14 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.14 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), join(complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), meet(composition(sk1, X), complement(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))))
% 213.11/27.15 = { by axiom 8 (maddux2_join_associativity_2) }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(composition(sk1, complement(Y)), join(join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))), meet(composition(sk1, X), complement(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))))))
% 213.11/27.15 = { by axiom 8 (maddux2_join_associativity_2) }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))), meet(composition(sk1, X), complement(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.15 = { by lemma 87 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(top, meet(composition(sk1, X), complement(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.15 = { by lemma 50 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(top)))
% 213.11/27.15 = { by lemma 18 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), zero))
% 213.11/27.15 = { by lemma 49 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), composition(sk1, X)), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))
% 213.11/27.15 = { by lemma 19 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), join(composition(sk1, complement(Y)), composition(sk1, X))))
% 213.11/27.15 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), join(composition(sk1, X), composition(sk1, complement(Y)))))
% 213.11/27.15 = { by lemma 83 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(X, complement(Y)))))
% 213.11/27.15 = { by lemma 73 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(join(meet(X, Y), X), complement(Y)))))
% 213.11/27.15 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), join(X, complement(Y))))))
% 213.11/27.15 = { by lemma 61 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), complement(meet(Y, complement(X)))))))
% 213.11/27.15 = { by lemma 74 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), complement(meet(meet(Y, complement(X)), join(meet(Y, complement(X)), meet(X, Y))))))))
% 213.11/27.15 = { by lemma 64 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), join(complement(meet(Y, complement(X))), complement(join(meet(Y, complement(X)), meet(X, Y))))))))
% 213.11/27.15 = { by axiom 8 (maddux2_join_associativity_2) }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(join(meet(X, Y), complement(meet(Y, complement(X)))), complement(join(meet(Y, complement(X)), meet(X, Y)))))))
% 213.11/27.15 = { by lemma 61 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, complement(meet(join(meet(Y, complement(X)), meet(X, Y)), complement(join(meet(X, Y), complement(meet(Y, complement(X))))))))))
% 213.11/27.15 = { by lemma 60 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, complement(meet(join(meet(Y, complement(X)), meet(X, Y)), meet(meet(Y, complement(X)), complement(meet(X, Y))))))))
% 213.11/27.15 = { by lemma 76 R->L }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, complement(meet(meet(Y, complement(X)), meet(complement(meet(X, Y)), join(meet(Y, complement(X)), meet(X, Y))))))))
% 213.11/27.15 = { by lemma 80 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, complement(meet(complement(meet(X, Y)), join(meet(Y, complement(X)), meet(X, Y)))))))
% 213.11/27.15 = { by lemma 62 }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), complement(join(meet(Y, complement(X)), meet(X, Y)))))))
% 213.11/27.15 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.15 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), complement(join(meet(X, Y), meet(Y, complement(X))))))))
% 213.11/27.16 = { by lemma 72 }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), composition(sk1, join(meet(X, Y), complement(Y)))))
% 213.11/27.16 = { by lemma 83 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y)))))
% 213.11/27.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))), join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))))
% 213.11/27.16 = { by lemma 19 }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), meet(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y))), join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))
% 213.11/27.16 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))))))
% 213.11/27.16 = { by lemma 38 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(zero, complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 18 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(top), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 30 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(meet(composition(sk1, X), composition(sk1, Y)), top)), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 87 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(meet(composition(sk1, X), composition(sk1, Y)), join(composition(sk1, meet(X, Y)), join(composition(sk1, complement(Y)), complement(join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y)))))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 70 R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(join(composition(sk1, complement(Y)), complement(join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y))))), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(composition(sk1, complement(Y)), join(complement(join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y)))), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y))))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(complement(join(composition(sk1, complement(Y)), join(join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y))), complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y))))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 85 }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(meet(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))), complement(join(complement(join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y)))), complement(join(composition(sk1, complement(Y)), join(meet(composition(sk1, X), composition(sk1, Y)), composition(sk1, meet(X, Y)))))))))
% 213.11/27.16 = { by lemma 33 }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(composition(sk1, complement(Y)), composition(sk1, meet(X, Y))))
% 213.11/27.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.16 meet(complement(composition(sk1, complement(Y))), join(composition(sk1, meet(X, Y)), composition(sk1, complement(Y))))
% 213.11/27.16 = { by lemma 82 }
% 213.11/27.16 meet(composition(sk1, meet(X, Y)), complement(composition(sk1, complement(Y))))
% 213.11/27.16
% 213.11/27.16 Goal 1 (goals_18): tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))) = tuple(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))).
% 213.11/27.16 Proof:
% 213.11/27.16 tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))
% 213.11/27.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 213.11/27.16 tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 213.11/27.16 = { by lemma 89 }
% 213.11/27.16 tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 90 }
% 213.11/27.16 tuple(meet(composition(sk1, meet(sk2, sk3)), complement(composition(sk1, complement(sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 19 }
% 213.11/27.16 tuple(meet(complement(composition(sk1, complement(sk3))), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 43 R->L }
% 213.11/27.16 tuple(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 41 R->L }
% 213.11/27.16 tuple(join(meet(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), composition(sk1, meet(sk2, sk3))), meet(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), complement(composition(sk1, meet(sk2, sk3))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 19 R->L }
% 213.11/27.16 tuple(join(meet(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), composition(sk1, meet(sk2, sk3))), meet(complement(composition(sk1, meet(sk2, sk3))), meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 66 }
% 213.11/27.16 tuple(join(meet(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), composition(sk1, meet(sk2, sk3))), zero), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 49 }
% 213.11/27.16 tuple(meet(meet(complement(composition(sk1, complement(sk3))), complement(complement(composition(sk1, meet(sk2, sk3))))), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 43 }
% 213.11/27.16 tuple(meet(meet(complement(composition(sk1, complement(sk3))), composition(sk1, meet(sk2, sk3))), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.16 = { by lemma 19 R->L }
% 213.11/27.16 tuple(meet(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, complement(sk3))), composition(sk1, meet(sk2, sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by lemma 19 R->L }
% 213.11/27.17 tuple(meet(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, meet(sk2, sk3)), complement(composition(sk1, complement(sk3))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by lemma 90 R->L }
% 213.11/27.17 tuple(meet(composition(sk1, meet(sk2, sk3)), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 213.11/27.17 tuple(complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by lemma 38 R->L }
% 213.11/27.17 tuple(join(zero, complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by lemma 77 R->L }
% 213.11/27.17 tuple(join(meet(composition(sk1, meet(sk2, sk3)), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))), complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 = { by lemma 33 }
% 213.11/27.17 tuple(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 213.11/27.17 % SZS output end Proof
% 213.11/27.17
% 213.11/27.17 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------