TSTP Solution File: REL028-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL028-2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n016.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:11 EDT 2023
% Result : Unsatisfiable 12.09s 1.96s
% Output : Proof 13.40s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL028-2 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.16/0.34 % Computer : n016.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Fri Aug 25 19:34:25 EDT 2023
% 0.16/0.34 % CPUTime :
% 12.09/1.96 Command-line arguments: --flatten
% 12.09/1.96
% 12.09/1.96 % SZS status Unsatisfiable
% 12.09/1.96
% 13.40/2.10 % SZS output start Proof
% 13.40/2.10 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 13.40/2.10 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 13.40/2.10 Axiom 3 (goals_17): join(sk1, one) = one.
% 13.40/2.10 Axiom 4 (goals_18): join(sk2, one) = one.
% 13.40/2.10 Axiom 5 (composition_identity_6): composition(X, one) = X.
% 13.40/2.10 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 13.40/2.10 Axiom 7 (def_top_12): top = join(X, complement(X)).
% 13.40/2.10 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 13.40/2.10 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 13.40/2.10 Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 13.40/2.10 Axiom 11 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 13.40/2.10 Axiom 12 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 13.40/2.10 Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 13.40/2.10 Axiom 14 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 13.40/2.10 Axiom 15 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 13.40/2.10 Axiom 16 (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).
% 13.40/2.10 Axiom 17 (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).
% 13.40/2.10
% 13.40/2.10 Lemma 18: complement(top) = zero.
% 13.40/2.10 Proof:
% 13.40/2.10 complement(top)
% 13.40/2.10 = { by axiom 7 (def_top_12) }
% 13.40/2.10 complement(join(complement(X), complement(complement(X))))
% 13.40/2.10 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.10 meet(X, complement(X))
% 13.40/2.10 = { by axiom 6 (def_zero_13) R->L }
% 13.40/2.12 zero
% 13.40/2.12
% 13.40/2.12 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 13.40/2.12 Proof:
% 13.40/2.12 converse(composition(converse(X), Y))
% 13.40/2.12 = { by axiom 10 (converse_multiplicativity_10) }
% 13.40/2.12 composition(converse(Y), converse(converse(X)))
% 13.40/2.12 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.12 composition(converse(Y), X)
% 13.40/2.12
% 13.40/2.12 Lemma 20: composition(X, join(sk1, one)) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 composition(X, join(sk1, one))
% 13.40/2.12 = { by axiom 3 (goals_17) }
% 13.40/2.12 composition(X, one)
% 13.40/2.12 = { by axiom 5 (composition_identity_6) }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 21: composition(converse(join(sk1, one)), X) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 composition(converse(join(sk1, one)), X)
% 13.40/2.12 = { by lemma 19 R->L }
% 13.40/2.12 converse(composition(converse(X), join(sk1, one)))
% 13.40/2.12 = { by lemma 20 }
% 13.40/2.12 converse(converse(X))
% 13.40/2.12 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 22: composition(join(sk1, one), X) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 composition(join(sk1, one), X)
% 13.40/2.12 = { by lemma 21 R->L }
% 13.40/2.12 composition(converse(join(sk1, one)), composition(join(sk1, one), X))
% 13.40/2.12 = { by axiom 11 (composition_associativity_5) }
% 13.40/2.12 composition(composition(converse(join(sk1, one)), join(sk1, one)), X)
% 13.40/2.12 = { by lemma 20 }
% 13.40/2.12 composition(converse(join(sk1, one)), X)
% 13.40/2.12 = { by lemma 21 }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 13.40/2.12 Proof:
% 13.40/2.12 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 13.40/2.12 = { by axiom 14 (converse_cancellativity_11) }
% 13.40/2.12 complement(X)
% 13.40/2.12
% 13.40/2.12 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 13.40/2.12 Proof:
% 13.40/2.12 join(complement(X), complement(X))
% 13.40/2.12 = { by lemma 21 R->L }
% 13.40/2.12 join(complement(X), composition(converse(join(sk1, one)), complement(X)))
% 13.40/2.12 = { by lemma 22 R->L }
% 13.40/2.12 join(complement(X), composition(converse(join(sk1, one)), complement(composition(join(sk1, one), X))))
% 13.40/2.12 = { by lemma 23 }
% 13.40/2.12 complement(X)
% 13.40/2.12
% 13.40/2.12 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 join(meet(X, Y), complement(join(complement(X), Y)))
% 13.40/2.12 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.12 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 13.40/2.12 = { by axiom 15 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 26: join(zero, meet(X, X)) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 join(zero, meet(X, X))
% 13.40/2.12 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.12 join(zero, complement(join(complement(X), complement(X))))
% 13.40/2.12 = { by axiom 6 (def_zero_13) }
% 13.40/2.12 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.40/2.12 = { by lemma 25 }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 27: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 13.40/2.12 Proof:
% 13.40/2.12 join(zero, join(X, complement(complement(Y))))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(zero, join(complement(complement(Y)), X))
% 13.40/2.12 = { by lemma 24 R->L }
% 13.40/2.12 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 13.40/2.12 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.12 join(zero, join(meet(Y, Y), X))
% 13.40/2.12 = { by axiom 9 (maddux2_join_associativity_2) }
% 13.40/2.12 join(join(zero, meet(Y, Y)), X)
% 13.40/2.12 = { by lemma 26 }
% 13.40/2.12 join(Y, X)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.12 join(X, Y)
% 13.40/2.12
% 13.40/2.12 Lemma 28: join(X, join(Y, complement(X))) = join(Y, top).
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, join(Y, complement(X)))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(X, join(complement(X), Y))
% 13.40/2.12 = { by axiom 9 (maddux2_join_associativity_2) }
% 13.40/2.12 join(join(X, complement(X)), Y)
% 13.40/2.12 = { by axiom 7 (def_top_12) R->L }
% 13.40/2.12 join(top, Y)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.12 join(Y, top)
% 13.40/2.12
% 13.40/2.12 Lemma 29: join(top, complement(X)) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 join(top, complement(X))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(complement(X), top)
% 13.40/2.12 = { by lemma 28 R->L }
% 13.40/2.12 join(X, join(complement(X), complement(X)))
% 13.40/2.12 = { by lemma 24 }
% 13.40/2.12 join(X, complement(X))
% 13.40/2.12 = { by axiom 7 (def_top_12) R->L }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 30: join(Y, top) = join(X, top).
% 13.40/2.12 Proof:
% 13.40/2.12 join(Y, top)
% 13.40/2.12 = { by lemma 29 R->L }
% 13.40/2.12 join(Y, join(top, complement(Y)))
% 13.40/2.12 = { by lemma 28 }
% 13.40/2.12 join(top, top)
% 13.40/2.12 = { by lemma 28 R->L }
% 13.40/2.12 join(X, join(top, complement(X)))
% 13.40/2.12 = { by lemma 29 }
% 13.40/2.12 join(X, top)
% 13.40/2.12
% 13.40/2.12 Lemma 31: join(X, top) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, top)
% 13.40/2.12 = { by lemma 30 }
% 13.40/2.12 join(join(zero, zero), top)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(top, join(zero, zero))
% 13.40/2.12 = { by lemma 18 R->L }
% 13.40/2.12 join(top, join(zero, complement(top)))
% 13.40/2.12 = { by lemma 18 R->L }
% 13.40/2.12 join(top, join(complement(top), complement(top)))
% 13.40/2.12 = { by lemma 24 }
% 13.40/2.12 join(top, complement(top))
% 13.40/2.12 = { by axiom 7 (def_top_12) R->L }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 32: join(X, join(complement(X), Y)) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, join(complement(X), Y))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(X, join(Y, complement(X)))
% 13.40/2.12 = { by lemma 28 }
% 13.40/2.12 join(Y, top)
% 13.40/2.12 = { by lemma 30 R->L }
% 13.40/2.12 join(Z, top)
% 13.40/2.12 = { by lemma 31 }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 33: join(X, complement(zero)) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, complement(zero))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(complement(zero), X)
% 13.40/2.12 = { by lemma 27 R->L }
% 13.40/2.12 join(zero, join(complement(zero), complement(complement(X))))
% 13.40/2.12 = { by lemma 32 }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 34: complement(zero) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 complement(zero)
% 13.40/2.12 = { by lemma 24 R->L }
% 13.40/2.12 join(complement(zero), complement(zero))
% 13.40/2.12 = { by lemma 33 }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 35: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 13.40/2.12 Proof:
% 13.40/2.12 converse(join(X, converse(Y)))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 converse(join(converse(Y), X))
% 13.40/2.12 = { by axiom 8 (converse_additivity_9) }
% 13.40/2.12 join(converse(converse(Y)), converse(X))
% 13.40/2.12 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.12 join(Y, converse(X))
% 13.40/2.12
% 13.40/2.12 Lemma 36: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 13.40/2.12 Proof:
% 13.40/2.12 converse(join(converse(X), Y))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 converse(join(Y, converse(X)))
% 13.40/2.12 = { by lemma 35 }
% 13.40/2.12 join(X, converse(Y))
% 13.40/2.12
% 13.40/2.12 Lemma 37: join(X, converse(complement(converse(X)))) = converse(top).
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, converse(complement(converse(X))))
% 13.40/2.12 = { by lemma 36 R->L }
% 13.40/2.12 converse(join(converse(X), complement(converse(X))))
% 13.40/2.12 = { by axiom 7 (def_top_12) R->L }
% 13.40/2.12 converse(top)
% 13.40/2.12
% 13.40/2.12 Lemma 38: join(X, converse(top)) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, converse(top))
% 13.40/2.12 = { by lemma 37 R->L }
% 13.40/2.12 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 13.40/2.12 = { by lemma 32 }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 39: converse(top) = top.
% 13.40/2.12 Proof:
% 13.40/2.12 converse(top)
% 13.40/2.12 = { by lemma 31 R->L }
% 13.40/2.12 converse(join(X, top))
% 13.40/2.12 = { by axiom 8 (converse_additivity_9) }
% 13.40/2.12 join(converse(X), converse(top))
% 13.40/2.12 = { by lemma 38 }
% 13.40/2.12 top
% 13.40/2.12
% 13.40/2.12 Lemma 40: join(zero, complement(complement(X))) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 join(zero, complement(complement(X)))
% 13.40/2.12 = { by axiom 6 (def_zero_13) }
% 13.40/2.12 join(meet(X, complement(X)), complement(complement(X)))
% 13.40/2.12 = { by lemma 24 R->L }
% 13.40/2.12 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 13.40/2.12 = { by lemma 25 }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 41: join(X, zero) = join(X, X).
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, zero)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(zero, X)
% 13.40/2.12 = { by lemma 40 R->L }
% 13.40/2.12 join(zero, join(zero, complement(complement(X))))
% 13.40/2.12 = { by lemma 24 R->L }
% 13.40/2.12 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 13.40/2.12 = { by lemma 27 }
% 13.40/2.12 join(zero, join(complement(complement(X)), X))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.12 join(zero, join(X, complement(complement(X))))
% 13.40/2.12 = { by lemma 27 }
% 13.40/2.12 join(X, X)
% 13.40/2.12
% 13.40/2.12 Lemma 42: join(zero, complement(X)) = complement(X).
% 13.40/2.12 Proof:
% 13.40/2.12 join(zero, complement(X))
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(complement(X), zero)
% 13.40/2.12 = { by lemma 41 }
% 13.40/2.12 join(complement(X), complement(X))
% 13.40/2.12 = { by lemma 24 }
% 13.40/2.12 complement(X)
% 13.40/2.12
% 13.40/2.12 Lemma 43: join(X, zero) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 join(X, zero)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(zero, X)
% 13.40/2.12 = { by lemma 27 R->L }
% 13.40/2.12 join(zero, join(zero, complement(complement(X))))
% 13.40/2.12 = { by lemma 42 }
% 13.40/2.12 join(zero, complement(complement(X)))
% 13.40/2.12 = { by lemma 40 }
% 13.40/2.12 X
% 13.40/2.12
% 13.40/2.12 Lemma 44: join(zero, X) = X.
% 13.40/2.12 Proof:
% 13.40/2.12 join(zero, X)
% 13.40/2.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.12 join(X, zero)
% 13.40/2.13 = { by lemma 43 }
% 13.40/2.13 X
% 13.40/2.13
% 13.40/2.13 Lemma 45: meet(Y, X) = meet(X, Y).
% 13.40/2.13 Proof:
% 13.40/2.13 meet(Y, X)
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.13 complement(join(complement(Y), complement(X)))
% 13.40/2.13 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.13 complement(join(complement(X), complement(Y)))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.13 meet(X, Y)
% 13.40/2.13
% 13.40/2.13 Lemma 46: complement(join(zero, complement(X))) = meet(X, top).
% 13.40/2.13 Proof:
% 13.40/2.13 complement(join(zero, complement(X)))
% 13.40/2.13 = { by lemma 18 R->L }
% 13.40/2.13 complement(join(complement(top), complement(X)))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.13 meet(top, X)
% 13.40/2.13 = { by lemma 45 R->L }
% 13.40/2.13 meet(X, top)
% 13.40/2.13
% 13.40/2.13 Lemma 47: meet(X, zero) = zero.
% 13.40/2.13 Proof:
% 13.40/2.13 meet(X, zero)
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.13 complement(join(complement(X), complement(zero)))
% 13.40/2.13 = { by lemma 33 }
% 13.40/2.13 complement(top)
% 13.40/2.13 = { by lemma 18 }
% 13.40/2.13 zero
% 13.40/2.13
% 13.40/2.13 Lemma 48: join(meet(X, Y), meet(X, complement(Y))) = X.
% 13.40/2.13 Proof:
% 13.40/2.13 join(meet(X, Y), meet(X, complement(Y)))
% 13.40/2.13 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.13 join(meet(X, complement(Y)), meet(X, Y))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.13 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 13.40/2.13 = { by lemma 25 }
% 13.40/2.13 X
% 13.40/2.13
% 13.40/2.13 Lemma 49: meet(X, top) = X.
% 13.40/2.13 Proof:
% 13.40/2.13 meet(X, top)
% 13.40/2.13 = { by lemma 46 R->L }
% 13.40/2.13 complement(join(zero, complement(X)))
% 13.40/2.13 = { by lemma 42 R->L }
% 13.40/2.13 join(zero, complement(join(zero, complement(X))))
% 13.40/2.13 = { by lemma 46 }
% 13.40/2.13 join(zero, meet(X, top))
% 13.40/2.13 = { by lemma 34 R->L }
% 13.40/2.13 join(zero, meet(X, complement(zero)))
% 13.40/2.13 = { by lemma 47 R->L }
% 13.40/2.13 join(meet(X, zero), meet(X, complement(zero)))
% 13.40/2.13 = { by lemma 48 }
% 13.40/2.13 X
% 13.40/2.13
% 13.40/2.13 Lemma 50: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 13.40/2.13 Proof:
% 13.40/2.13 join(meet(X, Y), meet(X, Y))
% 13.40/2.13 = { by lemma 45 }
% 13.40/2.13 join(meet(Y, X), meet(X, Y))
% 13.40/2.13 = { by lemma 45 }
% 13.40/2.13 join(meet(Y, X), meet(Y, X))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.13 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.13 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 13.40/2.13 = { by lemma 24 }
% 13.40/2.13 complement(join(complement(Y), complement(X)))
% 13.40/2.13 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.13 meet(Y, X)
% 13.40/2.13 = { by lemma 45 R->L }
% 13.40/2.13 meet(X, Y)
% 13.40/2.14
% 13.40/2.14 Lemma 51: converse(zero) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 converse(zero)
% 13.40/2.14 = { by lemma 44 R->L }
% 13.40/2.14 join(zero, converse(zero))
% 13.40/2.14 = { by lemma 36 R->L }
% 13.40/2.14 converse(join(converse(zero), zero))
% 13.40/2.14 = { by lemma 41 }
% 13.40/2.14 converse(join(converse(zero), converse(zero)))
% 13.40/2.14 = { by lemma 35 }
% 13.40/2.14 join(zero, converse(converse(zero)))
% 13.40/2.14 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.14 join(zero, zero)
% 13.40/2.14 = { by lemma 49 R->L }
% 13.40/2.14 join(zero, meet(zero, top))
% 13.40/2.14 = { by lemma 49 R->L }
% 13.40/2.14 join(meet(zero, top), meet(zero, top))
% 13.40/2.14 = { by lemma 50 }
% 13.40/2.14 meet(zero, top)
% 13.40/2.14 = { by lemma 49 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 52: join(top, X) = top.
% 13.40/2.14 Proof:
% 13.40/2.14 join(top, X)
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 join(X, top)
% 13.40/2.14 = { by lemma 30 R->L }
% 13.40/2.14 join(Y, top)
% 13.40/2.14 = { by lemma 31 }
% 13.40/2.14 top
% 13.40/2.14
% 13.40/2.14 Lemma 53: complement(complement(X)) = X.
% 13.40/2.14 Proof:
% 13.40/2.14 complement(complement(X))
% 13.40/2.14 = { by lemma 42 R->L }
% 13.40/2.14 join(zero, complement(complement(X)))
% 13.40/2.14 = { by lemma 40 }
% 13.40/2.14 X
% 13.40/2.14
% 13.40/2.14 Lemma 54: meet(zero, X) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 meet(zero, X)
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 meet(X, zero)
% 13.40/2.14 = { by lemma 47 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 55: composition(join(X, join(sk1, one)), Y) = join(Y, composition(X, Y)).
% 13.40/2.14 Proof:
% 13.40/2.14 composition(join(X, join(sk1, one)), Y)
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 composition(join(join(sk1, one), X), Y)
% 13.40/2.14 = { by axiom 13 (composition_distributivity_7) }
% 13.40/2.14 join(composition(join(sk1, one), Y), composition(X, Y))
% 13.40/2.14 = { by lemma 22 }
% 13.40/2.14 join(Y, composition(X, Y))
% 13.40/2.14
% 13.40/2.14 Lemma 56: composition(join(join(sk1, one), Y), X) = join(X, composition(Y, X)).
% 13.40/2.14 Proof:
% 13.40/2.14 composition(join(join(sk1, one), Y), X)
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 composition(join(Y, join(sk1, one)), X)
% 13.40/2.14 = { by lemma 55 }
% 13.40/2.14 join(X, composition(Y, X))
% 13.40/2.14
% 13.40/2.14 Lemma 57: composition(top, zero) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 composition(top, zero)
% 13.40/2.14 = { by lemma 39 R->L }
% 13.40/2.14 composition(converse(top), zero)
% 13.40/2.14 = { by lemma 44 R->L }
% 13.40/2.14 join(zero, composition(converse(top), zero))
% 13.40/2.14 = { by lemma 18 R->L }
% 13.40/2.14 join(complement(top), composition(converse(top), zero))
% 13.40/2.14 = { by lemma 18 R->L }
% 13.40/2.14 join(complement(top), composition(converse(top), complement(top)))
% 13.40/2.14 = { by lemma 52 R->L }
% 13.40/2.14 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 13.40/2.14 = { by lemma 39 R->L }
% 13.40/2.14 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 13.40/2.14 = { by lemma 56 R->L }
% 13.40/2.14 join(complement(top), composition(converse(top), complement(composition(join(join(sk1, one), converse(top)), top))))
% 13.40/2.14 = { by lemma 38 }
% 13.40/2.14 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 13.40/2.14 = { by lemma 23 }
% 13.40/2.14 complement(top)
% 13.40/2.14 = { by lemma 18 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 58: composition(X, zero) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 composition(X, zero)
% 13.40/2.14 = { by lemma 44 R->L }
% 13.40/2.14 join(zero, composition(X, zero))
% 13.40/2.14 = { by lemma 57 R->L }
% 13.40/2.14 join(composition(top, zero), composition(X, zero))
% 13.40/2.14 = { by axiom 13 (composition_distributivity_7) R->L }
% 13.40/2.14 composition(join(top, X), zero)
% 13.40/2.14 = { by lemma 52 }
% 13.40/2.14 composition(top, zero)
% 13.40/2.14 = { by lemma 57 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 59: composition(zero, X) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 composition(zero, X)
% 13.40/2.14 = { by lemma 51 R->L }
% 13.40/2.14 composition(converse(zero), X)
% 13.40/2.14 = { by lemma 19 R->L }
% 13.40/2.14 converse(composition(converse(X), zero))
% 13.40/2.14 = { by lemma 58 }
% 13.40/2.14 converse(zero)
% 13.40/2.14 = { by lemma 51 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 60: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 13.40/2.14 Proof:
% 13.40/2.14 meet(X, join(complement(Y), complement(Z)))
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 meet(X, join(complement(Z), complement(Y)))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 meet(join(complement(Z), complement(Y)), X)
% 13.40/2.14 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.14 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 13.40/2.14 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.14 complement(join(meet(Z, Y), complement(X)))
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.14 complement(join(complement(X), meet(Z, Y)))
% 13.40/2.14 = { by lemma 45 R->L }
% 13.40/2.14 complement(join(complement(X), meet(Y, Z)))
% 13.40/2.14
% 13.40/2.14 Lemma 61: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 13.40/2.14 Proof:
% 13.40/2.14 join(complement(X), complement(Y))
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 join(complement(Y), complement(X))
% 13.40/2.14 = { by lemma 26 R->L }
% 13.40/2.14 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 13.40/2.14 = { by lemma 60 }
% 13.40/2.14 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 13.40/2.14 = { by lemma 42 }
% 13.40/2.14 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 13.40/2.14 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.14 complement(join(meet(Y, X), meet(Y, X)))
% 13.40/2.14 = { by lemma 50 }
% 13.40/2.14 complement(meet(Y, X))
% 13.40/2.14 = { by lemma 45 R->L }
% 13.40/2.14 complement(meet(X, Y))
% 13.40/2.14
% 13.40/2.14 Lemma 62: join(join(sk1, one), X) = join(X, one).
% 13.40/2.14 Proof:
% 13.40/2.14 join(join(sk1, one), X)
% 13.40/2.14 = { by axiom 3 (goals_17) }
% 13.40/2.14 join(one, X)
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 join(X, one)
% 13.40/2.14
% 13.40/2.14 Lemma 63: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 13.40/2.14 Proof:
% 13.40/2.14 complement(join(X, complement(Y)))
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 complement(join(complement(Y), X))
% 13.40/2.14 = { by lemma 49 R->L }
% 13.40/2.14 complement(join(complement(Y), meet(X, top)))
% 13.40/2.14 = { by lemma 45 R->L }
% 13.40/2.14 complement(join(complement(Y), meet(top, X)))
% 13.40/2.14 = { by lemma 60 R->L }
% 13.40/2.14 meet(Y, join(complement(top), complement(X)))
% 13.40/2.14 = { by lemma 18 }
% 13.40/2.14 meet(Y, join(zero, complement(X)))
% 13.40/2.14 = { by lemma 42 }
% 13.40/2.14 meet(Y, complement(X))
% 13.40/2.14
% 13.40/2.14 Lemma 64: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 13.40/2.14 Proof:
% 13.40/2.14 complement(join(complement(X), Y))
% 13.40/2.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.14 complement(join(Y, complement(X)))
% 13.40/2.14 = { by lemma 63 }
% 13.40/2.14 meet(X, complement(Y))
% 13.40/2.14
% 13.40/2.14 Lemma 65: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 13.40/2.14 Proof:
% 13.40/2.14 complement(meet(X, complement(Y)))
% 13.40/2.14 = { by lemma 44 R->L }
% 13.40/2.14 complement(join(zero, meet(X, complement(Y))))
% 13.40/2.14 = { by lemma 63 R->L }
% 13.40/2.14 complement(join(zero, complement(join(Y, complement(X)))))
% 13.40/2.14 = { by lemma 46 }
% 13.40/2.14 meet(join(Y, complement(X)), top)
% 13.40/2.14 = { by lemma 49 }
% 13.40/2.14 join(Y, complement(X))
% 13.40/2.14
% 13.40/2.14 Lemma 66: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 13.40/2.14 Proof:
% 13.40/2.14 meet(complement(X), complement(Y))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 meet(complement(Y), complement(X))
% 13.40/2.14 = { by lemma 42 R->L }
% 13.40/2.14 meet(join(zero, complement(Y)), complement(X))
% 13.40/2.14 = { by lemma 63 R->L }
% 13.40/2.14 complement(join(X, complement(join(zero, complement(Y)))))
% 13.40/2.14 = { by lemma 46 }
% 13.40/2.14 complement(join(X, meet(Y, top)))
% 13.40/2.14 = { by lemma 49 }
% 13.40/2.14 complement(join(X, Y))
% 13.40/2.14
% 13.40/2.14 Lemma 67: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 13.40/2.14 Proof:
% 13.40/2.14 converse(composition(X, converse(Y)))
% 13.40/2.14 = { by axiom 10 (converse_multiplicativity_10) }
% 13.40/2.14 composition(converse(converse(Y)), converse(X))
% 13.40/2.14 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.14 composition(Y, converse(X))
% 13.40/2.14
% 13.40/2.14 Lemma 68: meet(X, meet(Y, complement(X))) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 meet(X, meet(Y, complement(X)))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 meet(X, meet(complement(X), Y))
% 13.40/2.14 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.14 complement(join(complement(X), complement(meet(complement(X), Y))))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 complement(join(complement(X), complement(meet(Y, complement(X)))))
% 13.40/2.14 = { by lemma 61 R->L }
% 13.40/2.14 complement(join(complement(X), join(complement(Y), complement(complement(X)))))
% 13.40/2.14 = { by lemma 28 }
% 13.40/2.14 complement(join(complement(Y), top))
% 13.40/2.14 = { by lemma 31 }
% 13.40/2.14 complement(top)
% 13.40/2.14 = { by lemma 18 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 69: composition(converse(X), complement(composition(X, top))) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 composition(converse(X), complement(composition(X, top)))
% 13.40/2.14 = { by lemma 44 R->L }
% 13.40/2.14 join(zero, composition(converse(X), complement(composition(X, top))))
% 13.40/2.14 = { by lemma 18 R->L }
% 13.40/2.14 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 13.40/2.14 = { by lemma 23 }
% 13.40/2.14 complement(top)
% 13.40/2.14 = { by lemma 18 }
% 13.40/2.14 zero
% 13.40/2.14
% 13.40/2.14 Lemma 70: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 13.40/2.14 Proof:
% 13.40/2.14 join(meet(X, Y), meet(Y, complement(X)))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 join(meet(Y, X), meet(Y, complement(X)))
% 13.40/2.14 = { by lemma 48 }
% 13.40/2.14 Y
% 13.40/2.14
% 13.40/2.14 Lemma 71: meet(join(sk1, one), composition(converse(complement(X)), X)) = zero.
% 13.40/2.14 Proof:
% 13.40/2.14 meet(join(sk1, one), composition(converse(complement(X)), X))
% 13.40/2.14 = { by lemma 45 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), join(sk1, one))
% 13.40/2.14 = { by lemma 53 R->L }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(complement(join(sk1, one))))
% 13.40/2.14 = { by lemma 23 R->L }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), join(sk1, one)))))))
% 13.40/2.14 = { by lemma 20 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 13.40/2.14 = { by lemma 46 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 13.40/2.14 = { by lemma 42 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(complement(X)), meet(X, top)))))
% 13.40/2.14 = { by lemma 49 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(complement(X)), X))))
% 13.40/2.14 = { by lemma 64 }
% 13.40/2.14 meet(composition(converse(complement(X)), X), meet(join(sk1, one), complement(composition(converse(complement(X)), X))))
% 13.40/2.14 = { by lemma 68 }
% 13.40/2.15 zero
% 13.40/2.15
% 13.40/2.15 Goal 1 (goals_19): composition(sk1, sk2) = meet(sk1, sk2).
% 13.40/2.15 Proof:
% 13.40/2.15 composition(sk1, sk2)
% 13.40/2.15 = { by lemma 53 R->L }
% 13.40/2.15 complement(complement(composition(sk1, sk2)))
% 13.40/2.15 = { by lemma 25 R->L }
% 13.40/2.15 complement(complement(join(meet(composition(sk1, sk2), meet(join(zero, complement(sk1)), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), meet(join(zero, complement(sk1)), complement(composition(sk1, sk2))))))))
% 13.40/2.15 = { by lemma 68 }
% 13.40/2.15 complement(complement(join(zero, complement(join(complement(composition(sk1, sk2)), meet(join(zero, complement(sk1)), complement(composition(sk1, sk2))))))))
% 13.40/2.15 = { by lemma 42 }
% 13.40/2.15 complement(complement(complement(join(complement(composition(sk1, sk2)), meet(join(zero, complement(sk1)), complement(composition(sk1, sk2)))))))
% 13.40/2.15 = { by lemma 64 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), complement(meet(join(zero, complement(sk1)), complement(composition(sk1, sk2)))))))
% 13.40/2.15 = { by lemma 65 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(composition(sk1, sk2), complement(join(zero, complement(sk1)))))))
% 13.40/2.15 = { by lemma 46 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(composition(sk1, sk2), meet(sk1, top)))))
% 13.40/2.15 = { by lemma 49 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(composition(sk1, sk2), sk1))))
% 13.40/2.15 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(sk1, composition(sk1, sk2)))))
% 13.40/2.15 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(sk1, composition(sk1, converse(converse(sk2)))))))
% 13.40/2.15 = { by lemma 67 R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), join(sk1, converse(composition(converse(sk2), converse(sk1)))))))
% 13.40/2.15 = { by lemma 36 R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), converse(join(converse(sk1), composition(converse(sk2), converse(sk1)))))))
% 13.40/2.15 = { by lemma 55 R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), converse(composition(join(converse(sk2), join(sk1, one)), converse(sk1))))))
% 13.40/2.15 = { by lemma 67 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, converse(join(converse(sk2), join(sk1, one)))))))
% 13.40/2.15 = { by lemma 36 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(sk2, converse(join(sk1, one)))))))
% 13.40/2.15 = { by lemma 20 R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(sk2, composition(converse(join(sk1, one)), join(sk1, one)))))))
% 13.40/2.15 = { by lemma 21 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(sk2, join(sk1, one))))))
% 13.40/2.15 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(join(sk1, one), sk2)))))
% 13.40/2.15 = { by lemma 62 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(sk2, one)))))
% 13.40/2.15 = { by axiom 4 (goals_18) }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, one))))
% 13.40/2.15 = { by axiom 3 (goals_17) R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), composition(sk1, join(sk1, one)))))
% 13.40/2.15 = { by lemma 20 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), sk1)))
% 13.40/2.15 = { by lemma 45 }
% 13.40/2.15 complement(complement(meet(sk1, composition(sk1, sk2))))
% 13.40/2.15 = { by lemma 25 R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(meet(composition(sk1, sk2), complement(sk2)), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 63 R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(sk2, complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 22 R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(composition(join(sk1, one), sk2), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 62 R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(composition(join(join(sk1, one), sk1), sk2), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(composition(join(sk1, join(sk1, one)), sk2), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by axiom 13 (composition_distributivity_7) }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(join(composition(sk1, sk2), composition(join(sk1, one), sk2)), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 22 }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(join(composition(sk1, sk2), sk2), complement(composition(sk1, sk2)))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(composition(sk1, sk2), join(sk2, complement(composition(sk1, sk2))))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(join(composition(sk1, sk2), join(complement(composition(sk1, sk2)), sk2))), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 32 }
% 13.40/2.15 complement(complement(meet(sk1, join(complement(top), complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 18 }
% 13.40/2.15 complement(complement(meet(sk1, join(zero, complement(join(complement(composition(sk1, sk2)), complement(sk2)))))))
% 13.40/2.15 = { by lemma 42 }
% 13.40/2.15 complement(complement(meet(sk1, complement(join(complement(composition(sk1, sk2)), complement(sk2))))))
% 13.40/2.15 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.15 complement(complement(meet(sk1, meet(composition(sk1, sk2), sk2))))
% 13.40/2.15 = { by lemma 45 }
% 13.40/2.15 complement(complement(meet(sk1, meet(sk2, composition(sk1, sk2)))))
% 13.40/2.15 = { by lemma 49 R->L }
% 13.40/2.15 complement(complement(meet(meet(sk1, top), meet(sk2, composition(sk1, sk2)))))
% 13.40/2.15 = { by lemma 46 R->L }
% 13.40/2.15 complement(complement(meet(complement(join(zero, complement(sk1))), meet(sk2, composition(sk1, sk2)))))
% 13.40/2.15 = { by lemma 45 }
% 13.40/2.15 complement(complement(meet(complement(join(zero, complement(sk1))), meet(composition(sk1, sk2), sk2))))
% 13.40/2.15 = { by lemma 45 }
% 13.40/2.15 complement(complement(meet(meet(composition(sk1, sk2), sk2), complement(join(zero, complement(sk1))))))
% 13.40/2.15 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 13.40/2.15 complement(complement(meet(complement(join(complement(composition(sk1, sk2)), complement(sk2))), complement(join(zero, complement(sk1))))))
% 13.40/2.15 = { by lemma 66 }
% 13.40/2.15 complement(complement(complement(join(join(complement(composition(sk1, sk2)), complement(sk2)), join(zero, complement(sk1))))))
% 13.40/2.15 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 13.40/2.15 complement(complement(complement(join(complement(composition(sk1, sk2)), join(complement(sk2), join(zero, complement(sk1)))))))
% 13.40/2.15 = { by lemma 64 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), complement(join(complement(sk2), join(zero, complement(sk1)))))))
% 13.40/2.15 = { by lemma 64 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), meet(sk2, complement(join(zero, complement(sk1)))))))
% 13.40/2.15 = { by lemma 46 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), meet(sk2, meet(sk1, top)))))
% 13.40/2.15 = { by lemma 49 }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), meet(sk2, sk1))))
% 13.40/2.15 = { by lemma 45 R->L }
% 13.40/2.15 complement(complement(meet(composition(sk1, sk2), meet(sk1, sk2))))
% 13.40/2.15 = { by lemma 45 R->L }
% 13.40/2.15 complement(complement(meet(meet(sk1, sk2), composition(sk1, sk2))))
% 13.40/2.15 = { by lemma 43 R->L }
% 13.40/2.15 complement(complement(meet(meet(sk1, sk2), join(composition(sk1, sk2), zero))))
% 13.40/2.15 = { by lemma 18 R->L }
% 13.40/2.15 complement(complement(meet(meet(sk1, sk2), join(composition(sk1, sk2), complement(top)))))
% 13.40/2.15 = { by lemma 45 }
% 13.40/2.15 complement(complement(meet(join(composition(sk1, sk2), complement(top)), meet(sk1, sk2))))
% 13.40/2.15 = { by lemma 61 R->L }
% 13.40/2.15 complement(join(complement(join(composition(sk1, sk2), complement(top))), complement(meet(sk1, sk2))))
% 13.40/2.15 = { by lemma 63 }
% 13.40/2.15 complement(join(meet(top, complement(composition(sk1, sk2))), complement(meet(sk1, sk2))))
% 13.40/2.15 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(top, complement(composition(sk1, sk2)))))
% 13.40/2.15 = { by lemma 45 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), top)))
% 13.40/2.15 = { by lemma 34 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(zero))))
% 13.40/2.15 = { by lemma 47 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, zero)))))
% 13.40/2.15 = { by lemma 54 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(zero, complement(composition(sk1, top))))))))
% 13.40/2.15 = { by lemma 58 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(composition(sk1, zero), complement(composition(sk1, top))))))))
% 13.40/2.15 = { by lemma 47 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(composition(sk1, meet(join(sk1, one), zero)), complement(composition(sk1, top))))))))
% 13.40/2.15 = { by lemma 69 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(composition(sk1, meet(join(sk1, one), composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top))))))))
% 13.40/2.15 = { by axiom 16 (modular_law_1_15) R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, join(meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))), meet(composition(sk1, meet(join(sk1, one), composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top)))))))))
% 13.40/2.15 = { by lemma 69 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, join(meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))), meet(composition(sk1, meet(join(sk1, one), zero)), complement(composition(sk1, top)))))))))
% 13.40/2.15 = { by lemma 47 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, join(meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))), meet(composition(sk1, zero), complement(composition(sk1, top)))))))))
% 13.40/2.15 = { by lemma 58 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, join(meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top)))))))))
% 13.40/2.15 = { by lemma 54 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, join(meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))), zero))))))
% 13.40/2.15 = { by lemma 43 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(composition(sk1, join(sk1, one)), complement(composition(sk1, top))))))))
% 13.40/2.15 = { by lemma 20 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, meet(sk1, complement(composition(sk1, top))))))))
% 13.40/2.15 = { by lemma 63 R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), complement(meet(sk2, complement(join(composition(sk1, top), complement(sk1))))))))
% 13.40/2.15 = { by lemma 65 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(join(composition(sk1, top), complement(sk1)), complement(sk2)))))
% 13.40/2.15 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(composition(sk1, top), join(complement(sk1), complement(sk2))))))
% 13.40/2.15 = { by lemma 61 }
% 13.40/2.15 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(composition(sk1, top), complement(meet(sk1, sk2))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, top)))))
% 13.40/2.16 = { by lemma 39 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(top))))))
% 13.40/2.16 = { by lemma 32 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(converse(meet(sk1, sk2)), join(complement(converse(meet(sk1, sk2))), converse(meet(sk2, complement(sk1)))))))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(converse(meet(sk1, sk2)), join(converse(meet(sk2, complement(sk1))), complement(converse(meet(sk1, sk2)))))))))))
% 13.40/2.16 = { by axiom 9 (maddux2_join_associativity_2) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(join(converse(meet(sk1, sk2)), converse(meet(sk2, complement(sk1)))), complement(converse(meet(sk1, sk2))))))))))
% 13.40/2.16 = { by axiom 8 (converse_additivity_9) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(converse(join(meet(sk1, sk2), meet(sk2, complement(sk1)))), complement(converse(meet(sk1, sk2))))))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(complement(converse(meet(sk1, sk2))), converse(join(meet(sk1, sk2), meet(sk2, complement(sk1)))))))))))
% 13.40/2.16 = { by lemma 70 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(complement(converse(meet(sk1, sk2))), converse(sk2))))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, converse(join(converse(sk2), complement(converse(meet(sk1, sk2))))))))))
% 13.40/2.16 = { by lemma 36 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, join(sk2, converse(complement(converse(meet(sk1, sk2))))))))))
% 13.40/2.16 = { by axiom 1 (converse_idempotence_8) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(converse(composition(sk1, join(sk2, converse(complement(converse(meet(sk1, sk2))))))))))))
% 13.40/2.16 = { by axiom 10 (converse_multiplicativity_10) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(composition(converse(join(sk2, converse(complement(converse(meet(sk1, sk2)))))), converse(sk1)))))))
% 13.40/2.16 = { by lemma 35 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(composition(join(complement(converse(meet(sk1, sk2))), converse(sk2)), converse(sk1)))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(composition(join(converse(sk2), complement(converse(meet(sk1, sk2)))), converse(sk1)))))))
% 13.40/2.16 = { by axiom 13 (composition_distributivity_7) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(join(composition(converse(sk2), converse(sk1)), composition(complement(converse(meet(sk1, sk2))), converse(sk1))))))))
% 13.40/2.16 = { by axiom 10 (converse_multiplicativity_10) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(join(converse(composition(sk1, sk2)), composition(complement(converse(meet(sk1, sk2))), converse(sk1))))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(join(composition(complement(converse(meet(sk1, sk2))), converse(sk1)), converse(composition(sk1, sk2))))))))
% 13.40/2.16 = { by lemma 67 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(join(converse(composition(sk1, converse(complement(converse(meet(sk1, sk2)))))), converse(composition(sk1, sk2))))))))
% 13.40/2.16 = { by axiom 8 (converse_additivity_9) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(converse(join(composition(sk1, converse(complement(converse(meet(sk1, sk2))))), composition(sk1, sk2))))))))
% 13.40/2.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), converse(converse(join(composition(sk1, sk2), composition(sk1, converse(complement(converse(meet(sk1, sk2))))))))))))
% 13.40/2.16 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(complement(converse(meet(sk1, sk2))))))))))
% 13.40/2.16 = { by lemma 42 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(zero, complement(converse(meet(sk1, sk2)))))))))))
% 13.40/2.16 = { by lemma 25 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))), complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))))
% 13.40/2.16 = { by lemma 64 R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))), complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))))
% 13.40/2.16 = { by lemma 37 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))))
% 13.40/2.16 = { by lemma 39 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(complement(top), complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))))
% 13.40/2.16 = { by lemma 18 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(zero, complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))))
% 13.40/2.16 = { by lemma 42 }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(complement(join(complement(join(zero, complement(converse(meet(sk1, sk2))))), complement(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))))
% 13.40/2.16 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.16 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))
% 13.40/2.17 = { by lemma 43 R->L }
% 13.40/2.17 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), zero))))))))
% 13.40/2.17 = { by lemma 54 R->L }
% 13.40/2.17 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))
% 13.40/2.17 = { by lemma 59 R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(composition(zero, complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))
% 13.40/2.18 = { by lemma 71 R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(composition(meet(join(sk1, one), composition(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))), converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))), complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))
% 13.40/2.18 = { by axiom 17 (modular_law_2_16) R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), join(meet(composition(join(sk1, one), complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(composition(meet(join(sk1, one), composition(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))), converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))), complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))))
% 13.40/2.18 = { by lemma 71 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), join(meet(composition(join(sk1, one), complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(composition(zero, complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))))
% 13.40/2.18 = { by lemma 22 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), join(meet(complement(join(zero, complement(converse(meet(sk1, sk2))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(composition(zero, complement(join(zero, complement(converse(meet(sk1, sk2)))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))))
% 13.40/2.18 = { by lemma 59 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), join(meet(complement(join(zero, complement(converse(meet(sk1, sk2))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))))))
% 13.40/2.18 = { by lemma 54 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), join(meet(complement(join(zero, complement(converse(meet(sk1, sk2))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), zero)))))))))
% 13.40/2.18 = { by lemma 43 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(complement(join(zero, complement(converse(meet(sk1, sk2))))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))))
% 13.40/2.18 = { by lemma 45 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(meet(sk1, sk2)))), converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2))))))))), meet(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))), complement(join(zero, complement(converse(meet(sk1, sk2))))))))))))))
% 13.40/2.18 = { by lemma 70 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(converse(complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))))
% 13.40/2.18 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(converse(complement(join(zero, complement(converse(meet(sk1, sk2)))))))))))))
% 13.40/2.18 = { by lemma 46 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(converse(meet(converse(meet(sk1, sk2)), top)))))))))
% 13.40/2.18 = { by lemma 49 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(converse(converse(meet(sk1, sk2))))))))))
% 13.40/2.18 = { by axiom 1 (converse_idempotence_8) }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(meet(sk1, sk2))))))))
% 13.40/2.18 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), join(composition(sk1, complement(meet(sk1, sk2))), composition(sk1, sk2))))))
% 13.40/2.18 = { by axiom 9 (maddux2_join_associativity_2) }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(join(complement(meet(sk1, sk2)), composition(sk1, complement(meet(sk1, sk2)))), composition(sk1, sk2)))))
% 13.40/2.18 = { by lemma 56 R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(composition(join(join(sk1, one), sk1), complement(meet(sk1, sk2))), composition(sk1, sk2)))))
% 13.40/2.18 = { by lemma 62 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(composition(join(sk1, one), complement(meet(sk1, sk2))), composition(sk1, sk2)))))
% 13.40/2.18 = { by lemma 22 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(complement(composition(sk1, sk2)), join(complement(meet(sk1, sk2)), composition(sk1, sk2)))))
% 13.40/2.18 = { by lemma 45 }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), meet(join(complement(meet(sk1, sk2)), composition(sk1, sk2)), complement(composition(sk1, sk2)))))
% 13.40/2.18 = { by lemma 63 R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), complement(join(composition(sk1, sk2), complement(join(complement(meet(sk1, sk2)), composition(sk1, sk2)))))))
% 13.40/2.18 = { by lemma 66 R->L }
% 13.40/2.18 complement(join(complement(meet(sk1, sk2)), complement(join(composition(sk1, sk2), meet(complement(complement(meet(sk1, sk2))), complement(composition(sk1, sk2)))))))
% 13.40/2.18 = { by lemma 65 R->L }
% 13.40/2.18 complement(complement(meet(join(composition(sk1, sk2), meet(complement(complement(meet(sk1, sk2))), complement(composition(sk1, sk2)))), complement(complement(meet(sk1, sk2))))))
% 13.40/2.18 = { by lemma 45 R->L }
% 13.40/2.18 complement(complement(meet(complement(complement(meet(sk1, sk2))), join(composition(sk1, sk2), meet(complement(complement(meet(sk1, sk2))), complement(composition(sk1, sk2)))))))
% 13.40/2.18 = { by lemma 42 R->L }
% 13.40/2.18 complement(complement(meet(complement(complement(meet(sk1, sk2))), join(composition(sk1, sk2), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2))))))))
% 13.40/2.18 = { by lemma 49 R->L }
% 13.40/2.18 complement(complement(meet(complement(complement(meet(sk1, sk2))), join(meet(composition(sk1, sk2), top), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2))))))))
% 13.40/2.18 = { by lemma 46 R->L }
% 13.40/2.18 complement(complement(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2))))))))
% 13.40/2.18 = { by lemma 43 R->L }
% 13.40/2.18 complement(complement(join(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))), zero)))
% 13.40/2.18 = { by lemma 18 R->L }
% 13.40/2.18 complement(complement(join(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))), complement(top))))
% 13.40/2.18 = { by axiom 7 (def_top_12) }
% 13.40/2.18 complement(complement(join(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))), complement(join(join(complement(complement(complement(meet(sk1, sk2)))), complement(join(zero, complement(composition(sk1, sk2))))), complement(join(complement(complement(complement(meet(sk1, sk2)))), complement(join(zero, complement(composition(sk1, sk2)))))))))))
% 13.40/2.18 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 13.40/2.18 complement(complement(join(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))), complement(join(join(complement(complement(complement(meet(sk1, sk2)))), complement(join(zero, complement(composition(sk1, sk2))))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))))))
% 13.40/2.18 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 13.40/2.18 complement(complement(join(meet(complement(complement(meet(sk1, sk2))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2)))))), complement(join(complement(complement(complement(meet(sk1, sk2)))), join(complement(join(zero, complement(composition(sk1, sk2)))), meet(complement(complement(meet(sk1, sk2))), join(zero, complement(composition(sk1, sk2))))))))))
% 13.40/2.18 = { by lemma 25 }
% 13.40/2.19 complement(complement(complement(complement(meet(sk1, sk2)))))
% 13.40/2.19 = { by lemma 53 }
% 13.40/2.19 complement(complement(meet(sk1, sk2)))
% 13.40/2.19 = { by lemma 53 }
% 13.40/2.19 meet(sk1, sk2)
% 13.40/2.19 % SZS output end Proof
% 13.40/2.19
% 13.40/2.19 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------