TSTP Solution File: REL038-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL038-1 : 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 : n024.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 13:44:24 EDT 2023
% Result : Unsatisfiable 83.70s 11.04s
% Output : Proof 84.69s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : REL038-1 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n024.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 19:09:54 EDT 2023
% 0.13/0.34 % CPUTime :
% 83.70/11.04 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 83.70/11.04
% 83.70/11.04 % SZS status Unsatisfiable
% 83.70/11.04
% 83.70/11.16 % SZS output start Proof
% 83.70/11.16 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 83.70/11.16 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 83.70/11.16 Axiom 3 (converse_idempotence_8): converse(converse(X)) = X.
% 83.70/11.16 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 83.70/11.16 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 83.70/11.16 Axiom 6 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 83.70/11.16 Axiom 7 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 83.70/11.16 Axiom 8 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 83.70/11.16 Axiom 9 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 83.70/11.16 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 83.70/11.16 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 83.70/11.16 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 83.70/11.16 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 83.70/11.16
% 83.70/11.16 Lemma 14: complement(top) = zero.
% 83.70/11.16 Proof:
% 83.70/11.16 complement(top)
% 83.70/11.16 = { by axiom 4 (def_top_12) }
% 83.70/11.16 complement(join(complement(X), complement(complement(X))))
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 83.70/11.16 meet(X, complement(X))
% 83.70/11.16 = { by axiom 5 (def_zero_13) R->L }
% 83.70/11.16 zero
% 83.70/11.16
% 83.70/11.16 Lemma 15: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 join(meet(X, Y), complement(join(complement(X), Y)))
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 83.70/11.16 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 83.70/11.16 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 16: join(meet(X, Y), meet(X, complement(Y))) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 join(meet(X, Y), meet(X, complement(Y)))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(meet(X, complement(Y)), meet(X, Y))
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 83.70/11.16 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 83.70/11.16 = { by lemma 15 }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 17: meet(Y, X) = meet(X, Y).
% 83.70/11.16 Proof:
% 83.70/11.16 meet(Y, X)
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 83.70/11.16 complement(join(complement(Y), complement(X)))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 complement(join(complement(X), complement(Y)))
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 83.70/11.16 meet(X, Y)
% 83.70/11.16
% 83.70/11.16 Lemma 18: complement(complement(X)) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 complement(complement(X))
% 83.70/11.16 = { by lemma 16 R->L }
% 83.70/11.16 join(meet(complement(complement(X)), X), meet(complement(complement(X)), complement(X)))
% 83.70/11.16 = { by lemma 17 R->L }
% 83.70/11.16 join(meet(complement(complement(X)), X), meet(complement(X), complement(complement(X))))
% 83.70/11.16 = { by axiom 5 (def_zero_13) R->L }
% 83.70/11.16 join(meet(complement(complement(X)), X), zero)
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 83.70/11.16 join(zero, meet(complement(complement(X)), X))
% 83.70/11.16 = { by lemma 17 R->L }
% 83.70/11.16 join(zero, meet(X, complement(complement(X))))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(meet(X, complement(complement(X))), zero)
% 83.70/11.16 = { by lemma 14 R->L }
% 83.70/11.16 join(meet(X, complement(complement(X))), complement(top))
% 83.70/11.16 = { by axiom 4 (def_top_12) }
% 83.70/11.16 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 83.70/11.16 = { by lemma 15 }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 19: composition(converse(one), X) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 composition(converse(one), X)
% 83.70/11.16 = { by axiom 3 (converse_idempotence_8) R->L }
% 83.70/11.16 composition(converse(one), converse(converse(X)))
% 83.70/11.16 = { by axiom 6 (converse_multiplicativity_10) R->L }
% 83.70/11.16 converse(composition(converse(X), one))
% 83.70/11.16 = { by axiom 1 (composition_identity_6) }
% 83.70/11.16 converse(converse(X))
% 83.70/11.16 = { by axiom 3 (converse_idempotence_8) }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 20: join(complement(X), composition(Y, complement(composition(converse(Y), X)))) = complement(X).
% 83.70/11.16 Proof:
% 83.70/11.16 join(complement(X), composition(Y, complement(composition(converse(Y), X))))
% 83.70/11.16 = { by axiom 3 (converse_idempotence_8) R->L }
% 83.70/11.16 join(complement(X), composition(converse(converse(Y)), complement(composition(converse(Y), X))))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(composition(converse(converse(Y)), complement(composition(converse(Y), X))), complement(X))
% 83.70/11.16 = { by axiom 12 (converse_cancellativity_11) }
% 83.70/11.16 complement(X)
% 83.70/11.16
% 83.70/11.16 Lemma 21: join(complement(X), complement(X)) = complement(X).
% 83.70/11.16 Proof:
% 83.70/11.16 join(complement(X), complement(X))
% 83.70/11.16 = { by lemma 19 R->L }
% 83.70/11.16 join(complement(X), complement(composition(converse(one), X)))
% 83.70/11.16 = { by axiom 1 (composition_identity_6) R->L }
% 83.70/11.16 join(complement(X), complement(composition(composition(converse(one), one), X)))
% 83.70/11.16 = { by axiom 7 (composition_associativity_5) R->L }
% 83.70/11.16 join(complement(X), complement(composition(converse(one), composition(one, X))))
% 83.70/11.16 = { by lemma 19 }
% 83.70/11.16 join(complement(X), complement(composition(one, X)))
% 83.70/11.16 = { by axiom 3 (converse_idempotence_8) R->L }
% 83.70/11.16 join(complement(X), complement(composition(converse(converse(one)), X)))
% 83.70/11.16 = { by lemma 19 R->L }
% 83.70/11.16 join(complement(X), composition(converse(one), complement(composition(converse(converse(one)), X))))
% 83.70/11.16 = { by lemma 20 }
% 83.70/11.16 complement(X)
% 83.70/11.16
% 83.70/11.16 Lemma 22: join(X, X) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 join(X, X)
% 83.70/11.16 = { by lemma 18 R->L }
% 83.70/11.16 join(X, complement(complement(X)))
% 83.70/11.16 = { by lemma 18 R->L }
% 83.70/11.16 join(complement(complement(X)), complement(complement(X)))
% 83.70/11.16 = { by lemma 21 }
% 83.70/11.16 complement(complement(X))
% 83.70/11.16 = { by lemma 18 }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 23: join(X, join(Y, complement(X))) = join(Y, top).
% 83.70/11.16 Proof:
% 83.70/11.16 join(X, join(Y, complement(X)))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(X, join(complement(X), Y))
% 83.70/11.16 = { by axiom 9 (maddux2_join_associativity_2) }
% 83.70/11.16 join(join(X, complement(X)), Y)
% 83.70/11.16 = { by axiom 4 (def_top_12) R->L }
% 83.70/11.16 join(top, Y)
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 83.70/11.16 join(Y, top)
% 83.70/11.16
% 83.70/11.16 Lemma 24: join(X, join(Y, complement(join(X, Y)))) = top.
% 83.70/11.16 Proof:
% 83.70/11.16 join(X, join(Y, complement(join(X, Y))))
% 83.70/11.16 = { by axiom 9 (maddux2_join_associativity_2) }
% 83.70/11.16 join(join(X, Y), complement(join(X, Y)))
% 83.70/11.16 = { by axiom 4 (def_top_12) R->L }
% 83.70/11.16 top
% 83.70/11.16
% 83.70/11.16 Lemma 25: join(X, top) = top.
% 83.70/11.16 Proof:
% 83.70/11.16 join(X, top)
% 83.70/11.16 = { by axiom 4 (def_top_12) }
% 83.70/11.16 join(X, join(complement(X), complement(complement(X))))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(X, join(complement(complement(X)), complement(X)))
% 83.70/11.16 = { by lemma 23 }
% 83.70/11.16 join(complement(complement(X)), top)
% 83.70/11.16 = { by lemma 23 R->L }
% 83.70/11.16 join(complement(complement(X)), join(complement(complement(X)), complement(complement(complement(X)))))
% 83.70/11.16 = { by lemma 21 R->L }
% 83.70/11.16 join(complement(complement(X)), join(complement(complement(X)), complement(join(complement(complement(X)), complement(complement(X))))))
% 83.70/11.16 = { by lemma 24 }
% 83.70/11.16 top
% 83.70/11.16
% 83.70/11.16 Lemma 26: complement(join(zero, complement(X))) = meet(X, top).
% 83.70/11.16 Proof:
% 83.70/11.16 complement(join(zero, complement(X)))
% 83.70/11.16 = { by lemma 14 R->L }
% 83.70/11.16 complement(join(complement(top), complement(X)))
% 83.70/11.16 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 83.70/11.16 meet(top, X)
% 83.70/11.16 = { by lemma 17 R->L }
% 83.70/11.16 meet(X, top)
% 83.70/11.16
% 83.70/11.16 Lemma 27: join(zero, complement(X)) = complement(X).
% 83.70/11.16 Proof:
% 83.70/11.16 join(zero, complement(X))
% 83.70/11.16 = { by lemma 22 R->L }
% 83.70/11.16 join(zero, complement(join(X, X)))
% 83.70/11.16 = { by lemma 18 R->L }
% 83.70/11.16 join(zero, complement(join(complement(complement(X)), X)))
% 83.70/11.16 = { by axiom 5 (def_zero_13) }
% 83.70/11.16 join(meet(X, complement(X)), complement(join(complement(complement(X)), X)))
% 83.70/11.16 = { by lemma 17 }
% 83.70/11.16 join(meet(complement(X), X), complement(join(complement(complement(X)), X)))
% 83.70/11.16 = { by lemma 15 }
% 83.70/11.16 complement(X)
% 83.70/11.16
% 83.70/11.16 Lemma 28: meet(X, top) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 meet(X, top)
% 83.70/11.16 = { by lemma 26 R->L }
% 83.70/11.16 complement(join(zero, complement(X)))
% 83.70/11.16 = { by lemma 27 }
% 83.70/11.16 complement(complement(X))
% 83.70/11.16 = { by lemma 18 }
% 83.70/11.16 X
% 83.70/11.16
% 83.70/11.16 Lemma 29: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 83.70/11.16 Proof:
% 83.70/11.16 join(X, join(Y, Z))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(join(Y, Z), X)
% 83.70/11.16 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 83.70/11.16 join(Y, join(Z, X))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) }
% 83.70/11.16 join(Y, join(X, Z))
% 83.70/11.16
% 83.70/11.16 Lemma 30: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 83.70/11.16 Proof:
% 83.70/11.16 join(meet(X, Y), complement(join(Y, complement(X))))
% 83.70/11.16 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.16 join(meet(X, Y), complement(join(complement(X), Y)))
% 83.70/11.16 = { by lemma 15 }
% 83.70/11.16 X
% 83.70/11.17
% 83.70/11.17 Lemma 31: join(meet(X, Y), join(Z, complement(join(Y, complement(X))))) = join(Z, X).
% 83.70/11.17 Proof:
% 83.70/11.17 join(meet(X, Y), join(Z, complement(join(Y, complement(X)))))
% 83.70/11.17 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.17 join(meet(X, Y), join(complement(join(Y, complement(X))), Z))
% 83.70/11.17 = { by axiom 9 (maddux2_join_associativity_2) }
% 83.70/11.17 join(join(meet(X, Y), complement(join(Y, complement(X)))), Z)
% 83.70/11.17 = { by lemma 30 }
% 83.70/11.17 join(X, Z)
% 83.70/11.17 = { by axiom 2 (maddux1_join_commutativity_1) }
% 83.70/11.17 join(Z, X)
% 83.70/11.17
% 83.70/11.17 Lemma 32: join(X, meet(X, Y)) = X.
% 83.70/11.17 Proof:
% 83.70/11.17 join(X, meet(X, Y))
% 83.70/11.17 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 83.70/11.17 join(meet(X, Y), X)
% 83.70/11.17 = { by lemma 15 R->L }
% 83.70/11.17 join(meet(X, Y), join(meet(X, Y), complement(join(complement(X), Y))))
% 83.70/11.17 = { by lemma 21 R->L }
% 83.70/11.17 join(meet(X, Y), join(meet(X, Y), join(complement(join(complement(X), Y)), complement(join(complement(X), Y)))))
% 83.70/11.17 = { by axiom 9 (maddux2_join_associativity_2) }
% 83.70/11.17 join(meet(X, Y), join(join(meet(X, Y), complement(join(complement(X), Y))), complement(join(complement(X), Y))))
% 83.70/11.17 = { by lemma 15 }
% 83.70/11.17 join(meet(X, Y), join(X, complement(join(complement(X), Y))))
% 83.70/11.17 = { by axiom 2 (maddux1_join_commutativity_1) }
% 83.70/11.17 join(meet(X, Y), join(X, complement(join(Y, complement(X)))))
% 83.70/11.17 = { by lemma 31 }
% 83.70/11.17 join(X, X)
% 83.70/11.17 = { by lemma 22 }
% 84.69/11.18 X
% 84.69/11.18
% 84.69/11.18 Lemma 33: join(X, meet(Y, X)) = X.
% 84.69/11.18 Proof:
% 84.69/11.18 join(X, meet(Y, X))
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 join(X, meet(X, Y))
% 84.69/11.18 = { by lemma 32 }
% 84.69/11.18 X
% 84.69/11.18
% 84.69/11.18 Lemma 34: join(zero, meet(X, Y)) = meet(X, Y).
% 84.69/11.18 Proof:
% 84.69/11.18 join(zero, meet(X, Y))
% 84.69/11.18 = { by lemma 17 }
% 84.69/11.18 join(zero, meet(Y, X))
% 84.69/11.18 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 84.69/11.18 join(zero, complement(join(complement(Y), complement(X))))
% 84.69/11.18 = { by lemma 27 }
% 84.69/11.18 complement(join(complement(Y), complement(X)))
% 84.69/11.18 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 84.69/11.18 meet(Y, X)
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 meet(X, Y)
% 84.69/11.18
% 84.69/11.18 Lemma 35: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 84.69/11.18 Proof:
% 84.69/11.18 join(complement(X), complement(Y))
% 84.69/11.18 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.18 join(complement(Y), complement(X))
% 84.69/11.18 = { by lemma 28 R->L }
% 84.69/11.18 meet(join(complement(Y), complement(X)), top)
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 meet(top, join(complement(Y), complement(X)))
% 84.69/11.18 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.18 meet(top, join(complement(X), complement(Y)))
% 84.69/11.18 = { by lemma 17 }
% 84.69/11.18 meet(join(complement(X), complement(Y)), top)
% 84.69/11.18 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 84.69/11.18 complement(join(complement(join(complement(X), complement(Y))), complement(top)))
% 84.69/11.18 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 84.69/11.18 complement(join(meet(X, Y), complement(top)))
% 84.69/11.18 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.18 complement(join(complement(top), meet(X, Y)))
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 complement(join(complement(top), meet(Y, X)))
% 84.69/11.18 = { by lemma 14 }
% 84.69/11.18 complement(join(zero, meet(Y, X)))
% 84.69/11.18 = { by lemma 34 }
% 84.69/11.18 complement(meet(Y, X))
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 complement(meet(X, Y))
% 84.69/11.18
% 84.69/11.18 Lemma 36: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 84.69/11.18 Proof:
% 84.69/11.18 complement(join(X, complement(Y)))
% 84.69/11.18 = { by lemma 18 R->L }
% 84.69/11.18 complement(join(complement(complement(X)), complement(Y)))
% 84.69/11.18 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 84.69/11.18 meet(complement(X), Y)
% 84.69/11.18 = { by lemma 17 R->L }
% 84.69/11.18 meet(Y, complement(X))
% 84.69/11.18
% 84.69/11.18 Lemma 37: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 84.69/11.18 Proof:
% 84.69/11.18 complement(join(complement(X), Y))
% 84.69/11.18 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.18 complement(join(Y, complement(X)))
% 84.69/11.18 = { by lemma 36 }
% 84.69/11.18 meet(X, complement(Y))
% 84.69/11.18
% 84.69/11.18 Lemma 38: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 84.69/11.18 Proof:
% 84.69/11.18 complement(meet(X, complement(Y)))
% 84.69/11.18 = { by lemma 17 }
% 84.69/11.18 complement(meet(complement(Y), X))
% 84.69/11.18 = { by lemma 27 R->L }
% 84.69/11.18 complement(meet(join(zero, complement(Y)), X))
% 84.69/11.18 = { by lemma 35 R->L }
% 84.69/11.18 join(complement(join(zero, complement(Y))), complement(X))
% 84.69/11.18 = { by lemma 26 }
% 84.69/11.18 join(meet(Y, top), complement(X))
% 84.69/11.18 = { by lemma 28 }
% 84.69/11.18 join(Y, complement(X))
% 84.69/11.18
% 84.69/11.18 Lemma 39: meet(X, join(X, join(Y, Z))) = X.
% 84.69/11.18 Proof:
% 84.69/11.18 meet(X, join(X, join(Y, Z)))
% 84.69/11.18 = { by lemma 29 R->L }
% 84.69/11.18 meet(X, join(Y, join(X, Z)))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 meet(X, join(Y, join(Z, X)))
% 84.69/11.19 = { by lemma 15 R->L }
% 84.69/11.19 join(meet(meet(X, join(Y, join(Z, X))), complement(meet(X, join(Y, join(Z, X))))), complement(join(complement(meet(X, join(Y, join(Z, X)))), complement(meet(X, join(Y, join(Z, X)))))))
% 84.69/11.19 = { by lemma 22 }
% 84.69/11.19 join(meet(meet(X, join(Y, join(Z, X))), complement(meet(X, join(Y, join(Z, X))))), complement(complement(meet(X, join(Y, join(Z, X))))))
% 84.69/11.19 = { by axiom 5 (def_zero_13) R->L }
% 84.69/11.19 join(zero, complement(complement(meet(X, join(Y, join(Z, X))))))
% 84.69/11.19 = { by lemma 18 }
% 84.69/11.19 join(zero, meet(X, join(Y, join(Z, X))))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.19 join(meet(X, join(Y, join(Z, X))), zero)
% 84.69/11.19 = { by lemma 14 R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, X))), complement(top))
% 84.69/11.19 = { by lemma 18 R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(top))
% 84.69/11.19 = { by lemma 25 R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, top)))
% 84.69/11.19 = { by lemma 25 R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(Y, join(Z, top))))
% 84.69/11.19 = { by axiom 9 (maddux2_join_associativity_2) }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(join(Y, Z), top)))
% 84.69/11.19 = { by lemma 23 R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(join(Y, Z), complement(complement(X))))))
% 84.69/11.19 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 84.69/11.19 join(meet(X, join(Y, join(Z, complement(complement(X))))), complement(join(complement(X), join(Y, join(Z, complement(complement(X)))))))
% 84.69/11.19 = { by lemma 15 }
% 84.69/11.19 X
% 84.69/11.19
% 84.69/11.19 Lemma 40: meet(X, join(X, Y)) = X.
% 84.69/11.19 Proof:
% 84.69/11.19 meet(X, join(X, Y))
% 84.69/11.19 = { by lemma 15 R->L }
% 84.69/11.19 meet(X, join(X, join(meet(Y, Z), complement(join(complement(Y), Z)))))
% 84.69/11.19 = { by lemma 39 }
% 84.69/11.19 X
% 84.69/11.19
% 84.69/11.19 Lemma 41: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 84.69/11.19 Proof:
% 84.69/11.19 converse(join(X, converse(Y)))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 converse(join(converse(Y), X))
% 84.69/11.19 = { by axiom 8 (converse_additivity_9) }
% 84.69/11.19 join(converse(converse(Y)), converse(X))
% 84.69/11.19 = { by axiom 3 (converse_idempotence_8) }
% 84.69/11.19 join(Y, converse(X))
% 84.69/11.19
% 84.69/11.19 Lemma 42: meet(meet(X, Y), join(X, Z)) = meet(X, Y).
% 84.69/11.19 Proof:
% 84.69/11.19 meet(meet(X, Y), join(X, Z))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 meet(meet(X, Y), join(Z, X))
% 84.69/11.19 = { by lemma 31 R->L }
% 84.69/11.19 meet(meet(X, Y), join(meet(X, Y), join(Z, complement(join(Y, complement(X))))))
% 84.69/11.19 = { by lemma 39 }
% 84.69/11.19 meet(X, Y)
% 84.69/11.19
% 84.69/11.19 Lemma 43: join(complement(X), meet(meet(X, Y), join(Y, complement(X)))) = join(Y, complement(X)).
% 84.69/11.19 Proof:
% 84.69/11.19 join(complement(X), meet(meet(X, Y), join(Y, complement(X))))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(complement(X), meet(meet(X, Y), join(complement(X), Y)))
% 84.69/11.19 = { by lemma 17 }
% 84.69/11.19 join(complement(X), meet(join(complement(X), Y), meet(X, Y)))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(meet(join(complement(X), Y), meet(X, Y)), complement(X))
% 84.69/11.19 = { by lemma 15 R->L }
% 84.69/11.19 join(meet(join(complement(X), Y), meet(X, Y)), complement(join(meet(X, Y), complement(join(complement(X), Y)))))
% 84.69/11.19 = { by lemma 30 }
% 84.69/11.19 join(complement(X), Y)
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.19 join(Y, complement(X))
% 84.69/11.19
% 84.69/11.19 Lemma 44: join(meet(X, Y), complement(Y)) = join(X, complement(Y)).
% 84.69/11.19 Proof:
% 84.69/11.19 join(meet(X, Y), complement(Y))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(complement(Y), meet(X, Y))
% 84.69/11.19 = { by lemma 42 R->L }
% 84.69/11.19 join(complement(Y), meet(meet(X, Y), join(X, complement(Y))))
% 84.69/11.19 = { by lemma 17 R->L }
% 84.69/11.19 join(complement(Y), meet(meet(Y, X), join(X, complement(Y))))
% 84.69/11.19 = { by lemma 43 }
% 84.69/11.19 join(X, complement(Y))
% 84.69/11.19
% 84.69/11.19 Lemma 45: join(converse(composition(X, Y)), composition(Z, converse(X))) = composition(join(Z, converse(Y)), converse(X)).
% 84.69/11.19 Proof:
% 84.69/11.19 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 84.69/11.19 = { by axiom 6 (converse_multiplicativity_10) }
% 84.69/11.19 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 84.69/11.19 = { by axiom 11 (composition_distributivity_7) R->L }
% 84.69/11.19 composition(join(converse(Y), Z), converse(X))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.19 composition(join(Z, converse(Y)), converse(X))
% 84.69/11.19
% 84.69/11.19 Lemma 46: join(composition(X, Y), composition(X, Z)) = composition(X, join(Z, Y)).
% 84.69/11.19 Proof:
% 84.69/11.19 join(composition(X, Y), composition(X, Z))
% 84.69/11.19 = { by axiom 3 (converse_idempotence_8) R->L }
% 84.69/11.19 join(converse(converse(composition(X, Y))), composition(X, Z))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(composition(X, Z), converse(converse(composition(X, Y))))
% 84.69/11.19 = { by lemma 41 R->L }
% 84.69/11.19 converse(join(converse(composition(X, Y)), converse(composition(X, Z))))
% 84.69/11.19 = { by axiom 6 (converse_multiplicativity_10) }
% 84.69/11.19 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 84.69/11.19 = { by lemma 45 }
% 84.69/11.19 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 84.69/11.19 = { by axiom 6 (converse_multiplicativity_10) }
% 84.69/11.19 composition(converse(converse(X)), converse(join(converse(Z), converse(Y))))
% 84.69/11.19 = { by axiom 3 (converse_idempotence_8) }
% 84.69/11.19 composition(X, converse(join(converse(Z), converse(Y))))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 composition(X, converse(join(converse(Y), converse(Z))))
% 84.69/11.19 = { by lemma 41 }
% 84.69/11.19 composition(X, join(Z, converse(converse(Y))))
% 84.69/11.19 = { by axiom 3 (converse_idempotence_8) }
% 84.69/11.19 composition(X, join(Z, Y))
% 84.69/11.19
% 84.69/11.19 Lemma 47: join(meet(X, complement(Y)), complement(join(X, Y))) = complement(Y).
% 84.69/11.19 Proof:
% 84.69/11.19 join(meet(X, complement(Y)), complement(join(X, Y)))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(meet(X, complement(Y)), complement(join(Y, X)))
% 84.69/11.19 = { by lemma 17 }
% 84.69/11.19 join(meet(complement(Y), X), complement(join(Y, X)))
% 84.69/11.19 = { by lemma 18 R->L }
% 84.69/11.19 join(meet(complement(Y), X), complement(join(complement(complement(Y)), X)))
% 84.69/11.19 = { by lemma 15 }
% 84.69/11.19 complement(Y)
% 84.69/11.19
% 84.69/11.19 Lemma 48: join(complement(X), join(Y, composition(Z, complement(composition(converse(Z), X))))) = join(Y, complement(X)).
% 84.69/11.19 Proof:
% 84.69/11.19 join(complement(X), join(Y, composition(Z, complement(composition(converse(Z), X)))))
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.19 join(complement(X), join(composition(Z, complement(composition(converse(Z), X))), Y))
% 84.69/11.19 = { by axiom 9 (maddux2_join_associativity_2) }
% 84.69/11.19 join(join(complement(X), composition(Z, complement(composition(converse(Z), X)))), Y)
% 84.69/11.19 = { by lemma 20 }
% 84.69/11.19 join(complement(X), Y)
% 84.69/11.19 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.20 join(Y, complement(X))
% 84.69/11.20
% 84.69/11.20 Goal 1 (goals_14): join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)) = meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3).
% 84.69/11.20 Proof:
% 84.69/11.20 join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))
% 84.69/11.20 = { by lemma 40 R->L }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), meet(sk3, complement(composition(sk1, sk2)))))
% 84.69/11.20 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), meet(composition(sk1, sk2), sk3)), meet(sk3, complement(composition(sk1, sk2)))))
% 84.69/11.20 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), meet(sk3, complement(composition(sk1, sk2))))))
% 84.69/11.20 = { by lemma 17 }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(sk3, composition(sk1, sk2)), meet(sk3, complement(composition(sk1, sk2))))))
% 84.69/11.20 = { by lemma 16 }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), sk3))
% 84.69/11.20 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))
% 84.69/11.20 = { by lemma 33 }
% 84.69/11.20 meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), sk3)
% 84.69/11.20 = { by lemma 17 R->L }
% 84.69/11.20 meet(sk3, join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))
% 84.69/11.20 = { by lemma 18 R->L }
% 84.69/11.20 meet(sk3, complement(complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))
% 84.69/11.20 = { by lemma 37 R->L }
% 84.69/11.20 complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))
% 84.69/11.20 = { by lemma 38 R->L }
% 84.69/11.20 complement(complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3)))))
% 84.69/11.20 = { by lemma 47 R->L }
% 84.69/11.20 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3)))))))
% 84.69/11.20 = { by lemma 17 }
% 84.69/11.20 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))))
% 84.69/11.20 = { by lemma 40 R->L }
% 84.69/11.20 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(complement(complement(sk3)), meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(sk3))))))))
% 84.69/11.20 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.20 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(complement(complement(sk3)), meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))))))
% 84.69/11.20 = { by lemma 17 }
% 84.69/11.20 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(complement(complement(sk3)), meet(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))
% 84.69/11.20 = { by lemma 18 R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(complement(meet(complement(complement(sk3)), meet(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by lemma 35 R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(complement(complement(complement(sk3))), complement(meet(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by lemma 35 R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(complement(complement(complement(sk3))), join(complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))), complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(complement(complement(complement(sk3))), join(complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))))))))
% 84.69/11.21 = { by axiom 9 (maddux2_join_associativity_2) }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(join(complement(complement(complement(sk3))), complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by lemma 35 }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(complement(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(join(complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))), complement(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by lemma 35 }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), complement(complement(meet(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))))
% 84.69/11.21 = { by lemma 18 }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))
% 84.69/11.21 = { by lemma 17 }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))
% 84.69/11.21 = { by lemma 18 R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), meet(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(complement(complement(complement(sk3))), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))
% 84.69/11.21 = { by lemma 18 R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(complement(complement(sk3))), meet(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(complement(complement(complement(sk3))), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))
% 84.69/11.21 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.21 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(complement(complement(sk3))), meet(meet(complement(complement(sk3)), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(complement(sk3)))))))))
% 84.69/11.22 = { by lemma 43 }
% 84.69/11.22 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(complement(sk3)))))))
% 84.69/11.22 = { by lemma 18 }
% 84.69/11.22 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(sk3)))))
% 84.69/11.22 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.22 complement(join(meet(complement(sk3), complement(meet(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(sk3))))), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))
% 84.69/11.22 = { by lemma 38 }
% 84.69/11.22 complement(join(meet(complement(sk3), join(complement(sk3), complement(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))
% 84.69/11.22 = { by lemma 40 }
% 84.69/11.22 complement(join(complement(sk3), complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))
% 84.69/11.22 = { by lemma 37 }
% 84.69/11.22 meet(sk3, complement(complement(join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))
% 84.69/11.22 = { by lemma 18 }
% 84.69/11.22 meet(sk3, join(complement(sk3), join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))
% 84.69/11.22 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.22 meet(sk3, join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(sk3)))
% 84.69/11.22 = { by lemma 33 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(sk3)))
% 84.69/11.22 = { by lemma 28 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), top)), complement(sk3)))
% 84.69/11.22 = { by lemma 25 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3)))), top))), complement(sk3)))
% 84.69/11.22 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(top, complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3))))))), complement(sk3)))
% 84.69/11.22 = { by lemma 24 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(composition(sk1, sk2))))))), complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3))))))), complement(sk3)))
% 84.69/11.22 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(join(complement(sk3), complement(composition(sk1, sk2))), complement(join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(composition(sk1, sk2)))))), complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3)))))))), complement(sk3)))
% 84.69/11.22 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), join(complement(join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(composition(sk1, sk2))))), complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3))))))))), complement(sk3)))
% 84.69/11.22 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), join(complement(composition(sk1, meet(sk2, composition(converse(sk1), sk3)))), complement(join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(composition(sk1, sk2)))))))))), complement(sk3)))
% 84.69/11.22 = { by lemma 35 }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(composition(sk1, sk2)))))))))), complement(sk3)))
% 84.69/11.22 = { by lemma 29 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(complement(sk3), join(meet(composition(sk1, sk2), sk3), complement(composition(sk1, sk2)))))))))), complement(sk3)))
% 84.69/11.22 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(complement(sk3), join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), sk3))))))))), complement(sk3)))
% 84.69/11.22 = { by lemma 29 R->L }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(complement(composition(sk1, sk2)), join(complement(sk3), meet(composition(sk1, sk2), sk3))))))))), complement(sk3)))
% 84.69/11.22 = { by axiom 9 (maddux2_join_associativity_2) }
% 84.69/11.22 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(join(complement(composition(sk1, sk2)), complement(sk3)), meet(composition(sk1, sk2), sk3)))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(join(complement(composition(sk1, sk2)), complement(sk3)), complement(join(complement(composition(sk1, sk2)), complement(sk3)))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 4 (def_top_12) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(join(complement(sk3), complement(composition(sk1, sk2))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), top)))))), complement(sk3)))
% 84.69/11.23 = { by axiom 9 (maddux2_join_associativity_2) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(complement(sk3), join(complement(composition(sk1, sk2)), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), top))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 35 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), join(complement(sk3), complement(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), top))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 35 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), top))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 28 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, sk2), composition(sk1, meet(sk2, composition(converse(sk1), sk3))))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 17 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), composition(sk1, sk2))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 3 (converse_idempotence_8) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(converse(composition(sk1, sk2))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 6 (converse_multiplicativity_10) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(composition(converse(sk2), converse(sk1))))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 32 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(composition(converse(join(sk2, meet(sk2, composition(converse(sk1), sk3)))), converse(sk1))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 8 (converse_additivity_9) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(composition(join(converse(sk2), converse(meet(sk2, composition(converse(sk1), sk3)))), converse(sk1))))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 45 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(join(converse(composition(sk1, meet(sk2, composition(converse(sk1), sk3)))), composition(converse(sk2), converse(sk1)))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 6 (converse_multiplicativity_10) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(join(converse(composition(sk1, meet(sk2, composition(converse(sk1), sk3)))), converse(composition(sk1, sk2)))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 8 (converse_additivity_9) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), converse(converse(join(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), composition(sk1, sk2)))))))))), complement(sk3)))
% 84.69/11.23 = { by axiom 3 (converse_idempotence_8) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), join(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), composition(sk1, sk2)))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 40 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(sk3, composition(sk1, meet(sk2, composition(converse(sk1), sk3)))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 17 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(meet(composition(sk1, sk2), sk3), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by lemma 38 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3)))))), complement(sk3)))
% 84.69/11.23 = { by lemma 35 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(meet(composition(sk1, sk2), sk3)))))), complement(sk3)))
% 84.69/11.23 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by lemma 17 }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), meet(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), complement(sk3)))
% 84.69/11.23 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), complement(join(complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), complement(join(meet(complement(meet(composition(sk1, sk2), sk3)), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), complement(join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), meet(complement(meet(composition(sk1, sk2), sk3)), meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by lemma 17 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(join(meet(composition(sk1, sk2), sk3), complement(join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3)))))), complement(sk3)))
% 84.69/11.23 = { by lemma 38 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(meet(join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3)))), complement(meet(composition(sk1, sk2), sk3)))), complement(sk3)))
% 84.69/11.23 = { by lemma 17 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3)))))), complement(sk3)))
% 84.69/11.23 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(meet(complement(meet(composition(sk1, sk2), sk3)), join(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))), complement(sk3)))
% 84.69/11.23 = { by lemma 38 R->L }
% 84.69/11.23 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(meet(complement(meet(composition(sk1, sk2), sk3)), complement(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(meet(composition(sk1, sk2), sk3)))))))), complement(sk3)))
% 84.69/11.23 = { by lemma 35 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(meet(complement(meet(composition(sk1, sk2), sk3)), complement(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(meet(composition(sk1, sk2), sk3)))))))), complement(sk3)))
% 84.69/11.24 = { by lemma 34 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(zero, meet(complement(meet(composition(sk1, sk2), sk3)), complement(meet(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(meet(composition(sk1, sk2), sk3))))))))), complement(sk3)))
% 84.69/11.24 = { by lemma 35 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(zero, meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(complement(meet(composition(sk1, sk2), sk3))))))))), complement(sk3)))
% 84.69/11.24 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(zero, meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))))), complement(sk3)))
% 84.69/11.24 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))), zero)), complement(sk3)))
% 84.69/11.24 = { by lemma 14 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))), complement(top))), complement(sk3)))
% 84.69/11.24 = { by lemma 24 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(join(meet(complement(meet(composition(sk1, sk2), sk3)), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), join(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))))))))), complement(sk3)))
% 84.69/11.24 = { by lemma 15 }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(complement(meet(composition(sk1, sk2), sk3))), complement(sk3)))
% 84.69/11.24 = { by lemma 18 }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(meet(composition(sk1, sk2), sk3), complement(sk3)))
% 84.69/11.24 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), meet(composition(sk1, sk2), sk3)))
% 84.69/11.24 = { by lemma 42 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), meet(meet(composition(sk1, sk2), sk3), join(composition(sk1, sk2), complement(sk3)))))
% 84.69/11.24 = { by lemma 17 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), meet(meet(sk3, composition(sk1, sk2)), join(composition(sk1, sk2), complement(sk3)))))
% 84.69/11.24 = { by lemma 43 }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(composition(sk1, sk2), complement(sk3)))
% 84.69/11.24 = { by lemma 48 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), join(composition(sk1, sk2), composition(sk1, complement(composition(converse(sk1), sk3))))))
% 84.69/11.24 = { by lemma 46 }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), composition(sk1, join(complement(composition(converse(sk1), sk3)), sk2))))
% 84.69/11.24 = { by axiom 2 (maddux1_join_commutativity_1) }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), composition(sk1, join(sk2, complement(composition(converse(sk1), sk3))))))
% 84.69/11.24 = { by lemma 44 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), composition(sk1, join(meet(sk2, composition(converse(sk1), sk3)), complement(composition(converse(sk1), sk3))))))
% 84.69/11.24 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), composition(sk1, join(complement(composition(converse(sk1), sk3)), meet(sk2, composition(converse(sk1), sk3))))))
% 84.69/11.24 = { by lemma 46 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(complement(sk3), join(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), composition(sk1, complement(composition(converse(sk1), sk3))))))
% 84.69/11.24 = { by lemma 48 }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), complement(sk3)))
% 84.69/11.24 = { by lemma 44 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), join(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3), complement(sk3)))
% 84.69/11.24 = { by lemma 38 R->L }
% 84.69/11.24 meet(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)), complement(meet(sk3, complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))
% 84.69/11.24 = { by lemma 36 R->L }
% 84.69/11.24 complement(join(meet(sk3, complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3))), complement(join(sk3, meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))))
% 84.69/11.24 = { by lemma 47 }
% 84.69/11.24 complement(complement(meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)))
% 84.69/11.24 = { by lemma 18 }
% 84.69/11.24 meet(composition(sk1, meet(sk2, composition(converse(sk1), sk3))), sk3)
% 84.69/11.24 % SZS output end Proof
% 84.69/11.24
% 84.69/11.24 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------