TSTP Solution File: REL032-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL032-2 : 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 : n019.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 13:44:17 EDT 2023
% Result : Unsatisfiable 5.02s 1.03s
% Output : Proof 5.47s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL032-2 : 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
% 0.14/0.34 % Computer : n019.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 21:42:13 EDT 2023
% 0.14/0.34 % CPUTime :
% 5.02/1.03 Command-line arguments: --flatten
% 5.02/1.03
% 5.02/1.03 % SZS status Unsatisfiable
% 5.02/1.03
% 5.47/1.10 % SZS output start Proof
% 5.47/1.10 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 5.47/1.11 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 5.47/1.11 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 5.47/1.11 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 5.47/1.11 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 5.47/1.11 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 5.47/1.11 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 5.47/1.11 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 5.47/1.11 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 5.47/1.11 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 5.47/1.11 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 5.47/1.11 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 5.47/1.11 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 5.47/1.11
% 5.47/1.11 Lemma 14: complement(top) = zero.
% 5.47/1.11 Proof:
% 5.47/1.11 complement(top)
% 5.47/1.11 = { by axiom 4 (def_top_12) }
% 5.47/1.11 complement(join(complement(X), complement(complement(X))))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 meet(X, complement(X))
% 5.47/1.11 = { by axiom 5 (def_zero_13) R->L }
% 5.47/1.11 zero
% 5.47/1.11
% 5.47/1.11 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 5.47/1.11 Proof:
% 5.47/1.11 join(X, join(Y, complement(X)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(X, join(complement(X), Y))
% 5.47/1.11 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.47/1.11 join(join(X, complement(X)), Y)
% 5.47/1.11 = { by axiom 4 (def_top_12) R->L }
% 5.47/1.11 join(top, Y)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.11 join(Y, top)
% 5.47/1.11
% 5.47/1.11 Lemma 16: composition(converse(one), X) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 composition(converse(one), X)
% 5.47/1.11 = { by axiom 1 (converse_idempotence_8) R->L }
% 5.47/1.11 composition(converse(one), converse(converse(X)))
% 5.47/1.11 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 5.47/1.11 converse(composition(converse(X), one))
% 5.47/1.11 = { by axiom 3 (composition_identity_6) }
% 5.47/1.11 converse(converse(X))
% 5.47/1.11 = { by axiom 1 (converse_idempotence_8) }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 17: join(complement(X), complement(X)) = complement(X).
% 5.47/1.11 Proof:
% 5.47/1.11 join(complement(X), complement(X))
% 5.47/1.11 = { by lemma 16 R->L }
% 5.47/1.11 join(complement(X), composition(converse(one), complement(X)))
% 5.47/1.11 = { by lemma 16 R->L }
% 5.47/1.11 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 5.47/1.11 = { by axiom 3 (composition_identity_6) R->L }
% 5.47/1.11 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 5.47/1.11 = { by axiom 9 (composition_associativity_5) R->L }
% 5.47/1.11 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 5.47/1.11 = { by lemma 16 }
% 5.47/1.11 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 5.47/1.11 = { by axiom 12 (converse_cancellativity_11) }
% 5.47/1.11 complement(X)
% 5.47/1.11
% 5.47/1.11 Lemma 18: join(top, complement(X)) = top.
% 5.47/1.11 Proof:
% 5.47/1.11 join(top, complement(X))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(complement(X), top)
% 5.47/1.11 = { by lemma 15 R->L }
% 5.47/1.11 join(X, join(complement(X), complement(X)))
% 5.47/1.11 = { by lemma 17 }
% 5.47/1.11 join(X, complement(X))
% 5.47/1.11 = { by axiom 4 (def_top_12) R->L }
% 5.47/1.11 top
% 5.47/1.11
% 5.47/1.11 Lemma 19: join(Y, top) = join(X, top).
% 5.47/1.11 Proof:
% 5.47/1.11 join(Y, top)
% 5.47/1.11 = { by lemma 18 R->L }
% 5.47/1.11 join(Y, join(top, complement(Y)))
% 5.47/1.11 = { by lemma 15 }
% 5.47/1.11 join(top, top)
% 5.47/1.11 = { by lemma 15 R->L }
% 5.47/1.11 join(X, join(top, complement(X)))
% 5.47/1.11 = { by lemma 18 }
% 5.47/1.11 join(X, top)
% 5.47/1.11
% 5.47/1.11 Lemma 20: join(X, top) = top.
% 5.47/1.11 Proof:
% 5.47/1.11 join(X, top)
% 5.47/1.11 = { by lemma 19 }
% 5.47/1.11 join(join(zero, zero), top)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(top, join(zero, zero))
% 5.47/1.11 = { by lemma 14 R->L }
% 5.47/1.11 join(top, join(zero, complement(top)))
% 5.47/1.11 = { by lemma 14 R->L }
% 5.47/1.11 join(top, join(complement(top), complement(top)))
% 5.47/1.11 = { by lemma 17 }
% 5.47/1.11 join(top, complement(top))
% 5.47/1.11 = { by axiom 4 (def_top_12) R->L }
% 5.47/1.11 top
% 5.47/1.11
% 5.47/1.11 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(meet(X, Y), complement(join(complement(X), Y)))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 5.47/1.11 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 22: join(zero, meet(X, X)) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(zero, meet(X, X))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 join(zero, complement(join(complement(X), complement(X))))
% 5.47/1.11 = { by axiom 5 (def_zero_13) }
% 5.47/1.11 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 5.47/1.11 = { by lemma 21 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 23: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 5.47/1.11 Proof:
% 5.47/1.11 join(zero, join(X, complement(complement(Y))))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(zero, join(complement(complement(Y)), X))
% 5.47/1.11 = { by lemma 17 R->L }
% 5.47/1.11 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 join(zero, join(meet(Y, Y), X))
% 5.47/1.11 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.47/1.11 join(join(zero, meet(Y, Y)), X)
% 5.47/1.11 = { by lemma 22 }
% 5.47/1.11 join(Y, X)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.11 join(X, Y)
% 5.47/1.11
% 5.47/1.11 Lemma 24: join(zero, complement(complement(X))) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(zero, complement(complement(X)))
% 5.47/1.11 = { by axiom 5 (def_zero_13) }
% 5.47/1.11 join(meet(X, complement(X)), complement(complement(X)))
% 5.47/1.11 = { by lemma 17 R->L }
% 5.47/1.11 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 5.47/1.11 = { by lemma 21 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 25: join(zero, complement(X)) = complement(X).
% 5.47/1.11 Proof:
% 5.47/1.11 join(zero, complement(X))
% 5.47/1.11 = { by lemma 24 R->L }
% 5.47/1.11 join(zero, join(zero, complement(complement(complement(X)))))
% 5.47/1.11 = { by lemma 17 R->L }
% 5.47/1.11 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 5.47/1.11 = { by lemma 23 }
% 5.47/1.11 join(zero, join(complement(complement(complement(X))), complement(X)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.11 join(zero, join(complement(X), complement(complement(complement(X)))))
% 5.47/1.11 = { by lemma 23 }
% 5.47/1.11 join(complement(X), complement(X))
% 5.47/1.11 = { by lemma 17 }
% 5.47/1.11 complement(X)
% 5.47/1.11
% 5.47/1.11 Lemma 26: join(X, zero) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(X, zero)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(zero, X)
% 5.47/1.11 = { by lemma 23 R->L }
% 5.47/1.11 join(zero, join(zero, complement(complement(X))))
% 5.47/1.11 = { by lemma 25 }
% 5.47/1.11 join(zero, complement(complement(X)))
% 5.47/1.11 = { by lemma 24 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 27: join(top, X) = top.
% 5.47/1.11 Proof:
% 5.47/1.11 join(top, X)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(X, top)
% 5.47/1.11 = { by lemma 19 R->L }
% 5.47/1.11 join(Y, top)
% 5.47/1.11 = { by lemma 20 }
% 5.47/1.11 top
% 5.47/1.11
% 5.47/1.11 Lemma 28: join(zero, X) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(zero, X)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(X, zero)
% 5.47/1.11 = { by lemma 26 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 29: complement(complement(X)) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 complement(complement(X))
% 5.47/1.11 = { by lemma 25 R->L }
% 5.47/1.11 join(zero, complement(complement(X)))
% 5.47/1.11 = { by lemma 24 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 30: meet(Y, X) = meet(X, Y).
% 5.47/1.11 Proof:
% 5.47/1.11 meet(Y, X)
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 complement(join(complement(Y), complement(X)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 complement(join(complement(X), complement(Y)))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 meet(X, Y)
% 5.47/1.11
% 5.47/1.11 Lemma 31: complement(join(zero, complement(X))) = meet(X, top).
% 5.47/1.11 Proof:
% 5.47/1.11 complement(join(zero, complement(X)))
% 5.47/1.11 = { by lemma 14 R->L }
% 5.47/1.11 complement(join(complement(top), complement(X)))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 meet(top, X)
% 5.47/1.11 = { by lemma 30 R->L }
% 5.47/1.11 meet(X, top)
% 5.47/1.11
% 5.47/1.11 Lemma 32: join(X, complement(zero)) = top.
% 5.47/1.11 Proof:
% 5.47/1.11 join(X, complement(zero))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(complement(zero), X)
% 5.47/1.11 = { by lemma 23 R->L }
% 5.47/1.11 join(zero, join(complement(zero), complement(complement(X))))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(zero, join(complement(complement(X)), complement(zero)))
% 5.47/1.11 = { by lemma 15 }
% 5.47/1.11 join(complement(complement(X)), top)
% 5.47/1.11 = { by lemma 20 }
% 5.47/1.11 top
% 5.47/1.11
% 5.47/1.11 Lemma 33: meet(X, zero) = zero.
% 5.47/1.11 Proof:
% 5.47/1.11 meet(X, zero)
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 complement(join(complement(X), complement(zero)))
% 5.47/1.11 = { by lemma 32 }
% 5.47/1.11 complement(top)
% 5.47/1.11 = { by lemma 14 }
% 5.47/1.11 zero
% 5.47/1.11
% 5.47/1.11 Lemma 34: join(meet(X, Y), meet(X, complement(Y))) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 join(meet(X, Y), meet(X, complement(Y)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(meet(X, complement(Y)), meet(X, Y))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 5.47/1.11 = { by lemma 21 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 35: meet(X, top) = X.
% 5.47/1.11 Proof:
% 5.47/1.11 meet(X, top)
% 5.47/1.11 = { by lemma 31 R->L }
% 5.47/1.11 complement(join(zero, complement(X)))
% 5.47/1.11 = { by lemma 25 R->L }
% 5.47/1.11 join(zero, complement(join(zero, complement(X))))
% 5.47/1.11 = { by lemma 31 }
% 5.47/1.11 join(zero, meet(X, top))
% 5.47/1.11 = { by lemma 32 R->L }
% 5.47/1.11 join(zero, meet(X, join(complement(zero), complement(zero))))
% 5.47/1.11 = { by lemma 17 }
% 5.47/1.11 join(zero, meet(X, complement(zero)))
% 5.47/1.11 = { by lemma 33 R->L }
% 5.47/1.11 join(meet(X, zero), meet(X, complement(zero)))
% 5.47/1.11 = { by lemma 34 }
% 5.47/1.11 X
% 5.47/1.11
% 5.47/1.11 Lemma 36: join(Z, join(Y, X)) = join(X, join(Y, Z)).
% 5.47/1.11 Proof:
% 5.47/1.11 join(Z, join(Y, X))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 join(join(Y, X), Z)
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.11 join(join(X, Y), Z)
% 5.47/1.11 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 5.47/1.11 join(X, join(Y, Z))
% 5.47/1.11
% 5.47/1.11 Lemma 37: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 5.47/1.11 Proof:
% 5.47/1.11 meet(X, join(complement(Y), complement(Z)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 meet(X, join(complement(Z), complement(Y)))
% 5.47/1.11 = { by lemma 30 }
% 5.47/1.11 meet(join(complement(Z), complement(Y)), X)
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 complement(join(meet(Z, Y), complement(X)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.11 complement(join(complement(X), meet(Z, Y)))
% 5.47/1.11 = { by lemma 30 R->L }
% 5.47/1.11 complement(join(complement(X), meet(Y, Z)))
% 5.47/1.11
% 5.47/1.11 Lemma 38: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 5.47/1.11 Proof:
% 5.47/1.11 complement(join(X, complement(Y)))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 complement(join(complement(Y), X))
% 5.47/1.11 = { by lemma 35 R->L }
% 5.47/1.11 complement(join(complement(Y), meet(X, top)))
% 5.47/1.11 = { by lemma 30 R->L }
% 5.47/1.11 complement(join(complement(Y), meet(top, X)))
% 5.47/1.11 = { by lemma 37 R->L }
% 5.47/1.11 meet(Y, join(complement(top), complement(X)))
% 5.47/1.11 = { by lemma 14 }
% 5.47/1.11 meet(Y, join(zero, complement(X)))
% 5.47/1.11 = { by lemma 25 }
% 5.47/1.11 meet(Y, complement(X))
% 5.47/1.11
% 5.47/1.11 Lemma 39: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 5.47/1.11 Proof:
% 5.47/1.11 complement(meet(X, complement(Y)))
% 5.47/1.11 = { by lemma 28 R->L }
% 5.47/1.11 complement(join(zero, meet(X, complement(Y))))
% 5.47/1.11 = { by lemma 38 R->L }
% 5.47/1.11 complement(join(zero, complement(join(Y, complement(X)))))
% 5.47/1.11 = { by lemma 31 }
% 5.47/1.11 meet(join(Y, complement(X)), top)
% 5.47/1.11 = { by lemma 35 }
% 5.47/1.11 join(Y, complement(X))
% 5.47/1.11
% 5.47/1.11 Lemma 40: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 5.47/1.11 Proof:
% 5.47/1.11 complement(meet(complement(X), Y))
% 5.47/1.11 = { by lemma 30 }
% 5.47/1.11 complement(meet(Y, complement(X)))
% 5.47/1.11 = { by lemma 39 }
% 5.47/1.11 join(X, complement(Y))
% 5.47/1.11
% 5.47/1.11 Lemma 41: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 5.47/1.11 Proof:
% 5.47/1.11 complement(join(complement(X), Y))
% 5.47/1.11 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.11 complement(join(Y, complement(X)))
% 5.47/1.11 = { by lemma 38 }
% 5.47/1.11 meet(X, complement(Y))
% 5.47/1.11
% 5.47/1.11 Lemma 42: join(X, complement(meet(X, Y))) = top.
% 5.47/1.11 Proof:
% 5.47/1.11 join(X, complement(meet(X, Y)))
% 5.47/1.11 = { by lemma 30 }
% 5.47/1.11 join(X, complement(meet(Y, X)))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.11 join(X, complement(complement(join(complement(Y), complement(X)))))
% 5.47/1.11 = { by lemma 17 R->L }
% 5.47/1.11 join(X, complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 join(X, complement(join(meet(Y, X), complement(join(complement(Y), complement(X))))))
% 5.47/1.11 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.11 join(X, complement(join(meet(Y, X), meet(Y, X))))
% 5.47/1.12 = { by lemma 30 R->L }
% 5.47/1.12 join(X, complement(join(meet(Y, X), meet(X, Y))))
% 5.47/1.12 = { by lemma 30 R->L }
% 5.47/1.12 join(X, complement(join(meet(X, Y), meet(X, Y))))
% 5.47/1.12 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.12 join(X, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
% 5.47/1.12 = { by lemma 25 R->L }
% 5.47/1.12 join(X, join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))))
% 5.47/1.12 = { by lemma 37 R->L }
% 5.47/1.12 join(X, join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y)))))
% 5.47/1.12 = { by lemma 22 }
% 5.47/1.12 join(X, join(complement(X), complement(Y)))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 join(X, join(complement(Y), complement(X)))
% 5.47/1.12 = { by lemma 15 }
% 5.47/1.12 join(complement(Y), top)
% 5.47/1.12 = { by lemma 20 }
% 5.47/1.12 top
% 5.47/1.12
% 5.47/1.12 Lemma 43: meet(X, join(X, complement(Y))) = X.
% 5.47/1.12 Proof:
% 5.47/1.12 meet(X, join(X, complement(Y)))
% 5.47/1.12 = { by lemma 39 R->L }
% 5.47/1.12 meet(X, complement(meet(Y, complement(X))))
% 5.47/1.12 = { by lemma 41 R->L }
% 5.47/1.12 complement(join(complement(X), meet(Y, complement(X))))
% 5.47/1.12 = { by lemma 25 R->L }
% 5.47/1.12 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 5.47/1.12 = { by lemma 14 R->L }
% 5.47/1.12 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 5.47/1.12 = { by lemma 42 R->L }
% 5.47/1.12 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 5.47/1.12 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.12 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 5.47/1.12 = { by lemma 30 R->L }
% 5.47/1.12 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 5.47/1.12 = { by lemma 21 }
% 5.47/1.12 X
% 5.47/1.12
% 5.47/1.12 Lemma 44: join(meet(X, Y), X) = X.
% 5.47/1.12 Proof:
% 5.47/1.12 join(meet(X, Y), X)
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.12 join(X, meet(X, Y))
% 5.47/1.12 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.12 join(X, complement(join(complement(X), complement(Y))))
% 5.47/1.12 = { by lemma 40 R->L }
% 5.47/1.12 complement(meet(complement(X), join(complement(X), complement(Y))))
% 5.47/1.12 = { by lemma 43 }
% 5.47/1.12 complement(complement(X))
% 5.47/1.12 = { by lemma 29 }
% 5.47/1.12 X
% 5.47/1.12
% 5.47/1.12 Lemma 45: meet(X, join(X, Y)) = X.
% 5.47/1.12 Proof:
% 5.47/1.12 meet(X, join(X, Y))
% 5.47/1.12 = { by lemma 35 R->L }
% 5.47/1.12 meet(X, join(X, meet(Y, top)))
% 5.47/1.12 = { by lemma 31 R->L }
% 5.47/1.12 meet(X, join(X, complement(join(zero, complement(Y)))))
% 5.47/1.12 = { by lemma 43 }
% 5.47/1.12 X
% 5.47/1.12
% 5.47/1.12 Lemma 46: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 5.47/1.12 Proof:
% 5.47/1.12 meet(complement(X), complement(Y))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 meet(complement(Y), complement(X))
% 5.47/1.12 = { by lemma 25 R->L }
% 5.47/1.12 meet(join(zero, complement(Y)), complement(X))
% 5.47/1.12 = { by lemma 38 R->L }
% 5.47/1.12 complement(join(X, complement(join(zero, complement(Y)))))
% 5.47/1.12 = { by lemma 31 }
% 5.47/1.12 complement(join(X, meet(Y, top)))
% 5.47/1.12 = { by lemma 35 }
% 5.47/1.12 complement(join(X, Y))
% 5.47/1.12
% 5.47/1.12 Lemma 47: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 5.47/1.12 Proof:
% 5.47/1.12 meet(complement(Z), meet(Y, X))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 meet(complement(Z), meet(X, Y))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 meet(meet(X, Y), complement(Z))
% 5.47/1.12 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.12 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 5.47/1.12 = { by lemma 46 }
% 5.47/1.12 complement(join(join(complement(X), complement(Y)), Z))
% 5.47/1.12 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 5.47/1.12 complement(join(complement(X), join(complement(Y), Z)))
% 5.47/1.12 = { by lemma 41 }
% 5.47/1.12 meet(X, complement(join(complement(Y), Z)))
% 5.47/1.12 = { by lemma 41 }
% 5.47/1.12 meet(X, meet(Y, complement(Z)))
% 5.47/1.12
% 5.47/1.12 Lemma 48: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 5.47/1.12 Proof:
% 5.47/1.12 meet(Y, meet(Z, X))
% 5.47/1.12 = { by lemma 35 R->L }
% 5.47/1.12 meet(meet(Y, top), meet(Z, X))
% 5.47/1.12 = { by lemma 31 R->L }
% 5.47/1.12 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 5.47/1.12 = { by lemma 47 }
% 5.47/1.12 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 5.47/1.12 = { by lemma 31 }
% 5.47/1.12 meet(X, meet(Z, meet(Y, top)))
% 5.47/1.12 = { by lemma 35 }
% 5.47/1.12 meet(X, meet(Z, Y))
% 5.47/1.12 = { by lemma 30 R->L }
% 5.47/1.12 meet(X, meet(Y, Z))
% 5.47/1.12
% 5.47/1.12 Lemma 49: meet(X, complement(join(X, Y))) = zero.
% 5.47/1.12 Proof:
% 5.47/1.12 meet(X, complement(join(X, Y)))
% 5.47/1.12 = { by lemma 38 R->L }
% 5.47/1.12 complement(join(join(X, Y), complement(X)))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.12 complement(join(complement(X), join(X, Y)))
% 5.47/1.12 = { by lemma 36 }
% 5.47/1.12 complement(join(Y, join(X, complement(X))))
% 5.47/1.12 = { by axiom 4 (def_top_12) R->L }
% 5.47/1.12 complement(join(Y, top))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 complement(join(top, Y))
% 5.47/1.12 = { by lemma 27 }
% 5.47/1.12 complement(top)
% 5.47/1.12 = { by lemma 14 }
% 5.47/1.12 zero
% 5.47/1.12
% 5.47/1.12 Lemma 50: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 5.47/1.12 Proof:
% 5.47/1.12 join(composition(X, Z), composition(X, Y))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) R->L }
% 5.47/1.12 join(composition(X, Z), composition(X, converse(converse(Y))))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) R->L }
% 5.47/1.12 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.12 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 5.47/1.12 = { by axiom 6 (converse_additivity_9) }
% 5.47/1.12 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 5.47/1.12 = { by axiom 8 (converse_multiplicativity_10) }
% 5.47/1.12 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) }
% 5.47/1.12 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 5.47/1.12 = { by axiom 8 (converse_multiplicativity_10) }
% 5.47/1.12 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 5.47/1.12 = { by axiom 11 (composition_distributivity_7) R->L }
% 5.47/1.12 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.12 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) R->L }
% 5.47/1.12 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 5.47/1.12 = { by axiom 6 (converse_additivity_9) R->L }
% 5.47/1.12 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 5.47/1.12 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 5.47/1.12 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) }
% 5.47/1.12 composition(X, join(Z, converse(converse(Y))))
% 5.47/1.12 = { by axiom 1 (converse_idempotence_8) }
% 5.47/1.12 composition(X, join(Z, Y))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 composition(X, join(Y, Z))
% 5.47/1.12
% 5.47/1.12 Goal 1 (goals_17): join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))) = meet(composition(sk1, sk2), composition(sk1, sk3)).
% 5.47/1.12 Proof:
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 5.47/1.12 = { by lemma 34 R->L }
% 5.47/1.12 join(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 join(meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.12 join(complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by lemma 25 R->L }
% 5.47/1.12 join(join(zero, complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by lemma 49 R->L }
% 5.47/1.12 join(join(meet(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))), complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by lemma 21 }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.12 = { by lemma 45 R->L }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3)))))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))
% 5.47/1.12 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))))
% 5.47/1.12 = { by lemma 30 }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(composition(sk1, meet(sk2, sk3)))))))
% 5.47/1.12 = { by lemma 38 R->L }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(join(composition(sk1, meet(sk2, sk3)), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))))))
% 5.47/1.12 = { by lemma 46 R->L }
% 5.47/1.12 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(join(composition(sk1, meet(sk2, sk3)), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))))))
% 5.47/1.13 = { by lemma 40 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))))))))
% 5.47/1.13 = { by lemma 25 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))))))
% 5.47/1.13 = { by lemma 35 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, sk3)), top), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))))))
% 5.47/1.13 = { by lemma 31 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))))))
% 5.47/1.13 = { by lemma 26 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(join(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))), zero))))
% 5.47/1.13 = { by lemma 14 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(join(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))), complement(top)))))
% 5.47/1.13 = { by axiom 4 (def_top_12) }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(join(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))), complement(join(join(complement(complement(meet(composition(sk1, sk2), composition(sk1, sk3)))), complement(join(zero, complement(composition(sk1, meet(sk2, sk3)))))), complement(join(complement(complement(meet(composition(sk1, sk2), composition(sk1, sk3)))), complement(join(zero, complement(composition(sk1, meet(sk2, sk3)))))))))))))
% 5.47/1.13 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(join(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))), complement(join(join(complement(complement(meet(composition(sk1, sk2), composition(sk1, sk3)))), complement(join(zero, complement(composition(sk1, meet(sk2, sk3)))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3)))))))))))
% 5.47/1.13 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(join(meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))), complement(join(complement(complement(meet(composition(sk1, sk2), composition(sk1, sk3)))), join(complement(join(zero, complement(composition(sk1, meet(sk2, sk3))))), meet(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(zero, complement(composition(sk1, meet(sk2, sk3))))))))))))
% 5.47/1.13 = { by lemma 21 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), complement(complement(meet(composition(sk1, sk2), composition(sk1, sk3))))))
% 5.47/1.13 = { by lemma 29 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3))))
% 5.47/1.13 = { by lemma 30 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, sk3))))))
% 5.47/1.13 = { by lemma 48 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), meet(complement(composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))))
% 5.47/1.13 = { by lemma 30 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, sk3))))))
% 5.47/1.13 = { by lemma 48 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), meet(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3)))))
% 5.47/1.13 = { by lemma 30 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))))
% 5.47/1.13 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))))))
% 5.47/1.13 = { by lemma 25 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 14 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(top), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 27 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(top, composition(sk1, meet(sk2, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(composition(sk1, meet(sk2, sk3)), top)), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 15 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), join(composition(sk1, meet(sk2, sk3)), complement(meet(composition(sk1, sk2), composition(sk1, sk3)))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 36 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by axiom 2 (maddux1_join_commutativity_1) }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(meet(composition(sk1, sk2), composition(sk1, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 38 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))))))))
% 5.47/1.13 = { by lemma 21 }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(complement(composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), composition(sk1, sk3))))
% 5.47/1.13 = { by lemma 30 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 41 R->L }
% 5.47/1.13 join(composition(sk1, meet(sk2, sk3)), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 21 R->L }
% 5.47/1.13 join(join(meet(composition(sk1, meet(sk2, sk3)), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))), complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 48 }
% 5.47/1.13 join(join(meet(complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))), meet(composition(sk1, meet(sk2, sk3)), join(zero, complement(composition(sk1, sk2))))), complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 47 }
% 5.47/1.13 join(join(meet(join(zero, complement(composition(sk1, sk2))), meet(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))), complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 49 }
% 5.47/1.13 join(join(meet(join(zero, complement(composition(sk1, sk2))), zero), complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 33 }
% 5.47/1.13 join(join(zero, complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.13 = { by lemma 25 }
% 5.47/1.13 join(complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 41 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), complement(meet(join(zero, complement(composition(sk1, sk2))), complement(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 39 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)), complement(join(zero, complement(composition(sk1, sk2)))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 31 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)), meet(composition(sk1, sk2), top))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 35 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)), composition(sk1, sk2))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, sk2), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), X)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 36 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(meet(composition(sk1, sk2), X), join(composition(sk1, meet(sk2, sk3)), composition(sk1, sk2)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(meet(composition(sk1, sk2), X), join(composition(sk1, sk2), composition(sk1, meet(sk2, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by axiom 7 (maddux2_join_associativity_2) }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(join(meet(composition(sk1, sk2), X), composition(sk1, sk2)), composition(sk1, meet(sk2, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 44 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, sk2), composition(sk1, meet(sk2, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 50 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), composition(sk1, join(meet(sk2, sk3), sk2))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 44 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), composition(sk1, sk2)), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 30 }
% 5.47/1.14 join(meet(composition(sk1, sk2), composition(sk1, meet(sk2, sk3))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 45 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, meet(sk2, sk3)), composition(sk1, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, sk3), composition(sk1, meet(sk2, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 50 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, join(meet(sk2, sk3), sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 35 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, meet(join(meet(sk2, sk3), sk3), top)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 31 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(join(meet(sk2, sk3), sk3))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(join(sk3, meet(sk2, sk3)))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 46 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, meet(complement(sk3), complement(meet(sk2, sk3)))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 28 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(zero, meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 14 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(complement(top), meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 42 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(complement(join(sk3, complement(meet(sk3, sk2)))), meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 30 R->L }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(complement(join(sk3, complement(meet(sk2, sk3)))), meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 38 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(meet(meet(sk2, sk3), complement(sk3)), meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 30 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(meet(complement(sk3), meet(sk2, sk3)), meet(complement(sk3), complement(meet(sk2, sk3))))))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 34 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(sk3)))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 31 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, meet(sk3, top)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 35 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 30 }
% 5.47/1.14 join(meet(composition(sk1, sk2), meet(composition(sk1, sk3), composition(sk1, meet(sk2, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 48 }
% 5.47/1.14 join(meet(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 30 R->L }
% 5.47/1.14 join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), composition(sk1, meet(sk2, sk3)))))
% 5.47/1.14 = { by lemma 21 }
% 5.47/1.14 meet(composition(sk1, sk2), composition(sk1, sk3))
% 5.47/1.14 % SZS output end Proof
% 5.47/1.14
% 5.47/1.14 RESULT: Unsatisfiable (the axioms are contradictory).
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