TSTP Solution File: REL017-3 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL017-3 : 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 : n026.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 13:43:58 EDT 2023
% Result : Unsatisfiable 11.76s 1.91s
% Output : Proof 12.95s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL017-3 : 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
% 0.13/0.35 % Computer : n026.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 22:26:33 EDT 2023
% 0.13/0.35 % CPUTime :
% 11.76/1.91 Command-line arguments: --flatten
% 11.76/1.91
% 11.76/1.91 % SZS status Unsatisfiable
% 11.76/1.91
% 12.59/1.99 % SZS output start Proof
% 12.59/1.99 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 12.59/1.99 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 12.59/1.99 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 12.59/1.99 Axiom 4 (def_zero_13): zero = meet(X, complement(X)).
% 12.59/1.99 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 12.59/1.99 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 12.59/1.99 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 12.59/1.99 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 12.59/1.99 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 12.59/1.99 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 12.59/1.99 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 12.59/1.99 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 12.59/1.99 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 12.59/1.99
% 12.59/1.99 Lemma 14: complement(top) = zero.
% 12.59/1.99 Proof:
% 12.59/1.99 complement(top)
% 12.59/1.99 = { by axiom 5 (def_top_12) }
% 12.59/1.99 complement(join(complement(X), complement(complement(X))))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.59/1.99 meet(X, complement(X))
% 12.59/1.99 = { by axiom 4 (def_zero_13) R->L }
% 12.59/1.99 zero
% 12.59/1.99
% 12.59/1.99 Lemma 15: composition(converse(one), X) = X.
% 12.59/1.99 Proof:
% 12.59/1.99 composition(converse(one), X)
% 12.59/1.99 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.59/1.99 composition(converse(one), converse(converse(X)))
% 12.59/1.99 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 12.59/1.99 converse(composition(converse(X), one))
% 12.59/1.99 = { by axiom 3 (composition_identity_6) }
% 12.59/1.99 converse(converse(X))
% 12.59/1.99 = { by axiom 1 (converse_idempotence_8) }
% 12.59/1.99 X
% 12.59/1.99
% 12.59/1.99 Lemma 16: converse(one) = one.
% 12.59/1.99 Proof:
% 12.59/1.99 converse(one)
% 12.59/1.99 = { by axiom 3 (composition_identity_6) R->L }
% 12.59/1.99 composition(converse(one), one)
% 12.59/1.99 = { by lemma 15 }
% 12.59/1.99 one
% 12.59/1.99
% 12.59/1.99 Lemma 17: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 12.59/1.99 Proof:
% 12.59/1.99 join(meet(X, Y), complement(join(complement(X), Y)))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.59/1.99 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 12.59/1.99 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 12.59/1.99 X
% 12.59/1.99
% 12.59/1.99 Lemma 18: join(zero, meet(X, X)) = X.
% 12.59/1.99 Proof:
% 12.59/1.99 join(zero, meet(X, X))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.59/1.99 join(zero, complement(join(complement(X), complement(X))))
% 12.59/1.99 = { by axiom 4 (def_zero_13) }
% 12.59/1.99 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 12.59/1.99 = { by lemma 17 }
% 12.59/1.99 X
% 12.59/1.99
% 12.59/1.99 Lemma 19: composition(one, X) = X.
% 12.59/1.99 Proof:
% 12.59/1.99 composition(one, X)
% 12.59/1.99 = { by lemma 15 R->L }
% 12.59/1.99 composition(converse(one), composition(one, X))
% 12.59/1.99 = { by axiom 9 (composition_associativity_5) }
% 12.59/1.99 composition(composition(converse(one), one), X)
% 12.59/1.99 = { by axiom 3 (composition_identity_6) }
% 12.59/1.99 composition(converse(one), X)
% 12.59/1.99 = { by lemma 15 }
% 12.59/1.99 X
% 12.59/1.99
% 12.59/1.99 Lemma 20: join(complement(X), complement(X)) = complement(X).
% 12.59/1.99 Proof:
% 12.59/1.99 join(complement(X), complement(X))
% 12.59/1.99 = { by lemma 15 R->L }
% 12.59/1.99 join(complement(X), composition(converse(one), complement(X)))
% 12.59/1.99 = { by lemma 19 R->L }
% 12.59/1.99 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 12.59/1.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/1.99 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 12.59/1.99 = { by axiom 12 (converse_cancellativity_11) }
% 12.59/1.99 complement(X)
% 12.59/1.99
% 12.59/1.99 Lemma 21: complement(complement(X)) = meet(X, X).
% 12.59/1.99 Proof:
% 12.59/1.99 complement(complement(X))
% 12.59/1.99 = { by lemma 20 R->L }
% 12.59/1.99 complement(join(complement(X), complement(X)))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.59/1.99 meet(X, X)
% 12.59/1.99
% 12.59/1.99 Lemma 22: meet(Y, X) = meet(X, Y).
% 12.59/1.99 Proof:
% 12.59/1.99 meet(Y, X)
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.59/1.99 complement(join(complement(Y), complement(X)))
% 12.59/1.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/1.99 complement(join(complement(X), complement(Y)))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.59/1.99 meet(X, Y)
% 12.59/1.99
% 12.59/1.99 Lemma 23: complement(join(zero, complement(X))) = meet(X, top).
% 12.59/1.99 Proof:
% 12.59/1.99 complement(join(zero, complement(X)))
% 12.59/1.99 = { by lemma 14 R->L }
% 12.59/1.99 complement(join(complement(top), complement(X)))
% 12.59/1.99 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.59/1.99 meet(top, X)
% 12.59/1.99 = { by lemma 22 R->L }
% 12.59/1.99 meet(X, top)
% 12.59/1.99
% 12.59/1.99 Lemma 24: join(X, join(Y, complement(X))) = join(Y, top).
% 12.59/1.99 Proof:
% 12.59/1.99 join(X, join(Y, complement(X)))
% 12.59/1.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/1.99 join(X, join(complement(X), Y))
% 12.59/1.99 = { by axiom 7 (maddux2_join_associativity_2) }
% 12.59/1.99 join(join(X, complement(X)), Y)
% 12.59/1.99 = { by axiom 5 (def_top_12) R->L }
% 12.59/1.99 join(top, Y)
% 12.59/1.99 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.59/1.99 join(Y, top)
% 12.59/1.99
% 12.59/1.99 Lemma 25: join(top, complement(X)) = top.
% 12.59/1.99 Proof:
% 12.59/1.99 join(top, complement(X))
% 12.59/1.99 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/1.99 join(complement(X), top)
% 12.59/1.99 = { by lemma 24 R->L }
% 12.59/1.99 join(X, join(complement(X), complement(X)))
% 12.59/1.99 = { by lemma 20 }
% 12.59/1.99 join(X, complement(X))
% 12.59/1.99 = { by axiom 5 (def_top_12) R->L }
% 12.59/1.99 top
% 12.59/1.99
% 12.59/1.99 Lemma 26: join(X, top) = top.
% 12.59/1.99 Proof:
% 12.59/2.00 join(X, top)
% 12.59/2.00 = { by lemma 25 R->L }
% 12.59/2.00 join(X, join(top, complement(X)))
% 12.59/2.00 = { by lemma 24 }
% 12.59/2.00 join(top, top)
% 12.59/2.00 = { by lemma 24 R->L }
% 12.59/2.00 join(complement(Y), join(top, complement(complement(Y))))
% 12.59/2.00 = { by lemma 25 }
% 12.59/2.00 join(complement(Y), top)
% 12.59/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/2.00 join(top, complement(Y))
% 12.59/2.00 = { by lemma 25 }
% 12.59/2.00 top
% 12.59/2.00
% 12.59/2.00 Lemma 27: join(X, complement(zero)) = top.
% 12.59/2.00 Proof:
% 12.59/2.00 join(X, complement(zero))
% 12.59/2.00 = { by lemma 18 R->L }
% 12.59/2.00 join(join(zero, meet(X, X)), complement(zero))
% 12.59/2.00 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 12.59/2.00 join(zero, join(meet(X, X), complement(zero)))
% 12.59/2.00 = { by lemma 24 }
% 12.59/2.00 join(meet(X, X), top)
% 12.59/2.00 = { by lemma 26 }
% 12.59/2.00 top
% 12.59/2.00
% 12.59/2.00 Lemma 28: join(meet(X, Y), meet(X, complement(Y))) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 join(meet(X, Y), meet(X, complement(Y)))
% 12.59/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/2.00 join(meet(X, complement(Y)), meet(X, Y))
% 12.59/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.59/2.00 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 12.59/2.00 = { by lemma 17 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 29: join(zero, meet(X, top)) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 join(zero, meet(X, top))
% 12.59/2.00 = { by lemma 27 R->L }
% 12.59/2.00 join(zero, meet(X, join(complement(zero), complement(zero))))
% 12.59/2.00 = { by lemma 20 }
% 12.59/2.00 join(zero, meet(X, complement(zero)))
% 12.59/2.00 = { by lemma 14 R->L }
% 12.59/2.00 join(complement(top), meet(X, complement(zero)))
% 12.59/2.00 = { by lemma 27 R->L }
% 12.59/2.00 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 12.59/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.59/2.00 join(meet(X, zero), meet(X, complement(zero)))
% 12.59/2.00 = { by lemma 28 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 30: join(zero, complement(X)) = complement(X).
% 12.59/2.00 Proof:
% 12.59/2.00 join(zero, complement(X))
% 12.59/2.00 = { by lemma 18 R->L }
% 12.59/2.00 join(zero, complement(join(zero, meet(X, X))))
% 12.59/2.00 = { by lemma 21 R->L }
% 12.59/2.00 join(zero, complement(join(zero, complement(complement(X)))))
% 12.59/2.00 = { by lemma 23 }
% 12.59/2.00 join(zero, meet(complement(X), top))
% 12.59/2.00 = { by lemma 29 }
% 12.59/2.00 complement(X)
% 12.59/2.00
% 12.59/2.00 Lemma 31: complement(complement(X)) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 complement(complement(X))
% 12.59/2.00 = { by lemma 30 R->L }
% 12.59/2.00 join(zero, complement(complement(X)))
% 12.59/2.00 = { by lemma 21 }
% 12.59/2.00 join(zero, meet(X, X))
% 12.59/2.00 = { by lemma 18 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 32: join(X, X) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 join(X, X)
% 12.59/2.00 = { by lemma 31 R->L }
% 12.59/2.00 join(X, complement(complement(X)))
% 12.59/2.00 = { by lemma 31 R->L }
% 12.59/2.00 join(complement(complement(X)), complement(complement(X)))
% 12.59/2.00 = { by lemma 20 }
% 12.59/2.00 complement(complement(X))
% 12.59/2.00 = { by lemma 31 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 33: join(X, zero) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 join(X, zero)
% 12.59/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.59/2.00 join(zero, X)
% 12.59/2.00 = { by lemma 31 R->L }
% 12.59/2.00 join(zero, complement(complement(X)))
% 12.59/2.00 = { by lemma 21 }
% 12.59/2.00 join(zero, meet(X, X))
% 12.59/2.00 = { by lemma 18 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 34: meet(X, X) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 meet(X, X)
% 12.59/2.00 = { by lemma 21 R->L }
% 12.59/2.00 complement(complement(X))
% 12.59/2.00 = { by lemma 31 }
% 12.59/2.00 X
% 12.59/2.00
% 12.59/2.00 Lemma 35: meet(X, top) = X.
% 12.59/2.00 Proof:
% 12.59/2.00 meet(X, top)
% 12.77/2.00 = { by lemma 23 R->L }
% 12.77/2.00 complement(join(zero, complement(X)))
% 12.77/2.00 = { by lemma 30 R->L }
% 12.77/2.00 join(zero, complement(join(zero, complement(X))))
% 12.77/2.00 = { by lemma 23 }
% 12.77/2.00 join(zero, meet(X, top))
% 12.77/2.00 = { by lemma 29 }
% 12.77/2.00 X
% 12.77/2.00
% 12.77/2.00 Lemma 36: meet(top, X) = X.
% 12.77/2.00 Proof:
% 12.77/2.00 meet(top, X)
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 meet(X, top)
% 12.77/2.00 = { by lemma 35 }
% 12.77/2.00 X
% 12.77/2.00
% 12.77/2.00 Lemma 37: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 12.77/2.00 Proof:
% 12.77/2.00 join(Y, join(X, Z))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.00 join(join(X, Z), Y)
% 12.77/2.00 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 12.77/2.00 join(X, join(Z, Y))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.00 join(X, join(Y, Z))
% 12.77/2.00
% 12.77/2.00 Lemma 38: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 12.77/2.00 Proof:
% 12.77/2.00 complement(join(complement(X), meet(Y, Z)))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 complement(join(complement(X), meet(Z, Y)))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.00 complement(join(meet(Z, Y), complement(X)))
% 12.77/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.77/2.00 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 12.77/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.77/2.00 meet(join(complement(Z), complement(Y)), X)
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 meet(X, join(complement(Z), complement(Y)))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.00 meet(X, join(complement(Y), complement(Z)))
% 12.77/2.00
% 12.77/2.00 Lemma 39: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 12.77/2.00 Proof:
% 12.77/2.00 join(complement(X), complement(Y))
% 12.77/2.00 = { by lemma 36 R->L }
% 12.77/2.00 meet(top, join(complement(X), complement(Y)))
% 12.77/2.00 = { by lemma 38 R->L }
% 12.77/2.00 complement(join(complement(top), meet(X, Y)))
% 12.77/2.00 = { by lemma 14 }
% 12.77/2.00 complement(join(zero, meet(X, Y)))
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 complement(join(zero, meet(Y, X)))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.00 complement(join(meet(Y, X), zero))
% 12.77/2.00 = { by lemma 33 }
% 12.77/2.00 complement(meet(Y, X))
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 complement(meet(X, Y))
% 12.77/2.00
% 12.77/2.00 Lemma 40: join(join(X, Y), Z) = join(join(Z, Y), X).
% 12.77/2.00 Proof:
% 12.77/2.00 join(join(X, Y), Z)
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.00 join(Z, join(X, Y))
% 12.77/2.00 = { by lemma 37 }
% 12.77/2.00 join(X, join(Z, Y))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.00 join(join(Z, Y), X)
% 12.77/2.00
% 12.77/2.00 Lemma 41: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 12.77/2.00 Proof:
% 12.77/2.00 complement(join(complement(X), complement(Y)))
% 12.77/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.77/2.00 meet(X, Y)
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 meet(Y, X)
% 12.77/2.00
% 12.77/2.00 Lemma 42: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 12.77/2.00 Proof:
% 12.77/2.00 complement(meet(X, complement(Y)))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 complement(meet(complement(Y), X))
% 12.77/2.00 = { by lemma 30 R->L }
% 12.77/2.00 complement(meet(join(zero, complement(Y)), X))
% 12.77/2.00 = { by lemma 39 R->L }
% 12.77/2.00 join(complement(join(zero, complement(Y))), complement(X))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.00 join(complement(join(complement(Y), zero)), complement(X))
% 12.77/2.00 = { by lemma 14 R->L }
% 12.77/2.00 join(complement(join(complement(Y), complement(top))), complement(X))
% 12.77/2.00 = { by lemma 41 }
% 12.77/2.00 join(meet(top, Y), complement(X))
% 12.77/2.00 = { by lemma 36 }
% 12.77/2.00 join(Y, complement(X))
% 12.77/2.00
% 12.77/2.00 Lemma 43: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 12.77/2.00 Proof:
% 12.77/2.00 complement(join(X, complement(Y)))
% 12.77/2.00 = { by lemma 42 R->L }
% 12.77/2.00 complement(complement(meet(Y, complement(X))))
% 12.77/2.00 = { by lemma 21 }
% 12.77/2.00 meet(meet(Y, complement(X)), meet(Y, complement(X)))
% 12.77/2.00 = { by lemma 34 }
% 12.77/2.00 meet(Y, complement(X))
% 12.77/2.00
% 12.77/2.00 Lemma 44: meet(X, join(X, complement(Y))) = X.
% 12.77/2.00 Proof:
% 12.77/2.00 meet(X, join(X, complement(Y)))
% 12.77/2.00 = { by lemma 42 R->L }
% 12.77/2.00 meet(X, complement(meet(Y, complement(X))))
% 12.77/2.00 = { by lemma 39 R->L }
% 12.77/2.00 meet(X, join(complement(Y), complement(complement(X))))
% 12.77/2.00 = { by lemma 38 R->L }
% 12.77/2.00 complement(join(complement(X), meet(Y, complement(X))))
% 12.77/2.00 = { by lemma 30 R->L }
% 12.77/2.00 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 14 R->L }
% 12.77/2.00 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 26 R->L }
% 12.77/2.00 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 24 R->L }
% 12.77/2.00 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 39 }
% 12.77/2.00 join(complement(join(complement(X), complement(meet(Y, complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 12.77/2.00 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 12.77/2.00 = { by lemma 17 }
% 12.77/2.00 X
% 12.77/2.00
% 12.77/2.00 Lemma 45: meet(X, join(X, Y)) = X.
% 12.77/2.00 Proof:
% 12.77/2.00 meet(X, join(X, Y))
% 12.77/2.00 = { by lemma 34 R->L }
% 12.77/2.00 meet(X, join(X, meet(Y, Y)))
% 12.77/2.00 = { by lemma 21 R->L }
% 12.77/2.00 meet(X, join(X, complement(complement(Y))))
% 12.77/2.00 = { by lemma 44 }
% 12.77/2.00 X
% 12.77/2.00
% 12.77/2.00 Lemma 46: meet(X, join(Y, X)) = X.
% 12.77/2.00 Proof:
% 12.77/2.00 meet(X, join(Y, X))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.00 meet(X, join(X, Y))
% 12.77/2.00 = { by lemma 45 }
% 12.77/2.00 X
% 12.77/2.00
% 12.77/2.00 Lemma 47: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
% 12.77/2.00 Proof:
% 12.77/2.00 meet(X, meet(Y, Z))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 meet(meet(Y, Z), X)
% 12.77/2.00 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 12.77/2.00 complement(join(complement(meet(Y, Z)), complement(X)))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 complement(join(complement(meet(Z, Y)), complement(X)))
% 12.77/2.00 = { by lemma 39 R->L }
% 12.77/2.00 complement(join(join(complement(Z), complement(Y)), complement(X)))
% 12.77/2.00 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 12.77/2.00 complement(join(complement(Z), join(complement(Y), complement(X))))
% 12.77/2.00 = { by lemma 39 }
% 12.77/2.00 complement(join(complement(Z), complement(meet(Y, X))))
% 12.77/2.00 = { by lemma 41 }
% 12.77/2.00 meet(meet(Y, X), Z)
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 meet(Z, meet(Y, X))
% 12.77/2.00 = { by lemma 22 R->L }
% 12.77/2.00 meet(Z, meet(X, Y))
% 12.77/2.00
% 12.77/2.00 Lemma 48: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 12.77/2.00 Proof:
% 12.77/2.00 meet(Y, meet(X, Z))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 meet(Y, meet(Z, X))
% 12.77/2.00 = { by lemma 47 }
% 12.77/2.00 meet(X, meet(Y, Z))
% 12.77/2.00
% 12.77/2.00 Lemma 49: meet(complement(Y), complement(X)) = complement(join(X, Y)).
% 12.77/2.00 Proof:
% 12.77/2.00 meet(complement(Y), complement(X))
% 12.77/2.00 = { by lemma 22 }
% 12.77/2.00 meet(complement(X), complement(Y))
% 12.77/2.00 = { by lemma 43 R->L }
% 12.77/2.00 complement(join(Y, complement(complement(X))))
% 12.77/2.00 = { by lemma 21 }
% 12.77/2.00 complement(join(Y, meet(X, X)))
% 12.77/2.00 = { by lemma 34 }
% 12.77/2.00 complement(join(Y, X))
% 12.77/2.00 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.00 complement(join(X, Y))
% 12.77/2.00
% 12.77/2.00 Lemma 50: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 12.77/2.00 Proof:
% 12.77/2.00 meet(complement(X), complement(Y))
% 12.77/2.00 = { by lemma 49 }
% 12.77/2.01 complement(join(Y, X))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 complement(join(X, Y))
% 12.77/2.01
% 12.77/2.01 Lemma 51: join(join(complement(X), Y), X) = join(top, Y).
% 12.77/2.01 Proof:
% 12.77/2.01 join(join(complement(X), Y), X)
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(X, join(complement(X), Y))
% 12.77/2.01 = { by lemma 37 }
% 12.77/2.01 join(complement(X), join(X, Y))
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) }
% 12.77/2.01 join(join(complement(X), X), Y)
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(join(X, complement(X)), Y)
% 12.77/2.01 = { by axiom 5 (def_top_12) R->L }
% 12.77/2.01 join(top, Y)
% 12.77/2.01
% 12.77/2.01 Lemma 52: join(converse(X), composition(Y, converse(Z))) = converse(join(X, composition(Z, converse(Y)))).
% 12.77/2.01 Proof:
% 12.77/2.01 join(converse(X), composition(Y, converse(Z)))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(composition(Y, converse(Z)), converse(X))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.77/2.01 join(composition(converse(converse(Y)), converse(Z)), converse(X))
% 12.77/2.01 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 12.77/2.01 join(converse(composition(Z, converse(Y))), converse(X))
% 12.77/2.01 = { by axiom 6 (converse_additivity_9) R->L }
% 12.77/2.01 converse(join(composition(Z, converse(Y)), X))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 converse(join(X, composition(Z, converse(Y))))
% 12.77/2.01
% 12.77/2.01 Lemma 53: join(join(W, Y), join(Z, X)) = join(join(X, Y), join(Z, W)).
% 12.77/2.01 Proof:
% 12.77/2.01 join(join(W, Y), join(Z, X))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(join(W, Y), join(X, Z))
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) }
% 12.77/2.01 join(join(join(W, Y), X), Z)
% 12.77/2.01 = { by lemma 40 R->L }
% 12.77/2.01 join(join(join(X, Y), W), Z)
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 12.77/2.01 join(join(X, Y), join(W, Z))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 join(join(X, Y), join(Z, W))
% 12.77/2.01
% 12.77/2.01 Lemma 54: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 12.77/2.01 Proof:
% 12.77/2.01 join(composition(X, Z), composition(X, Y))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.77/2.01 join(composition(X, Z), composition(X, converse(converse(Y))))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.77/2.01 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 12.77/2.01 = { by lemma 52 R->L }
% 12.77/2.01 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 12.77/2.01 = { by axiom 8 (converse_multiplicativity_10) }
% 12.77/2.01 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 12.77/2.01 = { by axiom 11 (composition_distributivity_7) R->L }
% 12.77/2.01 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.77/2.01 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 12.77/2.01 = { by axiom 6 (converse_additivity_9) R->L }
% 12.77/2.01 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 12.77/2.01 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 12.77/2.01 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) }
% 12.77/2.01 composition(X, join(Z, converse(converse(Y))))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) }
% 12.77/2.01 composition(X, join(Z, Y))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 composition(X, join(Y, Z))
% 12.77/2.01
% 12.77/2.01 Lemma 55: join(join(complement(X), Y), complement(X)) = join(complement(X), Y).
% 12.77/2.01 Proof:
% 12.77/2.01 join(join(complement(X), Y), complement(X))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(complement(X), join(complement(X), Y))
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) }
% 12.77/2.01 join(join(complement(X), complement(X)), Y)
% 12.77/2.01 = { by lemma 20 }
% 12.77/2.01 join(complement(X), Y)
% 12.77/2.01
% 12.77/2.01 Lemma 56: complement(join(Y, join(complement(X), Z))) = meet(X, complement(join(Y, Z))).
% 12.77/2.01 Proof:
% 12.77/2.01 complement(join(Y, join(complement(X), Z)))
% 12.77/2.01 = { by lemma 37 }
% 12.77/2.01 complement(join(complement(X), join(Y, Z)))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 complement(join(join(Y, Z), complement(X)))
% 12.77/2.01 = { by lemma 43 }
% 12.77/2.01 meet(X, complement(join(Y, Z)))
% 12.77/2.01
% 12.77/2.01 Lemma 57: join(join(X, Y), join(complement(Z), Y)) = join(join(X, Y), complement(Z)).
% 12.77/2.01 Proof:
% 12.77/2.01 join(join(X, Y), join(complement(Z), Y))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) R->L }
% 12.77/2.01 converse(converse(join(join(X, Y), join(complement(Z), Y))))
% 12.77/2.01 = { by lemma 55 R->L }
% 12.77/2.01 converse(converse(join(join(X, Y), join(join(complement(Z), Y), complement(Z)))))
% 12.77/2.01 = { by lemma 53 }
% 12.77/2.01 converse(converse(join(join(complement(Z), Y), join(join(complement(Z), Y), X))))
% 12.77/2.01 = { by axiom 6 (converse_additivity_9) }
% 12.77/2.01 converse(join(converse(join(complement(Z), Y)), converse(join(join(complement(Z), Y), X))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 converse(join(converse(join(complement(Z), Y)), converse(join(X, join(complement(Z), Y)))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 converse(join(converse(join(X, join(complement(Z), Y))), converse(join(complement(Z), Y))))
% 12.77/2.01 = { by axiom 6 (converse_additivity_9) }
% 12.77/2.01 converse(join(join(converse(X), converse(join(complement(Z), Y))), converse(join(complement(Z), Y))))
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 12.77/2.01 converse(join(converse(X), join(converse(join(complement(Z), Y)), converse(join(complement(Z), Y)))))
% 12.77/2.01 = { by lemma 19 R->L }
% 12.77/2.01 converse(join(converse(X), join(converse(join(complement(Z), Y)), composition(one, converse(join(complement(Z), Y))))))
% 12.77/2.01 = { by lemma 19 R->L }
% 12.77/2.01 converse(join(converse(X), join(composition(one, converse(join(complement(Z), Y))), composition(one, converse(join(complement(Z), Y))))))
% 12.77/2.01 = { by axiom 11 (composition_distributivity_7) R->L }
% 12.77/2.01 converse(join(converse(X), composition(join(one, one), converse(join(complement(Z), Y)))))
% 12.77/2.01 = { by lemma 52 }
% 12.77/2.01 converse(converse(join(X, composition(join(complement(Z), Y), converse(join(one, one))))))
% 12.77/2.01 = { by axiom 6 (converse_additivity_9) }
% 12.77/2.01 converse(converse(join(X, composition(join(complement(Z), Y), join(converse(one), converse(one))))))
% 12.77/2.01 = { by lemma 16 }
% 12.77/2.01 converse(converse(join(X, composition(join(complement(Z), Y), join(one, converse(one))))))
% 12.77/2.01 = { by lemma 16 }
% 12.77/2.01 converse(converse(join(X, composition(join(complement(Z), Y), join(one, one)))))
% 12.77/2.01 = { by lemma 32 }
% 12.77/2.01 converse(converse(join(X, composition(join(complement(Z), Y), one))))
% 12.77/2.01 = { by axiom 3 (composition_identity_6) }
% 12.77/2.01 converse(converse(join(X, join(complement(Z), Y))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 converse(converse(join(join(complement(Z), Y), X)))
% 12.77/2.01 = { by lemma 40 }
% 12.77/2.01 converse(converse(join(join(X, Y), complement(Z))))
% 12.77/2.01 = { by axiom 1 (converse_idempotence_8) }
% 12.77/2.01 join(join(X, Y), complement(Z))
% 12.77/2.01
% 12.77/2.01 Lemma 58: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 12.77/2.01 Proof:
% 12.77/2.01 join(meet(X, Y), meet(Y, complement(X)))
% 12.77/2.01 = { by lemma 22 }
% 12.77/2.01 join(meet(Y, X), meet(Y, complement(X)))
% 12.77/2.01 = { by lemma 28 }
% 12.77/2.01 Y
% 12.77/2.01
% 12.77/2.01 Lemma 59: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 12.77/2.01 Proof:
% 12.77/2.01 join(meet(X, complement(Y)), meet(X, Y))
% 12.77/2.01 = { by lemma 41 R->L }
% 12.77/2.01 join(meet(X, complement(Y)), complement(join(complement(Y), complement(X))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 12.77/2.01 = { by lemma 17 }
% 12.77/2.01 X
% 12.77/2.01
% 12.77/2.01 Lemma 60: meet(X, meet(Y, join(complement(Y), complement(X)))) = zero.
% 12.77/2.01 Proof:
% 12.77/2.01 meet(X, meet(Y, join(complement(Y), complement(X))))
% 12.77/2.01 = { by lemma 22 }
% 12.77/2.01 meet(X, meet(join(complement(Y), complement(X)), Y))
% 12.77/2.01 = { by lemma 47 }
% 12.77/2.01 meet(Y, meet(X, join(complement(Y), complement(X))))
% 12.77/2.01 = { by lemma 22 }
% 12.77/2.01 meet(Y, meet(join(complement(Y), complement(X)), X))
% 12.77/2.01 = { by lemma 47 R->L }
% 12.77/2.01 meet(join(complement(Y), complement(X)), meet(X, Y))
% 12.77/2.01 = { by lemma 41 R->L }
% 12.77/2.01 meet(join(complement(Y), complement(X)), complement(join(complement(Y), complement(X))))
% 12.77/2.01 = { by axiom 4 (def_zero_13) R->L }
% 12.77/2.01 zero
% 12.77/2.01
% 12.77/2.01 Lemma 61: join(meet(X, complement(Y)), join(Z, meet(X, Y))) = join(Z, X).
% 12.77/2.01 Proof:
% 12.77/2.01 join(meet(X, complement(Y)), join(Z, meet(X, Y)))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(meet(X, complement(Y)), join(meet(X, Y), Z))
% 12.77/2.01 = { by axiom 7 (maddux2_join_associativity_2) }
% 12.77/2.01 join(join(meet(X, complement(Y)), meet(X, Y)), Z)
% 12.77/2.01 = { by lemma 59 }
% 12.77/2.01 join(X, Z)
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.01 join(Z, X)
% 12.77/2.01
% 12.77/2.01 Goal 1 (goals_17): join(complement(composition(sk1, sk2)), composition(sk1, sk3)) = join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)).
% 12.77/2.01 Proof:
% 12.77/2.01 join(complement(composition(sk1, sk2)), composition(sk1, sk3))
% 12.77/2.01 = { by lemma 33 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), zero)
% 12.77/2.01 = { by lemma 14 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(top))
% 12.77/2.01 = { by lemma 26 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(join(join(complement(top), composition(sk1, sk3)), top)))
% 12.77/2.01 = { by lemma 51 }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(join(top, composition(sk1, sk3))))
% 12.77/2.01 = { by lemma 51 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), composition(sk1, sk2))))
% 12.77/2.01 = { by lemma 49 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(complement(composition(sk1, sk2)), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.01 = { by lemma 50 }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(join(composition(sk1, sk2), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.01 = { by lemma 56 }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(join(composition(sk1, sk2), composition(sk1, sk3)))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, sk2)))))
% 12.77/2.01 = { by lemma 54 }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(sk2, sk3)))))
% 12.77/2.01 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(sk3, sk2)))))
% 12.77/2.01 = { by lemma 61 R->L }
% 12.77/2.01 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), join(sk3, meet(sk2, sk3)))))))
% 12.77/2.02 = { by lemma 41 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), join(sk3, complement(join(complement(sk3), complement(sk2)))))))))
% 12.77/2.02 = { by lemma 42 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), complement(meet(join(complement(sk3), complement(sk2)), complement(sk3))))))))
% 12.77/2.02 = { by lemma 22 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), complement(meet(complement(sk3), join(complement(sk3), complement(sk2)))))))))
% 12.77/2.02 = { by lemma 44 }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), complement(complement(sk3)))))))
% 12.77/2.02 = { by lemma 31 }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), sk3)))))
% 12.77/2.02 = { by lemma 54 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, meet(sk2, complement(sk3)))))))
% 12.77/2.02 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 56 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(join(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 50 R->L }
% 12.77/2.02 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.77/2.02 join(meet(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))
% 12.77/2.02 = { by lemma 46 R->L }
% 12.77/2.02 join(meet(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 31 R->L }
% 12.77/2.02 join(meet(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 22 }
% 12.77/2.02 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), complement(composition(sk1, meet(sk2, complement(sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 45 R->L }
% 12.77/2.02 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), composition(sk1, sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 37 R->L }
% 12.77/2.02 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.77/2.02 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.77/2.02 = { by lemma 48 }
% 12.77/2.02 join(meet(complement(composition(sk1, meet(sk2, complement(sk3)))), meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 22 R->L }
% 12.95/2.02 join(meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 33 R->L }
% 12.95/2.02 join(join(meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3))))), zero), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.95/2.02 join(join(zero, meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 60 R->L }
% 12.95/2.02 join(join(meet(join(zero, complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(join(zero, complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 23 }
% 12.95/2.02 join(join(meet(join(zero, complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), meet(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), top)))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 30 }
% 12.95/2.02 join(join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), meet(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), top)))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 35 }
% 12.95/2.02 join(join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(complement(composition(sk1, sk2)), composition(sk1, sk3))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 48 }
% 12.95/2.02 join(join(meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), join(complement(composition(sk1, sk2)), composition(sk1, sk3))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 37 }
% 12.95/2.02 join(join(meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.95/2.02 join(join(meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.02 = { by lemma 57 R->L }
% 12.95/2.02 join(join(meet(composition(sk1, meet(sk2, complement(sk3))), meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3))))), meet(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), complement(composition(sk1, meet(sk2, complement(sk3)))))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.03 = { by lemma 58 }
% 12.95/2.03 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))))
% 12.95/2.03 = { by lemma 22 }
% 12.95/2.03 join(meet(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))), join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))), meet(join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3))), complement(complement(join(complement(composition(sk1, sk2)), composition(sk1, sk3))))))
% 12.95/2.03 = { by lemma 58 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3)))
% 12.95/2.03 = { by lemma 57 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, sk2)))
% 12.95/2.03 = { by lemma 40 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))
% 12.95/2.03 = { by lemma 20 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, sk2)), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, meet(sk2, complement(sk3))))))
% 12.95/2.03 = { by lemma 53 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 59 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))), meet(join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))))))
% 12.95/2.03 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(composition(sk1, meet(sk2, complement(sk3))))))))
% 12.95/2.03 = { by lemma 31 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3))))))))))
% 12.95/2.03 = { by lemma 28 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), join(meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))), complement(composition(sk1, meet(sk2, complement(sk3))))), meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))), complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), join(meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))), complement(composition(sk1, meet(sk2, complement(sk3))))), meet(complement(complement(composition(sk1, meet(sk2, complement(sk3))))), meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3))))))))))))
% 12.95/2.03 = { by lemma 60 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), join(meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))), complement(composition(sk1, meet(sk2, complement(sk3))))), zero)))
% 12.95/2.03 = { by lemma 33 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, complement(sk3)))))))), complement(composition(sk1, meet(sk2, complement(sk3)))))))
% 12.95/2.03 = { by lemma 31 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(composition(sk1, meet(sk2, complement(sk3)))))), complement(composition(sk1, meet(sk2, complement(sk3)))))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), meet(composition(sk1, sk2), join(complement(composition(sk1, sk2)), complement(composition(sk1, meet(sk2, complement(sk3)))))))))
% 12.95/2.03 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), meet(composition(sk1, sk2), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))))))
% 12.95/2.03 = { by lemma 48 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))))))
% 12.95/2.03 = { by lemma 45 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(composition(sk1, sk2), complement(composition(sk1, meet(sk2, complement(sk3)))))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), join(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2)))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 46 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 32 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(sk2, sk2))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 61 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(meet(sk2, complement(sk3)), join(sk2, meet(sk2, sk3))))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(meet(sk2, complement(sk3)), join(meet(sk2, sk3), sk2)))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 61 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(meet(sk2, complement(sk3)), join(meet(sk2, complement(sk3)), join(meet(sk2, sk3), meet(sk2, sk3)))))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 32 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(meet(sk2, complement(sk3)), join(meet(sk2, complement(sk3)), meet(sk2, sk3))))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 59 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(composition(sk1, join(meet(sk2, complement(sk3)), sk2))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 54 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(join(composition(sk1, sk2), composition(sk1, meet(sk2, complement(sk3))))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by axiom 2 (maddux1_join_commutativity_1) }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(complement(join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk2))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 49 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, sk2)), complement(composition(sk1, meet(sk2, complement(sk3))))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 22 R->L }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), join(meet(complement(composition(sk1, meet(sk2, complement(sk3)))), complement(composition(sk1, sk2))), meet(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk2))))
% 12.95/2.03 = { by lemma 59 }
% 12.95/2.03 join(join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3)), complement(composition(sk1, meet(sk2, complement(sk3)))))
% 12.95/2.03 = { by lemma 55 }
% 12.95/2.03 join(complement(composition(sk1, meet(sk2, complement(sk3)))), composition(sk1, sk3))
% 12.95/2.03 % SZS output end Proof
% 12.95/2.03
% 12.95/2.03 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------