TSTP Solution File: REL040-3 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL040-3 : 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 : n018.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 13:44:25 EDT 2023
% Result : Unsatisfiable 110.48s 14.56s
% Output : Proof 112.27s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL040-3 : 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
% 0.16/0.34 % Computer : n018.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Fri Aug 25 22:26:29 EDT 2023
% 0.16/0.34 % CPUTime :
% 110.48/14.56 Command-line arguments: --ground-connectedness --complete-subsets
% 110.48/14.56
% 110.48/14.56 % SZS status Unsatisfiable
% 110.48/14.56
% 111.92/14.69 % SZS output start Proof
% 111.92/14.69 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 111.92/14.69 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 111.92/14.69 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 111.92/14.69 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 111.92/14.69 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 111.92/14.69 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 111.92/14.69 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 111.92/14.69 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 111.92/14.69 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 111.92/14.69 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 111.92/14.69 Axiom 11 (goals_17): join(composition(converse(sk1), sk1), one) = one.
% 111.92/14.69 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 111.92/14.69 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 111.92/14.69 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 111.92/14.69 Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 111.92/14.69 Axiom 16 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 111.92/14.69 Axiom 17 (dedekind_law_14): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 111.92/14.69
% 111.92/14.69 Lemma 18: complement(top) = zero.
% 111.92/14.69 Proof:
% 111.92/14.69 complement(top)
% 111.92/14.69 = { by axiom 4 (def_top_12) }
% 111.92/14.69 complement(join(complement(X), complement(complement(X))))
% 111.92/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 111.92/14.69 meet(X, complement(X))
% 111.92/14.69 = { by axiom 5 (def_zero_13) R->L }
% 111.92/14.69 zero
% 111.92/14.69
% 111.92/14.69 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 111.92/14.69 Proof:
% 111.92/14.69 join(X, join(Y, complement(X)))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(X, join(complement(X), Y))
% 111.92/14.69 = { by axiom 7 (maddux2_join_associativity_2) }
% 111.92/14.69 join(join(X, complement(X)), Y)
% 111.92/14.69 = { by axiom 4 (def_top_12) R->L }
% 111.92/14.69 join(top, Y)
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) }
% 111.92/14.69 join(Y, top)
% 111.92/14.69
% 111.92/14.69 Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 111.92/14.69 Proof:
% 111.92/14.69 converse(composition(converse(X), Y))
% 111.92/14.69 = { by axiom 8 (converse_multiplicativity_10) }
% 111.92/14.69 composition(converse(Y), converse(converse(X)))
% 111.92/14.69 = { by axiom 1 (converse_idempotence_8) }
% 111.92/14.69 composition(converse(Y), X)
% 111.92/14.69
% 111.92/14.69 Lemma 21: composition(converse(one), X) = X.
% 111.92/14.69 Proof:
% 111.92/14.69 composition(converse(one), X)
% 111.92/14.69 = { by lemma 20 R->L }
% 111.92/14.69 converse(composition(converse(X), one))
% 111.92/14.69 = { by axiom 3 (composition_identity_6) }
% 111.92/14.69 converse(converse(X))
% 111.92/14.69 = { by axiom 1 (converse_idempotence_8) }
% 111.92/14.69 X
% 111.92/14.69
% 111.92/14.69 Lemma 22: composition(one, X) = X.
% 111.92/14.69 Proof:
% 111.92/14.69 composition(one, X)
% 111.92/14.69 = { by lemma 21 R->L }
% 111.92/14.69 composition(converse(one), composition(one, X))
% 111.92/14.69 = { by axiom 9 (composition_associativity_5) }
% 111.92/14.69 composition(composition(converse(one), one), X)
% 111.92/14.69 = { by axiom 3 (composition_identity_6) }
% 111.92/14.69 composition(converse(one), X)
% 111.92/14.69 = { by lemma 21 }
% 111.92/14.69 X
% 111.92/14.69
% 111.92/14.69 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 111.92/14.69 Proof:
% 111.92/14.69 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 111.92/14.69 = { by axiom 13 (converse_cancellativity_11) }
% 111.92/14.69 complement(X)
% 111.92/14.69
% 111.92/14.69 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 111.92/14.69 Proof:
% 111.92/14.69 join(complement(X), complement(X))
% 111.92/14.69 = { by lemma 21 R->L }
% 111.92/14.69 join(complement(X), composition(converse(one), complement(X)))
% 111.92/14.69 = { by lemma 22 R->L }
% 111.92/14.69 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 111.92/14.69 = { by lemma 23 }
% 111.92/14.69 complement(X)
% 111.92/14.69
% 111.92/14.69 Lemma 25: join(top, complement(X)) = top.
% 111.92/14.69 Proof:
% 111.92/14.69 join(top, complement(X))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(complement(X), top)
% 111.92/14.69 = { by lemma 19 R->L }
% 111.92/14.69 join(X, join(complement(X), complement(X)))
% 111.92/14.69 = { by lemma 24 }
% 111.92/14.69 join(X, complement(X))
% 111.92/14.69 = { by axiom 4 (def_top_12) R->L }
% 111.92/14.69 top
% 111.92/14.69
% 111.92/14.69 Lemma 26: join(Y, top) = join(X, top).
% 111.92/14.69 Proof:
% 111.92/14.69 join(Y, top)
% 111.92/14.69 = { by lemma 25 R->L }
% 111.92/14.69 join(Y, join(top, complement(Y)))
% 111.92/14.69 = { by lemma 19 }
% 111.92/14.69 join(top, top)
% 111.92/14.69 = { by lemma 19 R->L }
% 111.92/14.69 join(X, join(top, complement(X)))
% 111.92/14.69 = { by lemma 25 }
% 111.92/14.69 join(X, top)
% 111.92/14.69
% 111.92/14.69 Lemma 27: join(X, top) = top.
% 111.92/14.69 Proof:
% 111.92/14.69 join(X, top)
% 111.92/14.69 = { by lemma 26 }
% 111.92/14.69 join(join(zero, zero), top)
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(top, join(zero, zero))
% 111.92/14.69 = { by lemma 18 R->L }
% 111.92/14.69 join(top, join(zero, complement(top)))
% 111.92/14.69 = { by lemma 18 R->L }
% 111.92/14.69 join(top, join(complement(top), complement(top)))
% 111.92/14.69 = { by lemma 24 }
% 111.92/14.69 join(top, complement(top))
% 111.92/14.69 = { by axiom 4 (def_top_12) R->L }
% 111.92/14.69 top
% 111.92/14.69
% 111.92/14.69 Lemma 28: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 111.92/14.69 Proof:
% 111.92/14.69 converse(join(X, converse(Y)))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 converse(join(converse(Y), X))
% 111.92/14.69 = { by axiom 6 (converse_additivity_9) }
% 111.92/14.69 join(converse(converse(Y)), converse(X))
% 111.92/14.69 = { by axiom 1 (converse_idempotence_8) }
% 111.92/14.69 join(Y, converse(X))
% 111.92/14.69
% 111.92/14.69 Lemma 29: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 111.92/14.69 Proof:
% 111.92/14.69 converse(join(converse(X), Y))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 converse(join(Y, converse(X)))
% 111.92/14.69 = { by lemma 28 }
% 111.92/14.69 join(X, converse(Y))
% 111.92/14.69
% 111.92/14.69 Lemma 30: join(X, converse(complement(converse(X)))) = converse(top).
% 111.92/14.69 Proof:
% 111.92/14.69 join(X, converse(complement(converse(X))))
% 111.92/14.69 = { by lemma 29 R->L }
% 111.92/14.69 converse(join(converse(X), complement(converse(X))))
% 111.92/14.69 = { by axiom 4 (def_top_12) R->L }
% 111.92/14.69 converse(top)
% 111.92/14.69
% 111.92/14.69 Lemma 31: join(X, join(complement(X), Y)) = top.
% 111.92/14.69 Proof:
% 111.92/14.69 join(X, join(complement(X), Y))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(X, join(Y, complement(X)))
% 111.92/14.69 = { by lemma 19 }
% 111.92/14.69 join(Y, top)
% 111.92/14.69 = { by lemma 26 R->L }
% 111.92/14.69 join(Z, top)
% 111.92/14.69 = { by lemma 27 }
% 111.92/14.69 top
% 111.92/14.69
% 111.92/14.69 Lemma 32: join(X, converse(top)) = top.
% 111.92/14.69 Proof:
% 111.92/14.69 join(X, converse(top))
% 111.92/14.69 = { by lemma 30 R->L }
% 111.92/14.69 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 111.92/14.69 = { by lemma 31 }
% 111.92/14.69 top
% 111.92/14.69
% 111.92/14.69 Lemma 33: converse(top) = top.
% 111.92/14.69 Proof:
% 111.92/14.69 converse(top)
% 111.92/14.69 = { by lemma 27 R->L }
% 111.92/14.69 converse(join(X, top))
% 111.92/14.69 = { by axiom 6 (converse_additivity_9) }
% 111.92/14.69 join(converse(X), converse(top))
% 111.92/14.69 = { by lemma 32 }
% 111.92/14.69 top
% 111.92/14.69
% 111.92/14.69 Lemma 34: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 111.92/14.69 Proof:
% 111.92/14.69 join(meet(X, Y), complement(join(complement(X), Y)))
% 111.92/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 111.92/14.69 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 111.92/14.69 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 111.92/14.69 X
% 111.92/14.69
% 111.92/14.69 Lemma 35: join(zero, meet(X, X)) = X.
% 111.92/14.69 Proof:
% 111.92/14.69 join(zero, meet(X, X))
% 111.92/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 111.92/14.69 join(zero, complement(join(complement(X), complement(X))))
% 111.92/14.69 = { by axiom 5 (def_zero_13) }
% 111.92/14.69 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 111.92/14.69 = { by lemma 34 }
% 111.92/14.69 X
% 111.92/14.69
% 111.92/14.69 Lemma 36: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 111.92/14.69 Proof:
% 111.92/14.69 join(zero, join(X, complement(complement(Y))))
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 111.92/14.69 join(zero, join(complement(complement(Y)), X))
% 111.92/14.69 = { by lemma 24 R->L }
% 111.92/14.69 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 111.92/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 111.92/14.69 join(zero, join(meet(Y, Y), X))
% 111.92/14.69 = { by axiom 7 (maddux2_join_associativity_2) }
% 111.92/14.69 join(join(zero, meet(Y, Y)), X)
% 111.92/14.69 = { by lemma 35 }
% 111.92/14.69 join(Y, X)
% 111.92/14.69 = { by axiom 2 (maddux1_join_commutativity_1) }
% 111.92/14.69 join(X, Y)
% 111.92/14.69
% 111.92/14.69 Lemma 37: join(zero, complement(complement(X))) = X.
% 111.92/14.69 Proof:
% 112.27/14.69 join(zero, complement(complement(X)))
% 112.27/14.69 = { by axiom 5 (def_zero_13) }
% 112.27/14.69 join(meet(X, complement(X)), complement(complement(X)))
% 112.27/14.69 = { by lemma 24 R->L }
% 112.27/14.69 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 112.27/14.69 = { by lemma 34 }
% 112.27/14.69 X
% 112.27/14.69
% 112.27/14.69 Lemma 38: join(X, zero) = join(X, X).
% 112.27/14.69 Proof:
% 112.27/14.69 join(X, zero)
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 join(zero, X)
% 112.27/14.69 = { by lemma 37 R->L }
% 112.27/14.69 join(zero, join(zero, complement(complement(X))))
% 112.27/14.69 = { by lemma 24 R->L }
% 112.27/14.69 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 112.27/14.69 = { by lemma 36 }
% 112.27/14.69 join(zero, join(complement(complement(X)), X))
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.69 join(zero, join(X, complement(complement(X))))
% 112.27/14.69 = { by lemma 36 }
% 112.27/14.69 join(X, X)
% 112.27/14.69
% 112.27/14.69 Lemma 39: join(zero, complement(X)) = complement(X).
% 112.27/14.69 Proof:
% 112.27/14.69 join(zero, complement(X))
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 join(complement(X), zero)
% 112.27/14.69 = { by lemma 38 }
% 112.27/14.69 join(complement(X), complement(X))
% 112.27/14.69 = { by lemma 24 }
% 112.27/14.69 complement(X)
% 112.27/14.69
% 112.27/14.69 Lemma 40: join(X, zero) = X.
% 112.27/14.69 Proof:
% 112.27/14.69 join(X, zero)
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 join(zero, X)
% 112.27/14.69 = { by lemma 36 R->L }
% 112.27/14.69 join(zero, join(zero, complement(complement(X))))
% 112.27/14.69 = { by lemma 39 }
% 112.27/14.69 join(zero, complement(complement(X)))
% 112.27/14.69 = { by lemma 37 }
% 112.27/14.69 X
% 112.27/14.69
% 112.27/14.69 Lemma 41: join(zero, X) = X.
% 112.27/14.69 Proof:
% 112.27/14.69 join(zero, X)
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 join(X, zero)
% 112.27/14.69 = { by lemma 40 }
% 112.27/14.69 X
% 112.27/14.69
% 112.27/14.69 Lemma 42: meet(Y, X) = meet(X, Y).
% 112.27/14.69 Proof:
% 112.27/14.69 meet(Y, X)
% 112.27/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.69 complement(join(complement(Y), complement(X)))
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 complement(join(complement(X), complement(Y)))
% 112.27/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.69 meet(X, Y)
% 112.27/14.69
% 112.27/14.69 Lemma 43: complement(join(zero, complement(X))) = meet(X, top).
% 112.27/14.69 Proof:
% 112.27/14.69 complement(join(zero, complement(X)))
% 112.27/14.69 = { by lemma 18 R->L }
% 112.27/14.69 complement(join(complement(top), complement(X)))
% 112.27/14.69 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.69 meet(top, X)
% 112.27/14.69 = { by lemma 42 R->L }
% 112.27/14.69 meet(X, top)
% 112.27/14.69
% 112.27/14.69 Lemma 44: join(X, complement(zero)) = top.
% 112.27/14.69 Proof:
% 112.27/14.69 join(X, complement(zero))
% 112.27/14.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.69 join(complement(zero), X)
% 112.27/14.69 = { by lemma 36 R->L }
% 112.27/14.69 join(zero, join(complement(zero), complement(complement(X))))
% 112.27/14.69 = { by lemma 31 }
% 112.27/14.69 top
% 112.27/14.69
% 112.27/14.70 Lemma 45: meet(X, zero) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 meet(X, zero)
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 complement(join(complement(X), complement(zero)))
% 112.27/14.70 = { by lemma 44 }
% 112.27/14.70 complement(top)
% 112.27/14.70 = { by lemma 18 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 46: join(meet(X, Y), meet(X, complement(Y))) = X.
% 112.27/14.70 Proof:
% 112.27/14.70 join(meet(X, Y), meet(X, complement(Y)))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 join(meet(X, complement(Y)), meet(X, Y))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 112.27/14.70 = { by lemma 34 }
% 112.27/14.70 X
% 112.27/14.70
% 112.27/14.70 Lemma 47: meet(X, top) = X.
% 112.27/14.70 Proof:
% 112.27/14.70 meet(X, top)
% 112.27/14.70 = { by lemma 43 R->L }
% 112.27/14.70 complement(join(zero, complement(X)))
% 112.27/14.70 = { by lemma 39 R->L }
% 112.27/14.70 join(zero, complement(join(zero, complement(X))))
% 112.27/14.70 = { by lemma 43 }
% 112.27/14.70 join(zero, meet(X, top))
% 112.27/14.70 = { by lemma 44 R->L }
% 112.27/14.70 join(zero, meet(X, join(complement(zero), complement(zero))))
% 112.27/14.70 = { by lemma 24 }
% 112.27/14.70 join(zero, meet(X, complement(zero)))
% 112.27/14.70 = { by lemma 45 R->L }
% 112.27/14.70 join(meet(X, zero), meet(X, complement(zero)))
% 112.27/14.70 = { by lemma 46 }
% 112.27/14.70 X
% 112.27/14.70
% 112.27/14.70 Lemma 48: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 112.27/14.70 Proof:
% 112.27/14.70 join(meet(X, Y), meet(X, Y))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 join(meet(Y, X), meet(X, Y))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 join(meet(Y, X), meet(Y, X))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 112.27/14.70 = { by lemma 24 }
% 112.27/14.70 complement(join(complement(Y), complement(X)))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.70 meet(Y, X)
% 112.27/14.70 = { by lemma 42 R->L }
% 112.27/14.70 meet(X, Y)
% 112.27/14.70
% 112.27/14.70 Lemma 49: converse(zero) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 converse(zero)
% 112.27/14.70 = { by lemma 41 R->L }
% 112.27/14.70 join(zero, converse(zero))
% 112.27/14.70 = { by lemma 29 R->L }
% 112.27/14.70 converse(join(converse(zero), zero))
% 112.27/14.70 = { by lemma 38 }
% 112.27/14.70 converse(join(converse(zero), converse(zero)))
% 112.27/14.70 = { by lemma 28 }
% 112.27/14.70 join(zero, converse(converse(zero)))
% 112.27/14.70 = { by axiom 1 (converse_idempotence_8) }
% 112.27/14.70 join(zero, zero)
% 112.27/14.70 = { by lemma 47 R->L }
% 112.27/14.70 join(zero, meet(zero, top))
% 112.27/14.70 = { by lemma 47 R->L }
% 112.27/14.70 join(meet(zero, top), meet(zero, top))
% 112.27/14.70 = { by lemma 48 }
% 112.27/14.70 meet(zero, top)
% 112.27/14.70 = { by lemma 47 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 50: join(top, X) = top.
% 112.27/14.70 Proof:
% 112.27/14.70 join(top, X)
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 join(X, top)
% 112.27/14.70 = { by lemma 26 R->L }
% 112.27/14.70 join(Y, top)
% 112.27/14.70 = { by lemma 27 }
% 112.27/14.70 top
% 112.27/14.70
% 112.27/14.70 Lemma 51: complement(complement(X)) = X.
% 112.27/14.70 Proof:
% 112.27/14.70 complement(complement(X))
% 112.27/14.70 = { by lemma 39 R->L }
% 112.27/14.70 join(zero, complement(complement(X)))
% 112.27/14.70 = { by lemma 37 }
% 112.27/14.70 X
% 112.27/14.70
% 112.27/14.70 Lemma 52: meet(zero, X) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 meet(zero, X)
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 meet(X, zero)
% 112.27/14.70 = { by lemma 45 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 53: composition(top, zero) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 composition(top, zero)
% 112.27/14.70 = { by lemma 33 R->L }
% 112.27/14.70 composition(converse(top), zero)
% 112.27/14.70 = { by lemma 41 R->L }
% 112.27/14.70 join(zero, composition(converse(top), zero))
% 112.27/14.70 = { by lemma 18 R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), zero))
% 112.27/14.70 = { by lemma 18 R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(top)))
% 112.27/14.70 = { by lemma 50 R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 112.27/14.70 = { by lemma 33 R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 112.27/14.70 = { by lemma 22 R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 112.27/14.70 = { by axiom 12 (composition_distributivity_7) R->L }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 112.27/14.70 = { by lemma 32 }
% 112.27/14.70 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 112.27/14.70 = { by lemma 23 }
% 112.27/14.70 complement(top)
% 112.27/14.70 = { by lemma 18 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 54: composition(X, zero) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 composition(X, zero)
% 112.27/14.70 = { by lemma 41 R->L }
% 112.27/14.70 join(zero, composition(X, zero))
% 112.27/14.70 = { by lemma 53 R->L }
% 112.27/14.70 join(composition(top, zero), composition(X, zero))
% 112.27/14.70 = { by axiom 12 (composition_distributivity_7) R->L }
% 112.27/14.70 composition(join(top, X), zero)
% 112.27/14.70 = { by lemma 50 }
% 112.27/14.70 composition(top, zero)
% 112.27/14.70 = { by lemma 53 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 55: composition(zero, X) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 composition(zero, X)
% 112.27/14.70 = { by lemma 49 R->L }
% 112.27/14.70 composition(converse(zero), X)
% 112.27/14.70 = { by lemma 20 R->L }
% 112.27/14.70 converse(composition(converse(X), zero))
% 112.27/14.70 = { by lemma 54 }
% 112.27/14.70 converse(zero)
% 112.27/14.70 = { by lemma 49 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 56: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 112.27/14.70 Proof:
% 112.27/14.70 meet(X, join(complement(Y), complement(Z)))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 meet(X, join(complement(Z), complement(Y)))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 meet(join(complement(Z), complement(Y)), X)
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.70 complement(join(meet(Z, Y), complement(X)))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.70 complement(join(complement(X), meet(Z, Y)))
% 112.27/14.70 = { by lemma 42 R->L }
% 112.27/14.70 complement(join(complement(X), meet(Y, Z)))
% 112.27/14.70
% 112.27/14.70 Lemma 57: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 112.27/14.70 Proof:
% 112.27/14.70 complement(join(X, complement(Y)))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 complement(join(complement(Y), X))
% 112.27/14.70 = { by lemma 47 R->L }
% 112.27/14.70 complement(join(complement(Y), meet(X, top)))
% 112.27/14.70 = { by lemma 42 R->L }
% 112.27/14.70 complement(join(complement(Y), meet(top, X)))
% 112.27/14.70 = { by lemma 56 R->L }
% 112.27/14.70 meet(Y, join(complement(top), complement(X)))
% 112.27/14.70 = { by lemma 18 }
% 112.27/14.70 meet(Y, join(zero, complement(X)))
% 112.27/14.70 = { by lemma 39 }
% 112.27/14.70 meet(Y, complement(X))
% 112.27/14.70
% 112.27/14.70 Lemma 58: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 112.27/14.70 Proof:
% 112.27/14.70 complement(join(complement(X), Y))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 complement(join(Y, complement(X)))
% 112.27/14.70 = { by lemma 57 }
% 112.27/14.70 meet(X, complement(Y))
% 112.27/14.70
% 112.27/14.70 Lemma 59: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 112.27/14.70 Proof:
% 112.27/14.70 join(complement(X), complement(Y))
% 112.27/14.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.70 join(complement(Y), complement(X))
% 112.27/14.70 = { by lemma 35 R->L }
% 112.27/14.70 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 112.27/14.70 = { by lemma 56 }
% 112.27/14.70 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 112.27/14.70 = { by lemma 39 }
% 112.27/14.70 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.70 complement(join(meet(Y, X), meet(Y, X)))
% 112.27/14.70 = { by lemma 48 }
% 112.27/14.70 complement(meet(Y, X))
% 112.27/14.70 = { by lemma 42 R->L }
% 112.27/14.70 complement(meet(X, Y))
% 112.27/14.70
% 112.27/14.70 Lemma 60: join(X, complement(meet(X, Y))) = top.
% 112.27/14.70 Proof:
% 112.27/14.70 join(X, complement(meet(X, Y)))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 join(X, complement(meet(Y, X)))
% 112.27/14.70 = { by lemma 59 R->L }
% 112.27/14.70 join(X, join(complement(Y), complement(X)))
% 112.27/14.70 = { by lemma 19 }
% 112.27/14.70 join(complement(Y), top)
% 112.27/14.70 = { by lemma 27 }
% 112.27/14.70 top
% 112.27/14.70
% 112.27/14.70 Lemma 61: meet(X, meet(Y, complement(X))) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 meet(X, meet(Y, complement(X)))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 meet(X, meet(complement(X), Y))
% 112.27/14.70 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.70 complement(join(complement(X), complement(meet(complement(X), Y))))
% 112.27/14.70 = { by lemma 60 }
% 112.27/14.70 complement(top)
% 112.27/14.70 = { by lemma 18 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 62: meet(one, composition(converse(complement(X)), X)) = zero.
% 112.27/14.70 Proof:
% 112.27/14.70 meet(one, composition(converse(complement(X)), X))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 meet(composition(converse(complement(X)), X), one)
% 112.27/14.70 = { by lemma 51 R->L }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 112.27/14.70 = { by lemma 23 R->L }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 112.27/14.70 = { by axiom 3 (composition_identity_6) }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 112.27/14.70 = { by lemma 43 }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 112.27/14.70 = { by lemma 39 }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 112.27/14.70 = { by lemma 47 }
% 112.27/14.70 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 112.27/14.70 = { by lemma 58 }
% 112.27/14.70 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 112.27/14.70 = { by lemma 61 }
% 112.27/14.70 zero
% 112.27/14.70
% 112.27/14.70 Lemma 63: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 112.27/14.70 Proof:
% 112.27/14.70 join(meet(X, Y), meet(Y, complement(X)))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 join(meet(Y, X), meet(Y, complement(X)))
% 112.27/14.70 = { by lemma 46 }
% 112.27/14.70 Y
% 112.27/14.70
% 112.27/14.70 Lemma 64: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 112.27/14.70 Proof:
% 112.27/14.70 join(meet(X, Y), meet(complement(X), Y))
% 112.27/14.70 = { by lemma 42 }
% 112.27/14.70 join(meet(X, Y), meet(Y, complement(X)))
% 112.27/14.70 = { by lemma 63 }
% 112.27/14.70 Y
% 112.27/14.70
% 112.27/14.70 Lemma 65: converse(complement(X)) = complement(converse(X)).
% 112.27/14.70 Proof:
% 112.27/14.70 converse(complement(X))
% 112.27/14.70 = { by lemma 39 R->L }
% 112.27/14.70 converse(join(zero, complement(X)))
% 112.27/14.70 = { by lemma 34 R->L }
% 112.27/14.70 converse(join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 112.27/14.70 = { by lemma 58 R->L }
% 112.27/14.71 converse(join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 112.27/14.71 = { by lemma 30 }
% 112.27/14.71 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 112.27/14.71 = { by lemma 33 }
% 112.27/14.71 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 112.27/14.71 = { by lemma 18 }
% 112.27/14.71 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 112.27/14.71 = { by lemma 39 }
% 112.27/14.71 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 112.27/14.71 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 112.27/14.71 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 112.27/14.71 = { by lemma 40 R->L }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 112.27/14.71 = { by lemma 52 R->L }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X)))))))))
% 112.27/14.71 = { by lemma 55 R->L }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 112.27/14.71 = { by lemma 62 R->L }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 112.27/14.71 = { by axiom 16 (modular_law_2_16) R->L }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(one, composition(converse(complement(converse(complement(join(zero, complement(X)))))), converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 112.27/14.71 = { by lemma 62 }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(one, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 112.27/14.71 = { by lemma 22 }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))))
% 112.27/14.71 = { by lemma 55 }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(X))))))))))
% 112.27/14.71 = { by lemma 52 }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))), zero)))
% 112.27/14.71 = { by lemma 40 }
% 112.27/14.71 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
% 112.27/14.71 = { by lemma 64 }
% 112.27/14.71 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 112.27/14.71 = { by axiom 1 (converse_idempotence_8) }
% 112.27/14.71 complement(converse(complement(join(zero, complement(X)))))
% 112.27/14.71 = { by lemma 43 }
% 112.27/14.71 complement(converse(meet(X, top)))
% 112.27/14.71 = { by lemma 47 }
% 112.27/14.71 complement(converse(X))
% 112.27/14.71
% 112.27/14.71 Lemma 66: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 112.27/14.71 Proof:
% 112.27/14.71 complement(meet(X, complement(Y)))
% 112.27/14.71 = { by lemma 41 R->L }
% 112.27/14.71 complement(join(zero, meet(X, complement(Y))))
% 112.27/14.71 = { by lemma 57 R->L }
% 112.27/14.71 complement(join(zero, complement(join(Y, complement(X)))))
% 112.27/14.71 = { by lemma 43 }
% 112.27/14.71 meet(join(Y, complement(X)), top)
% 112.27/14.71 = { by lemma 47 }
% 112.27/14.71 join(Y, complement(X))
% 112.27/14.71
% 112.27/14.71 Lemma 67: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 112.27/14.71 Proof:
% 112.27/14.71 complement(meet(complement(X), Y))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 complement(meet(Y, complement(X)))
% 112.27/14.71 = { by lemma 66 }
% 112.27/14.71 join(X, complement(Y))
% 112.27/14.71
% 112.27/14.71 Lemma 68: meet(X, join(X, complement(Y))) = X.
% 112.27/14.71 Proof:
% 112.27/14.71 meet(X, join(X, complement(Y)))
% 112.27/14.71 = { by lemma 66 R->L }
% 112.27/14.71 meet(X, complement(meet(Y, complement(X))))
% 112.27/14.71 = { by lemma 58 R->L }
% 112.27/14.71 complement(join(complement(X), meet(Y, complement(X))))
% 112.27/14.71 = { by lemma 39 R->L }
% 112.27/14.71 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 112.27/14.71 = { by lemma 61 R->L }
% 112.27/14.71 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 112.27/14.71 = { by lemma 34 }
% 112.27/14.71 X
% 112.27/14.71
% 112.27/14.71 Lemma 69: join(X, meet(X, Y)) = X.
% 112.27/14.71 Proof:
% 112.27/14.71 join(X, meet(X, Y))
% 112.27/14.71 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.71 join(X, complement(join(complement(X), complement(Y))))
% 112.27/14.71 = { by lemma 67 R->L }
% 112.27/14.71 complement(meet(complement(X), join(complement(X), complement(Y))))
% 112.27/14.71 = { by lemma 68 }
% 112.27/14.71 complement(complement(X))
% 112.27/14.71 = { by lemma 51 }
% 112.27/14.71 X
% 112.27/14.71
% 112.27/14.71 Lemma 70: meet(X, join(X, Y)) = X.
% 112.27/14.71 Proof:
% 112.27/14.71 meet(X, join(X, Y))
% 112.27/14.71 = { by lemma 47 R->L }
% 112.27/14.71 meet(X, join(X, meet(Y, top)))
% 112.27/14.71 = { by lemma 43 R->L }
% 112.27/14.71 meet(X, join(X, complement(join(zero, complement(Y)))))
% 112.27/14.71 = { by lemma 68 }
% 112.27/14.71 X
% 112.27/14.71
% 112.27/14.71 Lemma 71: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 112.27/14.71 Proof:
% 112.27/14.71 meet(complement(X), complement(Y))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 meet(complement(Y), complement(X))
% 112.27/14.71 = { by lemma 39 R->L }
% 112.27/14.71 meet(join(zero, complement(Y)), complement(X))
% 112.27/14.71 = { by lemma 57 R->L }
% 112.27/14.71 complement(join(X, complement(join(zero, complement(Y)))))
% 112.27/14.71 = { by lemma 43 }
% 112.27/14.71 complement(join(X, meet(Y, top)))
% 112.27/14.71 = { by lemma 47 }
% 112.27/14.71 complement(join(X, Y))
% 112.27/14.71
% 112.27/14.71 Lemma 72: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 112.27/14.71 Proof:
% 112.27/14.71 meet(Y, meet(Z, X))
% 112.27/14.71 = { by lemma 47 R->L }
% 112.27/14.71 meet(meet(Y, top), meet(Z, X))
% 112.27/14.71 = { by lemma 43 R->L }
% 112.27/14.71 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 meet(complement(join(zero, complement(Y))), meet(X, Z))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 meet(meet(X, Z), complement(join(zero, complement(Y))))
% 112.27/14.71 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.71 meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
% 112.27/14.71 = { by lemma 71 }
% 112.27/14.71 complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
% 112.27/14.71 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 112.27/14.71 complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
% 112.27/14.71 = { by lemma 58 }
% 112.27/14.71 meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
% 112.27/14.71 = { by lemma 58 }
% 112.27/14.71 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 112.27/14.71 = { by lemma 43 }
% 112.27/14.71 meet(X, meet(Z, meet(Y, top)))
% 112.27/14.71 = { by lemma 47 }
% 112.27/14.71 meet(X, meet(Z, Y))
% 112.27/14.71 = { by lemma 42 R->L }
% 112.27/14.71 meet(X, meet(Y, Z))
% 112.27/14.71
% 112.27/14.71 Lemma 73: meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 112.27/14.71 Proof:
% 112.27/14.71 meet(meet(X, Y), Z)
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 meet(Z, meet(X, Y))
% 112.27/14.71 = { by lemma 72 R->L }
% 112.27/14.71 meet(X, meet(Y, Z))
% 112.27/14.71
% 112.27/14.71 Lemma 74: join(complement(X), meet(complement(Y), Z)) = complement(meet(X, join(Y, complement(Z)))).
% 112.27/14.71 Proof:
% 112.27/14.71 join(complement(X), meet(complement(Y), Z))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 join(complement(X), meet(Z, complement(Y)))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 join(meet(Z, complement(Y)), complement(X))
% 112.27/14.71 = { by lemma 57 R->L }
% 112.27/14.71 join(complement(join(Y, complement(Z))), complement(X))
% 112.27/14.71 = { by lemma 59 }
% 112.27/14.71 complement(meet(join(Y, complement(Z)), X))
% 112.27/14.71 = { by lemma 42 R->L }
% 112.27/14.71 complement(meet(X, join(Y, complement(Z))))
% 112.27/14.71
% 112.27/14.71 Lemma 75: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 112.27/14.71 Proof:
% 112.27/14.71 complement(meet(Y, join(X, complement(Y))))
% 112.27/14.71 = { by lemma 74 R->L }
% 112.27/14.71 join(complement(Y), meet(complement(X), Y))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 join(meet(complement(X), Y), complement(Y))
% 112.27/14.71 = { by lemma 46 R->L }
% 112.27/14.71 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(Y), complement(X))))
% 112.27/14.71 = { by lemma 42 R->L }
% 112.27/14.71 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(X), complement(Y))))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 join(meet(complement(X), Y), join(meet(complement(X), complement(Y)), meet(complement(Y), X)))
% 112.27/14.71 = { by axiom 7 (maddux2_join_associativity_2) }
% 112.27/14.71 join(join(meet(complement(X), Y), meet(complement(X), complement(Y))), meet(complement(Y), X))
% 112.27/14.71 = { by lemma 46 }
% 112.27/14.71 join(complement(X), meet(complement(Y), X))
% 112.27/14.71 = { by lemma 74 }
% 112.27/14.71 complement(meet(X, join(Y, complement(X))))
% 112.27/14.71
% 112.27/14.71 Lemma 76: meet(Y, join(X, complement(Y))) = meet(X, join(Y, complement(X))).
% 112.27/14.71 Proof:
% 112.27/14.71 meet(Y, join(X, complement(Y)))
% 112.27/14.71 = { by lemma 47 R->L }
% 112.27/14.71 meet(Y, meet(join(X, complement(Y)), top))
% 112.27/14.71 = { by lemma 73 R->L }
% 112.27/14.71 meet(meet(Y, join(X, complement(Y))), top)
% 112.27/14.71 = { by lemma 43 R->L }
% 112.27/14.71 complement(join(zero, complement(meet(Y, join(X, complement(Y))))))
% 112.27/14.71 = { by lemma 75 }
% 112.27/14.71 complement(join(zero, complement(meet(X, join(Y, complement(X))))))
% 112.27/14.71 = { by lemma 43 }
% 112.27/14.71 meet(meet(X, join(Y, complement(X))), top)
% 112.27/14.71 = { by lemma 73 }
% 112.27/14.71 meet(X, meet(join(Y, complement(X)), top))
% 112.27/14.71 = { by lemma 47 }
% 112.27/14.71 meet(X, join(Y, complement(X)))
% 112.27/14.71
% 112.27/14.71 Lemma 77: meet(X, join(Y, complement(X))) = meet(X, Y).
% 112.27/14.71 Proof:
% 112.27/14.71 meet(X, join(Y, complement(X)))
% 112.27/14.71 = { by lemma 76 }
% 112.27/14.71 meet(Y, join(X, complement(Y)))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 meet(Y, join(complement(Y), X))
% 112.27/14.71 = { by lemma 42 }
% 112.27/14.71 meet(join(complement(Y), X), Y)
% 112.27/14.71 = { by lemma 34 R->L }
% 112.27/14.71 meet(join(complement(Y), X), join(meet(Y, X), complement(join(complement(Y), X))))
% 112.27/14.71 = { by lemma 76 R->L }
% 112.27/14.71 meet(meet(Y, X), join(join(complement(Y), X), complement(meet(Y, X))))
% 112.27/14.71 = { by lemma 73 }
% 112.27/14.71 meet(Y, meet(X, join(join(complement(Y), X), complement(meet(Y, X)))))
% 112.27/14.71 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 112.27/14.71 meet(Y, meet(X, join(complement(Y), join(X, complement(meet(Y, X))))))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 meet(Y, meet(X, join(complement(Y), join(complement(meet(Y, X)), X))))
% 112.27/14.71 = { by axiom 7 (maddux2_join_associativity_2) }
% 112.27/14.71 meet(Y, meet(X, join(join(complement(Y), complement(meet(Y, X))), X)))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.71 meet(Y, meet(X, join(X, join(complement(Y), complement(meet(Y, X))))))
% 112.27/14.71 = { by lemma 70 }
% 112.27/14.71 meet(Y, X)
% 112.27/14.71 = { by lemma 42 R->L }
% 112.27/14.71 meet(X, Y)
% 112.27/14.71
% 112.27/14.71 Lemma 78: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 112.27/14.71 Proof:
% 112.27/14.71 join(X, complement(join(X, Y)))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 join(X, complement(join(Y, X)))
% 112.27/14.71 = { by lemma 47 R->L }
% 112.27/14.71 join(X, complement(join(Y, meet(X, top))))
% 112.27/14.71 = { by lemma 67 R->L }
% 112.27/14.71 complement(meet(complement(X), join(Y, meet(X, top))))
% 112.27/14.71 = { by lemma 39 R->L }
% 112.27/14.71 complement(meet(join(zero, complement(X)), join(Y, meet(X, top))))
% 112.27/14.71 = { by lemma 43 R->L }
% 112.27/14.71 complement(meet(join(zero, complement(X)), join(Y, complement(join(zero, complement(X))))))
% 112.27/14.71 = { by lemma 75 R->L }
% 112.27/14.71 complement(meet(Y, join(join(zero, complement(X)), complement(Y))))
% 112.27/14.71 = { by lemma 77 }
% 112.27/14.71 complement(meet(Y, join(zero, complement(X))))
% 112.27/14.71 = { by lemma 39 }
% 112.27/14.71 complement(meet(Y, complement(X)))
% 112.27/14.71 = { by lemma 66 }
% 112.27/14.71 join(X, complement(Y))
% 112.27/14.71
% 112.27/14.71 Lemma 79: join(composition(X, Y), composition(X, converse(Z))) = composition(X, join(Y, converse(Z))).
% 112.27/14.71 Proof:
% 112.27/14.71 join(composition(X, Y), composition(X, converse(Z)))
% 112.27/14.71 = { by axiom 1 (converse_idempotence_8) R->L }
% 112.27/14.71 converse(converse(join(composition(X, Y), composition(X, converse(Z)))))
% 112.27/14.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.71 converse(converse(join(composition(X, converse(Z)), composition(X, Y))))
% 112.27/14.71 = { by axiom 6 (converse_additivity_9) }
% 112.27/14.71 converse(join(converse(composition(X, converse(Z))), converse(composition(X, Y))))
% 112.27/14.71 = { by axiom 8 (converse_multiplicativity_10) }
% 112.27/14.71 converse(join(composition(converse(converse(Z)), converse(X)), converse(composition(X, Y))))
% 112.27/14.71 = { by axiom 1 (converse_idempotence_8) }
% 112.27/14.71 converse(join(composition(Z, converse(X)), converse(composition(X, Y))))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.72 converse(join(converse(composition(X, Y)), composition(Z, converse(X))))
% 112.27/14.72 = { by axiom 8 (converse_multiplicativity_10) }
% 112.27/14.72 converse(join(composition(converse(Y), converse(X)), composition(Z, converse(X))))
% 112.27/14.72 = { by axiom 12 (composition_distributivity_7) R->L }
% 112.27/14.72 converse(composition(join(converse(Y), Z), converse(X)))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.72 converse(composition(join(Z, converse(Y)), converse(X)))
% 112.27/14.72 = { by lemma 28 R->L }
% 112.27/14.72 converse(composition(converse(join(Y, converse(Z))), converse(X)))
% 112.27/14.72 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 112.27/14.72 converse(converse(composition(X, join(Y, converse(Z)))))
% 112.27/14.72 = { by axiom 1 (converse_idempotence_8) }
% 112.27/14.72 composition(X, join(Y, converse(Z)))
% 112.27/14.72
% 112.27/14.72 Lemma 80: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 112.27/14.72 Proof:
% 112.27/14.72 join(composition(X, Z), composition(X, Y))
% 112.27/14.72 = { by axiom 1 (converse_idempotence_8) R->L }
% 112.27/14.72 join(composition(X, Z), composition(X, converse(converse(Y))))
% 112.27/14.72 = { by lemma 79 }
% 112.27/14.72 composition(X, join(Z, converse(converse(Y))))
% 112.27/14.72 = { by axiom 1 (converse_idempotence_8) }
% 112.27/14.72 composition(X, join(Z, Y))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.72 composition(X, join(Y, Z))
% 112.27/14.72
% 112.27/14.72 Lemma 81: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 112.27/14.72 Proof:
% 112.27/14.72 join(composition(X, Y), composition(X, Z))
% 112.27/14.72 = { by lemma 80 }
% 112.27/14.72 composition(X, join(Z, Y))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.72 composition(X, join(Y, Z))
% 112.27/14.72
% 112.27/14.72 Lemma 82: composition(converse(X), complement(composition(X, top))) = zero.
% 112.27/14.72 Proof:
% 112.27/14.72 composition(converse(X), complement(composition(X, top)))
% 112.27/14.72 = { by lemma 41 R->L }
% 112.27/14.72 join(zero, composition(converse(X), complement(composition(X, top))))
% 112.27/14.72 = { by lemma 18 R->L }
% 112.27/14.72 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 112.27/14.72 = { by lemma 23 }
% 112.27/14.72 complement(top)
% 112.27/14.72 = { by lemma 18 }
% 112.27/14.72 zero
% 112.27/14.72
% 112.27/14.72 Lemma 83: meet(complement(X), composition(converse(sk1), composition(sk1, X))) = zero.
% 112.27/14.72 Proof:
% 112.27/14.72 meet(complement(X), composition(converse(sk1), composition(sk1, X)))
% 112.27/14.72 = { by lemma 42 }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(X))
% 112.27/14.72 = { by lemma 22 R->L }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(composition(one, X)))
% 112.27/14.72 = { by axiom 11 (goals_17) R->L }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(composition(join(composition(converse(sk1), sk1), one), X)))
% 112.27/14.72 = { by axiom 12 (composition_distributivity_7) }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(composition(converse(sk1), sk1), X), composition(one, X))))
% 112.27/14.72 = { by axiom 9 (composition_associativity_5) R->L }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(converse(sk1), composition(sk1, X)), composition(one, X))))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(join(composition(one, X), composition(converse(sk1), composition(sk1, X)))))
% 112.27/14.72 = { by lemma 22 }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), complement(join(X, composition(converse(sk1), composition(sk1, X)))))
% 112.27/14.72 = { by lemma 71 R->L }
% 112.27/14.72 meet(composition(converse(sk1), composition(sk1, X)), meet(complement(X), complement(composition(converse(sk1), composition(sk1, X)))))
% 112.27/14.72 = { by lemma 61 }
% 112.27/14.72 zero
% 112.27/14.72
% 112.27/14.72 Lemma 84: meet(composition(X, Y), meet(composition(X, top), Z)) = meet(Z, composition(X, Y)).
% 112.27/14.72 Proof:
% 112.27/14.72 meet(composition(X, Y), meet(composition(X, top), Z))
% 112.27/14.72 = { by lemma 72 }
% 112.27/14.72 meet(Z, meet(composition(X, Y), composition(X, top)))
% 112.27/14.72 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.72 meet(Z, complement(join(complement(composition(X, Y)), complement(composition(X, top)))))
% 112.27/14.72 = { by lemma 39 R->L }
% 112.27/14.72 meet(Z, join(zero, complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 52 R->L }
% 112.27/14.72 meet(Z, join(meet(zero, complement(composition(X, top))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 54 R->L }
% 112.27/14.72 meet(Z, join(meet(composition(X, zero), complement(composition(X, top))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 45 R->L }
% 112.27/14.72 meet(Z, join(meet(composition(X, meet(Y, zero)), complement(composition(X, top))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 82 R->L }
% 112.27/14.72 meet(Z, join(meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by axiom 15 (modular_law_1_15) R->L }
% 112.27/14.72 meet(Z, join(join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top)))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 82 }
% 112.27/14.72 meet(Z, join(join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, zero)), complement(composition(X, top)))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 45 }
% 112.27/14.72 meet(Z, join(join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, zero), complement(composition(X, top)))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 54 }
% 112.27/14.72 meet(Z, join(join(meet(composition(X, Y), complement(composition(X, top))), meet(zero, complement(composition(X, top)))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 52 }
% 112.27/14.72 meet(Z, join(join(meet(composition(X, Y), complement(composition(X, top))), zero), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 40 }
% 112.27/14.72 meet(Z, join(meet(composition(X, Y), complement(composition(X, top))), complement(join(complement(composition(X, Y)), complement(composition(X, top))))))
% 112.27/14.72 = { by lemma 34 }
% 112.27/14.72 meet(Z, composition(X, Y))
% 112.27/14.72
% 112.27/14.72 Goal 1 (goals_18): composition(sk1, meet(sk2, sk3)) = meet(composition(sk1, sk2), composition(sk1, sk3)).
% 112.27/14.72 Proof:
% 112.27/14.72 composition(sk1, meet(sk2, sk3))
% 112.27/14.72 = { by lemma 51 R->L }
% 112.27/14.72 complement(complement(composition(sk1, meet(sk2, sk3))))
% 112.27/14.72 = { by lemma 34 R->L }
% 112.27/14.72 complement(join(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))), complement(join(complement(complement(composition(sk1, meet(sk2, sk3)))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))))))
% 112.27/14.72 = { by axiom 7 (maddux2_join_associativity_2) }
% 112.27/14.72 complement(join(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))), complement(join(join(complement(complement(composition(sk1, meet(sk2, sk3)))), complement(join(zero, complement(composition(sk1, complement(sk2)))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2)))))))))
% 112.27/14.72 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 112.27/14.72 complement(join(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))), complement(join(join(complement(complement(composition(sk1, meet(sk2, sk3)))), complement(join(zero, complement(composition(sk1, complement(sk2)))))), complement(join(complement(complement(composition(sk1, meet(sk2, sk3)))), complement(join(zero, complement(composition(sk1, complement(sk2)))))))))))
% 112.27/14.72 = { by axiom 4 (def_top_12) R->L }
% 112.27/14.72 complement(join(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))), complement(top)))
% 112.27/14.72 = { by lemma 18 }
% 112.27/14.72 complement(join(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))), zero))
% 112.27/14.72 = { by lemma 40 }
% 112.27/14.72 complement(meet(complement(composition(sk1, meet(sk2, sk3))), join(complement(join(zero, complement(composition(sk1, complement(sk2))))), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))))
% 112.27/14.72 = { by lemma 43 }
% 112.27/14.72 complement(meet(complement(composition(sk1, meet(sk2, sk3))), join(meet(composition(sk1, complement(sk2)), top), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))))
% 112.27/14.72 = { by lemma 47 }
% 112.27/14.72 complement(meet(complement(composition(sk1, meet(sk2, sk3))), join(composition(sk1, complement(sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), join(zero, complement(composition(sk1, complement(sk2))))))))
% 112.27/14.72 = { by lemma 39 }
% 112.27/14.72 complement(meet(complement(composition(sk1, meet(sk2, sk3))), join(composition(sk1, complement(sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by lemma 67 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by lemma 71 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(join(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by lemma 81 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(meet(sk2, sk3), complement(sk2)))))))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(complement(sk2), meet(sk2, sk3)))))))
% 112.27/14.72 = { by lemma 42 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(complement(sk2), meet(sk3, sk2)))))))
% 112.27/14.72 = { by lemma 51 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, complement(complement(join(complement(sk2), meet(sk3, sk2)))))))))
% 112.27/14.72 = { by lemma 56 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, complement(meet(sk2, join(complement(sk3), complement(sk2)))))))))
% 112.27/14.72 = { by lemma 75 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, complement(meet(complement(sk3), join(sk2, complement(complement(sk3))))))))))
% 112.27/14.72 = { by lemma 67 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(sk3, complement(join(sk2, complement(complement(sk3))))))))))
% 112.27/14.72 = { by lemma 57 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(sk3, meet(complement(sk3), complement(sk2))))))))
% 112.27/14.72 = { by lemma 71 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(sk3, complement(join(sk3, sk2))))))))
% 112.27/14.72 = { by lemma 78 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, join(sk3, complement(sk2)))))))
% 112.27/14.72 = { by lemma 81 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(join(composition(sk1, sk3), composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(join(composition(sk1, complement(sk2)), composition(sk1, sk3))))))
% 112.27/14.72 = { by lemma 78 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(composition(sk1, complement(sk2)), complement(composition(sk1, sk3)))))
% 112.27/14.72 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), complement(join(complement(composition(sk1, sk3)), composition(sk1, complement(sk2)))))
% 112.27/14.72 = { by lemma 58 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(composition(sk1, complement(sk2)))))
% 112.27/14.72 = { by lemma 64 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, complement(sk2))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by lemma 42 }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(composition(sk1, complement(sk2)), complement(composition(sk1, sk2))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.72 = { by lemma 57 R->L }
% 112.27/14.72 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(complement(join(composition(sk1, sk2), complement(composition(sk1, complement(sk2))))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 78 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(complement(join(composition(sk1, sk2), complement(join(composition(sk1, sk2), composition(sk1, complement(sk2)))))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 57 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(join(composition(sk1, sk2), composition(sk1, complement(sk2))), complement(composition(sk1, sk2))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 42 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(sk2)))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by axiom 1 (converse_idempotence_8) R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, complement(converse(converse(sk2)))))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 65 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), join(composition(sk1, sk2), composition(sk1, converse(complement(converse(sk2)))))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 79 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, join(sk2, converse(complement(converse(sk2)))))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 30 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, converse(top))), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 33 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), meet(complement(complement(composition(sk1, sk2))), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 51 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), meet(composition(sk1, sk2), composition(sk1, complement(sk2)))))))
% 112.27/14.73 = { by lemma 42 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), meet(composition(sk1, complement(sk2)), composition(sk1, sk2))))))
% 112.27/14.73 = { by lemma 40 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), join(meet(composition(sk1, complement(sk2)), composition(sk1, sk2)), zero)))))
% 112.27/14.73 = { by lemma 54 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), join(meet(composition(sk1, complement(sk2)), composition(sk1, sk2)), composition(meet(sk1, composition(composition(sk1, sk2), converse(complement(sk2)))), zero))))))
% 112.27/14.73 = { by lemma 83 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), join(meet(composition(sk1, complement(sk2)), composition(sk1, sk2)), composition(meet(sk1, composition(composition(sk1, sk2), converse(complement(sk2)))), meet(complement(sk2), composition(converse(sk1), composition(sk1, sk2)))))))))
% 112.27/14.73 = { by axiom 17 (dedekind_law_14) }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), composition(meet(sk1, composition(composition(sk1, sk2), converse(complement(sk2)))), meet(complement(sk2), composition(converse(sk1), composition(sk1, sk2))))))))
% 112.27/14.73 = { by lemma 65 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), composition(meet(sk1, composition(composition(sk1, sk2), complement(converse(sk2)))), meet(complement(sk2), composition(converse(sk1), composition(sk1, sk2))))))))
% 112.27/14.73 = { by lemma 83 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), composition(meet(sk1, composition(composition(sk1, sk2), complement(converse(sk2)))), zero)))))
% 112.27/14.73 = { by lemma 54 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(join(meet(complement(composition(sk1, sk2)), composition(sk1, top)), zero))))
% 112.27/14.73 = { by lemma 40 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), complement(meet(complement(composition(sk1, sk2)), composition(sk1, top)))))
% 112.27/14.73 = { by lemma 67 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), join(composition(sk1, sk2), complement(composition(sk1, top)))))
% 112.27/14.73 = { by lemma 42 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(join(composition(sk1, sk2), complement(composition(sk1, top))), composition(sk1, sk3)))
% 112.27/14.73 = { by lemma 84 R->L }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), meet(composition(sk1, top), join(composition(sk1, sk2), complement(composition(sk1, top))))))
% 112.27/14.73 = { by lemma 77 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk3), meet(composition(sk1, top), composition(sk1, sk2))))
% 112.27/14.73 = { by lemma 84 }
% 112.27/14.73 join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 112.27/14.73 = { by axiom 2 (maddux1_join_commutativity_1) }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))
% 112.27/14.73 = { by lemma 70 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, meet(sk2, sk3)), composition(sk1, sk2))))
% 112.27/14.73 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, sk2), composition(sk1, meet(sk2, sk3)))))
% 112.27/14.73 = { by lemma 80 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, join(meet(sk2, sk3), sk2))))
% 112.27/14.73 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, join(sk2, meet(sk2, sk3)))))
% 112.27/14.73 = { by lemma 69 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, sk2)))
% 112.27/14.73 = { by lemma 42 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), composition(sk1, meet(sk2, sk3))))
% 112.27/14.73 = { by lemma 70 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, meet(sk2, sk3)), composition(sk1, sk3)))))
% 112.27/14.73 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), join(composition(sk1, sk3), composition(sk1, meet(sk2, sk3))))))
% 112.27/14.73 = { by lemma 80 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, join(meet(sk2, sk3), sk3)))))
% 112.27/14.73 = { by lemma 47 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, meet(join(meet(sk2, sk3), sk3), top)))))
% 112.27/14.73 = { by lemma 43 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(join(meet(sk2, sk3), sk3))))))))
% 112.27/14.73 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(join(sk3, meet(sk2, sk3)))))))))
% 112.27/14.73 = { by lemma 71 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, meet(complement(sk3), complement(meet(sk2, sk3)))))))))
% 112.27/14.73 = { by lemma 41 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, join(zero, meet(complement(sk3), complement(meet(sk2, sk3))))))))))
% 112.27/14.73 = { by lemma 18 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), 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))))))))))
% 112.27/14.73 = { by lemma 60 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), 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))))))))))
% 112.27/14.73 = { by lemma 42 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), 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))))))))))
% 112.27/14.73 = { by lemma 57 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), 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))))))))))
% 112.27/14.73 = { by lemma 63 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(join(zero, complement(sk3)))))))
% 112.27/14.73 = { by lemma 43 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, meet(sk3, top)))))
% 112.27/14.73 = { by lemma 47 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, meet(sk2, sk3)), composition(sk1, sk3))))
% 112.27/14.73 = { by lemma 42 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), meet(composition(sk1, sk3), composition(sk1, meet(sk2, sk3)))))
% 112.27/14.73 = { by lemma 72 }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))
% 112.27/14.73 = { by lemma 42 R->L }
% 112.27/14.73 join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 112.27/14.73 = { by lemma 69 }
% 112.27/14.73 meet(composition(sk1, sk2), composition(sk1, sk3))
% 112.27/14.73 % SZS output end Proof
% 112.27/14.73
% 112.27/14.73 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------