TSTP Solution File: REL022-2 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL022-2 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n023.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 13:44:03 EDT 2023
% Result : Unsatisfiable 27.35s 3.87s
% Output : Proof 28.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.07/0.12 % Problem : REL022-2 : TPTP v8.1.2. Released v4.0.0.
% 0.07/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n023.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 22:46:56 EDT 2023
% 0.13/0.35 % CPUTime :
% 27.35/3.87 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 27.35/3.87
% 27.35/3.87 % SZS status Unsatisfiable
% 27.35/3.87
% 28.04/3.97 % SZS output start Proof
% 28.04/3.97 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 28.04/3.97 Axiom 2 (goals_17): composition(sk1, top) = sk1.
% 28.04/3.97 Axiom 3 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 28.04/3.97 Axiom 4 (converse_idempotence_8): converse(converse(X)) = X.
% 28.04/3.97 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 28.04/3.97 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 28.04/3.97 Axiom 7 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 28.04/3.97 Axiom 8 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 28.04/3.97 Axiom 9 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 28.04/3.97 Axiom 10 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 28.04/3.97 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 28.04/3.97 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 28.04/3.97 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 28.04/3.97 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 28.04/3.97 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).
% 28.04/3.97 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).
% 28.04/3.97 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))).
% 28.04/3.97
% 28.04/3.97 Lemma 18: complement(top) = zero.
% 28.04/3.97 Proof:
% 28.04/3.97 complement(top)
% 28.04/3.97 = { by axiom 6 (def_top_12) }
% 28.04/3.97 complement(join(complement(X), complement(complement(X))))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.04/3.97 meet(X, complement(X))
% 28.04/3.97 = { by axiom 5 (def_zero_13) R->L }
% 28.04/3.97 zero
% 28.04/3.97
% 28.04/3.97 Lemma 19: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 join(meet(X, Y), complement(join(complement(X), Y)))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.04/3.97 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 28.04/3.97 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 20: join(zero, meet(X, X)) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 join(zero, meet(X, X))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.04/3.97 join(zero, complement(join(complement(X), complement(X))))
% 28.04/3.97 = { by axiom 5 (def_zero_13) }
% 28.04/3.97 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 28.04/3.97 = { by lemma 19 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 21: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 28.04/3.97 Proof:
% 28.04/3.97 converse(composition(converse(X), Y))
% 28.04/3.97 = { by axiom 7 (converse_multiplicativity_10) }
% 28.04/3.97 composition(converse(Y), converse(converse(X)))
% 28.04/3.97 = { by axiom 4 (converse_idempotence_8) }
% 28.04/3.97 composition(converse(Y), X)
% 28.04/3.97
% 28.04/3.97 Lemma 22: composition(converse(one), X) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 composition(converse(one), X)
% 28.04/3.97 = { by lemma 21 R->L }
% 28.04/3.97 converse(composition(converse(X), one))
% 28.04/3.97 = { by axiom 1 (composition_identity_6) }
% 28.04/3.97 converse(converse(X))
% 28.04/3.97 = { by axiom 4 (converse_idempotence_8) }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 23: composition(one, X) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 composition(one, X)
% 28.04/3.97 = { by lemma 22 R->L }
% 28.04/3.97 composition(converse(one), composition(one, X))
% 28.04/3.97 = { by axiom 8 (composition_associativity_5) }
% 28.04/3.97 composition(composition(converse(one), one), X)
% 28.04/3.97 = { by axiom 1 (composition_identity_6) }
% 28.04/3.97 composition(converse(one), X)
% 28.04/3.97 = { by lemma 22 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 24: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 28.04/3.97 Proof:
% 28.04/3.97 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 28.04/3.97 = { by axiom 13 (converse_cancellativity_11) }
% 28.04/3.97 complement(X)
% 28.04/3.97
% 28.04/3.97 Lemma 25: join(complement(X), complement(X)) = complement(X).
% 28.04/3.97 Proof:
% 28.04/3.97 join(complement(X), complement(X))
% 28.04/3.97 = { by lemma 22 R->L }
% 28.04/3.97 join(complement(X), composition(converse(one), complement(X)))
% 28.04/3.97 = { by lemma 23 R->L }
% 28.04/3.97 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 28.04/3.97 = { by lemma 24 }
% 28.04/3.97 complement(X)
% 28.04/3.97
% 28.04/3.97 Lemma 26: join(zero, zero) = zero.
% 28.04/3.97 Proof:
% 28.04/3.97 join(zero, zero)
% 28.04/3.97 = { by lemma 18 R->L }
% 28.04/3.97 join(zero, complement(top))
% 28.04/3.97 = { by lemma 18 R->L }
% 28.04/3.97 join(complement(top), complement(top))
% 28.04/3.97 = { by lemma 25 }
% 28.04/3.97 complement(top)
% 28.04/3.97 = { by lemma 18 }
% 28.04/3.97 zero
% 28.04/3.97
% 28.04/3.97 Lemma 27: join(zero, join(zero, X)) = join(X, zero).
% 28.04/3.97 Proof:
% 28.04/3.97 join(zero, join(zero, X))
% 28.04/3.97 = { by axiom 10 (maddux2_join_associativity_2) }
% 28.04/3.97 join(join(zero, zero), X)
% 28.04/3.97 = { by lemma 26 }
% 28.04/3.97 join(zero, X)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.97 join(X, zero)
% 28.04/3.97
% 28.04/3.97 Lemma 28: join(X, zero) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 join(X, zero)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(zero, X)
% 28.04/3.97 = { by lemma 20 R->L }
% 28.04/3.97 join(zero, join(zero, meet(X, X)))
% 28.04/3.97 = { by lemma 27 }
% 28.04/3.97 join(meet(X, X), zero)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.97 join(zero, meet(X, X))
% 28.04/3.97 = { by lemma 20 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 29: join(zero, X) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 join(zero, X)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(X, zero)
% 28.04/3.97 = { by lemma 28 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 30: complement(zero) = top.
% 28.04/3.97 Proof:
% 28.04/3.97 complement(zero)
% 28.04/3.97 = { by lemma 29 R->L }
% 28.04/3.97 join(zero, complement(zero))
% 28.04/3.97 = { by axiom 6 (def_top_12) R->L }
% 28.04/3.97 top
% 28.04/3.97
% 28.04/3.97 Lemma 31: converse(one) = one.
% 28.04/3.97 Proof:
% 28.04/3.97 converse(one)
% 28.04/3.97 = { by axiom 1 (composition_identity_6) R->L }
% 28.04/3.97 composition(converse(one), one)
% 28.04/3.97 = { by lemma 22 }
% 28.04/3.97 one
% 28.04/3.97
% 28.04/3.97 Lemma 32: join(X, join(Y, complement(X))) = join(Y, top).
% 28.04/3.97 Proof:
% 28.04/3.97 join(X, join(Y, complement(X)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(X, join(complement(X), Y))
% 28.04/3.97 = { by axiom 10 (maddux2_join_associativity_2) }
% 28.04/3.97 join(join(X, complement(X)), Y)
% 28.04/3.97 = { by axiom 6 (def_top_12) R->L }
% 28.04/3.97 join(top, Y)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.97 join(Y, top)
% 28.04/3.97
% 28.04/3.97 Lemma 33: join(top, complement(X)) = top.
% 28.04/3.97 Proof:
% 28.04/3.97 join(top, complement(X))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(complement(X), top)
% 28.04/3.97 = { by lemma 32 R->L }
% 28.04/3.97 join(X, join(complement(X), complement(X)))
% 28.04/3.97 = { by lemma 25 }
% 28.04/3.97 join(X, complement(X))
% 28.04/3.97 = { by axiom 6 (def_top_12) R->L }
% 28.04/3.97 top
% 28.04/3.97
% 28.04/3.97 Lemma 34: join(top, X) = join(Y, top).
% 28.04/3.97 Proof:
% 28.04/3.97 join(top, X)
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(X, top)
% 28.04/3.97 = { by lemma 33 R->L }
% 28.04/3.97 join(X, join(top, complement(X)))
% 28.04/3.97 = { by lemma 32 }
% 28.04/3.97 join(top, top)
% 28.04/3.97 = { by lemma 32 R->L }
% 28.04/3.97 join(Y, join(top, complement(Y)))
% 28.04/3.97 = { by lemma 33 }
% 28.04/3.97 join(Y, top)
% 28.04/3.97
% 28.04/3.97 Lemma 35: join(X, top) = top.
% 28.04/3.97 Proof:
% 28.04/3.97 join(X, top)
% 28.04/3.97 = { by lemma 34 R->L }
% 28.04/3.97 join(top, complement(top))
% 28.04/3.97 = { by axiom 6 (def_top_12) R->L }
% 28.04/3.97 top
% 28.04/3.97
% 28.04/3.97 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 28.04/3.97 Proof:
% 28.04/3.97 converse(join(X, converse(Y)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 converse(join(converse(Y), X))
% 28.04/3.97 = { by axiom 9 (converse_additivity_9) }
% 28.04/3.97 join(converse(converse(Y)), converse(X))
% 28.04/3.97 = { by axiom 4 (converse_idempotence_8) }
% 28.04/3.97 join(Y, converse(X))
% 28.04/3.97
% 28.04/3.97 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 28.04/3.97 Proof:
% 28.04/3.97 converse(join(converse(X), Y))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 converse(join(Y, converse(X)))
% 28.04/3.97 = { by lemma 36 }
% 28.04/3.97 join(X, converse(Y))
% 28.04/3.97
% 28.04/3.97 Lemma 38: converse(top) = top.
% 28.04/3.97 Proof:
% 28.04/3.97 converse(top)
% 28.04/3.97 = { by lemma 35 R->L }
% 28.04/3.97 converse(join(converse(top), top))
% 28.04/3.97 = { by lemma 37 }
% 28.04/3.97 join(top, converse(top))
% 28.04/3.97 = { by lemma 34 }
% 28.04/3.97 join(X, top)
% 28.04/3.97 = { by lemma 35 }
% 28.04/3.97 top
% 28.04/3.97
% 28.04/3.97 Lemma 39: converse(zero) = zero.
% 28.04/3.97 Proof:
% 28.04/3.97 converse(zero)
% 28.04/3.97 = { by lemma 28 R->L }
% 28.04/3.97 join(converse(zero), zero)
% 28.04/3.97 = { by lemma 27 R->L }
% 28.04/3.97 join(zero, join(zero, converse(zero)))
% 28.04/3.97 = { by lemma 37 R->L }
% 28.04/3.97 join(zero, converse(join(converse(zero), zero)))
% 28.04/3.97 = { by lemma 28 }
% 28.04/3.97 join(zero, converse(converse(zero)))
% 28.04/3.97 = { by axiom 4 (converse_idempotence_8) }
% 28.04/3.97 join(zero, zero)
% 28.04/3.97 = { by lemma 26 }
% 28.04/3.97 zero
% 28.04/3.97
% 28.04/3.97 Lemma 40: complement(complement(X)) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 complement(complement(X))
% 28.04/3.97 = { by lemma 29 R->L }
% 28.04/3.97 join(zero, complement(complement(X)))
% 28.04/3.97 = { by lemma 25 R->L }
% 28.04/3.97 join(zero, complement(join(complement(X), complement(X))))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.04/3.97 join(zero, meet(X, X))
% 28.04/3.97 = { by lemma 20 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 41: meet(Y, X) = meet(X, Y).
% 28.04/3.97 Proof:
% 28.04/3.97 meet(Y, X)
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.04/3.97 complement(join(complement(Y), complement(X)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 complement(join(complement(X), complement(Y)))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.04/3.97 meet(X, Y)
% 28.04/3.97
% 28.04/3.97 Lemma 42: join(meet(X, Y), meet(X, complement(Y))) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 join(meet(X, Y), meet(X, complement(Y)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(meet(X, complement(Y)), meet(X, Y))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.04/3.97 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 28.04/3.97 = { by lemma 19 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 43: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 28.04/3.97 Proof:
% 28.04/3.97 join(meet(X, Y), meet(Y, complement(X)))
% 28.04/3.97 = { by lemma 41 }
% 28.04/3.97 join(meet(Y, X), meet(Y, complement(X)))
% 28.04/3.97 = { by lemma 42 }
% 28.04/3.97 Y
% 28.04/3.97
% 28.04/3.97 Lemma 44: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 28.04/3.97 Proof:
% 28.04/3.97 join(meet(X, Y), meet(complement(X), Y))
% 28.04/3.97 = { by lemma 41 }
% 28.04/3.97 join(meet(X, Y), meet(Y, complement(X)))
% 28.04/3.97 = { by lemma 43 }
% 28.04/3.97 Y
% 28.04/3.97
% 28.04/3.97 Lemma 45: complement(join(zero, complement(X))) = meet(X, top).
% 28.04/3.97 Proof:
% 28.04/3.97 complement(join(zero, complement(X)))
% 28.04/3.97 = { by lemma 18 R->L }
% 28.04/3.97 complement(join(complement(top), complement(X)))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.04/3.97 meet(top, X)
% 28.04/3.97 = { by lemma 41 R->L }
% 28.04/3.97 meet(X, top)
% 28.04/3.97
% 28.04/3.97 Lemma 46: meet(X, top) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 meet(X, top)
% 28.04/3.97 = { by lemma 45 R->L }
% 28.04/3.97 complement(join(zero, complement(X)))
% 28.04/3.97 = { by lemma 29 }
% 28.04/3.97 complement(complement(X))
% 28.04/3.97 = { by lemma 40 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 47: meet(top, X) = X.
% 28.04/3.97 Proof:
% 28.04/3.97 meet(top, X)
% 28.04/3.97 = { by lemma 41 }
% 28.04/3.97 meet(X, top)
% 28.04/3.97 = { by lemma 46 }
% 28.04/3.97 X
% 28.04/3.97
% 28.04/3.97 Lemma 48: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 28.04/3.97 Proof:
% 28.04/3.97 complement(join(complement(X), meet(Y, Z)))
% 28.04/3.97 = { by lemma 41 }
% 28.04/3.97 complement(join(complement(X), meet(Z, Y)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 complement(join(meet(Z, Y), complement(X)))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.04/3.97 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 28.04/3.97 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.04/3.97 meet(join(complement(Z), complement(Y)), X)
% 28.04/3.97 = { by lemma 41 R->L }
% 28.04/3.97 meet(X, join(complement(Z), complement(Y)))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.97 meet(X, join(complement(Y), complement(Z)))
% 28.04/3.97
% 28.04/3.97 Lemma 49: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 28.04/3.97 Proof:
% 28.04/3.97 join(complement(X), complement(Y))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 join(complement(Y), complement(X))
% 28.04/3.97 = { by lemma 47 R->L }
% 28.04/3.97 meet(top, join(complement(Y), complement(X)))
% 28.04/3.97 = { by lemma 48 R->L }
% 28.04/3.97 complement(join(complement(top), meet(Y, X)))
% 28.04/3.97 = { by lemma 18 }
% 28.04/3.97 complement(join(zero, meet(Y, X)))
% 28.04/3.97 = { by lemma 29 }
% 28.04/3.97 complement(meet(Y, X))
% 28.04/3.97 = { by lemma 41 R->L }
% 28.04/3.97 complement(meet(X, Y))
% 28.04/3.97
% 28.04/3.97 Lemma 50: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 28.04/3.97 Proof:
% 28.04/3.97 complement(meet(X, complement(Y)))
% 28.04/3.97 = { by lemma 41 }
% 28.04/3.97 complement(meet(complement(Y), X))
% 28.04/3.97 = { by lemma 29 R->L }
% 28.04/3.97 complement(meet(join(zero, complement(Y)), X))
% 28.04/3.97 = { by lemma 49 R->L }
% 28.04/3.97 join(complement(join(zero, complement(Y))), complement(X))
% 28.04/3.97 = { by lemma 45 }
% 28.04/3.97 join(meet(Y, top), complement(X))
% 28.04/3.97 = { by lemma 46 }
% 28.04/3.97 join(Y, complement(X))
% 28.04/3.97
% 28.04/3.97 Lemma 51: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 28.04/3.97 Proof:
% 28.04/3.97 complement(join(X, complement(Y)))
% 28.04/3.97 = { by lemma 29 R->L }
% 28.04/3.97 complement(join(zero, join(X, complement(Y))))
% 28.04/3.97 = { by lemma 50 R->L }
% 28.04/3.97 complement(join(zero, complement(meet(Y, complement(X)))))
% 28.04/3.97 = { by lemma 45 }
% 28.04/3.97 meet(meet(Y, complement(X)), top)
% 28.04/3.97 = { by lemma 46 }
% 28.04/3.97 meet(Y, complement(X))
% 28.04/3.97
% 28.04/3.97 Lemma 52: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 28.04/3.97 Proof:
% 28.04/3.97 complement(join(complement(X), Y))
% 28.04/3.97 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.97 complement(join(Y, complement(X)))
% 28.04/3.97 = { by lemma 51 }
% 28.04/3.97 meet(X, complement(Y))
% 28.04/3.97
% 28.04/3.97 Lemma 53: join(X, converse(complement(converse(X)))) = top.
% 28.04/3.97 Proof:
% 28.04/3.97 join(X, converse(complement(converse(X))))
% 28.04/3.97 = { by lemma 37 R->L }
% 28.04/3.97 converse(join(converse(X), complement(converse(X))))
% 28.04/3.97 = { by axiom 6 (def_top_12) R->L }
% 28.04/3.97 converse(top)
% 28.04/3.97 = { by lemma 38 }
% 28.04/3.97 top
% 28.04/3.97
% 28.04/3.97 Lemma 54: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 28.04/3.97 Proof:
% 28.04/3.97 join(X, composition(Y, X))
% 28.04/3.97 = { by lemma 23 R->L }
% 28.04/3.97 join(composition(one, X), composition(Y, X))
% 28.04/3.98 = { by axiom 12 (composition_distributivity_7) R->L }
% 28.04/3.98 composition(join(one, Y), X)
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.98 composition(join(Y, one), X)
% 28.04/3.98
% 28.04/3.98 Lemma 55: composition(top, zero) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 composition(top, zero)
% 28.04/3.98 = { by lemma 18 R->L }
% 28.04/3.98 composition(top, complement(top))
% 28.04/3.98 = { by lemma 35 R->L }
% 28.04/3.98 composition(join(X, top), complement(top))
% 28.04/3.98 = { by lemma 34 R->L }
% 28.04/3.98 composition(join(top, one), complement(top))
% 28.04/3.98 = { by lemma 38 R->L }
% 28.04/3.98 composition(join(converse(top), one), complement(top))
% 28.04/3.98 = { by lemma 54 R->L }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(top)))
% 28.04/3.98 = { by lemma 35 R->L }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(join(Y, top))))
% 28.04/3.98 = { by lemma 34 R->L }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 28.04/3.98 = { by lemma 54 }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 28.04/3.98 = { by lemma 34 }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(composition(join(Z, top), top))))
% 28.04/3.98 = { by lemma 35 }
% 28.04/3.98 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 28.04/3.98 = { by lemma 24 }
% 28.04/3.98 complement(top)
% 28.04/3.98 = { by lemma 18 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 56: composition(X, zero) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 composition(X, zero)
% 28.04/3.98 = { by lemma 29 R->L }
% 28.04/3.98 join(zero, composition(X, zero))
% 28.04/3.98 = { by lemma 55 R->L }
% 28.04/3.98 join(composition(top, zero), composition(X, zero))
% 28.04/3.98 = { by axiom 12 (composition_distributivity_7) R->L }
% 28.04/3.98 composition(join(top, X), zero)
% 28.04/3.98 = { by lemma 34 }
% 28.04/3.98 composition(join(Y, top), zero)
% 28.04/3.98 = { by lemma 35 }
% 28.04/3.98 composition(top, zero)
% 28.04/3.98 = { by lemma 55 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 57: composition(zero, X) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 composition(zero, X)
% 28.04/3.98 = { by lemma 39 R->L }
% 28.04/3.98 composition(converse(zero), X)
% 28.04/3.98 = { by lemma 21 R->L }
% 28.04/3.98 converse(composition(converse(X), zero))
% 28.04/3.98 = { by lemma 56 }
% 28.04/3.98 converse(zero)
% 28.04/3.98 = { by lemma 39 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 58: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 28.04/3.98 Proof:
% 28.04/3.98 join(complement(one), composition(converse(X), complement(X)))
% 28.04/3.98 = { by axiom 1 (composition_identity_6) R->L }
% 28.04/3.98 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 28.04/3.98 = { by lemma 24 }
% 28.04/3.98 complement(one)
% 28.04/3.98
% 28.04/3.98 Lemma 59: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 28.04/3.98 Proof:
% 28.04/3.98 join(Y, join(X, Z))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.98 join(join(X, Z), Y)
% 28.04/3.98 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 28.04/3.98 join(X, join(Z, Y))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.98 join(X, join(Y, Z))
% 28.04/3.98
% 28.04/3.98 Lemma 60: meet(X, complement(join(X, Y))) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 meet(X, complement(join(X, Y)))
% 28.04/3.98 = { by lemma 51 R->L }
% 28.04/3.98 complement(join(join(X, Y), complement(X)))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.98 complement(join(complement(X), join(X, Y)))
% 28.04/3.98 = { by lemma 59 }
% 28.04/3.98 complement(join(X, join(complement(X), Y)))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.98 complement(join(X, join(Y, complement(X))))
% 28.04/3.98 = { by lemma 32 }
% 28.04/3.98 complement(join(Y, top))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.04/3.98 complement(join(top, Y))
% 28.04/3.98 = { by lemma 34 }
% 28.04/3.98 complement(join(Z, top))
% 28.04/3.98 = { by lemma 35 }
% 28.04/3.98 complement(top)
% 28.04/3.98 = { by lemma 18 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 61: meet(X, complement(join(Y, X))) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 meet(X, complement(join(Y, X)))
% 28.04/3.98 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.04/3.98 meet(X, complement(join(X, Y)))
% 28.04/3.98 = { by lemma 60 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 62: meet(one, composition(converse(complement(X)), X)) = zero.
% 28.04/3.98 Proof:
% 28.04/3.98 meet(one, composition(converse(complement(X)), X))
% 28.04/3.98 = { by lemma 41 }
% 28.04/3.98 meet(composition(converse(complement(X)), X), one)
% 28.04/3.98 = { by lemma 40 R->L }
% 28.04/3.98 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 28.04/3.98 = { by lemma 58 R->L }
% 28.04/3.98 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 28.04/3.98 = { by lemma 45 }
% 28.04/3.98 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 28.04/3.98 = { by lemma 29 }
% 28.04/3.98 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), meet(X, top)))))
% 28.04/3.98 = { by lemma 46 }
% 28.04/3.98 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 28.04/3.98 = { by lemma 61 }
% 28.04/3.98 zero
% 28.04/3.98
% 28.04/3.98 Lemma 63: converse(complement(converse(X))) = complement(X).
% 28.04/3.98 Proof:
% 28.04/3.98 converse(complement(converse(X)))
% 28.04/3.98 = { by lemma 40 R->L }
% 28.04/3.98 converse(complement(converse(complement(complement(X)))))
% 28.04/3.98 = { by lemma 44 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), converse(complement(converse(complement(complement(X)))))))
% 28.04/3.98 = { by lemma 40 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), complement(complement(converse(complement(converse(complement(complement(X)))))))))
% 28.04/3.98 = { by lemma 52 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X)))))))))
% 28.04/3.98 = { by lemma 47 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(top, join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 28.04/3.98 = { by lemma 53 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))), join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 28.04/3.98 = { by lemma 41 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))))))
% 28.04/3.98 = { by lemma 40 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 28.04/3.98 = { by lemma 50 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 28.04/3.98 = { by lemma 41 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))), complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))))))
% 28.04/3.98 = { by lemma 51 R->L }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), complement(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))))
% 28.04/3.98 = { by lemma 51 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))))))
% 28.04/3.98 = { by lemma 44 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(complement(complement(X))))))
% 28.04/3.98 = { by lemma 40 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))
% 28.04/3.98 = { by lemma 40 }
% 28.04/3.98 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(X))
% 28.04/3.98 = { by lemma 40 }
% 28.04/3.98 join(meet(X, converse(complement(converse(X)))), complement(X))
% 28.04/3.98 = { by lemma 28 R->L }
% 28.04/3.98 join(join(meet(X, converse(complement(converse(X)))), zero), complement(X))
% 28.04/3.98 = { by lemma 57 R->L }
% 28.49/3.98 join(join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 28.49/3.98 = { by lemma 23 R->L }
% 28.49/3.98 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 28.49/3.98 = { by lemma 62 R->L }
% 28.49/3.98 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 28.49/3.98 = { by axiom 17 (dedekind_law_14) }
% 28.49/3.98 join(composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 28.49/3.98 = { by lemma 62 }
% 28.49/3.98 join(composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 28.49/3.98 = { by lemma 57 }
% 28.49/3.98 join(zero, complement(X))
% 28.49/3.98 = { by lemma 29 }
% 28.49/3.98 complement(X)
% 28.49/3.98
% 28.49/3.98 Lemma 64: complement(converse(X)) = converse(complement(X)).
% 28.49/3.98 Proof:
% 28.49/3.98 complement(converse(X))
% 28.49/3.98 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/3.98 converse(converse(complement(converse(X))))
% 28.49/3.98 = { by lemma 63 }
% 28.49/3.98 converse(complement(X))
% 28.49/3.98
% 28.49/3.98 Lemma 65: meet(X, X) = X.
% 28.49/3.98 Proof:
% 28.49/3.98 meet(X, X)
% 28.49/3.98 = { by lemma 29 R->L }
% 28.49/3.98 join(zero, meet(X, X))
% 28.49/3.98 = { by lemma 20 }
% 28.49/3.98 X
% 28.49/3.98
% 28.49/3.98 Lemma 66: meet(X, zero) = zero.
% 28.49/3.98 Proof:
% 28.49/3.98 meet(X, zero)
% 28.49/3.98 = { by lemma 41 }
% 28.49/3.98 meet(zero, X)
% 28.49/3.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/3.98 complement(join(complement(zero), complement(X)))
% 28.49/3.98 = { by lemma 30 }
% 28.49/3.98 complement(join(top, complement(X)))
% 28.49/3.98 = { by lemma 33 }
% 28.49/3.98 complement(top)
% 28.49/3.98 = { by lemma 18 }
% 28.49/3.98 zero
% 28.49/3.98
% 28.49/3.98 Lemma 67: meet(zero, X) = zero.
% 28.49/3.98 Proof:
% 28.49/3.98 meet(zero, X)
% 28.49/3.98 = { by lemma 41 }
% 28.49/3.98 meet(X, zero)
% 28.49/3.98 = { by lemma 66 }
% 28.49/3.98 zero
% 28.49/3.98
% 28.49/3.98 Lemma 68: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 28.49/3.98 Proof:
% 28.49/3.98 complement(meet(complement(X), Y))
% 28.49/3.98 = { by lemma 41 }
% 28.49/3.98 complement(meet(Y, complement(X)))
% 28.49/3.98 = { by lemma 50 }
% 28.49/3.98 join(X, complement(Y))
% 28.49/3.98
% 28.49/3.98 Lemma 69: join(X, complement(meet(X, Y))) = top.
% 28.49/3.98 Proof:
% 28.49/3.98 join(X, complement(meet(X, Y)))
% 28.49/3.98 = { by lemma 41 }
% 28.49/3.98 join(X, complement(meet(Y, X)))
% 28.49/3.98 = { by lemma 49 R->L }
% 28.49/3.98 join(X, join(complement(Y), complement(X)))
% 28.49/3.98 = { by lemma 32 }
% 28.49/3.98 join(complement(Y), top)
% 28.49/3.98 = { by lemma 35 }
% 28.49/3.98 top
% 28.49/3.98
% 28.49/3.98 Lemma 70: meet(X, join(X, complement(Y))) = X.
% 28.49/3.98 Proof:
% 28.49/3.98 meet(X, join(X, complement(Y)))
% 28.49/3.98 = { by lemma 28 R->L }
% 28.49/3.98 join(meet(X, join(X, complement(Y))), zero)
% 28.49/3.98 = { by lemma 18 R->L }
% 28.49/3.98 join(meet(X, join(X, complement(Y))), complement(top))
% 28.49/3.98 = { by lemma 68 R->L }
% 28.49/3.98 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 28.49/3.98 = { by lemma 69 R->L }
% 28.49/3.98 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 28.49/3.98 = { by lemma 19 }
% 28.49/3.98 X
% 28.49/3.98
% 28.49/3.98 Lemma 71: join(X, meet(X, Y)) = X.
% 28.49/3.98 Proof:
% 28.49/3.98 join(X, meet(X, Y))
% 28.49/3.98 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/3.98 join(X, complement(join(complement(X), complement(Y))))
% 28.49/3.98 = { by lemma 68 R->L }
% 28.49/3.98 complement(meet(complement(X), join(complement(X), complement(Y))))
% 28.49/3.98 = { by lemma 70 }
% 28.49/3.98 complement(complement(X))
% 28.49/3.98 = { by lemma 40 }
% 28.49/3.98 X
% 28.49/3.98
% 28.49/3.98 Lemma 72: join(X, meet(Y, X)) = X.
% 28.49/3.98 Proof:
% 28.49/3.98 join(X, meet(Y, X))
% 28.49/3.98 = { by lemma 41 }
% 28.49/3.98 join(X, meet(X, Y))
% 28.49/3.98 = { by lemma 71 }
% 28.49/3.98 X
% 28.49/3.98
% 28.49/3.98 Lemma 73: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 28.49/3.98 Proof:
% 28.49/3.98 converse(composition(X, converse(Y)))
% 28.49/3.98 = { by axiom 7 (converse_multiplicativity_10) }
% 28.49/3.98 composition(converse(converse(Y)), converse(X))
% 28.49/3.98 = { by axiom 4 (converse_idempotence_8) }
% 28.49/3.98 composition(Y, converse(X))
% 28.49/3.98
% 28.49/3.98 Lemma 74: join(X, composition(X, Y)) = composition(X, join(Y, one)).
% 28.49/3.98 Proof:
% 28.49/3.98 join(X, composition(X, Y))
% 28.49/3.98 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/3.98 join(X, composition(X, converse(converse(Y))))
% 28.49/3.98 = { by lemma 73 R->L }
% 28.49/3.98 join(X, converse(composition(converse(Y), converse(X))))
% 28.49/3.98 = { by lemma 37 R->L }
% 28.49/3.98 converse(join(converse(X), composition(converse(Y), converse(X))))
% 28.49/3.98 = { by lemma 54 }
% 28.49/3.98 converse(composition(join(converse(Y), one), converse(X)))
% 28.49/3.98 = { by lemma 73 }
% 28.49/3.98 composition(X, converse(join(converse(Y), one)))
% 28.49/3.98 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/3.98 composition(X, converse(join(one, converse(Y))))
% 28.49/3.98 = { by axiom 9 (converse_additivity_9) }
% 28.49/3.98 composition(X, join(converse(one), converse(converse(Y))))
% 28.49/3.98 = { by lemma 31 }
% 28.49/3.98 composition(X, join(one, converse(converse(Y))))
% 28.49/3.98 = { by axiom 4 (converse_idempotence_8) }
% 28.49/3.98 composition(X, join(one, Y))
% 28.49/3.98 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/3.98 composition(X, join(Y, one))
% 28.49/3.98
% 28.49/3.99 Lemma 75: meet(X, join(X, Y)) = X.
% 28.49/3.99 Proof:
% 28.49/3.99 meet(X, join(X, Y))
% 28.49/3.99 = { by lemma 46 R->L }
% 28.49/3.99 meet(X, join(X, meet(Y, top)))
% 28.49/3.99 = { by lemma 45 R->L }
% 28.49/3.99 meet(X, join(X, complement(join(zero, complement(Y)))))
% 28.49/3.99 = { by lemma 70 }
% 28.49/3.99 X
% 28.49/3.99
% 28.49/3.99 Lemma 76: meet(X, join(Y, X)) = X.
% 28.49/3.99 Proof:
% 28.49/3.99 meet(X, join(Y, X))
% 28.49/3.99 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/3.99 meet(X, join(X, Y))
% 28.49/3.99 = { by lemma 75 }
% 28.49/3.99 X
% 28.49/3.99
% 28.49/3.99 Lemma 77: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 28.49/3.99 Proof:
% 28.49/3.99 meet(Y, meet(X, Z))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(Y, meet(Z, X))
% 28.49/3.99 = { by lemma 46 R->L }
% 28.49/3.99 meet(meet(Y, meet(Z, X)), top)
% 28.49/3.99 = { by lemma 45 R->L }
% 28.49/3.99 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 28.49/3.99 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/3.99 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 28.49/3.99 = { by lemma 50 }
% 28.49/3.99 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 28.49/3.99 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 28.49/3.99 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 28.49/3.99 = { by lemma 49 }
% 28.49/3.99 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 28.49/3.99 = { by lemma 49 }
% 28.49/3.99 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 28.49/3.99 = { by lemma 41 R->L }
% 28.49/3.99 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 28.49/3.99 = { by lemma 45 }
% 28.49/3.99 meet(meet(X, meet(Y, Z)), top)
% 28.49/3.99 = { by lemma 46 }
% 28.49/3.99 meet(X, meet(Y, Z))
% 28.49/3.99
% 28.49/3.99 Lemma 78: meet(Z, meet(X, Y)) = meet(X, meet(Y, Z)).
% 28.49/3.99 Proof:
% 28.49/3.99 meet(Z, meet(X, Y))
% 28.49/3.99 = { by lemma 77 R->L }
% 28.49/3.99 meet(X, meet(Z, Y))
% 28.49/3.99 = { by lemma 41 R->L }
% 28.49/3.99 meet(X, meet(Y, Z))
% 28.49/3.99
% 28.49/3.99 Lemma 79: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 28.49/3.99 Proof:
% 28.49/3.99 meet(complement(X), complement(Y))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(complement(Y), complement(X))
% 28.49/3.99 = { by lemma 29 R->L }
% 28.49/3.99 meet(join(zero, complement(Y)), complement(X))
% 28.49/3.99 = { by lemma 51 R->L }
% 28.49/3.99 complement(join(X, complement(join(zero, complement(Y)))))
% 28.49/3.99 = { by lemma 45 }
% 28.49/3.99 complement(join(X, meet(Y, top)))
% 28.49/3.99 = { by lemma 46 }
% 28.49/3.99 complement(join(X, Y))
% 28.49/3.99
% 28.49/3.99 Lemma 80: composition(converse(sk1), complement(sk1)) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 composition(converse(sk1), complement(sk1))
% 28.49/3.99 = { by lemma 29 R->L }
% 28.49/3.99 join(zero, composition(converse(sk1), complement(sk1)))
% 28.49/3.99 = { by lemma 18 R->L }
% 28.49/3.99 join(complement(top), composition(converse(sk1), complement(sk1)))
% 28.49/3.99 = { by axiom 2 (goals_17) R->L }
% 28.49/3.99 join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))
% 28.49/3.99 = { by lemma 24 }
% 28.49/3.99 complement(top)
% 28.49/3.99 = { by lemma 18 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 81: composition(converse(complement(sk1)), sk1) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 composition(converse(complement(sk1)), sk1)
% 28.49/3.99 = { by lemma 21 R->L }
% 28.49/3.99 converse(composition(converse(sk1), complement(sk1)))
% 28.49/3.99 = { by lemma 80 }
% 28.49/3.99 converse(zero)
% 28.49/3.99 = { by lemma 39 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 82: converse(complement(join(X, converse(Y)))) = complement(join(Y, converse(X))).
% 28.49/3.99 Proof:
% 28.49/3.99 converse(complement(join(X, converse(Y))))
% 28.49/3.99 = { by lemma 36 R->L }
% 28.49/3.99 converse(complement(converse(join(Y, converse(X)))))
% 28.49/3.99 = { by lemma 63 }
% 28.49/3.99 complement(join(Y, converse(X)))
% 28.49/3.99
% 28.49/3.99 Lemma 83: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
% 28.49/3.99 Proof:
% 28.49/3.99 converse(meet(X, converse(Y)))
% 28.49/3.99 = { by lemma 40 R->L }
% 28.49/3.99 converse(complement(complement(meet(X, converse(Y)))))
% 28.49/3.99 = { by lemma 49 R->L }
% 28.49/3.99 converse(complement(join(complement(X), complement(converse(Y)))))
% 28.49/3.99 = { by lemma 64 }
% 28.49/3.99 converse(complement(join(complement(X), converse(complement(Y)))))
% 28.49/3.99 = { by lemma 82 }
% 28.49/3.99 complement(join(complement(Y), converse(complement(X))))
% 28.49/3.99 = { by lemma 52 }
% 28.49/3.99 meet(Y, complement(converse(complement(X))))
% 28.49/3.99 = { by lemma 64 }
% 28.49/3.99 meet(Y, converse(complement(complement(X))))
% 28.49/3.99 = { by lemma 40 }
% 28.49/3.99 meet(Y, converse(X))
% 28.49/3.99
% 28.49/3.99 Lemma 84: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 28.49/3.99 Proof:
% 28.49/3.99 join(complement(X), meet(X, Y))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 join(complement(X), meet(Y, X))
% 28.49/3.99 = { by lemma 71 R->L }
% 28.49/3.99 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 28.49/3.99 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 28.49/3.99 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 28.49/3.99 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/3.99 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 28.49/3.99 = { by lemma 42 }
% 28.49/3.99 join(complement(X), Y)
% 28.49/3.99 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/3.99 join(Y, complement(X))
% 28.49/3.99
% 28.49/3.99 Lemma 85: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 28.49/3.99 Proof:
% 28.49/3.99 join(complement(X), meet(Y, X))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 join(complement(X), meet(X, Y))
% 28.49/3.99 = { by lemma 84 }
% 28.49/3.99 join(Y, complement(X))
% 28.49/3.99
% 28.49/3.99 Lemma 86: join(complement(one), converse(complement(one))) = complement(one).
% 28.49/3.99 Proof:
% 28.49/3.99 join(complement(one), converse(complement(one)))
% 28.49/3.99 = { by lemma 46 R->L }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), top)))
% 28.49/3.99 = { by lemma 53 R->L }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 28.49/3.99 = { by lemma 31 }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 28.49/3.99 = { by lemma 51 R->L }
% 28.49/3.99 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 28.49/3.99 = { by lemma 79 R->L }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 28.49/3.99 = { by lemma 79 R->L }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 28.49/3.99 = { by lemma 49 R->L }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), join(complement(complement(one)), complement(complement(converse(complement(one))))))))
% 28.49/3.99 = { by lemma 48 R->L }
% 28.49/3.99 join(complement(one), converse(complement(join(complement(complement(one)), meet(complement(one), complement(converse(complement(one))))))))
% 28.49/3.99 = { by lemma 84 }
% 28.49/3.99 join(complement(one), converse(complement(join(complement(converse(complement(one))), complement(complement(one))))))
% 28.49/3.99 = { by lemma 51 }
% 28.49/3.99 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 28.49/3.99 = { by lemma 79 }
% 28.49/3.99 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 28.49/3.99 = { by lemma 51 }
% 28.49/3.99 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 28.49/3.99 = { by lemma 37 R->L }
% 28.49/3.99 converse(join(converse(complement(one)), meet(converse(complement(one)), complement(one))))
% 28.49/3.99 = { by lemma 71 }
% 28.49/3.99 converse(converse(complement(one)))
% 28.49/3.99 = { by axiom 4 (converse_idempotence_8) }
% 28.49/3.99 complement(one)
% 28.49/3.99
% 28.49/3.99 Lemma 87: join(meet(X, Y), complement(X)) = join(Y, complement(X)).
% 28.49/3.99 Proof:
% 28.49/3.99 join(meet(X, Y), complement(X))
% 28.49/3.99 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/3.99 join(complement(X), meet(X, Y))
% 28.49/3.99 = { by lemma 84 }
% 28.49/3.99 join(Y, complement(X))
% 28.49/3.99
% 28.49/3.99 Lemma 88: meet(meet(X, one), converse(Y)) = meet(X, converse(meet(Y, one))).
% 28.49/3.99 Proof:
% 28.49/3.99 meet(meet(X, one), converse(Y))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(converse(Y), meet(X, one))
% 28.49/3.99 = { by lemma 78 }
% 28.49/3.99 meet(X, meet(one, converse(Y)))
% 28.49/3.99 = { by lemma 40 R->L }
% 28.49/3.99 meet(X, meet(one, converse(complement(complement(Y)))))
% 28.49/3.99 = { by lemma 64 R->L }
% 28.49/3.99 meet(X, meet(one, complement(converse(complement(Y)))))
% 28.49/3.99 = { by lemma 52 R->L }
% 28.49/3.99 meet(X, complement(join(complement(one), converse(complement(Y)))))
% 28.49/3.99 = { by lemma 82 R->L }
% 28.49/3.99 meet(X, converse(complement(join(complement(Y), converse(complement(one))))))
% 28.49/3.99 = { by lemma 86 R->L }
% 28.49/3.99 meet(X, converse(complement(join(complement(Y), converse(join(complement(one), converse(complement(one))))))))
% 28.49/3.99 = { by lemma 36 }
% 28.49/3.99 meet(X, converse(complement(join(complement(Y), join(complement(one), converse(complement(one)))))))
% 28.49/3.99 = { by lemma 86 }
% 28.49/3.99 meet(X, converse(complement(join(complement(Y), complement(one)))))
% 28.49/3.99 = { by lemma 51 }
% 28.49/3.99 meet(X, converse(meet(one, complement(complement(Y)))))
% 28.49/3.99 = { by lemma 40 }
% 28.49/3.99 meet(X, converse(meet(one, Y)))
% 28.49/3.99 = { by lemma 41 R->L }
% 28.49/3.99 meet(X, converse(meet(Y, one)))
% 28.49/3.99
% 28.49/3.99 Lemma 89: meet(meet(sk1, X), composition(complement(sk1), Y)) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 meet(meet(sk1, X), composition(complement(sk1), Y))
% 28.49/3.99 = { by lemma 43 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(meet(sk1, composition(complement(sk1), Y)), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(meet(sk1, X), join(meet(composition(complement(sk1), Y), sk1), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 28 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(join(meet(composition(complement(sk1), Y), sk1), zero), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 67 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(join(meet(composition(complement(sk1), Y), sk1), meet(zero, sk1)), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 56 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(join(meet(composition(complement(sk1), Y), sk1), meet(composition(complement(sk1), zero), sk1)), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 66 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(join(meet(composition(complement(sk1), Y), sk1), meet(composition(complement(sk1), meet(Y, zero)), sk1)), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 81 R->L }
% 28.49/3.99 meet(meet(sk1, X), join(join(meet(composition(complement(sk1), Y), sk1), meet(composition(complement(sk1), meet(Y, composition(converse(complement(sk1)), sk1))), sk1)), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by axiom 15 (modular_law_1_15) }
% 28.49/3.99 meet(meet(sk1, X), join(meet(composition(complement(sk1), meet(Y, composition(converse(complement(sk1)), sk1))), sk1), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 81 }
% 28.49/3.99 meet(meet(sk1, X), join(meet(composition(complement(sk1), meet(Y, zero)), sk1), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 66 }
% 28.49/3.99 meet(meet(sk1, X), join(meet(composition(complement(sk1), zero), sk1), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 56 }
% 28.49/3.99 meet(meet(sk1, X), join(meet(zero, sk1), meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 67 }
% 28.49/3.99 meet(meet(sk1, X), join(zero, meet(composition(complement(sk1), Y), complement(sk1))))
% 28.49/3.99 = { by lemma 29 }
% 28.49/3.99 meet(meet(sk1, X), meet(composition(complement(sk1), Y), complement(sk1)))
% 28.49/3.99 = { by lemma 77 }
% 28.49/3.99 meet(composition(complement(sk1), Y), meet(meet(sk1, X), complement(sk1)))
% 28.49/3.99 = { by lemma 51 R->L }
% 28.49/3.99 meet(composition(complement(sk1), Y), complement(join(sk1, complement(meet(sk1, X)))))
% 28.49/3.99 = { by lemma 69 }
% 28.49/3.99 meet(composition(complement(sk1), Y), complement(top))
% 28.49/3.99 = { by lemma 18 }
% 28.49/3.99 meet(composition(complement(sk1), Y), zero)
% 28.49/3.99 = { by lemma 66 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 90: meet(complement(sk1), composition(meet(sk1, X), Y)) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 meet(complement(sk1), composition(meet(sk1, X), Y))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(composition(meet(sk1, X), Y), complement(sk1))
% 28.49/3.99 = { by lemma 28 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, X), Y), complement(sk1)), zero)
% 28.49/3.99 = { by lemma 67 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, X), Y), complement(sk1)), meet(zero, complement(sk1)))
% 28.49/3.99 = { by lemma 57 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, X), Y), complement(sk1)), meet(composition(zero, Y), complement(sk1)))
% 28.49/3.99 = { by lemma 89 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, X), Y), complement(sk1)), meet(composition(meet(meet(sk1, X), composition(complement(sk1), converse(Y))), Y), complement(sk1)))
% 28.49/3.99 = { by axiom 16 (modular_law_2_16) }
% 28.49/3.99 meet(composition(meet(meet(sk1, X), composition(complement(sk1), converse(Y))), Y), complement(sk1))
% 28.49/3.99 = { by lemma 89 }
% 28.49/3.99 meet(composition(zero, Y), complement(sk1))
% 28.49/3.99 = { by lemma 57 }
% 28.49/3.99 meet(zero, complement(sk1))
% 28.49/3.99 = { by lemma 67 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 91: meet(meet(X, one), composition(Y, converse(complement(Y)))) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 meet(meet(X, one), composition(Y, converse(complement(Y))))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(composition(Y, converse(complement(Y))), meet(X, one))
% 28.49/3.99 = { by lemma 64 R->L }
% 28.49/3.99 meet(composition(Y, complement(converse(Y))), meet(X, one))
% 28.49/3.99 = { by lemma 78 }
% 28.49/3.99 meet(X, meet(one, composition(Y, complement(converse(Y)))))
% 28.49/3.99 = { by lemma 41 }
% 28.49/3.99 meet(X, meet(composition(Y, complement(converse(Y))), one))
% 28.49/3.99 = { by lemma 40 R->L }
% 28.49/3.99 meet(X, meet(composition(Y, complement(converse(Y))), complement(complement(one))))
% 28.49/3.99 = { by lemma 58 R->L }
% 28.49/3.99 meet(X, meet(composition(Y, complement(converse(Y))), complement(join(complement(one), composition(converse(converse(Y)), complement(converse(Y)))))))
% 28.49/3.99 = { by axiom 4 (converse_idempotence_8) }
% 28.49/3.99 meet(X, meet(composition(Y, complement(converse(Y))), complement(join(complement(one), composition(Y, complement(converse(Y)))))))
% 28.49/3.99 = { by lemma 61 }
% 28.49/3.99 meet(X, zero)
% 28.49/3.99 = { by lemma 66 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 92: composition(meet(sk1, one), complement(sk1)) = zero.
% 28.49/3.99 Proof:
% 28.49/3.99 composition(meet(sk1, one), complement(sk1))
% 28.49/3.99 = { by lemma 43 R->L }
% 28.49/3.99 join(meet(complement(sk1), composition(meet(sk1, one), complement(sk1))), meet(composition(meet(sk1, one), complement(sk1)), complement(complement(sk1))))
% 28.49/3.99 = { by lemma 90 }
% 28.49/3.99 join(zero, meet(composition(meet(sk1, one), complement(sk1)), complement(complement(sk1))))
% 28.49/3.99 = { by lemma 29 }
% 28.49/3.99 meet(composition(meet(sk1, one), complement(sk1)), complement(complement(sk1)))
% 28.49/3.99 = { by lemma 40 }
% 28.49/3.99 meet(composition(meet(sk1, one), complement(sk1)), sk1)
% 28.49/3.99 = { by lemma 28 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, one), complement(sk1)), sk1), zero)
% 28.49/3.99 = { by lemma 57 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, one), complement(sk1)), sk1), composition(zero, meet(complement(sk1), composition(converse(meet(sk1, one)), sk1))))
% 28.49/3.99 = { by lemma 91 R->L }
% 28.49/3.99 join(meet(composition(meet(sk1, one), complement(sk1)), sk1), composition(meet(meet(sk1, one), composition(sk1, converse(complement(sk1)))), meet(complement(sk1), composition(converse(meet(sk1, one)), sk1))))
% 28.49/3.99 = { by axiom 17 (dedekind_law_14) }
% 28.49/3.99 composition(meet(meet(sk1, one), composition(sk1, converse(complement(sk1)))), meet(complement(sk1), composition(converse(meet(sk1, one)), sk1)))
% 28.49/3.99 = { by lemma 91 }
% 28.49/3.99 composition(zero, meet(complement(sk1), composition(converse(meet(sk1, one)), sk1)))
% 28.49/3.99 = { by lemma 57 }
% 28.49/3.99 zero
% 28.49/3.99
% 28.49/3.99 Lemma 93: composition(join(sk1, converse(meet(sk1, one))), top) = sk1.
% 28.49/3.99 Proof:
% 28.49/3.99 composition(join(sk1, converse(meet(sk1, one))), top)
% 28.49/3.99 = { by axiom 12 (composition_distributivity_7) }
% 28.49/3.99 join(composition(sk1, top), composition(converse(meet(sk1, one)), top))
% 28.49/3.99 = { by axiom 2 (goals_17) }
% 28.49/3.99 join(sk1, composition(converse(meet(sk1, one)), top))
% 28.49/3.99 = { by lemma 30 R->L }
% 28.49/3.99 join(sk1, composition(converse(meet(sk1, one)), complement(zero)))
% 28.49/3.99 = { by lemma 40 R->L }
% 28.49/4.00 join(complement(complement(sk1)), composition(converse(meet(sk1, one)), complement(zero)))
% 28.49/4.00 = { by lemma 92 R->L }
% 28.49/4.00 join(complement(complement(sk1)), composition(converse(meet(sk1, one)), complement(composition(meet(sk1, one), complement(sk1)))))
% 28.49/4.00 = { by lemma 24 }
% 28.49/4.00 complement(complement(sk1))
% 28.49/4.00 = { by lemma 40 }
% 28.49/4.00 sk1
% 28.49/4.00
% 28.49/4.00 Lemma 94: meet(converse(X), converse(join(X, Y))) = converse(X).
% 28.49/4.00 Proof:
% 28.49/4.00 meet(converse(X), converse(join(X, Y)))
% 28.49/4.00 = { by axiom 9 (converse_additivity_9) }
% 28.49/4.00 meet(converse(X), join(converse(X), converse(Y)))
% 28.49/4.00 = { by lemma 75 }
% 28.49/4.00 converse(X)
% 28.49/4.00
% 28.49/4.00 Lemma 95: meet(sk1, converse(meet(sk1, one))) = meet(sk1, one).
% 28.49/4.00 Proof:
% 28.49/4.00 meet(sk1, converse(meet(sk1, one)))
% 28.49/4.00 = { by lemma 88 R->L }
% 28.49/4.00 meet(meet(sk1, one), converse(sk1))
% 28.49/4.00 = { by lemma 83 R->L }
% 28.49/4.00 converse(meet(sk1, converse(meet(sk1, one))))
% 28.49/4.00 = { by lemma 41 R->L }
% 28.49/4.00 converse(meet(converse(meet(sk1, one)), sk1))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/4.00 converse(meet(converse(meet(sk1, one)), converse(converse(sk1))))
% 28.49/4.00 = { by lemma 83 }
% 28.49/4.00 meet(converse(sk1), converse(converse(meet(sk1, one))))
% 28.49/4.00 = { by lemma 41 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(sk1))
% 28.49/4.00 = { by lemma 93 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(composition(join(sk1, converse(meet(sk1, one))), top)))
% 28.49/4.00 = { by lemma 35 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(composition(join(sk1, converse(meet(sk1, one))), join(one, top))))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(composition(join(sk1, converse(meet(sk1, one))), join(top, one))))
% 28.49/4.00 = { by lemma 74 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(sk1, converse(meet(sk1, one))), composition(join(sk1, converse(meet(sk1, one))), top))))
% 28.49/4.00 = { by lemma 93 }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(sk1, converse(meet(sk1, one))), sk1)))
% 28.49/4.00 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(sk1, join(converse(meet(sk1, one)), sk1))))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(sk1, join(sk1, converse(meet(sk1, one))))))
% 28.49/4.00 = { by axiom 10 (maddux2_join_associativity_2) }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(sk1, sk1), converse(meet(sk1, one)))))
% 28.49/4.00 = { by lemma 65 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(sk1, meet(sk1, sk1)), converse(meet(sk1, one)))))
% 28.49/4.00 = { by lemma 65 R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(meet(sk1, sk1), meet(sk1, sk1)), converse(meet(sk1, one)))))
% 28.49/4.00 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(meet(sk1, sk1), complement(join(complement(sk1), complement(sk1)))), converse(meet(sk1, one)))))
% 28.49/4.00 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(join(complement(join(complement(sk1), complement(sk1))), complement(join(complement(sk1), complement(sk1)))), converse(meet(sk1, one)))))
% 28.49/4.00 = { by lemma 25 }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(complement(join(complement(sk1), complement(sk1))), converse(meet(sk1, one)))))
% 28.49/4.00 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(meet(sk1, sk1), converse(meet(sk1, one)))))
% 28.49/4.00 = { by lemma 65 }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(sk1, converse(meet(sk1, one)))))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 meet(converse(converse(meet(sk1, one))), converse(join(converse(meet(sk1, one)), sk1)))
% 28.49/4.00 = { by lemma 94 }
% 28.49/4.00 converse(converse(meet(sk1, one)))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) }
% 28.49/4.00 meet(sk1, one)
% 28.49/4.00
% 28.49/4.00 Lemma 96: join(X, composition(meet(Y, one), X)) = X.
% 28.49/4.00 Proof:
% 28.49/4.00 join(X, composition(meet(Y, one), X))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 join(composition(meet(Y, one), X), X)
% 28.49/4.00 = { by lemma 22 R->L }
% 28.49/4.00 join(composition(meet(Y, one), X), composition(converse(one), X))
% 28.49/4.00 = { by axiom 12 (composition_distributivity_7) R->L }
% 28.49/4.00 composition(join(meet(Y, one), converse(one)), X)
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 composition(join(converse(one), meet(Y, one)), X)
% 28.49/4.00 = { by lemma 31 }
% 28.49/4.00 composition(join(one, meet(Y, one)), X)
% 28.49/4.00 = { by lemma 72 }
% 28.49/4.00 composition(one, X)
% 28.49/4.00 = { by lemma 23 }
% 28.49/4.00 X
% 28.49/4.00
% 28.49/4.00 Lemma 97: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 28.49/4.00 Proof:
% 28.49/4.00 join(composition(X, Y), composition(X, Z))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/4.00 join(composition(X, Y), composition(converse(converse(X)), Z))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/4.00 join(converse(converse(composition(X, Y))), composition(converse(converse(X)), Z))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 join(composition(converse(converse(X)), Z), converse(converse(composition(X, Y))))
% 28.49/4.00 = { by lemma 21 R->L }
% 28.49/4.00 join(converse(composition(converse(Z), converse(X))), converse(converse(composition(X, Y))))
% 28.49/4.00 = { by axiom 9 (converse_additivity_9) R->L }
% 28.49/4.00 converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 converse(join(converse(composition(X, Y)), composition(converse(Z), converse(X))))
% 28.49/4.00 = { by axiom 7 (converse_multiplicativity_10) }
% 28.49/4.00 converse(join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X))))
% 28.49/4.00 = { by axiom 12 (composition_distributivity_7) R->L }
% 28.49/4.00 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 28.49/4.00 = { by axiom 7 (converse_multiplicativity_10) }
% 28.49/4.00 composition(converse(converse(X)), converse(join(converse(Y), converse(Z))))
% 28.49/4.00 = { by lemma 36 }
% 28.49/4.00 composition(converse(converse(X)), join(Z, converse(converse(Y))))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) }
% 28.49/4.00 composition(X, join(Z, converse(converse(Y))))
% 28.49/4.00 = { by axiom 4 (converse_idempotence_8) }
% 28.49/4.00 composition(X, join(Z, Y))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 composition(X, join(Y, Z))
% 28.49/4.00
% 28.49/4.00 Lemma 98: meet(converse(X), converse(meet(X, Y))) = converse(meet(X, Y)).
% 28.49/4.00 Proof:
% 28.49/4.00 meet(converse(X), converse(meet(X, Y)))
% 28.49/4.00 = { by lemma 41 }
% 28.49/4.00 meet(converse(meet(X, Y)), converse(X))
% 28.49/4.00 = { by lemma 19 R->L }
% 28.49/4.00 meet(converse(meet(X, Y)), converse(join(meet(X, Y), complement(join(complement(X), Y)))))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 meet(converse(meet(X, Y)), converse(join(meet(X, Y), complement(join(Y, complement(X))))))
% 28.49/4.00 = { by lemma 94 }
% 28.49/4.00 converse(meet(X, Y))
% 28.49/4.00
% 28.49/4.00 Lemma 99: composition(meet(sk1, one), join(X, complement(sk1))) = composition(meet(sk1, one), X).
% 28.49/4.00 Proof:
% 28.49/4.00 composition(meet(sk1, one), join(X, complement(sk1)))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.00 composition(meet(sk1, one), join(complement(sk1), X))
% 28.49/4.00 = { by lemma 97 R->L }
% 28.49/4.00 join(composition(meet(sk1, one), complement(sk1)), composition(meet(sk1, one), X))
% 28.49/4.00 = { by lemma 92 }
% 28.49/4.00 join(zero, composition(meet(sk1, one), X))
% 28.49/4.00 = { by lemma 29 }
% 28.49/4.00 composition(meet(sk1, one), X)
% 28.49/4.00
% 28.49/4.00 Lemma 100: meet(X, join(meet(X, Y), composition(meet(Z, one), Y))) = meet(X, Y).
% 28.49/4.00 Proof:
% 28.49/4.00 meet(X, join(meet(X, Y), composition(meet(Z, one), Y)))
% 28.49/4.00 = { by lemma 76 R->L }
% 28.49/4.00 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(Y, join(meet(X, Y), composition(meet(Z, one), Y)))))
% 28.49/4.00 = { by lemma 59 }
% 28.49/4.00 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(meet(X, Y), join(Y, composition(meet(Z, one), Y)))))
% 28.49/4.00 = { by lemma 96 }
% 28.49/4.00 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(meet(X, Y), Y)))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), join(Y, meet(X, Y))))
% 28.49/4.00 = { by lemma 72 }
% 28.49/4.00 meet(X, meet(join(meet(X, Y), composition(meet(Z, one), Y)), Y))
% 28.49/4.00 = { by lemma 41 R->L }
% 28.49/4.00 meet(X, meet(Y, join(meet(X, Y), composition(meet(Z, one), Y))))
% 28.49/4.00 = { by lemma 78 R->L }
% 28.49/4.00 meet(join(meet(X, Y), composition(meet(Z, one), Y)), meet(X, Y))
% 28.49/4.00 = { by lemma 41 }
% 28.49/4.00 meet(meet(X, Y), join(meet(X, Y), composition(meet(Z, one), Y)))
% 28.49/4.00 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 28.49/4.00 complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y)))))
% 28.49/4.00 = { by lemma 29 R->L }
% 28.49/4.00 join(zero, complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y))))))
% 28.49/4.00 = { by lemma 60 R->L }
% 28.49/4.00 join(meet(meet(X, Y), complement(join(meet(X, Y), composition(meet(Z, one), Y)))), complement(join(complement(meet(X, Y)), complement(join(meet(X, Y), composition(meet(Z, one), Y))))))
% 28.49/4.00 = { by lemma 19 }
% 28.49/4.00 meet(X, Y)
% 28.49/4.00
% 28.49/4.00 Lemma 101: composition(converse(join(meet(sk1, X), composition(meet(sk1, one), X))), complement(sk1)) = zero.
% 28.49/4.00 Proof:
% 28.49/4.00 composition(converse(join(meet(sk1, X), composition(meet(sk1, one), X))), complement(sk1))
% 28.49/4.00 = { by lemma 43 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), meet(join(meet(sk1, X), composition(meet(sk1, one), X)), complement(sk1)))), complement(sk1))
% 28.49/4.00 = { by lemma 51 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(sk1, complement(join(meet(sk1, X), composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 85 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(join(meet(sk1, X), composition(meet(sk1, one), X))), meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 100 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(join(meet(sk1, X), composition(meet(sk1, one), X))), meet(sk1, X))))), complement(sk1))
% 28.49/4.00 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(meet(sk1, X), complement(join(meet(sk1, X), composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 68 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(complement(meet(complement(meet(sk1, X)), join(meet(sk1, X), composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 49 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(complement(meet(sk1, X))), complement(join(meet(sk1, X), composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 79 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(complement(meet(sk1, X))), meet(complement(meet(sk1, X)), complement(composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 84 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), complement(complement(meet(sk1, X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 49 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(complement(meet(composition(meet(sk1, one), X), complement(meet(sk1, X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 50 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(meet(sk1, X), complement(composition(meet(sk1, one), X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 85 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(meet(sk1, X), composition(meet(sk1, one), X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 41 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(composition(meet(sk1, one), X), meet(sk1, X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 78 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(sk1, meet(X, composition(meet(sk1, one), X))))))), complement(sk1))
% 28.49/4.00 = { by lemma 41 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(sk1, meet(composition(meet(sk1, one), X), X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 96 R->L }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(sk1, meet(composition(meet(sk1, one), X), join(X, composition(meet(sk1, one), X)))))))), complement(sk1))
% 28.49/4.00 = { by lemma 76 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(complement(composition(meet(sk1, one), X)), meet(sk1, composition(meet(sk1, one), X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 85 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), complement(join(sk1, complement(composition(meet(sk1, one), X)))))), complement(sk1))
% 28.49/4.00 = { by lemma 51 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), meet(composition(meet(sk1, one), X), complement(sk1)))), complement(sk1))
% 28.49/4.00 = { by lemma 41 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), meet(complement(sk1), composition(meet(sk1, one), X)))), complement(sk1))
% 28.49/4.00 = { by lemma 90 }
% 28.49/4.00 composition(converse(join(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X))), zero)), complement(sk1))
% 28.49/4.00 = { by lemma 28 }
% 28.49/4.01 composition(converse(meet(sk1, join(meet(sk1, X), composition(meet(sk1, one), X)))), complement(sk1))
% 28.49/4.01 = { by lemma 100 }
% 28.49/4.01 composition(converse(meet(sk1, X)), complement(sk1))
% 28.49/4.01 = { by lemma 29 R->L }
% 28.49/4.01 join(zero, composition(converse(meet(sk1, X)), complement(sk1)))
% 28.49/4.01 = { by lemma 80 R->L }
% 28.49/4.01 join(composition(converse(sk1), complement(sk1)), composition(converse(meet(sk1, X)), complement(sk1)))
% 28.49/4.01 = { by axiom 12 (composition_distributivity_7) R->L }
% 28.49/4.01 composition(join(converse(sk1), converse(meet(sk1, X))), complement(sk1))
% 28.49/4.01 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.01 composition(join(converse(meet(sk1, X)), converse(sk1)), complement(sk1))
% 28.49/4.01 = { by axiom 9 (converse_additivity_9) R->L }
% 28.49/4.01 composition(converse(join(meet(sk1, X), sk1)), complement(sk1))
% 28.49/4.01 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.01 composition(converse(join(sk1, meet(sk1, X))), complement(sk1))
% 28.49/4.01 = { by lemma 71 }
% 28.49/4.01 composition(converse(sk1), complement(sk1))
% 28.49/4.01 = { by lemma 80 }
% 28.49/4.01 zero
% 28.49/4.01
% 28.49/4.01 Goal 1 (goals_18): join(meet(sk1, sk2), composition(meet(sk1, one), sk2)) = composition(meet(sk1, one), sk2).
% 28.49/4.01 Proof:
% 28.49/4.01 join(meet(sk1, sk2), composition(meet(sk1, one), sk2))
% 28.49/4.01 = { by axiom 4 (converse_idempotence_8) R->L }
% 28.49/4.01 converse(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))))
% 28.49/4.01 = { by lemma 28 R->L }
% 28.49/4.01 converse(join(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), zero))
% 28.49/4.01 = { by lemma 101 R->L }
% 28.49/4.01 converse(join(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), complement(sk1))))
% 28.49/4.01 = { by lemma 74 }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), join(complement(sk1), one)))
% 28.49/4.01 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), join(one, complement(sk1))))
% 28.49/4.01 = { by lemma 87 R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), join(meet(sk1, one), complement(sk1))))
% 28.49/4.01 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), join(complement(sk1), meet(sk1, one))))
% 28.49/4.01 = { by lemma 97 R->L }
% 28.49/4.01 converse(join(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), complement(sk1)), composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(sk1, one))))
% 28.49/4.01 = { by lemma 101 }
% 28.49/4.01 converse(join(zero, composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(sk1, one))))
% 28.49/4.01 = { by lemma 29 }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(sk1, one)))
% 28.49/4.01 = { by lemma 95 R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(sk1, converse(meet(sk1, one)))))
% 28.49/4.01 = { by lemma 88 R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(meet(sk1, one), converse(sk1))))
% 28.49/4.01 = { by lemma 83 R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), converse(meet(sk1, converse(meet(sk1, one))))))
% 28.49/4.01 = { by lemma 98 R->L }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(converse(sk1), converse(meet(sk1, converse(meet(sk1, one)))))))
% 28.49/4.01 = { by lemma 95 }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), meet(converse(sk1), converse(meet(sk1, one)))))
% 28.49/4.01 = { by lemma 98 }
% 28.49/4.01 converse(composition(converse(join(meet(sk1, sk2), composition(meet(sk1, one), sk2))), converse(meet(sk1, one))))
% 28.49/4.01 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(meet(sk1, sk2), composition(meet(sk1, one), sk2)))))
% 28.49/4.01 = { by lemma 99 R->L }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(join(meet(sk1, sk2), composition(meet(sk1, one), sk2)), complement(sk1)))))
% 28.49/4.01 = { by lemma 84 R->L }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(complement(sk1), meet(sk1, join(meet(sk1, sk2), composition(meet(sk1, one), sk2)))))))
% 28.49/4.01 = { by lemma 100 }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(complement(sk1), meet(sk1, sk2)))))
% 28.49/4.01 = { by axiom 3 (maddux1_join_commutativity_1) }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(meet(sk1, sk2), complement(sk1)))))
% 28.49/4.01 = { by lemma 87 }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), join(sk2, complement(sk1)))))
% 28.49/4.01 = { by lemma 99 }
% 28.49/4.01 converse(converse(composition(meet(sk1, one), sk2)))
% 28.49/4.01 = { by axiom 4 (converse_idempotence_8) }
% 28.49/4.01 composition(meet(sk1, one), sk2)
% 28.49/4.01 % SZS output end Proof
% 28.49/4.01
% 28.49/4.01 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------