TSTP Solution File: REL020-2 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL020-2 : 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 13:44:01 EDT 2023
% Result : Unsatisfiable 22.76s 3.26s
% Output : Proof 25.10s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.11/0.12 % Problem : REL020-2 : TPTP v8.1.2. Released v4.0.0.
% 0.11/0.12 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n023.cluster.edu
% 0.13/0.33 % Model : x86_64 x86_64
% 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33 % Memory : 8042.1875MB
% 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33 % CPULimit : 300
% 0.13/0.33 % WCLimit : 300
% 0.13/0.33 % DateTime : Fri Aug 25 19:30:11 EDT 2023
% 0.13/0.33 % CPUTime :
% 22.76/3.26 Command-line arguments: --flatten
% 22.76/3.26
% 22.76/3.26 % SZS status Unsatisfiable
% 22.76/3.26
% 24.32/3.45 % SZS output start Proof
% 24.32/3.45 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 24.32/3.45 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 24.32/3.45 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 24.32/3.45 Axiom 4 (goals_17): composition(sk1, top) = sk1.
% 24.32/3.45 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 24.32/3.45 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 24.32/3.45 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 24.32/3.45 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 24.32/3.45 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 24.32/3.45 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 24.32/3.45 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 24.32/3.45 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 24.32/3.45 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 24.32/3.45 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 24.32/3.45 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).
% 24.32/3.45 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).
% 24.32/3.45
% 24.32/3.45 Lemma 17: complement(top) = zero.
% 24.32/3.45 Proof:
% 24.32/3.45 complement(top)
% 24.32/3.45 = { by axiom 5 (def_top_12) }
% 24.32/3.45 complement(join(complement(X), complement(complement(X))))
% 24.32/3.45 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.45 meet(X, complement(X))
% 24.32/3.45 = { by axiom 6 (def_zero_13) R->L }
% 24.32/3.45 zero
% 24.32/3.45
% 24.32/3.45 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 24.32/3.45 Proof:
% 24.32/3.45 converse(composition(converse(X), Y))
% 24.32/3.45 = { by axiom 9 (converse_multiplicativity_10) }
% 24.32/3.45 composition(converse(Y), converse(converse(X)))
% 24.32/3.45 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.45 composition(converse(Y), X)
% 24.32/3.45
% 24.32/3.45 Lemma 19: composition(converse(one), X) = X.
% 24.32/3.45 Proof:
% 24.32/3.45 composition(converse(one), X)
% 24.32/3.45 = { by lemma 18 R->L }
% 24.32/3.45 converse(composition(converse(X), one))
% 24.32/3.45 = { by axiom 3 (composition_identity_6) }
% 24.32/3.45 converse(converse(X))
% 24.32/3.45 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.45 X
% 24.32/3.45
% 24.32/3.45 Lemma 20: composition(one, X) = X.
% 24.32/3.45 Proof:
% 24.32/3.45 composition(one, X)
% 24.32/3.45 = { by lemma 19 R->L }
% 24.32/3.45 composition(converse(one), composition(one, X))
% 24.32/3.45 = { by axiom 10 (composition_associativity_5) }
% 24.32/3.45 composition(composition(converse(one), one), X)
% 24.32/3.45 = { by axiom 3 (composition_identity_6) }
% 24.32/3.45 composition(converse(one), X)
% 24.32/3.45 = { by lemma 19 }
% 24.32/3.45 X
% 24.32/3.45
% 24.32/3.45 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 24.32/3.45 Proof:
% 24.32/3.45 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 24.32/3.45 = { by axiom 13 (converse_cancellativity_11) }
% 24.32/3.45 complement(X)
% 24.32/3.45
% 24.32/3.45 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 24.32/3.45 Proof:
% 24.32/3.45 join(complement(X), complement(X))
% 24.32/3.45 = { by lemma 19 R->L }
% 24.32/3.45 join(complement(X), composition(converse(one), complement(X)))
% 24.32/3.45 = { by lemma 20 R->L }
% 24.32/3.45 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 24.32/3.45 = { by lemma 21 }
% 24.32/3.45 complement(X)
% 24.32/3.45
% 24.32/3.45 Lemma 23: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 24.32/3.45 Proof:
% 24.32/3.45 join(meet(X, Y), complement(join(complement(X), Y)))
% 24.32/3.45 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.45 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 24.32/3.45 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 24.32/3.45 X
% 24.32/3.45
% 24.32/3.45 Lemma 24: join(zero, meet(X, X)) = X.
% 24.32/3.45 Proof:
% 24.32/3.45 join(zero, meet(X, X))
% 24.32/3.45 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.45 join(zero, complement(join(complement(X), complement(X))))
% 24.32/3.45 = { by axiom 6 (def_zero_13) }
% 24.32/3.45 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 24.32/3.45 = { by lemma 23 }
% 24.32/3.45 X
% 24.32/3.45
% 24.32/3.45 Lemma 25: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 24.32/3.45 Proof:
% 24.32/3.45 join(zero, join(X, complement(complement(Y))))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(zero, join(complement(complement(Y)), X))
% 24.32/3.45 = { by lemma 22 R->L }
% 24.32/3.45 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 24.32/3.45 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.45 join(zero, join(meet(Y, Y), X))
% 24.32/3.45 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.32/3.45 join(join(zero, meet(Y, Y)), X)
% 24.32/3.45 = { by lemma 24 }
% 24.32/3.45 join(Y, X)
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.45 join(X, Y)
% 24.32/3.45
% 24.32/3.45 Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
% 24.32/3.45 Proof:
% 24.32/3.45 join(X, join(Y, complement(X)))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(X, join(complement(X), Y))
% 24.32/3.45 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.32/3.45 join(join(X, complement(X)), Y)
% 24.32/3.45 = { by axiom 5 (def_top_12) R->L }
% 24.32/3.45 join(top, Y)
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.45 join(Y, top)
% 24.32/3.45
% 24.32/3.45 Lemma 27: join(top, complement(X)) = top.
% 24.32/3.45 Proof:
% 24.32/3.45 join(top, complement(X))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(complement(X), top)
% 24.32/3.45 = { by lemma 26 R->L }
% 24.32/3.45 join(X, join(complement(X), complement(X)))
% 24.32/3.45 = { by lemma 22 }
% 24.32/3.45 join(X, complement(X))
% 24.32/3.45 = { by axiom 5 (def_top_12) R->L }
% 24.32/3.45 top
% 24.32/3.45
% 24.32/3.45 Lemma 28: join(Y, top) = join(X, top).
% 24.32/3.45 Proof:
% 24.32/3.45 join(Y, top)
% 24.32/3.45 = { by lemma 27 R->L }
% 24.32/3.45 join(Y, join(top, complement(Y)))
% 24.32/3.45 = { by lemma 26 }
% 24.32/3.45 join(top, top)
% 24.32/3.45 = { by lemma 26 R->L }
% 24.32/3.45 join(X, join(top, complement(X)))
% 24.32/3.45 = { by lemma 27 }
% 24.32/3.45 join(X, top)
% 24.32/3.45
% 24.32/3.45 Lemma 29: join(X, top) = top.
% 24.32/3.45 Proof:
% 24.32/3.45 join(X, top)
% 24.32/3.45 = { by lemma 28 }
% 24.32/3.45 join(zero, top)
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(top, zero)
% 24.32/3.45 = { by lemma 17 R->L }
% 24.32/3.45 join(top, complement(top))
% 24.32/3.45 = { by axiom 5 (def_top_12) R->L }
% 24.32/3.45 top
% 24.32/3.45
% 24.32/3.45 Lemma 30: join(X, join(complement(X), Y)) = top.
% 24.32/3.45 Proof:
% 24.32/3.45 join(X, join(complement(X), Y))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(X, join(Y, complement(X)))
% 24.32/3.45 = { by lemma 26 }
% 24.32/3.45 join(Y, top)
% 24.32/3.45 = { by lemma 28 R->L }
% 24.32/3.45 join(Z, top)
% 24.32/3.45 = { by lemma 29 }
% 24.32/3.45 top
% 24.32/3.45
% 24.32/3.45 Lemma 31: join(X, complement(zero)) = top.
% 24.32/3.45 Proof:
% 24.32/3.45 join(X, complement(zero))
% 24.32/3.45 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.45 join(complement(zero), X)
% 24.32/3.45 = { by lemma 25 R->L }
% 24.32/3.46 join(zero, join(complement(zero), complement(complement(X))))
% 24.32/3.46 = { by lemma 30 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 32: complement(zero) = top.
% 24.32/3.46 Proof:
% 24.32/3.46 complement(zero)
% 24.32/3.46 = { by lemma 22 R->L }
% 24.32/3.46 join(complement(zero), complement(zero))
% 24.32/3.46 = { by lemma 31 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 33: converse(one) = one.
% 24.32/3.46 Proof:
% 24.32/3.46 converse(one)
% 24.32/3.46 = { by axiom 3 (composition_identity_6) R->L }
% 24.32/3.46 composition(converse(one), one)
% 24.32/3.46 = { by lemma 19 }
% 24.32/3.46 one
% 24.32/3.46
% 24.32/3.46 Lemma 34: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 24.32/3.46 Proof:
% 24.32/3.46 converse(join(X, converse(Y)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 converse(join(converse(Y), X))
% 24.32/3.46 = { by axiom 7 (converse_additivity_9) }
% 24.32/3.46 join(converse(converse(Y)), converse(X))
% 24.32/3.46 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.46 join(Y, converse(X))
% 24.32/3.46
% 24.32/3.46 Lemma 35: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 24.32/3.46 Proof:
% 24.32/3.46 converse(join(converse(X), Y))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 converse(join(Y, converse(X)))
% 24.32/3.46 = { by lemma 34 }
% 24.32/3.46 join(X, converse(Y))
% 24.32/3.46
% 24.32/3.46 Lemma 36: join(X, converse(complement(converse(X)))) = converse(top).
% 24.32/3.46 Proof:
% 24.32/3.46 join(X, converse(complement(converse(X))))
% 24.32/3.46 = { by lemma 35 R->L }
% 24.32/3.46 converse(join(converse(X), complement(converse(X))))
% 24.32/3.46 = { by axiom 5 (def_top_12) R->L }
% 24.32/3.46 converse(top)
% 24.32/3.46
% 24.32/3.46 Lemma 37: join(X, converse(top)) = top.
% 24.32/3.46 Proof:
% 24.32/3.46 join(X, converse(top))
% 24.32/3.46 = { by lemma 36 R->L }
% 24.32/3.46 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 24.32/3.46 = { by lemma 30 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 38: converse(top) = top.
% 24.32/3.46 Proof:
% 24.32/3.46 converse(top)
% 24.32/3.46 = { by lemma 29 R->L }
% 24.32/3.46 converse(join(X, top))
% 24.32/3.46 = { by axiom 7 (converse_additivity_9) }
% 24.32/3.46 join(converse(X), converse(top))
% 24.32/3.46 = { by lemma 37 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 39: join(zero, complement(complement(X))) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 join(zero, complement(complement(X)))
% 24.32/3.46 = { by axiom 6 (def_zero_13) }
% 24.32/3.46 join(meet(X, complement(X)), complement(complement(X)))
% 24.32/3.46 = { by lemma 22 R->L }
% 24.32/3.46 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 24.32/3.46 = { by lemma 23 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 40: join(X, zero) = join(X, X).
% 24.32/3.46 Proof:
% 24.32/3.46 join(X, zero)
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(zero, X)
% 24.32/3.46 = { by lemma 39 R->L }
% 24.32/3.46 join(zero, join(zero, complement(complement(X))))
% 24.32/3.46 = { by lemma 22 R->L }
% 24.32/3.46 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 24.32/3.46 = { by lemma 25 }
% 24.32/3.46 join(zero, join(complement(complement(X)), X))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.46 join(zero, join(X, complement(complement(X))))
% 24.32/3.46 = { by lemma 25 }
% 24.32/3.46 join(X, X)
% 24.32/3.46
% 24.32/3.46 Lemma 41: join(zero, complement(X)) = complement(X).
% 24.32/3.46 Proof:
% 24.32/3.46 join(zero, complement(X))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(complement(X), zero)
% 24.32/3.46 = { by lemma 40 }
% 24.32/3.46 join(complement(X), complement(X))
% 24.32/3.46 = { by lemma 22 }
% 24.32/3.46 complement(X)
% 24.32/3.46
% 24.32/3.46 Lemma 42: join(X, zero) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 join(X, zero)
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(zero, X)
% 24.32/3.46 = { by lemma 25 R->L }
% 24.32/3.46 join(zero, join(zero, complement(complement(X))))
% 24.32/3.46 = { by lemma 41 }
% 24.32/3.46 join(zero, complement(complement(X)))
% 24.32/3.46 = { by lemma 39 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 43: join(zero, X) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 join(zero, X)
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(X, zero)
% 24.32/3.46 = { by lemma 42 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 44: meet(Y, X) = meet(X, Y).
% 24.32/3.46 Proof:
% 24.32/3.46 meet(Y, X)
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 complement(join(complement(Y), complement(X)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 complement(join(complement(X), complement(Y)))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.46 meet(X, Y)
% 24.32/3.46
% 24.32/3.46 Lemma 45: complement(join(zero, complement(X))) = meet(X, top).
% 24.32/3.46 Proof:
% 24.32/3.46 complement(join(zero, complement(X)))
% 24.32/3.46 = { by lemma 17 R->L }
% 24.32/3.46 complement(join(complement(top), complement(X)))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.46 meet(top, X)
% 24.32/3.46 = { by lemma 44 R->L }
% 24.32/3.46 meet(X, top)
% 24.32/3.46
% 24.32/3.46 Lemma 46: meet(X, zero) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 meet(X, zero)
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 complement(join(complement(X), complement(zero)))
% 24.32/3.46 = { by lemma 31 }
% 24.32/3.46 complement(top)
% 24.32/3.46 = { by lemma 17 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 47: join(meet(X, Y), meet(X, complement(Y))) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 join(meet(X, Y), meet(X, complement(Y)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(meet(X, complement(Y)), meet(X, Y))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 24.32/3.46 = { by lemma 23 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 48: meet(X, top) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 meet(X, top)
% 24.32/3.46 = { by lemma 45 R->L }
% 24.32/3.46 complement(join(zero, complement(X)))
% 24.32/3.46 = { by lemma 41 R->L }
% 24.32/3.46 join(zero, complement(join(zero, complement(X))))
% 24.32/3.46 = { by lemma 45 }
% 24.32/3.46 join(zero, meet(X, top))
% 24.32/3.46 = { by lemma 32 R->L }
% 24.32/3.46 join(zero, meet(X, complement(zero)))
% 24.32/3.46 = { by lemma 46 R->L }
% 24.32/3.46 join(meet(X, zero), meet(X, complement(zero)))
% 24.32/3.46 = { by lemma 47 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 49: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 24.32/3.46 Proof:
% 24.32/3.46 join(meet(X, Y), meet(X, Y))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 join(meet(Y, X), meet(X, Y))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 join(meet(Y, X), meet(Y, X))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 24.32/3.46 = { by lemma 22 }
% 24.32/3.46 complement(join(complement(Y), complement(X)))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.46 meet(Y, X)
% 24.32/3.46 = { by lemma 44 R->L }
% 24.32/3.46 meet(X, Y)
% 24.32/3.46
% 24.32/3.46 Lemma 50: converse(zero) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 converse(zero)
% 24.32/3.46 = { by lemma 43 R->L }
% 24.32/3.46 join(zero, converse(zero))
% 24.32/3.46 = { by lemma 35 R->L }
% 24.32/3.46 converse(join(converse(zero), zero))
% 24.32/3.46 = { by lemma 40 }
% 24.32/3.46 converse(join(converse(zero), converse(zero)))
% 24.32/3.46 = { by lemma 34 }
% 24.32/3.46 join(zero, converse(converse(zero)))
% 24.32/3.46 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.46 join(zero, zero)
% 24.32/3.46 = { by lemma 48 R->L }
% 24.32/3.46 join(zero, meet(zero, top))
% 24.32/3.46 = { by lemma 48 R->L }
% 24.32/3.46 join(meet(zero, top), meet(zero, top))
% 24.32/3.46 = { by lemma 49 }
% 24.32/3.46 meet(zero, top)
% 24.32/3.46 = { by lemma 48 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 51: join(top, X) = top.
% 24.32/3.46 Proof:
% 24.32/3.46 join(top, X)
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(X, top)
% 24.32/3.46 = { by lemma 28 R->L }
% 24.32/3.46 join(Y, top)
% 24.32/3.46 = { by lemma 29 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 52: complement(complement(X)) = X.
% 24.32/3.46 Proof:
% 24.32/3.46 complement(complement(X))
% 24.32/3.46 = { by lemma 41 R->L }
% 24.32/3.46 join(zero, complement(complement(X)))
% 24.32/3.46 = { by lemma 39 }
% 24.32/3.46 X
% 24.32/3.46
% 24.32/3.46 Lemma 53: meet(zero, X) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 meet(zero, X)
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 meet(X, zero)
% 24.32/3.46 = { by lemma 46 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 54: composition(join(X, one), Y) = join(Y, composition(X, Y)).
% 24.32/3.46 Proof:
% 24.32/3.46 composition(join(X, one), Y)
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 composition(join(one, X), Y)
% 24.32/3.46 = { by axiom 12 (composition_distributivity_7) }
% 24.32/3.46 join(composition(one, Y), composition(X, Y))
% 24.32/3.46 = { by lemma 20 }
% 24.32/3.46 join(Y, composition(X, Y))
% 24.32/3.46
% 24.32/3.46 Lemma 55: composition(top, zero) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 composition(top, zero)
% 24.32/3.46 = { by lemma 38 R->L }
% 24.32/3.46 composition(converse(top), zero)
% 24.32/3.46 = { by lemma 43 R->L }
% 24.32/3.46 join(zero, composition(converse(top), zero))
% 24.32/3.46 = { by lemma 17 R->L }
% 24.32/3.46 join(complement(top), composition(converse(top), zero))
% 24.32/3.46 = { by lemma 17 R->L }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(top)))
% 24.32/3.46 = { by lemma 51 R->L }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 24.32/3.46 = { by lemma 38 R->L }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 24.32/3.46 = { by lemma 54 R->L }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(composition(join(converse(top), one), top))))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 24.32/3.46 = { by lemma 37 }
% 24.32/3.46 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 24.32/3.46 = { by lemma 21 }
% 24.32/3.46 complement(top)
% 24.32/3.46 = { by lemma 17 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 56: composition(X, zero) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 composition(X, zero)
% 24.32/3.46 = { by lemma 43 R->L }
% 24.32/3.46 join(zero, composition(X, zero))
% 24.32/3.46 = { by lemma 55 R->L }
% 24.32/3.46 join(composition(top, zero), composition(X, zero))
% 24.32/3.46 = { by axiom 12 (composition_distributivity_7) R->L }
% 24.32/3.46 composition(join(top, X), zero)
% 24.32/3.46 = { by lemma 51 }
% 24.32/3.46 composition(top, zero)
% 24.32/3.46 = { by lemma 55 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 57: composition(zero, X) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 composition(zero, X)
% 24.32/3.46 = { by lemma 50 R->L }
% 24.32/3.46 composition(converse(zero), X)
% 24.32/3.46 = { by lemma 18 R->L }
% 24.32/3.46 converse(composition(converse(X), zero))
% 24.32/3.46 = { by lemma 56 }
% 24.32/3.46 converse(zero)
% 24.32/3.46 = { by lemma 50 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 58: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 24.32/3.46 Proof:
% 24.32/3.46 meet(X, join(complement(Y), complement(Z)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 meet(X, join(complement(Z), complement(Y)))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 meet(join(complement(Z), complement(Y)), X)
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.46 complement(join(meet(Z, Y), complement(X)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.46 complement(join(complement(X), meet(Z, Y)))
% 24.32/3.46 = { by lemma 44 R->L }
% 24.32/3.46 complement(join(complement(X), meet(Y, Z)))
% 24.32/3.46
% 24.32/3.46 Lemma 59: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 24.32/3.46 Proof:
% 24.32/3.46 complement(join(X, complement(Y)))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 complement(join(complement(Y), X))
% 24.32/3.46 = { by lemma 48 R->L }
% 24.32/3.46 complement(join(complement(Y), meet(X, top)))
% 24.32/3.46 = { by lemma 44 R->L }
% 24.32/3.46 complement(join(complement(Y), meet(top, X)))
% 24.32/3.46 = { by lemma 58 R->L }
% 24.32/3.46 meet(Y, join(complement(top), complement(X)))
% 24.32/3.46 = { by lemma 17 }
% 24.32/3.46 meet(Y, join(zero, complement(X)))
% 24.32/3.46 = { by lemma 41 }
% 24.32/3.46 meet(Y, complement(X))
% 24.32/3.46
% 24.32/3.46 Lemma 60: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 24.32/3.46 Proof:
% 24.32/3.46 complement(join(complement(X), Y))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 complement(join(Y, complement(X)))
% 24.32/3.46 = { by lemma 59 }
% 24.32/3.46 meet(X, complement(Y))
% 24.32/3.46
% 24.32/3.46 Lemma 61: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 24.32/3.46 Proof:
% 24.32/3.46 join(complement(one), composition(converse(X), complement(X)))
% 24.32/3.46 = { by axiom 3 (composition_identity_6) R->L }
% 24.32/3.46 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 24.32/3.46 = { by lemma 21 }
% 24.32/3.46 complement(one)
% 24.32/3.46
% 24.32/3.46 Lemma 62: join(complement(one), composition(converse(complement(X)), X)) = complement(one).
% 24.32/3.46 Proof:
% 24.32/3.46 join(complement(one), composition(converse(complement(X)), X))
% 24.32/3.46 = { by lemma 48 R->L }
% 24.32/3.46 join(complement(one), composition(converse(complement(X)), meet(X, top)))
% 24.32/3.46 = { by lemma 41 R->L }
% 24.32/3.46 join(complement(one), composition(converse(join(zero, complement(X))), meet(X, top)))
% 24.32/3.46 = { by lemma 45 R->L }
% 24.32/3.46 join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))
% 24.32/3.46 = { by lemma 61 }
% 24.32/3.46 complement(one)
% 24.32/3.46
% 24.32/3.46 Lemma 63: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 24.32/3.46 Proof:
% 24.32/3.46 join(complement(X), complement(Y))
% 24.32/3.46 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.46 join(complement(Y), complement(X))
% 24.32/3.46 = { by lemma 24 R->L }
% 24.32/3.46 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 24.32/3.46 = { by lemma 58 }
% 24.32/3.46 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 24.32/3.46 = { by lemma 41 }
% 24.32/3.46 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.46 complement(join(meet(Y, X), meet(Y, X)))
% 24.32/3.46 = { by lemma 49 }
% 24.32/3.46 complement(meet(Y, X))
% 24.32/3.46 = { by lemma 44 R->L }
% 24.32/3.46 complement(meet(X, Y))
% 24.32/3.46
% 24.32/3.46 Lemma 64: join(X, complement(meet(X, Y))) = top.
% 24.32/3.46 Proof:
% 24.32/3.46 join(X, complement(meet(X, Y)))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 join(X, complement(meet(Y, X)))
% 24.32/3.46 = { by lemma 63 R->L }
% 24.32/3.46 join(X, join(complement(Y), complement(X)))
% 24.32/3.46 = { by lemma 26 }
% 24.32/3.46 join(complement(Y), top)
% 24.32/3.46 = { by lemma 29 }
% 24.32/3.46 top
% 24.32/3.46
% 24.32/3.46 Lemma 65: meet(X, meet(Y, complement(X))) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 meet(X, meet(Y, complement(X)))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 meet(X, meet(complement(X), Y))
% 24.32/3.46 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.46 complement(join(complement(X), complement(meet(complement(X), Y))))
% 24.32/3.46 = { by lemma 64 }
% 24.32/3.46 complement(top)
% 24.32/3.46 = { by lemma 17 }
% 24.32/3.46 zero
% 24.32/3.46
% 24.32/3.46 Lemma 66: meet(one, composition(converse(complement(X)), X)) = zero.
% 24.32/3.46 Proof:
% 24.32/3.46 meet(one, composition(converse(complement(X)), X))
% 24.32/3.46 = { by lemma 44 }
% 24.32/3.46 meet(composition(converse(complement(X)), X), one)
% 24.32/3.46 = { by lemma 52 R->L }
% 24.32/3.46 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 24.32/3.46 = { by lemma 62 R->L }
% 24.32/3.46 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 24.32/3.46 = { by lemma 60 }
% 24.32/3.46 meet(composition(converse(complement(X)), X), meet(one, complement(composition(converse(complement(X)), X))))
% 24.32/3.47 = { by lemma 65 }
% 24.32/3.47 zero
% 24.32/3.47
% 24.32/3.47 Lemma 67: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 24.32/3.47 Proof:
% 24.32/3.47 join(meet(X, Y), meet(Y, complement(X)))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 join(meet(Y, X), meet(Y, complement(X)))
% 24.32/3.47 = { by lemma 47 }
% 24.32/3.47 Y
% 24.32/3.47
% 24.32/3.47 Lemma 68: converse(complement(X)) = complement(converse(X)).
% 24.32/3.47 Proof:
% 24.32/3.47 converse(complement(X))
% 24.32/3.47 = { by lemma 41 R->L }
% 24.32/3.47 converse(join(zero, complement(X)))
% 24.32/3.47 = { by lemma 23 R->L }
% 24.32/3.47 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)))))))))))
% 24.32/3.47 = { by lemma 60 R->L }
% 24.32/3.47 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)))))))))))
% 24.32/3.47 = { by lemma 36 }
% 24.32/3.47 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 24.32/3.47 = { by lemma 38 }
% 24.32/3.47 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 24.32/3.47 = { by lemma 17 }
% 24.32/3.47 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 24.32/3.47 = { by lemma 41 }
% 24.32/3.47 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 24.32/3.47 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.47 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 24.32/3.47 = { by lemma 42 R->L }
% 24.32/3.47 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 24.32/3.47 = { by lemma 53 R->L }
% 24.32/3.47 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)))))))))
% 24.32/3.47 = { by lemma 57 R->L }
% 24.32/3.47 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)))))))))
% 24.32/3.47 = { by lemma 66 R->L }
% 24.32/3.47 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)))))))))
% 24.32/3.47 = { by axiom 16 (modular_law_2_16) R->L }
% 24.32/3.47 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))))))))))
% 24.32/3.47 = { by lemma 66 }
% 24.32/3.47 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))))))))))
% 24.32/3.47 = { by lemma 20 }
% 24.32/3.47 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))))))))))
% 24.32/3.47 = { by lemma 57 }
% 24.32/3.47 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))))))))))
% 24.32/3.47 = { by lemma 53 }
% 24.32/3.47 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)))
% 24.32/3.47 = { by lemma 42 }
% 24.32/3.47 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)))))))))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
% 24.32/3.47 = { by lemma 67 }
% 24.32/3.47 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 24.32/3.47 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.47 complement(converse(complement(join(zero, complement(X)))))
% 24.32/3.47 = { by lemma 45 }
% 24.32/3.47 complement(converse(meet(X, top)))
% 24.32/3.47 = { by lemma 48 }
% 24.32/3.47 complement(converse(X))
% 24.32/3.47
% 24.32/3.47 Lemma 69: composition(converse(X), top) = converse(composition(top, X)).
% 24.32/3.47 Proof:
% 24.32/3.47 composition(converse(X), top)
% 24.32/3.47 = { by lemma 38 R->L }
% 24.32/3.47 composition(converse(X), converse(top))
% 24.32/3.47 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 24.32/3.47 converse(composition(top, X))
% 24.32/3.47
% 24.32/3.47 Lemma 70: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 24.32/3.47 Proof:
% 24.32/3.47 complement(meet(X, complement(Y)))
% 24.32/3.47 = { by lemma 43 R->L }
% 24.32/3.47 complement(join(zero, meet(X, complement(Y))))
% 24.32/3.47 = { by lemma 59 R->L }
% 24.32/3.47 complement(join(zero, complement(join(Y, complement(X)))))
% 24.32/3.47 = { by lemma 45 }
% 24.32/3.47 meet(join(Y, complement(X)), top)
% 24.32/3.47 = { by lemma 48 }
% 24.32/3.47 join(Y, complement(X))
% 24.32/3.47
% 24.32/3.47 Lemma 71: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 24.32/3.47 Proof:
% 24.32/3.47 complement(meet(complement(X), Y))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 complement(meet(Y, complement(X)))
% 24.32/3.47 = { by lemma 70 }
% 24.32/3.47 join(X, complement(Y))
% 24.32/3.47
% 24.32/3.47 Lemma 72: meet(X, join(X, complement(Y))) = X.
% 24.32/3.47 Proof:
% 24.32/3.47 meet(X, join(X, complement(Y)))
% 24.32/3.47 = { by lemma 70 R->L }
% 24.32/3.47 meet(X, complement(meet(Y, complement(X))))
% 24.32/3.47 = { by lemma 60 R->L }
% 24.32/3.47 complement(join(complement(X), meet(Y, complement(X))))
% 24.32/3.47 = { by lemma 41 R->L }
% 24.32/3.47 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 24.32/3.47 = { by lemma 65 R->L }
% 24.32/3.47 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 24.32/3.47 = { by lemma 23 }
% 24.32/3.47 X
% 24.32/3.47
% 24.32/3.47 Lemma 73: join(X, meet(X, Y)) = X.
% 24.32/3.47 Proof:
% 24.32/3.47 join(X, meet(X, Y))
% 24.32/3.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.47 join(X, complement(join(complement(X), complement(Y))))
% 24.32/3.47 = { by lemma 71 R->L }
% 24.32/3.47 complement(meet(complement(X), join(complement(X), complement(Y))))
% 24.32/3.47 = { by lemma 72 }
% 24.32/3.47 complement(complement(X))
% 24.32/3.47 = { by lemma 52 }
% 24.32/3.47 X
% 24.32/3.47
% 24.32/3.47 Lemma 74: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 24.32/3.47 Proof:
% 24.32/3.47 meet(complement(X), complement(Y))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 meet(complement(Y), complement(X))
% 24.32/3.47 = { by lemma 41 R->L }
% 24.32/3.47 meet(join(zero, complement(Y)), complement(X))
% 24.32/3.47 = { by lemma 59 R->L }
% 24.32/3.47 complement(join(X, complement(join(zero, complement(Y)))))
% 24.32/3.47 = { by lemma 45 }
% 24.32/3.47 complement(join(X, meet(Y, top)))
% 24.32/3.47 = { by lemma 48 }
% 24.32/3.47 complement(join(X, Y))
% 24.32/3.47
% 24.32/3.47 Lemma 75: join(meet(X, Y), Y) = Y.
% 24.32/3.47 Proof:
% 24.32/3.47 join(meet(X, Y), Y)
% 24.32/3.47 = { by lemma 48 R->L }
% 24.32/3.47 meet(join(meet(X, Y), Y), top)
% 24.32/3.47 = { by lemma 45 R->L }
% 24.32/3.47 complement(join(zero, complement(join(meet(X, Y), Y))))
% 24.32/3.47 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.47 complement(join(zero, complement(join(Y, meet(X, Y)))))
% 24.32/3.47 = { by lemma 74 R->L }
% 24.32/3.47 complement(join(zero, meet(complement(Y), complement(meet(X, Y)))))
% 24.32/3.47 = { by lemma 43 R->L }
% 24.32/3.47 complement(join(zero, join(zero, meet(complement(Y), complement(meet(X, Y))))))
% 24.32/3.47 = { by lemma 17 R->L }
% 24.32/3.47 complement(join(zero, join(complement(top), meet(complement(Y), complement(meet(X, Y))))))
% 24.32/3.47 = { by lemma 64 R->L }
% 24.32/3.47 complement(join(zero, join(complement(join(Y, complement(meet(Y, X)))), meet(complement(Y), complement(meet(X, Y))))))
% 24.32/3.47 = { by lemma 44 R->L }
% 24.32/3.47 complement(join(zero, join(complement(join(Y, complement(meet(X, Y)))), meet(complement(Y), complement(meet(X, Y))))))
% 24.32/3.47 = { by lemma 59 }
% 24.32/3.47 complement(join(zero, join(meet(meet(X, Y), complement(Y)), meet(complement(Y), complement(meet(X, Y))))))
% 24.32/3.47 = { by lemma 67 }
% 24.32/3.47 complement(join(zero, complement(Y)))
% 24.32/3.47 = { by lemma 45 }
% 24.32/3.47 meet(Y, top)
% 24.32/3.47 = { by lemma 48 }
% 24.32/3.47 Y
% 24.32/3.47
% 24.32/3.47 Lemma 76: meet(X, join(X, Y)) = X.
% 24.32/3.47 Proof:
% 24.32/3.47 meet(X, join(X, Y))
% 24.32/3.47 = { by lemma 48 R->L }
% 24.32/3.47 meet(X, join(X, meet(Y, top)))
% 24.32/3.47 = { by lemma 45 R->L }
% 24.32/3.47 meet(X, join(X, complement(join(zero, complement(Y)))))
% 24.32/3.47 = { by lemma 72 }
% 24.32/3.47 X
% 24.32/3.47
% 24.32/3.47 Lemma 77: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 24.32/3.47 Proof:
% 24.32/3.47 meet(complement(Z), meet(Y, X))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 meet(complement(Z), meet(X, Y))
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 meet(meet(X, Y), complement(Z))
% 24.32/3.47 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 24.32/3.47 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 24.32/3.47 = { by lemma 74 }
% 24.32/3.47 complement(join(join(complement(X), complement(Y)), Z))
% 24.32/3.47 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.32/3.47 complement(join(complement(X), join(complement(Y), Z)))
% 24.32/3.47 = { by lemma 60 }
% 24.32/3.47 meet(X, complement(join(complement(Y), Z)))
% 24.32/3.47 = { by lemma 60 }
% 24.32/3.47 meet(X, meet(Y, complement(Z)))
% 24.32/3.47
% 24.32/3.47 Lemma 78: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 24.32/3.47 Proof:
% 24.32/3.47 meet(Y, meet(Z, X))
% 24.32/3.47 = { by lemma 48 R->L }
% 24.32/3.47 meet(meet(Y, top), meet(Z, X))
% 24.32/3.47 = { by lemma 45 R->L }
% 24.32/3.47 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 24.32/3.47 = { by lemma 77 }
% 24.32/3.47 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 24.32/3.47 = { by lemma 45 }
% 24.32/3.47 meet(X, meet(Z, meet(Y, top)))
% 24.32/3.47 = { by lemma 48 }
% 24.32/3.47 meet(X, meet(Z, Y))
% 24.32/3.47 = { by lemma 44 R->L }
% 24.32/3.47 meet(X, meet(Y, Z))
% 24.32/3.47
% 24.32/3.47 Lemma 79: meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 24.32/3.47 Proof:
% 24.32/3.47 meet(meet(X, Y), Z)
% 24.32/3.47 = { by lemma 44 }
% 24.32/3.47 meet(Z, meet(X, Y))
% 24.32/3.47 = { by lemma 78 R->L }
% 24.32/3.47 meet(X, meet(Y, Z))
% 24.32/3.47
% 24.32/3.47 Lemma 80: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 24.32/3.47 Proof:
% 24.32/3.47 converse(composition(X, converse(Y)))
% 24.32/3.47 = { by axiom 9 (converse_multiplicativity_10) }
% 24.32/3.47 composition(converse(converse(Y)), converse(X))
% 24.32/3.47 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.47 composition(Y, converse(X))
% 24.32/3.47
% 24.32/3.47 Lemma 81: converse(join(composition(X, Y), composition(X, converse(Z)))) = converse(composition(X, join(Y, converse(Z)))).
% 24.32/3.47 Proof:
% 24.32/3.47 converse(join(composition(X, Y), composition(X, converse(Z))))
% 24.32/3.47 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.47 converse(join(composition(X, converse(Z)), composition(X, Y)))
% 24.32/3.47 = { by axiom 7 (converse_additivity_9) }
% 24.32/3.47 join(converse(composition(X, converse(Z))), converse(composition(X, Y)))
% 24.32/3.47 = { by lemma 80 }
% 24.32/3.47 join(composition(Z, converse(X)), converse(composition(X, Y)))
% 24.32/3.47 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.47 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 24.32/3.47 = { by axiom 9 (converse_multiplicativity_10) }
% 24.32/3.47 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 24.32/3.47 = { by axiom 12 (composition_distributivity_7) R->L }
% 24.32/3.47 composition(join(converse(Y), Z), converse(X))
% 24.32/3.47 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.47 composition(join(Z, converse(Y)), converse(X))
% 24.32/3.47 = { by lemma 34 R->L }
% 24.32/3.47 composition(converse(join(Y, converse(Z))), converse(X))
% 24.32/3.47 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 24.32/3.47 converse(composition(X, join(Y, converse(Z))))
% 24.32/3.47
% 24.32/3.47 Lemma 82: composition(converse(X), complement(composition(X, top))) = zero.
% 24.32/3.47 Proof:
% 24.32/3.47 composition(converse(X), complement(composition(X, top)))
% 24.32/3.47 = { by lemma 43 R->L }
% 24.32/3.47 join(zero, composition(converse(X), complement(composition(X, top))))
% 24.32/3.47 = { by lemma 17 R->L }
% 24.32/3.47 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 24.32/3.47 = { by lemma 21 }
% 24.32/3.47 complement(top)
% 24.32/3.47 = { by lemma 17 }
% 24.32/3.47 zero
% 24.32/3.47
% 24.32/3.47 Lemma 83: composition(join(X, complement(converse(composition(Y, top)))), Y) = composition(X, Y).
% 24.32/3.47 Proof:
% 24.32/3.47 composition(join(X, complement(converse(composition(Y, top)))), Y)
% 24.32/3.47 = { by lemma 68 R->L }
% 24.32/3.47 composition(join(X, converse(complement(composition(Y, top)))), Y)
% 24.32/3.47 = { by lemma 34 R->L }
% 24.32/3.47 composition(converse(join(complement(composition(Y, top)), converse(X))), Y)
% 24.32/3.47 = { by lemma 18 R->L }
% 24.32/3.47 converse(composition(converse(Y), join(complement(composition(Y, top)), converse(X))))
% 24.32/3.47 = { by lemma 81 R->L }
% 24.32/3.47 converse(join(composition(converse(Y), complement(composition(Y, top))), composition(converse(Y), converse(X))))
% 24.32/3.47 = { by lemma 82 }
% 24.32/3.47 converse(join(zero, composition(converse(Y), converse(X))))
% 24.32/3.47 = { by lemma 43 }
% 24.32/3.47 converse(composition(converse(Y), converse(X)))
% 24.32/3.47 = { by lemma 18 }
% 24.32/3.47 composition(converse(converse(X)), Y)
% 24.32/3.47 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.47 composition(X, Y)
% 24.32/3.47
% 24.32/3.47 Lemma 84: composition(converse(composition(sk1, top)), complement(composition(sk1, top))) = zero.
% 24.32/3.47 Proof:
% 24.32/3.47 composition(converse(composition(sk1, top)), complement(composition(sk1, top)))
% 24.32/3.47 = { by lemma 42 R->L }
% 24.32/3.47 join(composition(converse(composition(sk1, top)), complement(composition(sk1, top))), zero)
% 24.32/3.47 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.47 join(zero, composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 24.32/3.47 = { by lemma 17 R->L }
% 24.32/3.47 join(complement(top), composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 24.32/3.47 = { by axiom 4 (goals_17) R->L }
% 24.32/3.47 join(complement(top), composition(converse(composition(sk1, top)), complement(composition(composition(sk1, top), top))))
% 24.32/3.47 = { by lemma 21 }
% 24.32/3.47 complement(top)
% 24.32/3.47 = { by lemma 17 }
% 24.32/3.47 zero
% 24.32/3.47
% 24.32/3.47 Lemma 85: composition(converse(complement(composition(sk1, top))), composition(sk1, top)) = zero.
% 24.32/3.47 Proof:
% 24.32/3.47 composition(converse(complement(composition(sk1, top))), composition(sk1, top))
% 24.32/3.47 = { by lemma 18 R->L }
% 24.32/3.47 converse(composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 24.32/3.47 = { by lemma 84 }
% 24.32/3.47 converse(zero)
% 24.32/3.47 = { by lemma 50 }
% 24.32/3.47 zero
% 24.32/3.47
% 24.32/3.47 Lemma 86: meet(meet(sk1, X), complement(composition(sk1, top))) = zero.
% 24.32/3.47 Proof:
% 24.32/3.47 meet(meet(sk1, X), complement(composition(sk1, top)))
% 24.32/3.47 = { by lemma 59 R->L }
% 24.32/3.47 complement(join(composition(sk1, top), complement(meet(sk1, X))))
% 24.32/3.47 = { by axiom 4 (goals_17) }
% 24.32/3.47 complement(join(sk1, complement(meet(sk1, X))))
% 24.32/3.48 = { by lemma 64 }
% 24.32/3.48 complement(top)
% 24.32/3.48 = { by lemma 17 }
% 24.32/3.48 zero
% 24.32/3.48
% 24.32/3.48 Lemma 87: meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y)) = zero.
% 24.32/3.48 Proof:
% 24.32/3.48 meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y))
% 24.32/3.48 = { by lemma 67 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(meet(composition(sk1, top), composition(complement(composition(sk1, top)), Y)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 44 }
% 24.32/3.48 meet(meet(sk1, X), join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 42 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), zero), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 53 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), meet(zero, composition(sk1, top))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 56 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), zero), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 46 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), meet(Y, zero)), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 85 R->L }
% 24.32/3.48 meet(meet(sk1, X), join(join(meet(composition(complement(composition(sk1, top)), Y), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), meet(Y, composition(converse(complement(composition(sk1, top))), composition(sk1, top)))), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by axiom 15 (modular_law_1_15) }
% 24.32/3.48 meet(meet(sk1, X), join(meet(composition(complement(composition(sk1, top)), meet(Y, composition(converse(complement(composition(sk1, top))), composition(sk1, top)))), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 85 }
% 24.32/3.48 meet(meet(sk1, X), join(meet(composition(complement(composition(sk1, top)), meet(Y, zero)), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 46 }
% 24.32/3.48 meet(meet(sk1, X), join(meet(composition(complement(composition(sk1, top)), zero), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 56 }
% 24.32/3.48 meet(meet(sk1, X), join(meet(zero, composition(sk1, top)), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 53 }
% 24.32/3.48 meet(meet(sk1, X), join(zero, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 24.32/3.48 = { by lemma 43 }
% 24.32/3.48 meet(meet(sk1, X), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))
% 24.32/3.48 = { by lemma 78 }
% 24.32/3.48 meet(complement(composition(sk1, top)), meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y)))
% 24.32/3.48 = { by lemma 77 }
% 24.32/3.48 meet(composition(complement(composition(sk1, top)), Y), meet(meet(sk1, X), complement(composition(sk1, top))))
% 24.32/3.48 = { by lemma 86 }
% 24.32/3.48 meet(composition(complement(composition(sk1, top)), Y), zero)
% 24.32/3.48 = { by lemma 46 }
% 24.32/3.48 zero
% 24.32/3.48
% 24.32/3.48 Lemma 88: meet(complement(composition(sk1, top)), composition(meet(sk1, X), Y)) = zero.
% 24.32/3.48 Proof:
% 24.32/3.48 meet(complement(composition(sk1, top)), composition(meet(sk1, X), Y))
% 24.32/3.48 = { by lemma 44 }
% 24.32/3.48 meet(composition(meet(sk1, X), Y), complement(composition(sk1, top)))
% 24.32/3.48 = { by lemma 42 R->L }
% 24.32/3.48 join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), zero)
% 24.32/3.48 = { by lemma 53 R->L }
% 24.32/3.48 join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top))))
% 24.32/3.48 = { by lemma 57 R->L }
% 24.32/3.48 join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(composition(zero, Y), complement(composition(sk1, top))))
% 24.32/3.48 = { by lemma 87 R->L }
% 24.32/3.48 join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(composition(meet(meet(sk1, X), composition(complement(composition(sk1, top)), converse(Y))), Y), complement(composition(sk1, top))))
% 24.32/3.48 = { by axiom 16 (modular_law_2_16) }
% 24.32/3.48 meet(composition(meet(meet(sk1, X), composition(complement(composition(sk1, top)), converse(Y))), Y), complement(composition(sk1, top)))
% 24.32/3.48 = { by lemma 87 }
% 24.32/3.48 meet(composition(zero, Y), complement(composition(sk1, top)))
% 24.32/3.48 = { by lemma 57 }
% 24.32/3.48 meet(zero, complement(composition(sk1, top)))
% 24.32/3.48 = { by lemma 53 }
% 24.32/3.48 zero
% 24.32/3.48
% 24.32/3.48 Lemma 89: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 24.32/3.48 Proof:
% 24.32/3.48 join(composition(X, Y), composition(X, Z))
% 24.32/3.48 = { by axiom 1 (converse_idempotence_8) R->L }
% 24.32/3.48 join(composition(X, Y), composition(X, converse(converse(Z))))
% 24.32/3.48 = { by axiom 1 (converse_idempotence_8) R->L }
% 24.32/3.48 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 24.32/3.48 = { by lemma 81 }
% 24.32/3.48 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 24.32/3.48 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.48 composition(X, join(Y, converse(converse(Z))))
% 24.32/3.48 = { by axiom 1 (converse_idempotence_8) }
% 24.32/3.48 composition(X, join(Y, Z))
% 24.32/3.48
% 24.32/3.48 Lemma 90: composition(converse(composition(sk1, top)), join(X, complement(composition(sk1, top)))) = composition(converse(composition(sk1, top)), X).
% 24.32/3.48 Proof:
% 24.32/3.48 composition(converse(composition(sk1, top)), join(X, complement(composition(sk1, top))))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(complement(composition(sk1, top)), X))
% 24.32/3.48 = { by lemma 89 R->L }
% 24.32/3.48 join(composition(converse(composition(sk1, top)), complement(composition(sk1, top))), composition(converse(composition(sk1, top)), X))
% 24.32/3.48 = { by lemma 84 }
% 24.32/3.48 join(zero, composition(converse(composition(sk1, top)), X))
% 24.32/3.48 = { by lemma 43 }
% 24.32/3.48 composition(converse(composition(sk1, top)), X)
% 24.32/3.48
% 24.32/3.48 Lemma 91: join(complement(X), meet(complement(Y), Z)) = complement(meet(X, join(Y, complement(Z)))).
% 24.32/3.48 Proof:
% 24.32/3.48 join(complement(X), meet(complement(Y), Z))
% 24.32/3.48 = { by lemma 44 }
% 24.32/3.48 join(complement(X), meet(Z, complement(Y)))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 join(meet(Z, complement(Y)), complement(X))
% 24.32/3.48 = { by lemma 59 R->L }
% 24.32/3.48 join(complement(join(Y, complement(Z))), complement(X))
% 24.32/3.48 = { by lemma 63 }
% 24.32/3.48 complement(meet(join(Y, complement(Z)), X))
% 24.32/3.48 = { by lemma 44 R->L }
% 24.32/3.48 complement(meet(X, join(Y, complement(Z))))
% 24.32/3.48
% 24.32/3.48 Lemma 92: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 24.32/3.48 Proof:
% 24.32/3.48 complement(meet(Y, join(X, complement(Y))))
% 24.32/3.48 = { by lemma 91 R->L }
% 24.32/3.48 join(complement(Y), meet(complement(X), Y))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 join(meet(complement(X), Y), complement(Y))
% 24.32/3.48 = { by lemma 47 R->L }
% 24.32/3.48 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(Y), complement(X))))
% 24.32/3.48 = { by lemma 44 R->L }
% 24.32/3.48 join(meet(complement(X), Y), join(meet(complement(Y), X), meet(complement(X), complement(Y))))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 join(meet(complement(X), Y), join(meet(complement(X), complement(Y)), meet(complement(Y), X)))
% 24.32/3.48 = { by axiom 8 (maddux2_join_associativity_2) }
% 24.32/3.48 join(join(meet(complement(X), Y), meet(complement(X), complement(Y))), meet(complement(Y), X))
% 24.32/3.48 = { by lemma 47 }
% 24.32/3.48 join(complement(X), meet(complement(Y), X))
% 24.32/3.48 = { by lemma 91 }
% 24.32/3.48 complement(meet(X, join(Y, complement(X))))
% 24.32/3.48
% 24.32/3.48 Lemma 93: meet(Y, join(X, complement(Y))) = meet(X, join(Y, complement(X))).
% 24.32/3.48 Proof:
% 24.32/3.48 meet(Y, join(X, complement(Y)))
% 24.32/3.48 = { by lemma 48 R->L }
% 24.32/3.48 meet(Y, meet(join(X, complement(Y)), top))
% 24.32/3.48 = { by lemma 79 R->L }
% 24.32/3.48 meet(meet(Y, join(X, complement(Y))), top)
% 24.32/3.48 = { by lemma 45 R->L }
% 24.32/3.48 complement(join(zero, complement(meet(Y, join(X, complement(Y))))))
% 24.32/3.48 = { by lemma 92 }
% 24.32/3.48 complement(join(zero, complement(meet(X, join(Y, complement(X))))))
% 24.32/3.48 = { by lemma 45 }
% 24.32/3.48 meet(meet(X, join(Y, complement(X))), top)
% 24.32/3.48 = { by lemma 79 }
% 24.32/3.48 meet(X, meet(join(Y, complement(X)), top))
% 24.32/3.48 = { by lemma 48 }
% 24.32/3.48 meet(X, join(Y, complement(X)))
% 24.32/3.48
% 24.32/3.48 Lemma 94: meet(X, join(Y, complement(X))) = meet(X, Y).
% 24.32/3.48 Proof:
% 24.32/3.48 meet(X, join(Y, complement(X)))
% 24.32/3.48 = { by lemma 93 }
% 24.32/3.48 meet(Y, join(X, complement(Y)))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 meet(Y, join(complement(Y), X))
% 24.32/3.48 = { by lemma 44 }
% 24.32/3.48 meet(join(complement(Y), X), Y)
% 24.32/3.48 = { by lemma 23 R->L }
% 24.32/3.48 meet(join(complement(Y), X), join(meet(Y, X), complement(join(complement(Y), X))))
% 24.32/3.48 = { by lemma 93 R->L }
% 24.32/3.48 meet(meet(Y, X), join(join(complement(Y), X), complement(meet(Y, X))))
% 24.32/3.48 = { by lemma 79 }
% 24.32/3.48 meet(Y, meet(X, join(join(complement(Y), X), complement(meet(Y, X)))))
% 24.32/3.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.32/3.48 meet(Y, meet(X, join(complement(Y), join(X, complement(meet(Y, X))))))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 meet(Y, meet(X, join(join(X, complement(meet(Y, X))), complement(Y))))
% 24.32/3.48 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.32/3.48 meet(Y, meet(X, join(X, join(complement(meet(Y, X)), complement(Y)))))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.48 meet(Y, meet(X, join(X, join(complement(Y), complement(meet(Y, X))))))
% 24.32/3.48 = { by lemma 76 }
% 24.32/3.48 meet(Y, X)
% 24.32/3.48 = { by lemma 44 R->L }
% 24.32/3.48 meet(X, Y)
% 24.32/3.48
% 24.32/3.48 Lemma 95: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 24.32/3.48 Proof:
% 24.32/3.48 join(X, complement(join(X, Y)))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 join(X, complement(join(Y, X)))
% 24.32/3.48 = { by lemma 48 R->L }
% 24.32/3.48 join(X, complement(join(Y, meet(X, top))))
% 24.32/3.48 = { by lemma 71 R->L }
% 24.32/3.48 complement(meet(complement(X), join(Y, meet(X, top))))
% 24.32/3.48 = { by lemma 41 R->L }
% 24.32/3.48 complement(meet(join(zero, complement(X)), join(Y, meet(X, top))))
% 24.32/3.48 = { by lemma 45 R->L }
% 24.32/3.48 complement(meet(join(zero, complement(X)), join(Y, complement(join(zero, complement(X))))))
% 24.32/3.48 = { by lemma 92 R->L }
% 24.32/3.48 complement(meet(Y, join(join(zero, complement(X)), complement(Y))))
% 24.32/3.48 = { by lemma 94 }
% 24.32/3.48 complement(meet(Y, join(zero, complement(X))))
% 24.32/3.48 = { by lemma 41 }
% 24.32/3.48 complement(meet(Y, complement(X)))
% 24.32/3.48 = { by lemma 70 }
% 24.32/3.48 join(X, complement(Y))
% 24.32/3.48
% 24.32/3.48 Lemma 96: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 24.32/3.48 Proof:
% 24.32/3.48 join(complement(X), meet(X, Y))
% 24.32/3.48 = { by lemma 44 }
% 24.32/3.48 join(complement(X), meet(Y, X))
% 24.32/3.48 = { by lemma 52 R->L }
% 24.32/3.48 complement(complement(join(complement(X), meet(Y, X))))
% 24.32/3.48 = { by lemma 58 R->L }
% 24.32/3.48 complement(meet(X, join(complement(Y), complement(X))))
% 24.32/3.48 = { by lemma 92 R->L }
% 24.32/3.48 complement(meet(complement(Y), join(X, complement(complement(Y)))))
% 24.32/3.48 = { by lemma 71 }
% 24.32/3.48 join(Y, complement(join(X, complement(complement(Y)))))
% 24.32/3.48 = { by lemma 59 }
% 24.32/3.48 join(Y, meet(complement(Y), complement(X)))
% 24.32/3.48 = { by lemma 74 }
% 24.32/3.48 join(Y, complement(join(Y, X)))
% 24.32/3.48 = { by lemma 95 }
% 24.32/3.48 join(Y, complement(X))
% 24.32/3.48
% 24.32/3.48 Lemma 97: composition(top, meet(sk1, one)) = converse(composition(sk1, top)).
% 24.32/3.48 Proof:
% 24.32/3.48 composition(top, meet(sk1, one))
% 24.32/3.48 = { by lemma 38 R->L }
% 24.32/3.48 composition(converse(top), meet(sk1, one))
% 24.32/3.48 = { by lemma 30 R->L }
% 24.32/3.48 composition(converse(join(converse(complement(converse(composition(meet(sk1, one), top)))), join(complement(converse(complement(converse(composition(meet(sk1, one), top))))), composition(sk1, top)))), meet(sk1, one))
% 24.32/3.48 = { by lemma 35 }
% 24.32/3.48 composition(join(complement(converse(composition(meet(sk1, one), top))), converse(join(complement(converse(complement(converse(composition(meet(sk1, one), top))))), composition(sk1, top)))), meet(sk1, one))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.48 composition(join(complement(converse(composition(meet(sk1, one), top))), converse(join(composition(sk1, top), complement(converse(complement(converse(composition(meet(sk1, one), top)))))))), meet(sk1, one))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 composition(join(converse(join(composition(sk1, top), complement(converse(complement(converse(composition(meet(sk1, one), top))))))), complement(converse(composition(meet(sk1, one), top)))), meet(sk1, one))
% 24.32/3.48 = { by lemma 83 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), complement(converse(complement(converse(composition(meet(sk1, one), top))))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 68 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), complement(complement(converse(converse(composition(meet(sk1, one), top))))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 52 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), converse(converse(composition(meet(sk1, one), top))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 34 }
% 24.32/3.48 composition(join(converse(composition(meet(sk1, one), top)), converse(composition(sk1, top))), meet(sk1, one))
% 24.32/3.48 = { by axiom 7 (converse_additivity_9) R->L }
% 24.32/3.48 composition(converse(join(composition(meet(sk1, one), top), composition(sk1, top))), meet(sk1, one))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.48 composition(converse(join(composition(sk1, top), composition(meet(sk1, one), top))), meet(sk1, one))
% 24.32/3.48 = { by lemma 67 R->L }
% 24.32/3.48 composition(converse(join(composition(sk1, top), join(meet(complement(composition(sk1, top)), composition(meet(sk1, one), top)), meet(composition(meet(sk1, one), top), complement(complement(composition(sk1, top))))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 88 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), join(zero, meet(composition(meet(sk1, one), top), complement(complement(composition(sk1, top))))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 43 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), meet(composition(meet(sk1, one), top), complement(complement(composition(sk1, top)))))), meet(sk1, one))
% 24.32/3.48 = { by lemma 52 }
% 24.32/3.48 composition(converse(join(composition(sk1, top), meet(composition(meet(sk1, one), top), composition(sk1, top)))), meet(sk1, one))
% 24.32/3.48 = { by lemma 44 R->L }
% 24.32/3.48 composition(converse(join(composition(sk1, top), meet(composition(sk1, top), composition(meet(sk1, one), top)))), meet(sk1, one))
% 24.32/3.48 = { by lemma 73 }
% 24.32/3.48 composition(converse(composition(sk1, top)), meet(sk1, one))
% 24.32/3.48 = { by lemma 90 R->L }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(meet(sk1, one), complement(composition(sk1, top))))
% 24.32/3.48 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(complement(composition(sk1, top)), meet(sk1, one)))
% 24.32/3.48 = { by axiom 4 (goals_17) }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(complement(sk1), meet(sk1, one)))
% 24.32/3.48 = { by lemma 96 }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(one, complement(sk1)))
% 24.32/3.48 = { by axiom 4 (goals_17) R->L }
% 24.32/3.48 composition(converse(composition(sk1, top)), join(one, complement(composition(sk1, top))))
% 24.32/3.48 = { by lemma 90 }
% 24.32/3.48 composition(converse(composition(sk1, top)), one)
% 24.32/3.48 = { by axiom 3 (composition_identity_6) }
% 24.32/3.48 converse(composition(sk1, top))
% 24.32/3.48
% 24.32/3.48 Lemma 98: join(complement(one), converse(complement(one))) = complement(one).
% 24.32/3.48 Proof:
% 24.32/3.48 join(complement(one), converse(complement(one)))
% 24.32/3.48 = { by axiom 3 (composition_identity_6) R->L }
% 24.32/3.48 join(complement(one), composition(converse(complement(one)), one))
% 24.32/3.48 = { by lemma 62 }
% 24.32/3.48 complement(one)
% 24.32/3.48
% 24.32/3.48 Lemma 99: join(meet(sk1, X), meet(X, complement(composition(sk1, top)))) = X.
% 24.32/3.48 Proof:
% 24.32/3.48 join(meet(sk1, X), meet(X, complement(composition(sk1, top))))
% 24.32/3.48 = { by axiom 4 (goals_17) }
% 24.32/3.48 join(meet(sk1, X), meet(X, complement(sk1)))
% 24.32/3.48 = { by lemma 67 }
% 24.32/3.49 X
% 24.32/3.49
% 24.32/3.49 Lemma 100: join(complement(converse(X)), converse(join(X, Y))) = top.
% 24.32/3.49 Proof:
% 24.32/3.49 join(complement(converse(X)), converse(join(X, Y)))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.49 join(converse(join(X, Y)), complement(converse(X)))
% 24.32/3.49 = { by axiom 7 (converse_additivity_9) }
% 24.32/3.49 join(join(converse(X), converse(Y)), complement(converse(X)))
% 24.32/3.49 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.32/3.49 join(converse(X), join(converse(Y), complement(converse(X))))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.49 join(converse(X), join(complement(converse(X)), converse(Y)))
% 24.32/3.49 = { by lemma 30 }
% 24.32/3.49 top
% 24.32/3.49
% 24.32/3.49 Lemma 101: join(one, complement(converse(meet(sk1, one)))) = top.
% 24.32/3.49 Proof:
% 24.32/3.49 join(one, complement(converse(meet(sk1, one))))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.49 join(complement(converse(meet(sk1, one))), one)
% 24.32/3.49 = { by lemma 33 R->L }
% 24.32/3.49 join(complement(converse(meet(sk1, one))), converse(one))
% 24.32/3.49 = { by lemma 99 R->L }
% 24.32/3.49 join(complement(converse(meet(sk1, one))), converse(join(meet(sk1, one), meet(one, complement(composition(sk1, top))))))
% 24.32/3.49 = { by lemma 100 }
% 24.32/3.49 top
% 24.32/3.49
% 24.32/3.49 Lemma 102: meet(composition(sk1, top), meet(X, Y)) = meet(Y, meet(sk1, X)).
% 24.32/3.49 Proof:
% 24.32/3.49 meet(composition(sk1, top), meet(X, Y))
% 24.32/3.49 = { by lemma 78 }
% 24.32/3.49 meet(Y, meet(composition(sk1, top), X))
% 24.32/3.49 = { by axiom 4 (goals_17) }
% 24.32/3.49 meet(Y, meet(sk1, X))
% 24.32/3.49
% 24.32/3.49 Lemma 103: composition(meet(sk1, one), composition(sk1, top)) = composition(meet(sk1, one), top).
% 24.32/3.49 Proof:
% 24.32/3.49 composition(meet(sk1, one), composition(sk1, top))
% 24.32/3.49 = { by axiom 1 (converse_idempotence_8) R->L }
% 24.32/3.49 composition(meet(sk1, one), converse(converse(composition(sk1, top))))
% 24.32/3.49 = { by lemma 97 R->L }
% 24.32/3.49 composition(meet(sk1, one), converse(composition(top, meet(sk1, one))))
% 24.32/3.49 = { by lemma 69 R->L }
% 24.32/3.49 composition(meet(sk1, one), composition(converse(meet(sk1, one)), top))
% 24.32/3.49 = { by axiom 1 (converse_idempotence_8) R->L }
% 24.32/3.49 composition(meet(sk1, one), converse(converse(composition(converse(meet(sk1, one)), top))))
% 24.32/3.49 = { by lemma 80 R->L }
% 24.32/3.49 converse(composition(converse(composition(converse(meet(sk1, one)), top)), converse(meet(sk1, one))))
% 24.32/3.49 = { by lemma 83 R->L }
% 24.32/3.49 converse(composition(join(converse(composition(converse(meet(sk1, one)), top)), complement(converse(composition(converse(meet(sk1, one)), top)))), converse(meet(sk1, one))))
% 24.32/3.49 = { by axiom 5 (def_top_12) R->L }
% 24.32/3.49 converse(composition(top, converse(meet(sk1, one))))
% 24.32/3.49 = { by lemma 80 }
% 24.32/3.49 composition(meet(sk1, one), converse(top))
% 24.32/3.49 = { by lemma 38 }
% 24.32/3.49 composition(meet(sk1, one), top)
% 24.32/3.49
% 24.32/3.49 Lemma 104: meet(meet(sk1, X), complement(meet(X, Y))) = meet(meet(sk1, X), complement(Y)).
% 24.32/3.49 Proof:
% 24.32/3.49 meet(meet(sk1, X), complement(meet(X, Y)))
% 24.32/3.49 = { by lemma 44 }
% 24.32/3.49 meet(complement(meet(X, Y)), meet(sk1, X))
% 24.32/3.49 = { by lemma 102 R->L }
% 24.32/3.49 meet(composition(sk1, top), meet(X, complement(meet(X, Y))))
% 24.32/3.49 = { by lemma 60 R->L }
% 24.32/3.49 meet(composition(sk1, top), complement(join(complement(X), meet(X, Y))))
% 24.32/3.49 = { by lemma 96 }
% 24.32/3.49 meet(composition(sk1, top), complement(join(Y, complement(X))))
% 24.32/3.49 = { by lemma 59 }
% 24.32/3.49 meet(composition(sk1, top), meet(X, complement(Y)))
% 24.32/3.49 = { by lemma 102 }
% 24.32/3.49 meet(complement(Y), meet(sk1, X))
% 24.32/3.49 = { by lemma 44 R->L }
% 24.32/3.49 meet(meet(sk1, X), complement(Y))
% 24.32/3.49
% 24.32/3.49 Goal 1 (goals_18): composition(meet(sk1, one), sk2) = meet(sk1, sk2).
% 24.32/3.49 Proof:
% 24.32/3.49 composition(meet(sk1, one), sk2)
% 24.32/3.49 = { by lemma 23 R->L }
% 24.32/3.49 join(meet(composition(meet(sk1, one), sk2), complement(composition(sk1, top))), complement(join(complement(composition(meet(sk1, one), sk2)), complement(composition(sk1, top)))))
% 24.32/3.49 = { by lemma 44 }
% 24.32/3.49 join(meet(complement(composition(sk1, top)), composition(meet(sk1, one), sk2)), complement(join(complement(composition(meet(sk1, one), sk2)), complement(composition(sk1, top)))))
% 24.32/3.49 = { by lemma 88 }
% 24.32/3.49 join(zero, complement(join(complement(composition(meet(sk1, one), sk2)), complement(composition(sk1, top)))))
% 24.32/3.49 = { by lemma 41 }
% 24.32/3.49 complement(join(complement(composition(meet(sk1, one), sk2)), complement(composition(sk1, top))))
% 24.32/3.49 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.49 meet(composition(meet(sk1, one), sk2), composition(sk1, top))
% 24.32/3.49 = { by lemma 44 }
% 24.32/3.49 meet(composition(sk1, top), composition(meet(sk1, one), sk2))
% 24.32/3.49 = { by lemma 76 R->L }
% 24.32/3.49 meet(composition(sk1, top), meet(composition(meet(sk1, one), sk2), join(composition(meet(sk1, one), sk2), sk2)))
% 24.32/3.49 = { by lemma 20 R->L }
% 24.32/3.49 meet(composition(sk1, top), meet(composition(meet(sk1, one), sk2), join(composition(meet(sk1, one), sk2), composition(one, sk2))))
% 24.32/3.49 = { by axiom 12 (composition_distributivity_7) R->L }
% 24.32/3.49 meet(composition(sk1, top), meet(composition(meet(sk1, one), sk2), composition(join(meet(sk1, one), one), sk2)))
% 24.32/3.49 = { by lemma 75 }
% 24.32/3.49 meet(composition(sk1, top), meet(composition(meet(sk1, one), sk2), composition(one, sk2)))
% 24.32/3.49 = { by lemma 20 }
% 24.32/3.49 meet(composition(sk1, top), meet(composition(meet(sk1, one), sk2), sk2))
% 24.32/3.49 = { by lemma 44 }
% 24.32/3.49 meet(composition(sk1, top), meet(sk2, composition(meet(sk1, one), sk2)))
% 24.32/3.49 = { by lemma 102 }
% 24.32/3.49 meet(composition(meet(sk1, one), sk2), meet(sk1, sk2))
% 24.32/3.49 = { by lemma 44 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(sk1, one), sk2))
% 24.32/3.49 = { by lemma 94 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(composition(meet(sk1, one), sk2), complement(meet(sk1, sk2))))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), sk2)))
% 24.32/3.49 = { by lemma 20 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(composition(one, complement(meet(sk1, sk2))), composition(meet(sk1, one), sk2)))
% 24.32/3.49 = { by lemma 75 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(composition(join(meet(sk1, one), one), complement(meet(sk1, sk2))), composition(meet(sk1, one), sk2)))
% 24.32/3.49 = { by lemma 54 }
% 24.32/3.49 meet(meet(sk1, sk2), join(join(complement(meet(sk1, sk2)), composition(meet(sk1, one), complement(meet(sk1, sk2)))), composition(meet(sk1, one), sk2)))
% 24.32/3.49 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), join(composition(meet(sk1, one), complement(meet(sk1, sk2))), composition(meet(sk1, one), sk2))))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), join(composition(meet(sk1, one), sk2), composition(meet(sk1, one), complement(meet(sk1, sk2))))))
% 24.32/3.49 = { by axiom 1 (converse_idempotence_8) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), join(composition(meet(sk1, one), sk2), composition(meet(sk1, one), complement(converse(converse(meet(sk1, sk2))))))))
% 24.32/3.49 = { by lemma 68 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), join(composition(meet(sk1, one), sk2), composition(meet(sk1, one), converse(complement(converse(meet(sk1, sk2))))))))
% 24.32/3.49 = { by lemma 89 }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), join(sk2, converse(complement(converse(meet(sk1, sk2))))))))
% 24.32/3.49 = { by lemma 35 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), converse(join(converse(sk2), complement(converse(meet(sk1, sk2))))))))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), converse(join(complement(converse(meet(sk1, sk2))), converse(sk2))))))
% 24.32/3.49 = { by lemma 99 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), converse(join(complement(converse(meet(sk1, sk2))), converse(join(meet(sk1, sk2), meet(sk2, complement(composition(sk1, top))))))))))
% 24.32/3.49 = { by lemma 100 }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), converse(top))))
% 24.32/3.49 = { by lemma 38 }
% 24.32/3.49 meet(meet(sk1, sk2), join(complement(meet(sk1, sk2)), composition(meet(sk1, one), top)))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) }
% 24.32/3.49 meet(meet(sk1, sk2), join(composition(meet(sk1, one), top), complement(meet(sk1, sk2))))
% 24.32/3.49 = { by lemma 103 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), join(composition(meet(sk1, one), composition(sk1, top)), complement(meet(sk1, sk2))))
% 24.32/3.49 = { by lemma 94 }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(sk1, one), composition(sk1, top)))
% 24.32/3.49 = { by lemma 103 }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(sk1, one), top))
% 24.32/3.49 = { by lemma 52 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(complement(complement(meet(sk1, one))), top))
% 24.32/3.49 = { by lemma 22 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(complement(join(complement(meet(sk1, one)), complement(meet(sk1, one)))), top))
% 24.32/3.49 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), meet(sk1, one)), top))
% 24.32/3.49 = { by lemma 52 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(meet(sk1, one)))), top))
% 24.32/3.49 = { by lemma 104 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(meet(one, complement(meet(sk1, one))))), top))
% 24.32/3.49 = { by lemma 59 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(meet(sk1, one), complement(one))))), top))
% 24.32/3.49 = { by lemma 73 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(meet(sk1, one), join(complement(one), meet(complement(one), composition(sk1, top))))))), top))
% 24.32/3.49 = { by lemma 44 R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(meet(sk1, one), join(complement(one), meet(composition(sk1, top), complement(one))))))), top))
% 24.32/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 24.32/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(meet(sk1, one), join(meet(composition(sk1, top), complement(one)), complement(one)))))), top))
% 25.10/3.49 = { by axiom 8 (maddux2_join_associativity_2) }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(join(meet(sk1, one), meet(composition(sk1, top), complement(one))), complement(one))))), top))
% 25.10/3.49 = { by axiom 4 (goals_17) }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(join(meet(sk1, one), meet(sk1, complement(one))), complement(one))))), top))
% 25.10/3.49 = { by lemma 47 }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(sk1, complement(one))))), top))
% 25.10/3.49 = { by axiom 4 (goals_17) R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(composition(sk1, top), complement(one))))), top))
% 25.10/3.49 = { by lemma 33 R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(composition(sk1, top), complement(converse(one)))))), top))
% 25.10/3.49 = { by lemma 68 R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(composition(sk1, top), converse(complement(one)))))), top))
% 25.10/3.49 = { by lemma 35 R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(converse(join(converse(composition(sk1, top)), complement(one)))))), top))
% 25.10/3.49 = { by lemma 97 R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(converse(join(composition(top, meet(sk1, one)), complement(one)))))), top))
% 25.10/3.49 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(converse(join(complement(one), composition(top, meet(sk1, one))))))), top))
% 25.10/3.49 = { by axiom 7 (converse_additivity_9) }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(converse(complement(one)), converse(composition(top, meet(sk1, one))))))), top))
% 25.10/3.49 = { by lemma 98 R->L }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(converse(join(complement(one), converse(complement(one)))), converse(composition(top, meet(sk1, one))))))), top))
% 25.10/3.49 = { by lemma 34 }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(join(complement(one), converse(complement(one))), converse(composition(top, meet(sk1, one))))))), top))
% 25.10/3.49 = { by lemma 98 }
% 25.10/3.49 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), converse(composition(top, meet(sk1, one))))))), top))
% 25.10/3.49 = { by lemma 69 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), composition(converse(meet(sk1, one)), top))))), top))
% 25.10/3.50 = { by lemma 38 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), composition(converse(meet(sk1, one)), converse(top)))))), top))
% 25.10/3.50 = { by lemma 101 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), composition(converse(meet(sk1, one)), converse(join(one, complement(converse(meet(sk1, one)))))))))), top))
% 25.10/3.50 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), composition(converse(meet(sk1, one)), converse(join(complement(converse(meet(sk1, one))), one))))))), top))
% 25.10/3.50 = { by lemma 80 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), converse(composition(join(complement(converse(meet(sk1, one))), one), converse(converse(meet(sk1, one))))))))), top))
% 25.10/3.50 = { by lemma 54 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), converse(join(converse(converse(meet(sk1, one))), composition(complement(converse(meet(sk1, one))), converse(converse(meet(sk1, one)))))))))), top))
% 25.10/3.50 = { by lemma 35 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), join(converse(meet(sk1, one)), converse(composition(complement(converse(meet(sk1, one))), converse(converse(meet(sk1, one)))))))))), top))
% 25.10/3.50 = { by lemma 80 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), join(converse(meet(sk1, one)), composition(converse(meet(sk1, one)), converse(complement(converse(meet(sk1, one)))))))))), top))
% 25.10/3.50 = { by lemma 68 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), join(converse(meet(sk1, one)), composition(converse(meet(sk1, one)), complement(converse(converse(meet(sk1, one)))))))))), top))
% 25.10/3.50 = { by axiom 1 (converse_idempotence_8) }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), join(converse(meet(sk1, one)), composition(converse(meet(sk1, one)), complement(meet(sk1, one)))))))), top))
% 25.10/3.50 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), join(composition(converse(meet(sk1, one)), complement(meet(sk1, one))), converse(meet(sk1, one))))))), top))
% 25.10/3.50 = { by axiom 8 (maddux2_join_associativity_2) }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(join(complement(one), composition(converse(meet(sk1, one)), complement(meet(sk1, one)))), converse(meet(sk1, one)))))), top))
% 25.10/3.50 = { by lemma 61 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(join(complement(one), converse(meet(sk1, one)))))), top))
% 25.10/3.50 = { by lemma 60 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(meet(one, complement(converse(meet(sk1, one)))))), top))
% 25.10/3.50 = { by lemma 104 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), complement(complement(converse(meet(sk1, one))))), top))
% 25.10/3.50 = { by lemma 52 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(meet(sk1, one), converse(meet(sk1, one))), top))
% 25.10/3.50 = { by lemma 44 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(meet(sk1, one)), meet(sk1, one)), top))
% 25.10/3.50 = { by lemma 102 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), meet(one, converse(meet(sk1, one)))), top))
% 25.10/3.50 = { by lemma 44 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), meet(converse(meet(sk1, one)), one)), top))
% 25.10/3.50 = { by lemma 52 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), meet(converse(meet(sk1, one)), complement(complement(one)))), top))
% 25.10/3.50 = { by lemma 43 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), join(zero, meet(converse(meet(sk1, one)), complement(complement(one))))), top))
% 25.10/3.50 = { by lemma 17 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), join(complement(top), meet(converse(meet(sk1, one)), complement(complement(one))))), top))
% 25.10/3.50 = { by lemma 101 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), join(complement(join(one, complement(converse(meet(sk1, one))))), meet(converse(meet(sk1, one)), complement(complement(one))))), top))
% 25.10/3.50 = { by lemma 59 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), join(meet(converse(meet(sk1, one)), complement(one)), meet(converse(meet(sk1, one)), complement(complement(one))))), top))
% 25.10/3.50 = { by lemma 44 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), join(meet(complement(one), converse(meet(sk1, one))), meet(converse(meet(sk1, one)), complement(complement(one))))), top))
% 25.10/3.50 = { by lemma 67 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(composition(sk1, top), converse(meet(sk1, one))), top))
% 25.10/3.50 = { by lemma 44 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(meet(sk1, one)), composition(sk1, top)), top))
% 25.10/3.50 = { by axiom 1 (converse_idempotence_8) R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(meet(sk1, one)), converse(converse(composition(sk1, top)))), top))
% 25.10/3.50 = { by lemma 97 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(meet(sk1, one)), converse(composition(top, meet(sk1, one)))), top))
% 25.10/3.50 = { by lemma 44 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(composition(top, meet(sk1, one))), converse(meet(sk1, one))), top))
% 25.10/3.50 = { by lemma 52 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(converse(composition(top, meet(sk1, one))), complement(complement(converse(meet(sk1, one))))), top))
% 25.10/3.50 = { by lemma 59 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(complement(join(complement(converse(meet(sk1, one))), complement(converse(composition(top, meet(sk1, one)))))), top))
% 25.10/3.50 = { by lemma 95 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(complement(join(complement(converse(meet(sk1, one))), complement(join(complement(converse(meet(sk1, one))), converse(composition(top, meet(sk1, one))))))), top))
% 25.10/3.50 = { by lemma 59 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(join(complement(converse(meet(sk1, one))), converse(composition(top, meet(sk1, one)))), complement(complement(converse(meet(sk1, one))))), top))
% 25.10/3.50 = { by lemma 44 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(complement(converse(meet(sk1, one))), converse(composition(top, meet(sk1, one))))), top))
% 25.10/3.50 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(converse(composition(top, meet(sk1, one))), complement(converse(meet(sk1, one))))), top))
% 25.10/3.50 = { by lemma 69 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), complement(converse(meet(sk1, one))))), top))
% 25.10/3.50 = { by lemma 23 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), complement(join(meet(converse(meet(sk1, one)), converse(join(meet(sk1, one), X))), complement(join(complement(converse(meet(sk1, one))), converse(join(meet(sk1, one), X)))))))), top))
% 25.10/3.50 = { by lemma 100 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), complement(join(meet(converse(meet(sk1, one)), converse(join(meet(sk1, one), X))), complement(top))))), top))
% 25.10/3.50 = { by lemma 17 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), complement(join(meet(converse(meet(sk1, one)), converse(join(meet(sk1, one), X))), zero)))), top))
% 25.10/3.50 = { by lemma 42 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), complement(meet(converse(meet(sk1, one)), converse(join(meet(sk1, one), X)))))), top))
% 25.10/3.50 = { by lemma 63 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(composition(converse(meet(sk1, one)), top), join(complement(converse(meet(sk1, one))), complement(converse(join(meet(sk1, one), X)))))), top))
% 25.10/3.50 = { by axiom 8 (maddux2_join_associativity_2) }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), join(join(composition(converse(meet(sk1, one)), top), complement(converse(meet(sk1, one)))), complement(converse(join(meet(sk1, one), X))))), top))
% 25.10/3.50 = { by lemma 70 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), complement(join(composition(converse(meet(sk1, one)), top), complement(converse(meet(sk1, one)))))))), top))
% 25.10/3.50 = { by lemma 59 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(converse(meet(sk1, one)), complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by axiom 3 (composition_identity_6) R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by lemma 42 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), join(meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top))), zero)))), top))
% 25.10/3.50 = { by lemma 53 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), join(meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top))), meet(zero, complement(composition(converse(meet(sk1, one)), top))))))), top))
% 25.10/3.50 = { by lemma 56 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), join(meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top))), meet(composition(converse(meet(sk1, one)), zero), complement(composition(converse(meet(sk1, one)), top))))))), top))
% 25.10/3.50 = { by lemma 46 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), join(meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top))), meet(composition(converse(meet(sk1, one)), meet(one, zero)), complement(composition(converse(meet(sk1, one)), top))))))), top))
% 25.10/3.50 = { by lemma 82 R->L }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), join(meet(composition(converse(meet(sk1, one)), one), complement(composition(converse(meet(sk1, one)), top))), meet(composition(converse(meet(sk1, one)), meet(one, composition(converse(converse(meet(sk1, one))), complement(composition(converse(meet(sk1, one)), top))))), complement(composition(converse(meet(sk1, one)), top))))))), top))
% 25.10/3.50 = { by axiom 15 (modular_law_1_15) }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(composition(converse(meet(sk1, one)), meet(one, composition(converse(converse(meet(sk1, one))), complement(composition(converse(meet(sk1, one)), top))))), complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by lemma 82 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(composition(converse(meet(sk1, one)), meet(one, zero)), complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by lemma 46 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(composition(converse(meet(sk1, one)), zero), complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by lemma 56 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), meet(zero, complement(composition(converse(meet(sk1, one)), top)))))), top))
% 25.10/3.50 = { by lemma 53 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(meet(converse(join(meet(sk1, one), X)), zero))), top))
% 25.10/3.50 = { by lemma 46 }
% 25.10/3.50 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), complement(zero)), top))
% 25.10/3.51 = { by lemma 32 }
% 25.10/3.51 meet(meet(sk1, sk2), composition(meet(complement(complement(converse(meet(sk1, one)))), top), top))
% 25.10/3.51 = { by lemma 48 }
% 25.10/3.51 meet(meet(sk1, sk2), composition(complement(complement(converse(meet(sk1, one)))), top))
% 25.10/3.51 = { by lemma 52 }
% 25.10/3.51 meet(meet(sk1, sk2), composition(converse(meet(sk1, one)), top))
% 25.10/3.51 = { by lemma 69 }
% 25.10/3.51 meet(meet(sk1, sk2), converse(composition(top, meet(sk1, one))))
% 25.10/3.51 = { by lemma 97 }
% 25.10/3.51 meet(meet(sk1, sk2), converse(converse(composition(sk1, top))))
% 25.10/3.51 = { by axiom 1 (converse_idempotence_8) }
% 25.10/3.51 meet(meet(sk1, sk2), composition(sk1, top))
% 25.10/3.51 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 25.10/3.51 complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top))))
% 25.10/3.51 = { by lemma 41 R->L }
% 25.10/3.51 join(zero, complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top)))))
% 25.10/3.51 = { by lemma 86 R->L }
% 25.10/3.51 join(meet(meet(sk1, sk2), complement(composition(sk1, top))), complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top)))))
% 25.10/3.51 = { by lemma 23 }
% 25.10/3.51 meet(sk1, sk2)
% 25.10/3.51 % SZS output end Proof
% 25.10/3.51
% 25.10/3.51 RESULT: Unsatisfiable (the axioms are contradictory).
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