TSTP Solution File: REL030-1 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL030-1 : 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 : n029.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 13:44:13 EDT 2023
% Result : Unsatisfiable 42.29s 5.89s
% Output : Proof 44.66s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL030-1 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.35 % Computer : n029.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 20:29:24 EDT 2023
% 0.14/0.35 % CPUTime :
% 42.29/5.89 Command-line arguments: --no-flatten-goal
% 42.29/5.89
% 42.29/5.89 % SZS status Unsatisfiable
% 42.29/5.89
% 43.62/6.04 % SZS output start Proof
% 43.62/6.04 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 43.62/6.04 Axiom 2 (composition_identity_6): composition(X, one) = X.
% 43.62/6.04 Axiom 3 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 43.62/6.04 Axiom 4 (goals_14): join(sk1, one) = one.
% 43.62/6.04 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 43.62/6.04 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 43.62/6.04 Axiom 7 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 43.62/6.04 Axiom 8 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 43.62/6.04 Axiom 9 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 43.62/6.04 Axiom 10 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 43.62/6.04 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 43.62/6.04 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 43.62/6.04 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 43.62/6.04 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 43.62/6.04
% 43.62/6.04 Lemma 15: complement(top) = zero.
% 43.62/6.04 Proof:
% 43.62/6.04 complement(top)
% 43.62/6.04 = { by axiom 6 (def_top_12) }
% 43.62/6.04 complement(join(complement(X), complement(complement(X))))
% 43.62/6.04 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.04 meet(X, complement(X))
% 43.62/6.04 = { by axiom 5 (def_zero_13) R->L }
% 43.62/6.04 zero
% 43.62/6.04
% 43.62/6.04 Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 43.62/6.04 Proof:
% 43.62/6.04 converse(composition(converse(X), Y))
% 43.62/6.04 = { by axiom 7 (converse_multiplicativity_10) }
% 43.62/6.04 composition(converse(Y), converse(converse(X)))
% 43.62/6.04 = { by axiom 1 (converse_idempotence_8) }
% 43.62/6.04 composition(converse(Y), X)
% 43.62/6.04
% 43.62/6.04 Lemma 17: composition(converse(one), X) = X.
% 43.62/6.04 Proof:
% 43.62/6.04 composition(converse(one), X)
% 43.62/6.04 = { by lemma 16 R->L }
% 43.62/6.04 converse(composition(converse(X), one))
% 43.62/6.04 = { by axiom 2 (composition_identity_6) }
% 43.62/6.04 converse(converse(X))
% 43.62/6.04 = { by axiom 1 (converse_idempotence_8) }
% 43.62/6.04 X
% 43.62/6.04
% 43.62/6.04 Lemma 18: converse(one) = one.
% 43.62/6.04 Proof:
% 43.62/6.04 converse(one)
% 43.62/6.04 = { by axiom 2 (composition_identity_6) R->L }
% 43.62/6.04 composition(converse(one), one)
% 43.62/6.04 = { by lemma 17 }
% 43.62/6.04 one
% 43.62/6.04
% 43.62/6.04 Lemma 19: join(X, join(Y, complement(X))) = join(Y, top).
% 43.62/6.04 Proof:
% 43.62/6.04 join(X, join(Y, complement(X)))
% 43.62/6.04 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.04 join(X, join(complement(X), Y))
% 43.62/6.04 = { by axiom 10 (maddux2_join_associativity_2) }
% 43.62/6.04 join(join(X, complement(X)), Y)
% 43.62/6.04 = { by axiom 6 (def_top_12) R->L }
% 43.62/6.04 join(top, Y)
% 43.62/6.04 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.04 join(Y, top)
% 43.62/6.04
% 43.62/6.04 Lemma 20: composition(one, X) = X.
% 43.62/6.04 Proof:
% 43.62/6.04 composition(one, X)
% 43.62/6.04 = { by lemma 17 R->L }
% 43.62/6.04 composition(converse(one), composition(one, X))
% 43.62/6.04 = { by axiom 8 (composition_associativity_5) }
% 43.62/6.04 composition(composition(converse(one), one), X)
% 43.62/6.04 = { by axiom 2 (composition_identity_6) }
% 43.62/6.04 composition(converse(one), X)
% 43.62/6.04 = { by lemma 17 }
% 43.62/6.04 X
% 43.62/6.04
% 43.62/6.04 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 43.62/6.04 Proof:
% 43.62/6.04 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 43.62/6.04 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.04 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 43.62/6.04 = { by axiom 13 (converse_cancellativity_11) }
% 43.62/6.04 complement(X)
% 43.62/6.04
% 43.62/6.04 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 43.62/6.04 Proof:
% 43.62/6.04 join(complement(X), complement(X))
% 43.62/6.04 = { by lemma 17 R->L }
% 43.62/6.04 join(complement(X), composition(converse(one), complement(X)))
% 43.62/6.04 = { by lemma 20 R->L }
% 43.62/6.04 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 43.62/6.04 = { by lemma 21 }
% 43.62/6.04 complement(X)
% 43.62/6.04
% 43.62/6.04 Lemma 23: join(top, complement(X)) = top.
% 43.62/6.04 Proof:
% 43.62/6.04 join(top, complement(X))
% 43.62/6.04 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.04 join(complement(X), top)
% 43.62/6.04 = { by lemma 19 R->L }
% 43.62/6.04 join(X, join(complement(X), complement(X)))
% 43.62/6.04 = { by lemma 22 }
% 43.62/6.04 join(X, complement(X))
% 43.62/6.04 = { by axiom 6 (def_top_12) R->L }
% 43.62/6.04 top
% 43.62/6.04
% 43.62/6.05 Lemma 24: join(Y, top) = join(X, top).
% 43.62/6.05 Proof:
% 43.62/6.05 join(Y, top)
% 43.62/6.05 = { by lemma 23 R->L }
% 43.62/6.05 join(Y, join(top, complement(Y)))
% 43.62/6.05 = { by lemma 19 }
% 43.62/6.05 join(top, top)
% 43.62/6.05 = { by lemma 19 R->L }
% 43.62/6.05 join(X, join(top, complement(X)))
% 43.62/6.05 = { by lemma 23 }
% 43.62/6.05 join(X, top)
% 43.62/6.05
% 43.62/6.05 Lemma 25: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, composition(Y, X))
% 43.62/6.05 = { by lemma 20 R->L }
% 43.62/6.05 join(composition(one, X), composition(Y, X))
% 43.62/6.05 = { by axiom 12 (composition_distributivity_7) R->L }
% 43.62/6.05 composition(join(one, Y), X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 composition(join(Y, one), X)
% 43.62/6.05
% 43.62/6.05 Lemma 26: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 43.62/6.05 Proof:
% 43.62/6.05 join(meet(X, Y), complement(join(complement(X), Y)))
% 43.62/6.05 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.05 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 43.62/6.05 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 43.62/6.05 X
% 43.62/6.05
% 43.62/6.05 Lemma 27: join(zero, meet(X, X)) = X.
% 43.62/6.05 Proof:
% 43.62/6.05 join(zero, meet(X, X))
% 43.62/6.05 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.05 join(zero, complement(join(complement(X), complement(X))))
% 43.62/6.05 = { by axiom 5 (def_zero_13) }
% 43.62/6.05 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 43.62/6.05 = { by lemma 26 }
% 43.62/6.05 X
% 43.62/6.05
% 43.62/6.05 Lemma 28: join(zero, join(X, meet(Y, Y))) = join(X, Y).
% 43.62/6.05 Proof:
% 43.62/6.05 join(zero, join(X, meet(Y, Y)))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(zero, join(meet(Y, Y), X))
% 43.62/6.05 = { by axiom 10 (maddux2_join_associativity_2) }
% 43.62/6.05 join(join(zero, meet(Y, Y)), X)
% 43.62/6.05 = { by lemma 27 }
% 43.62/6.05 join(Y, X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 join(X, Y)
% 43.62/6.05
% 43.62/6.05 Lemma 29: join(X, zero) = join(X, X).
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, zero)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(zero, X)
% 43.62/6.05 = { by lemma 27 R->L }
% 43.62/6.05 join(zero, join(zero, meet(X, X)))
% 43.62/6.05 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.05 join(zero, join(zero, complement(join(complement(X), complement(X)))))
% 43.62/6.05 = { by lemma 22 R->L }
% 43.62/6.05 join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
% 43.62/6.05 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.05 join(zero, join(zero, join(meet(X, X), complement(join(complement(X), complement(X))))))
% 43.62/6.05 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.05 join(zero, join(zero, join(meet(X, X), meet(X, X))))
% 43.62/6.05 = { by lemma 28 }
% 43.62/6.05 join(zero, join(meet(X, X), X))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 join(zero, join(X, meet(X, X)))
% 43.62/6.05 = { by lemma 28 }
% 43.62/6.05 join(X, X)
% 43.62/6.05
% 43.62/6.05 Lemma 30: join(one, sk1) = one.
% 43.62/6.05 Proof:
% 43.62/6.05 join(one, sk1)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(sk1, one)
% 43.62/6.05 = { by axiom 4 (goals_14) }
% 43.62/6.05 one
% 43.62/6.05
% 43.62/6.05 Lemma 31: join(one, join(X, sk1)) = join(X, one).
% 43.62/6.05 Proof:
% 43.62/6.05 join(one, join(X, sk1))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(one, join(sk1, X))
% 43.62/6.05 = { by axiom 10 (maddux2_join_associativity_2) }
% 43.62/6.05 join(join(one, sk1), X)
% 43.62/6.05 = { by lemma 30 }
% 43.62/6.05 join(one, X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 join(X, one)
% 43.62/6.05
% 43.62/6.05 Lemma 32: join(one, join(sk1, X)) = join(X, one).
% 43.62/6.05 Proof:
% 43.62/6.05 join(one, join(sk1, X))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(one, join(X, sk1))
% 43.62/6.05 = { by lemma 31 }
% 43.62/6.05 join(X, one)
% 43.62/6.05
% 43.62/6.05 Lemma 33: join(X, X) = X.
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, X)
% 43.62/6.05 = { by lemma 17 R->L }
% 43.62/6.05 join(X, composition(converse(one), X))
% 43.62/6.05 = { by lemma 25 }
% 43.62/6.05 composition(join(converse(one), one), X)
% 43.62/6.05 = { by lemma 18 }
% 43.62/6.05 composition(join(one, one), X)
% 43.62/6.05 = { by lemma 29 R->L }
% 43.62/6.05 composition(join(one, zero), X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 composition(join(zero, one), X)
% 43.62/6.05 = { by lemma 32 R->L }
% 43.62/6.05 composition(join(one, join(sk1, zero)), X)
% 43.62/6.05 = { by lemma 29 }
% 43.62/6.05 composition(join(one, join(sk1, sk1)), X)
% 43.62/6.05 = { by lemma 31 }
% 43.62/6.05 composition(join(sk1, one), X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.05 composition(join(one, sk1), X)
% 43.62/6.05 = { by lemma 30 }
% 43.62/6.05 composition(one, X)
% 43.62/6.05 = { by lemma 20 }
% 43.62/6.05 X
% 43.62/6.05
% 43.62/6.05 Lemma 34: join(X, zero) = X.
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, zero)
% 43.62/6.05 = { by lemma 29 }
% 43.62/6.05 join(X, X)
% 43.62/6.05 = { by lemma 33 }
% 43.62/6.05 X
% 43.62/6.05
% 43.62/6.05 Lemma 35: join(X, top) = top.
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, top)
% 43.62/6.05 = { by lemma 24 }
% 43.62/6.05 join(join(zero, zero), top)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(top, join(zero, zero))
% 43.62/6.05 = { by lemma 33 }
% 43.62/6.05 join(top, zero)
% 43.62/6.05 = { by lemma 34 }
% 43.62/6.05 top
% 43.62/6.05
% 43.62/6.05 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 43.62/6.05 Proof:
% 43.62/6.05 converse(join(X, converse(Y)))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 converse(join(converse(Y), X))
% 43.62/6.05 = { by axiom 9 (converse_additivity_9) }
% 43.62/6.05 join(converse(converse(Y)), converse(X))
% 43.62/6.05 = { by axiom 1 (converse_idempotence_8) }
% 43.62/6.05 join(Y, converse(X))
% 43.62/6.05
% 43.62/6.05 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 43.62/6.05 Proof:
% 43.62/6.05 converse(join(converse(X), Y))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 converse(join(Y, converse(X)))
% 43.62/6.05 = { by lemma 36 }
% 43.62/6.05 join(X, converse(Y))
% 43.62/6.05
% 43.62/6.05 Lemma 38: join(X, converse(complement(converse(X)))) = converse(top).
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, converse(complement(converse(X))))
% 43.62/6.05 = { by lemma 37 R->L }
% 43.62/6.05 converse(join(converse(X), complement(converse(X))))
% 43.62/6.05 = { by axiom 6 (def_top_12) R->L }
% 43.62/6.05 converse(top)
% 43.62/6.05
% 43.62/6.05 Lemma 39: join(X, join(complement(X), Y)) = top.
% 43.62/6.05 Proof:
% 43.62/6.05 join(X, join(complement(X), Y))
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(X, join(Y, complement(X)))
% 43.62/6.05 = { by lemma 19 }
% 43.62/6.05 join(Y, top)
% 43.62/6.05 = { by lemma 24 R->L }
% 43.62/6.05 join(Z, top)
% 43.62/6.05 = { by lemma 35 }
% 43.62/6.05 top
% 43.62/6.05
% 43.62/6.05 Lemma 40: converse(top) = top.
% 43.62/6.05 Proof:
% 43.62/6.05 converse(top)
% 43.62/6.05 = { by lemma 35 R->L }
% 43.62/6.05 converse(join(X, top))
% 43.62/6.05 = { by axiom 9 (converse_additivity_9) }
% 43.62/6.05 join(converse(X), converse(top))
% 43.62/6.05 = { by lemma 38 R->L }
% 43.62/6.05 join(converse(X), join(complement(converse(X)), converse(complement(converse(complement(converse(X)))))))
% 43.62/6.05 = { by lemma 39 }
% 43.62/6.05 top
% 43.62/6.05
% 43.62/6.05 Lemma 41: join(top, X) = top.
% 43.62/6.05 Proof:
% 43.62/6.05 join(top, X)
% 43.62/6.05 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.05 join(X, top)
% 43.62/6.05 = { by lemma 24 R->L }
% 43.62/6.05 join(Y, top)
% 43.62/6.05 = { by lemma 35 }
% 43.62/6.05 top
% 43.62/6.06
% 43.62/6.06 Lemma 42: join(zero, X) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 join(zero, X)
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 join(X, zero)
% 43.62/6.06 = { by lemma 34 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 43: complement(complement(X)) = meet(X, X).
% 43.62/6.06 Proof:
% 43.62/6.06 complement(complement(X))
% 43.62/6.06 = { by lemma 22 R->L }
% 43.62/6.06 complement(join(complement(X), complement(X)))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.06 meet(X, X)
% 43.62/6.06
% 43.62/6.06 Lemma 44: complement(complement(X)) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 complement(complement(X))
% 43.62/6.06 = { by lemma 42 R->L }
% 43.62/6.06 join(zero, complement(complement(X)))
% 43.62/6.06 = { by lemma 43 }
% 43.62/6.06 join(zero, meet(X, X))
% 43.62/6.06 = { by lemma 27 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 45: meet(X, meet(X, X)) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 meet(X, meet(X, X))
% 43.62/6.06 = { by lemma 43 R->L }
% 43.62/6.06 meet(X, complement(complement(X)))
% 43.62/6.06 = { by lemma 34 R->L }
% 43.62/6.06 join(meet(X, complement(complement(X))), zero)
% 43.62/6.06 = { by lemma 15 R->L }
% 43.62/6.06 join(meet(X, complement(complement(X))), complement(top))
% 43.62/6.06 = { by axiom 6 (def_top_12) }
% 43.62/6.06 join(meet(X, complement(complement(X))), complement(join(complement(X), complement(complement(X)))))
% 43.62/6.06 = { by lemma 26 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 46: meet(Y, X) = meet(X, Y).
% 43.62/6.06 Proof:
% 43.62/6.06 meet(Y, X)
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.06 complement(join(complement(Y), complement(X)))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 complement(join(complement(X), complement(Y)))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.06 meet(X, Y)
% 43.62/6.06
% 43.62/6.06 Lemma 47: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 43.62/6.06 Proof:
% 43.62/6.06 complement(join(complement(X), meet(Y, Z)))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 complement(join(complement(X), meet(Z, Y)))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 complement(join(meet(Z, Y), complement(X)))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.06 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.06 meet(join(complement(Z), complement(Y)), X)
% 43.62/6.06 = { by lemma 46 R->L }
% 43.62/6.06 meet(X, join(complement(Z), complement(Y)))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.06 meet(X, join(complement(Y), complement(Z)))
% 43.62/6.06
% 43.62/6.06 Lemma 48: complement(join(zero, complement(X))) = meet(X, top).
% 43.62/6.06 Proof:
% 43.62/6.06 complement(join(zero, complement(X)))
% 43.62/6.06 = { by lemma 15 R->L }
% 43.62/6.06 complement(join(complement(top), complement(X)))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.06 meet(top, X)
% 43.62/6.06 = { by lemma 46 R->L }
% 43.62/6.06 meet(X, top)
% 43.62/6.06
% 43.62/6.06 Lemma 49: meet(X, top) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 meet(X, top)
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 meet(top, X)
% 43.62/6.06 = { by lemma 44 R->L }
% 43.62/6.06 meet(top, complement(complement(X)))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 meet(complement(complement(X)), top)
% 43.62/6.06 = { by lemma 48 R->L }
% 43.62/6.06 complement(join(zero, complement(complement(complement(X)))))
% 43.62/6.06 = { by lemma 43 }
% 43.62/6.06 complement(join(zero, meet(complement(X), complement(X))))
% 43.62/6.06 = { by lemma 27 }
% 43.62/6.06 complement(complement(X))
% 43.62/6.06 = { by lemma 44 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 50: meet(top, X) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 meet(top, X)
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 meet(X, top)
% 43.62/6.06 = { by lemma 49 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 51: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 43.62/6.06 Proof:
% 43.62/6.06 join(complement(X), complement(Y))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 join(complement(Y), complement(X))
% 43.62/6.06 = { by lemma 50 R->L }
% 43.62/6.06 meet(top, join(complement(Y), complement(X)))
% 43.62/6.06 = { by lemma 47 R->L }
% 43.62/6.06 complement(join(complement(top), meet(Y, X)))
% 43.62/6.06 = { by lemma 15 }
% 43.62/6.06 complement(join(zero, meet(Y, X)))
% 43.62/6.06 = { by lemma 42 }
% 43.62/6.06 complement(meet(Y, X))
% 43.62/6.06 = { by lemma 46 R->L }
% 43.62/6.06 complement(meet(X, Y))
% 43.62/6.06
% 43.62/6.06 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 43.62/6.06 Proof:
% 43.62/6.06 complement(join(X, complement(Y)))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 complement(join(complement(Y), X))
% 43.62/6.06 = { by lemma 45 R->L }
% 43.62/6.06 complement(join(complement(Y), meet(X, meet(X, X))))
% 43.62/6.06 = { by lemma 47 }
% 43.62/6.06 meet(Y, join(complement(X), complement(meet(X, X))))
% 43.62/6.06 = { by lemma 51 }
% 43.62/6.06 meet(Y, complement(meet(X, meet(X, X))))
% 43.62/6.06 = { by lemma 45 }
% 43.62/6.06 meet(Y, complement(X))
% 43.62/6.06
% 43.62/6.06 Lemma 53: join(meet(X, Y), meet(X, complement(Y))) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 join(meet(X, Y), meet(X, complement(Y)))
% 43.62/6.06 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.06 join(meet(X, complement(Y)), meet(X, Y))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.06 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 43.62/6.06 = { by lemma 26 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 54: join(meet(X, Y), meet(complement(Y), X)) = X.
% 43.62/6.06 Proof:
% 43.62/6.06 join(meet(X, Y), meet(complement(Y), X))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 join(meet(X, Y), meet(X, complement(Y)))
% 43.62/6.06 = { by lemma 53 }
% 43.62/6.06 X
% 43.62/6.06
% 43.62/6.06 Lemma 55: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 43.62/6.06 Proof:
% 43.62/6.06 complement(meet(X, complement(Y)))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 complement(meet(complement(Y), X))
% 43.62/6.06 = { by lemma 42 R->L }
% 43.62/6.06 complement(meet(join(zero, complement(Y)), X))
% 43.62/6.06 = { by lemma 51 R->L }
% 43.62/6.06 join(complement(join(zero, complement(Y))), complement(X))
% 43.62/6.06 = { by lemma 48 }
% 43.62/6.06 join(meet(Y, top), complement(X))
% 43.62/6.06 = { by lemma 49 }
% 43.62/6.06 join(Y, complement(X))
% 43.62/6.06
% 43.62/6.06 Lemma 56: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 43.62/6.06 Proof:
% 43.62/6.06 meet(Y, meet(X, Z))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 meet(Y, meet(Z, X))
% 43.62/6.06 = { by lemma 49 R->L }
% 43.62/6.06 meet(meet(Y, meet(Z, X)), top)
% 43.62/6.06 = { by lemma 48 R->L }
% 43.62/6.06 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 43.62/6.06 = { by lemma 46 }
% 43.62/6.06 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 43.62/6.06 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 43.62/6.06 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 43.62/6.06 = { by lemma 55 }
% 43.62/6.06 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 43.62/6.06 = { by axiom 10 (maddux2_join_associativity_2) R->L }
% 43.62/6.07 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 43.62/6.07 = { by lemma 51 }
% 43.62/6.07 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 43.62/6.07 = { by lemma 51 }
% 43.62/6.07 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 43.62/6.07 = { by lemma 46 R->L }
% 43.62/6.07 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 43.62/6.07 = { by lemma 48 }
% 43.62/6.07 meet(meet(X, meet(Y, Z)), top)
% 43.62/6.07 = { by lemma 49 }
% 43.62/6.07 meet(X, meet(Y, Z))
% 43.62/6.07
% 43.62/6.07 Lemma 57: meet(meet(X, Y), Z) = meet(X, meet(Z, Y)).
% 43.62/6.07 Proof:
% 43.62/6.07 meet(meet(X, Y), Z)
% 43.62/6.07 = { by lemma 46 }
% 43.62/6.07 meet(Z, meet(X, Y))
% 43.62/6.07 = { by lemma 56 R->L }
% 43.62/6.07 meet(X, meet(Z, Y))
% 43.62/6.07
% 43.62/6.07 Lemma 58: meet(X, X) = X.
% 43.62/6.07 Proof:
% 43.62/6.07 meet(X, X)
% 43.62/6.07 = { by lemma 43 R->L }
% 43.62/6.07 complement(complement(X))
% 43.62/6.07 = { by lemma 44 }
% 43.62/6.07 X
% 43.62/6.07
% 43.62/6.07 Lemma 59: meet(X, join(X, Y)) = X.
% 43.62/6.07 Proof:
% 43.62/6.07 meet(X, join(X, Y))
% 43.62/6.07 = { by lemma 58 R->L }
% 43.62/6.07 meet(X, join(X, meet(Y, Y)))
% 43.62/6.07 = { by lemma 43 R->L }
% 43.62/6.07 meet(X, join(X, complement(complement(Y))))
% 43.62/6.07 = { by lemma 55 R->L }
% 43.62/6.07 meet(X, complement(meet(complement(Y), complement(X))))
% 43.62/6.07 = { by lemma 51 R->L }
% 43.62/6.07 meet(X, join(complement(complement(Y)), complement(complement(X))))
% 43.62/6.07 = { by lemma 47 R->L }
% 43.62/6.07 complement(join(complement(X), meet(complement(Y), complement(X))))
% 43.62/6.07 = { by lemma 42 R->L }
% 43.62/6.07 join(zero, complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 15 R->L }
% 43.62/6.07 join(complement(top), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 35 R->L }
% 43.62/6.07 join(complement(join(complement(complement(Y)), top)), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 19 R->L }
% 43.62/6.07 join(complement(join(complement(X), join(complement(complement(Y)), complement(complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 51 }
% 43.62/6.07 join(complement(join(complement(X), complement(meet(complement(Y), complement(X))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 46 R->L }
% 43.62/6.07 join(complement(join(complement(X), complement(meet(complement(X), complement(Y))))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.07 join(meet(X, meet(complement(X), complement(Y))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 46 R->L }
% 43.62/6.07 join(meet(X, meet(complement(Y), complement(X))), complement(join(complement(X), meet(complement(Y), complement(X)))))
% 43.62/6.07 = { by lemma 26 }
% 43.62/6.07 X
% 43.62/6.07
% 43.62/6.07 Lemma 60: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
% 43.62/6.07 Proof:
% 43.62/6.07 meet(complement(X), join(X, Y))
% 43.62/6.07 = { by lemma 44 R->L }
% 43.62/6.07 meet(complement(X), join(X, complement(complement(Y))))
% 43.62/6.07 = { by lemma 54 R->L }
% 43.62/6.07 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
% 43.62/6.07 = { by lemma 46 }
% 43.62/6.07 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
% 43.62/6.07 = { by lemma 56 }
% 43.62/6.07 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
% 43.62/6.07 = { by lemma 52 R->L }
% 43.62/6.07 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
% 43.62/6.07 = { by axiom 5 (def_zero_13) R->L }
% 43.62/6.07 join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
% 43.62/6.07 = { by lemma 34 }
% 43.62/6.07 meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
% 43.62/6.07 = { by lemma 57 }
% 43.62/6.07 meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
% 43.62/6.07 = { by lemma 44 }
% 43.62/6.07 meet(complement(X), meet(Y, join(X, Y)))
% 43.62/6.07 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.07 meet(complement(X), meet(Y, join(Y, X)))
% 43.62/6.07 = { by lemma 59 }
% 43.62/6.07 meet(complement(X), Y)
% 43.62/6.07 = { by lemma 46 R->L }
% 43.62/6.07 meet(Y, complement(X))
% 43.62/6.07
% 43.62/6.07 Lemma 61: meet(X, join(Y, complement(X))) = meet(X, Y).
% 43.62/6.07 Proof:
% 43.62/6.07 meet(X, join(Y, complement(X)))
% 43.62/6.07 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.07 meet(X, join(complement(X), Y))
% 43.62/6.07 = { by lemma 42 R->L }
% 43.62/6.07 meet(X, join(zero, join(complement(X), Y)))
% 43.62/6.07 = { by axiom 10 (maddux2_join_associativity_2) }
% 43.62/6.07 meet(X, join(join(zero, complement(X)), Y))
% 43.62/6.07 = { by lemma 49 R->L }
% 43.62/6.07 meet(X, meet(join(join(zero, complement(X)), Y), top))
% 43.62/6.07 = { by lemma 57 R->L }
% 43.62/6.07 meet(meet(X, top), join(join(zero, complement(X)), Y))
% 43.62/6.07 = { by lemma 48 R->L }
% 43.62/6.07 meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
% 43.62/6.07 = { by lemma 60 }
% 43.62/6.07 meet(Y, complement(join(zero, complement(X))))
% 43.62/6.07 = { by lemma 48 }
% 43.62/6.07 meet(Y, meet(X, top))
% 43.62/6.07 = { by lemma 49 }
% 43.62/6.07 meet(Y, X)
% 43.62/6.07 = { by lemma 46 R->L }
% 43.62/6.07 meet(X, Y)
% 43.62/6.07
% 43.62/6.07 Lemma 62: meet(X, join(complement(X), Y)) = meet(X, Y).
% 43.62/6.07 Proof:
% 43.62/6.07 meet(X, join(complement(X), Y))
% 43.62/6.07 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 43.62/6.07 meet(X, join(Y, complement(X)))
% 43.62/6.07 = { by lemma 61 }
% 43.62/6.07 meet(X, Y)
% 43.62/6.07
% 43.62/6.07 Lemma 63: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 43.62/6.07 Proof:
% 43.62/6.07 join(meet(X, Y), meet(Y, complement(X)))
% 43.62/6.07 = { by lemma 46 }
% 43.62/6.07 join(meet(Y, X), meet(Y, complement(X)))
% 43.62/6.07 = { by lemma 53 }
% 43.62/6.07 Y
% 43.62/6.07
% 43.62/6.07 Lemma 64: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 43.62/6.07 Proof:
% 43.62/6.07 join(meet(X, Y), meet(complement(X), Y))
% 43.62/6.07 = { by lemma 46 }
% 43.62/6.07 join(meet(X, Y), meet(Y, complement(X)))
% 43.62/6.07 = { by lemma 63 }
% 43.62/6.07 Y
% 43.62/6.07
% 43.62/6.08 Lemma 65: complement(converse(X)) = converse(complement(X)).
% 43.62/6.08 Proof:
% 43.62/6.08 complement(converse(X))
% 43.62/6.08 = { by lemma 26 R->L }
% 43.62/6.08 complement(converse(join(meet(X, complement(converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by lemma 52 R->L }
% 43.62/6.08 complement(converse(join(complement(join(converse(complement(converse(complement(X)))), complement(X))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by axiom 3 (maddux1_join_commutativity_1) }
% 43.62/6.08 complement(converse(join(complement(join(complement(X), converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by lemma 38 }
% 43.62/6.08 complement(converse(join(complement(converse(top)), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by lemma 40 }
% 43.62/6.08 complement(converse(join(complement(top), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by lemma 15 }
% 43.62/6.08 complement(converse(join(zero, complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))))
% 43.62/6.08 = { by lemma 42 }
% 43.62/6.08 complement(converse(complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))))
% 43.62/6.08 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 43.62/6.08 complement(converse(meet(X, converse(complement(converse(complement(X)))))))
% 43.62/6.08 = { by lemma 34 R->L }
% 43.62/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), zero)))
% 43.62/6.08 = { by axiom 5 (def_zero_13) }
% 43.62/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), complement(complement(X))))))
% 44.39/6.08 = { by axiom 1 (converse_idempotence_8) R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(converse(complement(complement(X))))))))
% 44.39/6.08 = { by lemma 44 R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(complement(converse(complement(complement(X))))))))))
% 44.39/6.08 = { by lemma 54 R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(join(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))), meet(complement(converse(complement(converse(converse(complement(complement(X))))))), complement(converse(complement(complement(X))))))))))))
% 44.39/6.08 = { by lemma 52 R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(join(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))), complement(join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))))
% 44.39/6.08 = { by lemma 52 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X))))))))), complement(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X))))))))))))))
% 44.39/6.08 = { by lemma 46 R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(complement(meet(complement(converse(complement(complement(X)))), converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))
% 44.39/6.08 = { by lemma 46 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(complement(meet(converse(complement(converse(converse(complement(complement(X)))))), complement(converse(complement(complement(X)))))), join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))
% 44.39/6.08 = { by lemma 55 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), complement(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))))
% 44.39/6.08 = { by lemma 44 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))), join(converse(complement(complement(X))), converse(complement(converse(converse(complement(complement(X)))))))))))))
% 44.39/6.08 = { by lemma 46 R->L }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(join(converse(complement(complement(X))), converse(complement(converse(converse(complement(complement(X))))))), join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 44.39/6.08 = { by lemma 38 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(converse(top), join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 44.39/6.08 = { by lemma 40 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(meet(top, join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X))))))))))))))
% 44.39/6.08 = { by lemma 50 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(join(converse(complement(complement(X))), complement(converse(complement(converse(converse(complement(complement(X)))))))))))))
% 44.39/6.08 = { by lemma 37 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), join(complement(complement(X)), converse(complement(converse(complement(converse(converse(complement(complement(X)))))))))))))
% 44.39/6.08 = { by axiom 1 (converse_idempotence_8) }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), join(complement(complement(X)), converse(complement(converse(complement(complement(complement(X)))))))))))
% 44.39/6.08 = { by lemma 62 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(complement(complement(X))))))))))
% 44.39/6.08 = { by lemma 44 }
% 44.39/6.08 complement(converse(join(meet(X, converse(complement(converse(complement(X))))), meet(complement(X), converse(complement(converse(complement(X))))))))
% 44.39/6.09 = { by lemma 64 }
% 44.39/6.09 complement(converse(converse(complement(converse(complement(X))))))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) }
% 44.39/6.09 complement(complement(converse(complement(X))))
% 44.39/6.09 = { by lemma 43 }
% 44.39/6.09 meet(converse(complement(X)), converse(complement(X)))
% 44.39/6.09 = { by lemma 58 }
% 44.39/6.09 converse(complement(X))
% 44.39/6.09
% 44.39/6.09 Lemma 66: join(one, converse(X)) = converse(join(X, one)).
% 44.39/6.09 Proof:
% 44.39/6.09 join(one, converse(X))
% 44.39/6.09 = { by lemma 18 R->L }
% 44.39/6.09 join(converse(one), converse(X))
% 44.39/6.09 = { by axiom 9 (converse_additivity_9) R->L }
% 44.39/6.09 converse(join(one, X))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.39/6.09 converse(join(X, one))
% 44.39/6.09
% 44.39/6.09 Lemma 67: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 44.39/6.09 Proof:
% 44.39/6.09 converse(composition(X, converse(Y)))
% 44.39/6.09 = { by axiom 7 (converse_multiplicativity_10) }
% 44.39/6.09 composition(converse(converse(Y)), converse(X))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) }
% 44.39/6.09 composition(Y, converse(X))
% 44.39/6.09
% 44.39/6.09 Lemma 68: converse(join(composition(X, Y), composition(X, converse(Z)))) = converse(composition(X, join(Y, converse(Z)))).
% 44.39/6.09 Proof:
% 44.39/6.09 converse(join(composition(X, Y), composition(X, converse(Z))))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.39/6.09 converse(join(composition(X, converse(Z)), composition(X, Y)))
% 44.39/6.09 = { by axiom 9 (converse_additivity_9) }
% 44.39/6.09 join(converse(composition(X, converse(Z))), converse(composition(X, Y)))
% 44.39/6.09 = { by lemma 67 }
% 44.39/6.09 join(composition(Z, converse(X)), converse(composition(X, Y)))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.39/6.09 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 44.39/6.09 = { by axiom 7 (converse_multiplicativity_10) }
% 44.39/6.09 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 44.39/6.09 = { by axiom 12 (composition_distributivity_7) R->L }
% 44.39/6.09 composition(join(converse(Y), Z), converse(X))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.39/6.09 composition(join(Z, converse(Y)), converse(X))
% 44.39/6.09 = { by lemma 36 R->L }
% 44.39/6.09 composition(converse(join(Y, converse(Z))), converse(X))
% 44.39/6.09 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 44.39/6.09 converse(composition(X, join(Y, converse(Z))))
% 44.39/6.09
% 44.39/6.09 Lemma 69: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 44.39/6.09 Proof:
% 44.39/6.09 join(composition(X, Y), composition(X, Z))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) R->L }
% 44.39/6.09 join(composition(X, Y), composition(X, converse(converse(Z))))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) R->L }
% 44.39/6.09 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 44.39/6.09 = { by lemma 68 }
% 44.39/6.09 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) }
% 44.39/6.09 composition(X, join(Y, converse(converse(Z))))
% 44.39/6.09 = { by axiom 1 (converse_idempotence_8) }
% 44.39/6.09 composition(X, join(Y, Z))
% 44.39/6.09
% 44.39/6.09 Lemma 70: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
% 44.39/6.09 Proof:
% 44.39/6.09 join(composition(X, Y), join(Z, composition(W, Y)))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.39/6.09 join(composition(X, Y), join(composition(W, Y), Z))
% 44.39/6.09 = { by axiom 10 (maddux2_join_associativity_2) }
% 44.39/6.09 join(join(composition(X, Y), composition(W, Y)), Z)
% 44.39/6.09 = { by axiom 12 (composition_distributivity_7) R->L }
% 44.39/6.09 join(composition(join(X, W), Y), Z)
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.39/6.09 join(Z, composition(join(X, W), Y))
% 44.39/6.09
% 44.39/6.09 Lemma 71: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
% 44.39/6.09 Proof:
% 44.39/6.09 join(complement(composition(X, Y)), composition(join(X, Z), Y))
% 44.39/6.09 = { by lemma 70 R->L }
% 44.39/6.09 join(composition(X, Y), join(complement(composition(X, Y)), composition(Z, Y)))
% 44.39/6.09 = { by lemma 39 }
% 44.39/6.09 top
% 44.39/6.09
% 44.39/6.09 Lemma 72: meet(X, composition(sk1, top)) = composition(sk1, X).
% 44.39/6.09 Proof:
% 44.39/6.09 meet(X, composition(sk1, top))
% 44.39/6.09 = { by lemma 62 R->L }
% 44.39/6.09 meet(X, join(complement(X), composition(sk1, top)))
% 44.39/6.09 = { by axiom 6 (def_top_12) }
% 44.39/6.09 meet(X, join(complement(X), composition(sk1, join(complement(X), complement(complement(X))))))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.39/6.09 meet(X, join(complement(X), composition(sk1, join(complement(complement(X)), complement(X)))))
% 44.39/6.09 = { by lemma 69 R->L }
% 44.39/6.09 meet(X, join(complement(X), join(composition(sk1, complement(complement(X))), composition(sk1, complement(X)))))
% 44.39/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.39/6.09 meet(X, join(complement(X), join(composition(sk1, complement(X)), composition(sk1, complement(complement(X))))))
% 44.50/6.09 = { by axiom 10 (maddux2_join_associativity_2) }
% 44.50/6.09 meet(X, join(join(complement(X), composition(sk1, complement(X))), composition(sk1, complement(complement(X)))))
% 44.50/6.09 = { by lemma 25 }
% 44.50/6.09 meet(X, join(composition(join(sk1, one), complement(X)), composition(sk1, complement(complement(X)))))
% 44.50/6.09 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.50/6.09 meet(X, join(composition(join(one, sk1), complement(X)), composition(sk1, complement(complement(X)))))
% 44.50/6.09 = { by lemma 30 }
% 44.50/6.09 meet(X, join(composition(one, complement(X)), composition(sk1, complement(complement(X)))))
% 44.50/6.09 = { by lemma 20 }
% 44.50/6.09 meet(X, join(complement(X), composition(sk1, complement(complement(X)))))
% 44.50/6.09 = { by lemma 62 }
% 44.50/6.09 meet(X, composition(sk1, complement(complement(X))))
% 44.50/6.09 = { by lemma 44 }
% 44.50/6.09 meet(X, composition(sk1, X))
% 44.50/6.09 = { by lemma 46 }
% 44.50/6.09 meet(composition(sk1, X), X)
% 44.50/6.09 = { by lemma 61 R->L }
% 44.50/6.09 meet(composition(sk1, X), join(X, complement(composition(sk1, X))))
% 44.50/6.09 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), X))
% 44.50/6.09 = { by lemma 20 R->L }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(one, X)))
% 44.50/6.09 = { by lemma 53 R->L }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(meet(one, sk1), meet(one, complement(sk1))), X)))
% 44.50/6.09 = { by lemma 46 }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(meet(sk1, one), meet(one, complement(sk1))), X)))
% 44.50/6.09 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(complement(join(complement(sk1), complement(one))), meet(one, complement(sk1))), X)))
% 44.50/6.09 = { by lemma 42 R->L }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(zero, complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.09 = { by lemma 15 R->L }
% 44.50/6.09 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(top), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by lemma 41 R->L }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(join(top, one)), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(join(one, top)), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by axiom 6 (def_top_12) }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(join(one, join(sk1, complement(sk1)))), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by lemma 32 }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(join(complement(sk1), one)), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(complement(join(one, complement(sk1))), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by lemma 52 }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(join(meet(sk1, complement(one)), complement(join(complement(sk1), complement(one)))), meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by lemma 26 }
% 44.50/6.10 meet(composition(sk1, X), join(complement(composition(sk1, X)), composition(join(sk1, meet(one, complement(sk1))), X)))
% 44.50/6.10 = { by lemma 71 }
% 44.50/6.10 meet(composition(sk1, X), top)
% 44.50/6.10 = { by lemma 49 }
% 44.50/6.10 composition(sk1, X)
% 44.50/6.10
% 44.50/6.10 Lemma 73: meet(Y, composition(sk1, X)) = meet(X, composition(sk1, Y)).
% 44.50/6.10 Proof:
% 44.50/6.10 meet(Y, composition(sk1, X))
% 44.50/6.10 = { by lemma 72 R->L }
% 44.50/6.10 meet(Y, meet(X, composition(sk1, top)))
% 44.50/6.10 = { by lemma 56 R->L }
% 44.50/6.10 meet(X, meet(Y, composition(sk1, top)))
% 44.50/6.10 = { by lemma 72 }
% 44.50/6.10 meet(X, composition(sk1, Y))
% 44.50/6.10
% 44.50/6.10 Lemma 74: composition(join(X, converse(complement(composition(Y, top)))), Y) = composition(X, Y).
% 44.50/6.10 Proof:
% 44.50/6.10 composition(join(X, converse(complement(composition(Y, top)))), Y)
% 44.50/6.10 = { by lemma 36 R->L }
% 44.50/6.10 composition(converse(join(complement(composition(Y, top)), converse(X))), Y)
% 44.50/6.10 = { by lemma 16 R->L }
% 44.50/6.10 converse(composition(converse(Y), join(complement(composition(Y, top)), converse(X))))
% 44.50/6.10 = { by lemma 68 R->L }
% 44.50/6.10 converse(join(composition(converse(Y), complement(composition(Y, top))), composition(converse(Y), converse(X))))
% 44.50/6.10 = { by lemma 42 R->L }
% 44.50/6.10 converse(join(join(zero, composition(converse(Y), complement(composition(Y, top)))), composition(converse(Y), converse(X))))
% 44.50/6.10 = { by lemma 15 R->L }
% 44.50/6.10 converse(join(join(complement(top), composition(converse(Y), complement(composition(Y, top)))), composition(converse(Y), converse(X))))
% 44.50/6.10 = { by lemma 21 }
% 44.50/6.10 converse(join(complement(top), composition(converse(Y), converse(X))))
% 44.50/6.10 = { by lemma 15 }
% 44.50/6.10 converse(join(zero, composition(converse(Y), converse(X))))
% 44.50/6.10 = { by lemma 42 }
% 44.50/6.10 converse(composition(converse(Y), converse(X)))
% 44.50/6.10 = { by lemma 16 }
% 44.50/6.10 composition(converse(converse(X)), Y)
% 44.50/6.10 = { by axiom 1 (converse_idempotence_8) }
% 44.50/6.10 composition(X, Y)
% 44.50/6.10
% 44.50/6.10 Lemma 75: composition(X, converse(join(complement(composition(top, X)), Y))) = composition(X, converse(Y)).
% 44.50/6.10 Proof:
% 44.50/6.10 composition(X, converse(join(complement(composition(top, X)), Y)))
% 44.50/6.10 = { by axiom 9 (converse_additivity_9) }
% 44.50/6.10 composition(X, join(converse(complement(composition(top, X))), converse(Y)))
% 44.50/6.10 = { by lemma 65 R->L }
% 44.50/6.10 composition(X, join(complement(converse(composition(top, X))), converse(Y)))
% 44.50/6.10 = { by axiom 7 (converse_multiplicativity_10) }
% 44.50/6.10 composition(X, join(complement(composition(converse(X), converse(top))), converse(Y)))
% 44.50/6.10 = { by lemma 40 }
% 44.50/6.10 composition(X, join(complement(composition(converse(X), top)), converse(Y)))
% 44.50/6.10 = { by lemma 36 R->L }
% 44.50/6.10 composition(X, converse(join(Y, converse(complement(composition(converse(X), top))))))
% 44.50/6.10 = { by lemma 67 R->L }
% 44.50/6.10 converse(composition(join(Y, converse(complement(composition(converse(X), top)))), converse(X)))
% 44.50/6.10 = { by lemma 74 }
% 44.50/6.10 converse(composition(Y, converse(X)))
% 44.50/6.10 = { by lemma 67 }
% 44.50/6.10 composition(X, converse(Y))
% 44.50/6.10
% 44.50/6.10 Lemma 76: composition(sk1, converse(composition(X, sk1))) = composition(sk1, converse(X)).
% 44.50/6.10 Proof:
% 44.50/6.10 composition(sk1, converse(composition(X, sk1)))
% 44.50/6.10 = { by lemma 49 R->L }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), top)))
% 44.50/6.10 = { by lemma 40 R->L }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(top))))
% 44.50/6.10 = { by lemma 71 R->L }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(join(complement(composition(meet(converse(sk1), one), converse(X))), composition(join(meet(converse(sk1), one), meet(one, complement(converse(sk1)))), converse(X)))))))
% 44.50/6.10 = { by lemma 63 }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(join(complement(composition(meet(converse(sk1), one), converse(X))), composition(one, converse(X)))))))
% 44.50/6.10 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(meet(converse(sk1), one), converse(X))))))))
% 44.50/6.10 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(complement(join(complement(converse(sk1)), complement(one))), converse(X))))))))
% 44.50/6.10 = { by lemma 42 R->L }
% 44.50/6.10 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(zero, complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 15 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(top), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 40 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(top)), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 40 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(converse(top))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 41 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(converse(join(top, one)))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 66 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(join(one, converse(top)))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 38 R->L }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(join(one, join(sk1, converse(complement(converse(sk1))))))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 32 }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(join(converse(complement(converse(sk1))), one))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(join(one, converse(complement(converse(sk1)))))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by lemma 66 }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(converse(join(complement(converse(sk1)), one)))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.50/6.11 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.50/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(converse(converse(join(one, complement(converse(sk1)))))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.64/6.11 = { by axiom 1 (converse_idempotence_8) }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(complement(join(one, complement(converse(sk1)))), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.64/6.11 = { by lemma 52 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(join(meet(converse(sk1), complement(one)), complement(join(complement(converse(sk1)), complement(one)))), converse(X))))))))
% 44.64/6.11 = { by lemma 26 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(composition(one, converse(X)), complement(composition(converse(sk1), converse(X))))))))
% 44.64/6.11 = { by lemma 20 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), converse(join(converse(X), complement(composition(converse(sk1), converse(X))))))))
% 44.64/6.11 = { by lemma 37 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), join(X, converse(complement(composition(converse(sk1), converse(X))))))))
% 44.64/6.11 = { by axiom 7 (converse_multiplicativity_10) R->L }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), join(X, converse(complement(converse(composition(X, sk1))))))))
% 44.64/6.11 = { by lemma 65 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), join(X, converse(converse(complement(composition(X, sk1))))))))
% 44.64/6.11 = { by axiom 1 (converse_idempotence_8) }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), join(X, complement(composition(X, sk1))))))
% 44.64/6.11 = { by lemma 61 }
% 44.64/6.11 composition(sk1, converse(meet(composition(X, sk1), X)))
% 44.64/6.11 = { by lemma 46 R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, composition(X, sk1))))
% 44.64/6.11 = { by lemma 44 R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, composition(complement(complement(X)), sk1))))
% 44.64/6.11 = { by lemma 62 R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(complement(X), composition(complement(complement(X)), sk1)))))
% 44.64/6.11 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(complement(X)), sk1), complement(X)))))
% 44.64/6.11 = { by axiom 2 (composition_identity_6) R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(complement(X)), sk1), composition(complement(X), one)))))
% 44.64/6.11 = { by lemma 30 R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(complement(X)), sk1), composition(complement(X), join(one, sk1))))))
% 44.64/6.11 = { by lemma 69 R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(complement(X)), sk1), join(composition(complement(X), one), composition(complement(X), sk1))))))
% 44.64/6.11 = { by lemma 70 }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(X), one), composition(join(complement(complement(X)), complement(X)), sk1)))))
% 44.64/6.11 = { by axiom 3 (maddux1_join_commutativity_1) }
% 44.64/6.11 composition(sk1, converse(meet(X, join(composition(complement(X), one), composition(join(complement(X), complement(complement(X))), sk1)))))
% 44.64/6.11 = { by axiom 2 (composition_identity_6) }
% 44.64/6.11 composition(sk1, converse(meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), sk1)))))
% 44.64/6.11 = { by axiom 6 (def_top_12) R->L }
% 44.64/6.11 composition(sk1, converse(meet(X, join(complement(X), composition(top, sk1)))))
% 44.64/6.11 = { by lemma 62 }
% 44.64/6.11 composition(sk1, converse(meet(X, composition(top, sk1))))
% 44.66/6.11 = { by lemma 44 R->L }
% 44.66/6.11 composition(sk1, converse(meet(X, complement(complement(composition(top, sk1))))))
% 44.66/6.11 = { by lemma 75 R->L }
% 44.66/6.11 composition(sk1, converse(join(complement(composition(top, sk1)), meet(X, complement(complement(composition(top, sk1)))))))
% 44.66/6.11 = { by axiom 3 (maddux1_join_commutativity_1) R->L }
% 44.66/6.11 composition(sk1, converse(join(meet(X, complement(complement(composition(top, sk1)))), complement(composition(top, sk1)))))
% 44.66/6.11 = { by lemma 59 R->L }
% 44.66/6.12 composition(sk1, converse(join(meet(X, complement(complement(composition(top, sk1)))), meet(complement(composition(top, sk1)), join(complement(composition(top, sk1)), X)))))
% 44.66/6.12 = { by lemma 44 R->L }
% 44.66/6.12 composition(sk1, converse(join(meet(X, complement(complement(composition(top, sk1)))), meet(complement(complement(complement(composition(top, sk1)))), join(complement(composition(top, sk1)), X)))))
% 44.66/6.12 = { by lemma 60 R->L }
% 44.66/6.12 composition(sk1, converse(join(meet(complement(complement(composition(top, sk1))), join(complement(composition(top, sk1)), X)), meet(complement(complement(complement(composition(top, sk1)))), join(complement(composition(top, sk1)), X)))))
% 44.66/6.12 = { by lemma 64 }
% 44.66/6.12 composition(sk1, converse(join(complement(composition(top, sk1)), X)))
% 44.66/6.12 = { by lemma 75 }
% 44.66/6.12 composition(sk1, converse(X))
% 44.66/6.12
% 44.66/6.12 Goal 1 (goals_15): meet(composition(sk1, sk2), complement(sk3)) = meet(composition(sk1, sk2), complement(composition(sk1, sk3))).
% 44.66/6.12 Proof:
% 44.66/6.12 meet(composition(sk1, sk2), complement(sk3))
% 44.66/6.12 = { by lemma 46 R->L }
% 44.66/6.12 meet(complement(sk3), composition(sk1, sk2))
% 44.66/6.12 = { by lemma 73 R->L }
% 44.66/6.12 meet(sk2, composition(sk1, complement(sk3)))
% 44.66/6.12 = { by axiom 1 (converse_idempotence_8) R->L }
% 44.66/6.12 meet(sk2, composition(sk1, converse(converse(complement(sk3)))))
% 44.66/6.12 = { by lemma 76 R->L }
% 44.66/6.12 meet(sk2, composition(sk1, converse(composition(converse(complement(sk3)), sk1))))
% 44.66/6.12 = { by lemma 74 R->L }
% 44.66/6.12 meet(sk2, composition(sk1, converse(composition(join(converse(complement(sk3)), converse(complement(composition(sk1, top)))), sk1))))
% 44.66/6.12 = { by axiom 9 (converse_additivity_9) R->L }
% 44.66/6.12 meet(sk2, composition(sk1, converse(composition(converse(join(complement(sk3), complement(composition(sk1, top)))), sk1))))
% 44.66/6.12 = { by lemma 51 }
% 44.66/6.12 meet(sk2, composition(sk1, converse(composition(converse(complement(meet(sk3, composition(sk1, top)))), sk1))))
% 44.66/6.12 = { by lemma 76 }
% 44.66/6.12 meet(sk2, composition(sk1, converse(converse(complement(meet(sk3, composition(sk1, top)))))))
% 44.66/6.12 = { by axiom 1 (converse_idempotence_8) }
% 44.66/6.12 meet(sk2, composition(sk1, complement(meet(sk3, composition(sk1, top)))))
% 44.66/6.12 = { by lemma 72 }
% 44.66/6.12 meet(sk2, composition(sk1, complement(composition(sk1, sk3))))
% 44.66/6.12 = { by lemma 73 }
% 44.66/6.12 meet(complement(composition(sk1, sk3)), composition(sk1, sk2))
% 44.66/6.12 = { by lemma 46 R->L }
% 44.66/6.12 meet(composition(sk1, sk2), complement(composition(sk1, sk3)))
% 44.66/6.12 % SZS output end Proof
% 44.66/6.12
% 44.66/6.12 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------