TSTP Solution File: REL040-10 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL040-10 : TPTP v8.1.2. Released v7.5.0.
% 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 : n002.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 13:44:24 EDT 2023
% Result : Unsatisfiable 173.10s 22.43s
% Output : Proof 174.74s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.12/0.12 % Problem : REL040-10 : TPTP v8.1.2. Released v7.5.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.33 % Computer : n002.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 22:26:32 EDT 2023
% 0.13/0.34 % CPUTime :
% 173.10/22.43 Command-line arguments: --flatten
% 173.10/22.43
% 173.10/22.43 % SZS status Unsatisfiable
% 173.10/22.43
% 174.74/22.58 % SZS output start Proof
% 174.74/22.58 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 174.74/22.58 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 174.74/22.58 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 174.74/22.58 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 174.74/22.58 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 174.74/22.58 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 174.74/22.58 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 174.74/22.58 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 174.74/22.58 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 174.74/22.58 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 174.74/22.58 Axiom 11 (goals_17): join(composition(converse(sk1), sk1), one) = one.
% 174.74/22.58 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 174.74/22.58 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 174.74/22.58 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 174.74/22.58 Axiom 15 (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).
% 174.74/22.58
% 174.74/22.58 Lemma 16: complement(top) = zero.
% 174.74/22.58 Proof:
% 174.74/22.58 complement(top)
% 174.74/22.58 = { by axiom 4 (def_top_12) }
% 174.74/22.58 complement(join(complement(X), complement(complement(X))))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.58 meet(X, complement(X))
% 174.74/22.58 = { by axiom 5 (def_zero_13) R->L }
% 174.74/22.58 zero
% 174.74/22.58
% 174.74/22.58 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, join(Y, complement(X)))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(X, join(complement(X), Y))
% 174.74/22.58 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.58 join(join(X, complement(X)), Y)
% 174.74/22.58 = { by axiom 4 (def_top_12) R->L }
% 174.74/22.58 join(top, Y)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.58 join(Y, top)
% 174.74/22.58
% 174.74/22.58 Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 174.74/22.58 Proof:
% 174.74/22.58 converse(composition(converse(X), Y))
% 174.74/22.58 = { by axiom 8 (converse_multiplicativity_10) }
% 174.74/22.58 composition(converse(Y), converse(converse(X)))
% 174.74/22.58 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.58 composition(converse(Y), X)
% 174.74/22.58
% 174.74/22.58 Lemma 19: composition(converse(join(composition(converse(sk1), sk1), one)), X) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 composition(converse(join(composition(converse(sk1), sk1), one)), X)
% 174.74/22.58 = { by axiom 11 (goals_17) }
% 174.74/22.58 composition(converse(one), X)
% 174.74/22.58 = { by lemma 18 R->L }
% 174.74/22.58 converse(composition(converse(X), one))
% 174.74/22.58 = { by axiom 3 (composition_identity_6) }
% 174.74/22.58 converse(converse(X))
% 174.74/22.58 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 20: composition(join(composition(converse(sk1), sk1), one), X) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 composition(join(composition(converse(sk1), sk1), one), X)
% 174.74/22.58 = { by lemma 19 R->L }
% 174.74/22.58 composition(converse(join(composition(converse(sk1), sk1), one)), composition(join(composition(converse(sk1), sk1), one), X))
% 174.74/22.58 = { by axiom 11 (goals_17) }
% 174.74/22.58 composition(converse(join(composition(converse(sk1), sk1), one)), composition(one, X))
% 174.74/22.58 = { by axiom 9 (composition_associativity_5) }
% 174.74/22.58 composition(composition(converse(join(composition(converse(sk1), sk1), one)), one), X)
% 174.74/22.58 = { by axiom 3 (composition_identity_6) }
% 174.74/22.58 composition(converse(join(composition(converse(sk1), sk1), one)), X)
% 174.74/22.58 = { by lemma 19 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 174.74/22.58 Proof:
% 174.74/22.58 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 174.74/22.58 = { by axiom 13 (converse_cancellativity_11) }
% 174.74/22.58 complement(X)
% 174.74/22.58
% 174.74/22.58 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 174.74/22.58 Proof:
% 174.74/22.58 join(complement(X), complement(X))
% 174.74/22.58 = { by lemma 19 R->L }
% 174.74/22.58 join(complement(X), composition(converse(join(composition(converse(sk1), sk1), one)), complement(X)))
% 174.74/22.58 = { by lemma 20 R->L }
% 174.74/22.58 join(complement(X), composition(converse(join(composition(converse(sk1), sk1), one)), complement(composition(join(composition(converse(sk1), sk1), one), X))))
% 174.74/22.58 = { by lemma 21 }
% 174.74/22.58 complement(X)
% 174.74/22.58
% 174.74/22.58 Lemma 23: join(top, complement(X)) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(top, complement(X))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(complement(X), top)
% 174.74/22.58 = { by lemma 17 R->L }
% 174.74/22.58 join(X, join(complement(X), complement(X)))
% 174.74/22.58 = { by lemma 22 }
% 174.74/22.58 join(X, complement(X))
% 174.74/22.58 = { by axiom 4 (def_top_12) R->L }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 24: join(Y, top) = join(X, top).
% 174.74/22.58 Proof:
% 174.74/22.58 join(Y, top)
% 174.74/22.58 = { by lemma 23 R->L }
% 174.74/22.58 join(Y, join(top, complement(Y)))
% 174.74/22.58 = { by lemma 17 }
% 174.74/22.58 join(top, top)
% 174.74/22.58 = { by lemma 17 R->L }
% 174.74/22.58 join(X, join(top, complement(X)))
% 174.74/22.58 = { by lemma 23 }
% 174.74/22.58 join(X, top)
% 174.74/22.58
% 174.74/22.58 Lemma 25: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(meet(X, Y), complement(join(complement(X), Y)))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 174.74/22.58 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 26: join(zero, meet(X, X)) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(zero, meet(X, X))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 join(zero, complement(join(complement(X), complement(X))))
% 174.74/22.58 = { by axiom 5 (def_zero_13) }
% 174.74/22.58 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 174.74/22.58 = { by lemma 25 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 27: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 174.74/22.58 Proof:
% 174.74/22.58 join(zero, join(X, complement(complement(Y))))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(zero, join(complement(complement(Y)), X))
% 174.74/22.58 = { by lemma 22 R->L }
% 174.74/22.58 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.58 join(zero, join(meet(Y, Y), X))
% 174.74/22.58 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.58 join(join(zero, meet(Y, Y)), X)
% 174.74/22.58 = { by lemma 26 }
% 174.74/22.58 join(Y, X)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.58 join(X, Y)
% 174.74/22.58
% 174.74/22.58 Lemma 28: join(zero, complement(complement(X))) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(zero, complement(complement(X)))
% 174.74/22.58 = { by axiom 5 (def_zero_13) }
% 174.74/22.58 join(meet(X, complement(X)), complement(complement(X)))
% 174.74/22.58 = { by lemma 22 R->L }
% 174.74/22.58 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 174.74/22.58 = { by lemma 25 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 29: join(X, zero) = join(X, X).
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, zero)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(zero, X)
% 174.74/22.58 = { by lemma 28 R->L }
% 174.74/22.58 join(zero, join(zero, complement(complement(X))))
% 174.74/22.58 = { by lemma 22 R->L }
% 174.74/22.58 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 174.74/22.58 = { by lemma 27 }
% 174.74/22.58 join(zero, join(complement(complement(X)), X))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.58 join(zero, join(X, complement(complement(X))))
% 174.74/22.58 = { by lemma 27 }
% 174.74/22.58 join(X, X)
% 174.74/22.58
% 174.74/22.58 Lemma 30: join(zero, complement(X)) = complement(X).
% 174.74/22.58 Proof:
% 174.74/22.58 join(zero, complement(X))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(complement(X), zero)
% 174.74/22.58 = { by lemma 29 }
% 174.74/22.58 join(complement(X), complement(X))
% 174.74/22.58 = { by lemma 22 }
% 174.74/22.58 complement(X)
% 174.74/22.58
% 174.74/22.58 Lemma 31: join(X, zero) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, zero)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(zero, X)
% 174.74/22.58 = { by lemma 27 R->L }
% 174.74/22.58 join(zero, join(zero, complement(complement(X))))
% 174.74/22.58 = { by lemma 30 }
% 174.74/22.58 join(zero, complement(complement(X)))
% 174.74/22.58 = { by lemma 28 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 32: join(X, top) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, top)
% 174.74/22.58 = { by lemma 24 }
% 174.74/22.58 join(join(zero, zero), top)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(top, join(zero, zero))
% 174.74/22.58 = { by lemma 31 }
% 174.74/22.58 join(top, zero)
% 174.74/22.58 = { by lemma 31 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 33: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 174.74/22.58 Proof:
% 174.74/22.58 converse(join(X, converse(Y)))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 converse(join(converse(Y), X))
% 174.74/22.58 = { by axiom 6 (converse_additivity_9) }
% 174.74/22.58 join(converse(converse(Y)), converse(X))
% 174.74/22.58 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.58 join(Y, converse(X))
% 174.74/22.58
% 174.74/22.58 Lemma 34: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 174.74/22.58 Proof:
% 174.74/22.58 converse(join(converse(X), Y))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 converse(join(Y, converse(X)))
% 174.74/22.58 = { by lemma 33 }
% 174.74/22.58 join(X, converse(Y))
% 174.74/22.58
% 174.74/22.58 Lemma 35: join(X, converse(complement(converse(X)))) = converse(top).
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, converse(complement(converse(X))))
% 174.74/22.58 = { by lemma 34 R->L }
% 174.74/22.58 converse(join(converse(X), complement(converse(X))))
% 174.74/22.58 = { by axiom 4 (def_top_12) R->L }
% 174.74/22.58 converse(top)
% 174.74/22.58
% 174.74/22.58 Lemma 36: join(X, join(complement(X), Y)) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, join(complement(X), Y))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(X, join(Y, complement(X)))
% 174.74/22.58 = { by lemma 17 }
% 174.74/22.58 join(Y, top)
% 174.74/22.58 = { by lemma 24 R->L }
% 174.74/22.58 join(Z, top)
% 174.74/22.58 = { by lemma 32 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 37: join(X, converse(top)) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, converse(top))
% 174.74/22.58 = { by lemma 35 R->L }
% 174.74/22.58 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 174.74/22.58 = { by lemma 36 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 38: converse(top) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 converse(top)
% 174.74/22.58 = { by lemma 32 R->L }
% 174.74/22.58 converse(join(X, top))
% 174.74/22.58 = { by axiom 6 (converse_additivity_9) }
% 174.74/22.58 join(converse(X), converse(top))
% 174.74/22.58 = { by lemma 37 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 39: join(zero, X) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(zero, X)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(X, zero)
% 174.74/22.58 = { by lemma 31 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 40: meet(Y, X) = meet(X, Y).
% 174.74/22.58 Proof:
% 174.74/22.58 meet(Y, X)
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 complement(join(complement(Y), complement(X)))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 complement(join(complement(X), complement(Y)))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.58 meet(X, Y)
% 174.74/22.58
% 174.74/22.58 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 174.74/22.58 Proof:
% 174.74/22.58 complement(join(zero, complement(X)))
% 174.74/22.58 = { by lemma 16 R->L }
% 174.74/22.58 complement(join(complement(top), complement(X)))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.58 meet(top, X)
% 174.74/22.58 = { by lemma 40 R->L }
% 174.74/22.58 meet(X, top)
% 174.74/22.58
% 174.74/22.58 Lemma 42: join(X, complement(zero)) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(X, complement(zero))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(complement(zero), X)
% 174.74/22.58 = { by lemma 27 R->L }
% 174.74/22.58 join(zero, join(complement(zero), complement(complement(X))))
% 174.74/22.58 = { by lemma 36 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 43: meet(X, zero) = zero.
% 174.74/22.58 Proof:
% 174.74/22.58 meet(X, zero)
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 complement(join(complement(X), complement(zero)))
% 174.74/22.58 = { by lemma 42 }
% 174.74/22.58 complement(top)
% 174.74/22.58 = { by lemma 16 }
% 174.74/22.58 zero
% 174.74/22.58
% 174.74/22.58 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 join(meet(X, Y), meet(X, complement(Y)))
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(meet(X, complement(Y)), meet(X, Y))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 174.74/22.58 = { by lemma 25 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 45: meet(X, top) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 meet(X, top)
% 174.74/22.58 = { by lemma 41 R->L }
% 174.74/22.58 complement(join(zero, complement(X)))
% 174.74/22.58 = { by lemma 30 R->L }
% 174.74/22.58 join(zero, complement(join(zero, complement(X))))
% 174.74/22.58 = { by lemma 41 }
% 174.74/22.58 join(zero, meet(X, top))
% 174.74/22.58 = { by lemma 42 R->L }
% 174.74/22.58 join(zero, meet(X, join(complement(zero), complement(zero))))
% 174.74/22.58 = { by lemma 22 }
% 174.74/22.58 join(zero, meet(X, complement(zero)))
% 174.74/22.58 = { by lemma 43 R->L }
% 174.74/22.58 join(meet(X, zero), meet(X, complement(zero)))
% 174.74/22.58 = { by lemma 44 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.58 Lemma 46: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 174.74/22.58 Proof:
% 174.74/22.58 join(meet(X, Y), meet(X, Y))
% 174.74/22.58 = { by lemma 40 }
% 174.74/22.58 join(meet(Y, X), meet(X, Y))
% 174.74/22.58 = { by lemma 40 }
% 174.74/22.58 join(meet(Y, X), meet(Y, X))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.58 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 174.74/22.58 = { by lemma 22 }
% 174.74/22.58 complement(join(complement(Y), complement(X)))
% 174.74/22.58 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.58 meet(Y, X)
% 174.74/22.58 = { by lemma 40 R->L }
% 174.74/22.58 meet(X, Y)
% 174.74/22.58
% 174.74/22.58 Lemma 47: converse(zero) = zero.
% 174.74/22.58 Proof:
% 174.74/22.58 converse(zero)
% 174.74/22.58 = { by lemma 39 R->L }
% 174.74/22.58 join(zero, converse(zero))
% 174.74/22.58 = { by lemma 34 R->L }
% 174.74/22.58 converse(join(converse(zero), zero))
% 174.74/22.58 = { by lemma 29 }
% 174.74/22.58 converse(join(converse(zero), converse(zero)))
% 174.74/22.58 = { by lemma 33 }
% 174.74/22.58 join(zero, converse(converse(zero)))
% 174.74/22.58 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.58 join(zero, zero)
% 174.74/22.58 = { by lemma 45 R->L }
% 174.74/22.58 join(zero, meet(zero, top))
% 174.74/22.58 = { by lemma 45 R->L }
% 174.74/22.58 join(meet(zero, top), meet(zero, top))
% 174.74/22.58 = { by lemma 46 }
% 174.74/22.58 meet(zero, top)
% 174.74/22.58 = { by lemma 45 }
% 174.74/22.58 zero
% 174.74/22.58
% 174.74/22.58 Lemma 48: join(top, X) = top.
% 174.74/22.58 Proof:
% 174.74/22.58 join(top, X)
% 174.74/22.58 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.58 join(X, top)
% 174.74/22.58 = { by lemma 24 R->L }
% 174.74/22.58 join(Y, top)
% 174.74/22.58 = { by lemma 32 }
% 174.74/22.58 top
% 174.74/22.58
% 174.74/22.58 Lemma 49: complement(complement(X)) = X.
% 174.74/22.58 Proof:
% 174.74/22.58 complement(complement(X))
% 174.74/22.58 = { by lemma 30 R->L }
% 174.74/22.58 join(zero, complement(complement(X)))
% 174.74/22.58 = { by lemma 28 }
% 174.74/22.58 X
% 174.74/22.58
% 174.74/22.59 Lemma 50: meet(zero, X) = zero.
% 174.74/22.59 Proof:
% 174.74/22.59 meet(zero, X)
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 meet(X, zero)
% 174.74/22.59 = { by lemma 43 }
% 174.74/22.59 zero
% 174.74/22.59
% 174.74/22.59 Lemma 51: composition(join(join(composition(converse(sk1), sk1), one), Y), X) = join(X, composition(Y, X)).
% 174.74/22.59 Proof:
% 174.74/22.59 composition(join(join(composition(converse(sk1), sk1), one), Y), X)
% 174.74/22.59 = { by axiom 12 (composition_distributivity_7) }
% 174.74/22.59 join(composition(join(composition(converse(sk1), sk1), one), X), composition(Y, X))
% 174.74/22.59 = { by lemma 20 }
% 174.74/22.59 join(X, composition(Y, X))
% 174.74/22.59
% 174.74/22.59 Lemma 52: composition(top, zero) = zero.
% 174.74/22.59 Proof:
% 174.74/22.59 composition(top, zero)
% 174.74/22.59 = { by lemma 38 R->L }
% 174.74/22.59 composition(converse(top), zero)
% 174.74/22.59 = { by lemma 39 R->L }
% 174.74/22.59 join(zero, composition(converse(top), zero))
% 174.74/22.59 = { by lemma 16 R->L }
% 174.74/22.59 join(complement(top), composition(converse(top), zero))
% 174.74/22.59 = { by lemma 16 R->L }
% 174.74/22.59 join(complement(top), composition(converse(top), complement(top)))
% 174.74/22.59 = { by lemma 48 R->L }
% 174.74/22.59 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 174.74/22.59 = { by lemma 38 R->L }
% 174.74/22.59 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 174.74/22.59 = { by lemma 51 R->L }
% 174.74/22.59 join(complement(top), composition(converse(top), complement(composition(join(join(composition(converse(sk1), sk1), one), converse(top)), top))))
% 174.74/22.59 = { by lemma 37 }
% 174.74/22.59 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 174.74/22.59 = { by lemma 21 }
% 174.74/22.59 complement(top)
% 174.74/22.59 = { by lemma 16 }
% 174.74/22.59 zero
% 174.74/22.59
% 174.74/22.59 Lemma 53: composition(zero, X) = zero.
% 174.74/22.59 Proof:
% 174.74/22.59 composition(zero, X)
% 174.74/22.59 = { by lemma 47 R->L }
% 174.74/22.59 composition(converse(zero), X)
% 174.74/22.59 = { by lemma 18 R->L }
% 174.74/22.59 converse(composition(converse(X), zero))
% 174.74/22.59 = { by lemma 39 R->L }
% 174.74/22.59 converse(join(zero, composition(converse(X), zero)))
% 174.74/22.59 = { by lemma 52 R->L }
% 174.74/22.59 converse(join(composition(top, zero), composition(converse(X), zero)))
% 174.74/22.59 = { by axiom 12 (composition_distributivity_7) R->L }
% 174.74/22.59 converse(composition(join(top, converse(X)), zero))
% 174.74/22.59 = { by lemma 48 }
% 174.74/22.59 converse(composition(top, zero))
% 174.74/22.59 = { by lemma 52 }
% 174.74/22.59 converse(zero)
% 174.74/22.59 = { by lemma 47 }
% 174.74/22.59 zero
% 174.74/22.59
% 174.74/22.59 Lemma 54: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 174.74/22.59 Proof:
% 174.74/22.59 meet(X, join(complement(Y), complement(Z)))
% 174.74/22.59 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.59 meet(X, join(complement(Z), complement(Y)))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 meet(join(complement(Z), complement(Y)), X)
% 174.74/22.59 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.59 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 174.74/22.59 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.59 complement(join(meet(Z, Y), complement(X)))
% 174.74/22.59 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.59 complement(join(complement(X), meet(Z, Y)))
% 174.74/22.59 = { by lemma 40 R->L }
% 174.74/22.59 complement(join(complement(X), meet(Y, Z)))
% 174.74/22.59
% 174.74/22.59 Lemma 55: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 174.74/22.59 Proof:
% 174.74/22.59 complement(join(X, complement(Y)))
% 174.74/22.59 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.59 complement(join(complement(Y), X))
% 174.74/22.59 = { by lemma 45 R->L }
% 174.74/22.59 complement(join(complement(Y), meet(X, top)))
% 174.74/22.59 = { by lemma 40 R->L }
% 174.74/22.59 complement(join(complement(Y), meet(top, X)))
% 174.74/22.59 = { by lemma 54 R->L }
% 174.74/22.59 meet(Y, join(complement(top), complement(X)))
% 174.74/22.59 = { by lemma 16 }
% 174.74/22.59 meet(Y, join(zero, complement(X)))
% 174.74/22.59 = { by lemma 30 }
% 174.74/22.59 meet(Y, complement(X))
% 174.74/22.59
% 174.74/22.59 Lemma 56: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 174.74/22.59 Proof:
% 174.74/22.59 complement(join(complement(X), Y))
% 174.74/22.59 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.59 complement(join(Y, complement(X)))
% 174.74/22.59 = { by lemma 55 }
% 174.74/22.59 meet(X, complement(Y))
% 174.74/22.59
% 174.74/22.59 Lemma 57: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 174.74/22.59 Proof:
% 174.74/22.59 join(complement(X), complement(Y))
% 174.74/22.59 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.59 join(complement(Y), complement(X))
% 174.74/22.59 = { by lemma 26 R->L }
% 174.74/22.59 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 174.74/22.59 = { by lemma 54 }
% 174.74/22.59 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 174.74/22.59 = { by lemma 30 }
% 174.74/22.59 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 174.74/22.59 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.59 complement(join(meet(Y, X), meet(Y, X)))
% 174.74/22.59 = { by lemma 46 }
% 174.74/22.59 complement(meet(Y, X))
% 174.74/22.59 = { by lemma 40 R->L }
% 174.74/22.59 complement(meet(X, Y))
% 174.74/22.59
% 174.74/22.59 Lemma 58: join(X, complement(meet(X, Y))) = top.
% 174.74/22.59 Proof:
% 174.74/22.59 join(X, complement(meet(X, Y)))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 join(X, complement(meet(Y, X)))
% 174.74/22.59 = { by lemma 57 R->L }
% 174.74/22.59 join(X, join(complement(Y), complement(X)))
% 174.74/22.59 = { by lemma 17 }
% 174.74/22.59 join(complement(Y), top)
% 174.74/22.59 = { by lemma 32 }
% 174.74/22.59 top
% 174.74/22.59
% 174.74/22.59 Lemma 59: meet(X, meet(Y, complement(X))) = zero.
% 174.74/22.59 Proof:
% 174.74/22.59 meet(X, meet(Y, complement(X)))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 meet(X, meet(complement(X), Y))
% 174.74/22.59 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.59 complement(join(complement(X), complement(meet(complement(X), Y))))
% 174.74/22.59 = { by lemma 58 }
% 174.74/22.59 complement(top)
% 174.74/22.59 = { by lemma 16 }
% 174.74/22.59 zero
% 174.74/22.59
% 174.74/22.59 Lemma 60: meet(join(composition(converse(sk1), sk1), one), composition(converse(complement(X)), X)) = zero.
% 174.74/22.59 Proof:
% 174.74/22.59 meet(join(composition(converse(sk1), sk1), one), composition(converse(complement(X)), X))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 meet(composition(converse(complement(X)), X), join(composition(converse(sk1), sk1), one))
% 174.74/22.59 = { by lemma 49 R->L }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(complement(join(composition(converse(sk1), sk1), one))))
% 174.74/22.59 = { by axiom 11 (goals_17) }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 174.74/22.59 = { by lemma 21 R->L }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), one))))))
% 174.74/22.59 = { by axiom 3 (composition_identity_6) }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 174.74/22.59 = { by axiom 11 (goals_17) R->L }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(sk1), sk1), one)), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 174.74/22.59 = { by lemma 41 }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(sk1), sk1), one)), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 174.74/22.59 = { by lemma 30 }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(sk1), sk1), one)), composition(converse(complement(X)), meet(X, top)))))
% 174.74/22.59 = { by lemma 45 }
% 174.74/22.59 meet(composition(converse(complement(X)), X), complement(join(complement(join(composition(converse(sk1), sk1), one)), composition(converse(complement(X)), X))))
% 174.74/22.59 = { by lemma 56 }
% 174.74/22.59 meet(composition(converse(complement(X)), X), meet(join(composition(converse(sk1), sk1), one), complement(composition(converse(complement(X)), X))))
% 174.74/22.59 = { by lemma 59 }
% 174.74/22.59 zero
% 174.74/22.59
% 174.74/22.59 Lemma 61: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 174.74/22.59 Proof:
% 174.74/22.59 join(meet(X, Y), meet(Y, complement(X)))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 join(meet(Y, X), meet(Y, complement(X)))
% 174.74/22.59 = { by lemma 44 }
% 174.74/22.59 Y
% 174.74/22.59
% 174.74/22.59 Lemma 62: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 174.74/22.59 Proof:
% 174.74/22.59 join(meet(X, Y), meet(complement(X), Y))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 join(meet(X, Y), meet(Y, complement(X)))
% 174.74/22.59 = { by lemma 61 }
% 174.74/22.59 Y
% 174.74/22.59
% 174.74/22.59 Lemma 63: converse(complement(X)) = complement(converse(X)).
% 174.74/22.59 Proof:
% 174.74/22.59 converse(complement(X))
% 174.74/22.59 = { by lemma 30 R->L }
% 174.74/22.59 converse(join(zero, complement(X)))
% 174.74/22.59 = { by lemma 25 R->L }
% 174.74/22.59 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)))))))))))
% 174.74/22.59 = { by lemma 56 R->L }
% 174.74/22.59 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)))))))))))
% 174.74/22.59 = { by lemma 35 }
% 174.74/22.59 converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 174.74/22.59 = { by lemma 38 }
% 174.74/22.59 converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 174.74/22.59 = { by lemma 16 }
% 174.74/22.59 converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
% 174.74/22.59 = { by lemma 30 }
% 174.74/22.59 converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
% 174.74/22.59 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.59 converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
% 174.74/22.59 = { by lemma 31 R->L }
% 174.74/22.59 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), zero))
% 174.74/22.59 = { by lemma 50 R->L }
% 174.74/22.59 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)))))))))
% 174.74/22.59 = { by lemma 53 R->L }
% 174.74/22.59 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)))))))))
% 174.74/22.59 = { by lemma 60 R->L }
% 174.74/22.59 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(join(composition(converse(sk1), sk1), 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)))))))))
% 174.74/22.59 = { by axiom 15 (modular_law_2_16) R->L }
% 174.74/22.59 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(join(composition(converse(sk1), sk1), one), complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(composition(meet(join(composition(converse(sk1), sk1), 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))))))))))
% 174.74/22.59 = { by lemma 60 }
% 174.74/22.59 converse(join(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(meet(composition(join(composition(converse(sk1), sk1), 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))))))))))
% 174.74/22.59 = { by lemma 20 }
% 174.74/22.59 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))))))))))
% 174.74/22.59 = { by lemma 53 }
% 174.74/22.59 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))))))))))
% 174.74/22.59 = { by lemma 50 }
% 174.74/22.59 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)))
% 174.74/22.59 = { by lemma 31 }
% 174.74/22.59 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)))))))))
% 174.74/22.59 = { by lemma 62 }
% 174.74/22.59 converse(converse(complement(converse(complement(join(zero, complement(X)))))))
% 174.74/22.59 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.59 complement(converse(complement(join(zero, complement(X)))))
% 174.74/22.59 = { by lemma 41 }
% 174.74/22.59 complement(converse(meet(X, top)))
% 174.74/22.59 = { by lemma 45 }
% 174.74/22.59 complement(converse(X))
% 174.74/22.59
% 174.74/22.59 Lemma 64: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 174.74/22.59 Proof:
% 174.74/22.59 complement(meet(X, complement(Y)))
% 174.74/22.59 = { by lemma 39 R->L }
% 174.74/22.59 complement(join(zero, meet(X, complement(Y))))
% 174.74/22.59 = { by lemma 55 R->L }
% 174.74/22.59 complement(join(zero, complement(join(Y, complement(X)))))
% 174.74/22.59 = { by lemma 41 }
% 174.74/22.59 meet(join(Y, complement(X)), top)
% 174.74/22.59 = { by lemma 45 }
% 174.74/22.59 join(Y, complement(X))
% 174.74/22.59
% 174.74/22.59 Lemma 65: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 174.74/22.59 Proof:
% 174.74/22.59 complement(meet(complement(X), Y))
% 174.74/22.59 = { by lemma 40 }
% 174.74/22.59 complement(meet(Y, complement(X)))
% 174.74/22.59 = { by lemma 64 }
% 174.74/22.59 join(X, complement(Y))
% 174.74/22.59
% 174.74/22.59 Lemma 66: meet(X, join(X, complement(Y))) = X.
% 174.74/22.60 Proof:
% 174.74/22.60 meet(X, join(X, complement(Y)))
% 174.74/22.60 = { by lemma 64 R->L }
% 174.74/22.60 meet(X, complement(meet(Y, complement(X))))
% 174.74/22.60 = { by lemma 56 R->L }
% 174.74/22.60 complement(join(complement(X), meet(Y, complement(X))))
% 174.74/22.60 = { by lemma 30 R->L }
% 174.74/22.60 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 174.74/22.60 = { by lemma 59 R->L }
% 174.74/22.60 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 174.74/22.60 = { by lemma 25 }
% 174.74/22.60 X
% 174.74/22.60
% 174.74/22.60 Lemma 67: join(X, meet(X, Y)) = X.
% 174.74/22.60 Proof:
% 174.74/22.60 join(X, meet(X, Y))
% 174.74/22.60 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.60 join(X, complement(join(complement(X), complement(Y))))
% 174.74/22.60 = { by lemma 65 R->L }
% 174.74/22.60 complement(meet(complement(X), join(complement(X), complement(Y))))
% 174.74/22.60 = { by lemma 66 }
% 174.74/22.60 complement(complement(X))
% 174.74/22.60 = { by lemma 49 }
% 174.74/22.60 X
% 174.74/22.60
% 174.74/22.60 Lemma 68: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(Y, join(X, Z))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 join(join(X, Z), Y)
% 174.74/22.60 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 174.74/22.60 join(X, join(Z, Y))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 join(X, join(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 69: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(Z, join(X, Y))
% 174.74/22.60 = { by lemma 68 }
% 174.74/22.60 join(X, join(Z, Y))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 join(X, join(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 70: join(meet(X, Y), X) = X.
% 174.74/22.60 Proof:
% 174.74/22.60 join(meet(X, Y), X)
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 join(X, meet(X, Y))
% 174.74/22.60 = { by lemma 67 }
% 174.74/22.60 X
% 174.74/22.60
% 174.74/22.60 Lemma 71: meet(X, join(X, Y)) = X.
% 174.74/22.60 Proof:
% 174.74/22.60 meet(X, join(X, Y))
% 174.74/22.60 = { by lemma 45 R->L }
% 174.74/22.60 meet(X, join(X, meet(Y, top)))
% 174.74/22.60 = { by lemma 41 R->L }
% 174.74/22.60 meet(X, join(X, complement(join(zero, complement(Y)))))
% 174.74/22.60 = { by lemma 66 }
% 174.74/22.60 X
% 174.74/22.60
% 174.74/22.60 Lemma 72: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(complement(X), complement(Y))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 meet(complement(Y), complement(X))
% 174.74/22.60 = { by lemma 30 R->L }
% 174.74/22.60 meet(join(zero, complement(Y)), complement(X))
% 174.74/22.60 = { by lemma 55 R->L }
% 174.74/22.60 complement(join(X, complement(join(zero, complement(Y)))))
% 174.74/22.60 = { by lemma 41 }
% 174.74/22.60 complement(join(X, meet(Y, top)))
% 174.74/22.60 = { by lemma 45 }
% 174.74/22.60 complement(join(X, Y))
% 174.74/22.60
% 174.74/22.60 Lemma 73: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(complement(Z), meet(Y, X))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 meet(complement(Z), meet(X, Y))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 meet(meet(X, Y), complement(Z))
% 174.74/22.60 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 174.74/22.60 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 174.74/22.60 = { by lemma 72 }
% 174.74/22.60 complement(join(join(complement(X), complement(Y)), Z))
% 174.74/22.60 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 174.74/22.60 complement(join(complement(X), join(complement(Y), Z)))
% 174.74/22.60 = { by lemma 56 }
% 174.74/22.60 meet(X, complement(join(complement(Y), Z)))
% 174.74/22.60 = { by lemma 56 }
% 174.74/22.60 meet(X, meet(Y, complement(Z)))
% 174.74/22.60
% 174.74/22.60 Lemma 74: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(Y, meet(Z, X))
% 174.74/22.60 = { by lemma 45 R->L }
% 174.74/22.60 meet(meet(Y, top), meet(Z, X))
% 174.74/22.60 = { by lemma 41 R->L }
% 174.74/22.60 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 174.74/22.60 = { by lemma 73 }
% 174.74/22.60 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 174.74/22.60 = { by lemma 41 }
% 174.74/22.60 meet(X, meet(Z, meet(Y, top)))
% 174.74/22.60 = { by lemma 45 }
% 174.74/22.60 meet(X, meet(Z, Y))
% 174.74/22.60 = { by lemma 40 R->L }
% 174.74/22.60 meet(X, meet(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 75: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(Y, meet(X, Z))
% 174.74/22.60 = { by lemma 74 R->L }
% 174.74/22.60 meet(X, meet(Z, Y))
% 174.74/22.60 = { by lemma 40 R->L }
% 174.74/22.60 meet(X, meet(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 76: converse(composition(Y, converse(X))) = composition(X, converse(Y)).
% 174.74/22.60 Proof:
% 174.74/22.60 converse(composition(Y, converse(X)))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) }
% 174.74/22.60 composition(converse(converse(X)), converse(Y))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.60 composition(X, converse(Y))
% 174.74/22.60
% 174.74/22.60 Lemma 77: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 174.74/22.60 Proof:
% 174.74/22.60 join(complement(X), meet(Y, complement(Z)))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 join(meet(Y, complement(Z)), complement(X))
% 174.74/22.60 = { by lemma 55 R->L }
% 174.74/22.60 join(complement(join(Z, complement(Y))), complement(X))
% 174.74/22.60 = { by lemma 57 }
% 174.74/22.60 complement(meet(join(Z, complement(Y)), X))
% 174.74/22.60 = { by lemma 40 R->L }
% 174.74/22.60 complement(meet(X, join(Z, complement(Y))))
% 174.74/22.60
% 174.74/22.60 Lemma 78: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 174.74/22.60 Proof:
% 174.74/22.60 complement(meet(Y, join(X, complement(Y))))
% 174.74/22.60 = { by lemma 77 R->L }
% 174.74/22.60 join(complement(Y), meet(Y, complement(X)))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 join(meet(Y, complement(X)), complement(Y))
% 174.74/22.60 = { by lemma 62 R->L }
% 174.74/22.60 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 174.74/22.60 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.60 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 174.74/22.60 = { by lemma 61 }
% 174.74/22.60 join(complement(X), meet(X, complement(Y)))
% 174.74/22.60 = { by lemma 77 }
% 174.74/22.60 complement(meet(X, join(Y, complement(X))))
% 174.74/22.60
% 174.74/22.60 Lemma 79: join(complement(Y), meet(Y, X)) = join(X, complement(join(X, Y))).
% 174.74/22.60 Proof:
% 174.74/22.60 join(complement(Y), meet(Y, X))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(complement(Y), meet(X, Y))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 join(meet(X, Y), complement(Y))
% 174.74/22.60 = { by lemma 64 R->L }
% 174.74/22.60 complement(meet(Y, complement(meet(X, Y))))
% 174.74/22.60 = { by lemma 57 R->L }
% 174.74/22.60 complement(meet(Y, join(complement(X), complement(Y))))
% 174.74/22.60 = { by lemma 78 R->L }
% 174.74/22.60 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 174.74/22.60 = { by lemma 65 }
% 174.74/22.60 join(X, complement(join(Y, complement(complement(X)))))
% 174.74/22.60 = { by lemma 55 }
% 174.74/22.60 join(X, meet(complement(X), complement(Y)))
% 174.74/22.60 = { by lemma 72 }
% 174.74/22.60 join(X, complement(join(X, Y)))
% 174.74/22.60
% 174.74/22.60 Lemma 80: join(X, complement(meet(Y, X))) = top.
% 174.74/22.60 Proof:
% 174.74/22.60 join(X, complement(meet(Y, X)))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(X, complement(meet(X, Y)))
% 174.74/22.60 = { by lemma 58 }
% 174.74/22.60 top
% 174.74/22.60
% 174.74/22.60 Lemma 81: join(X, complement(join(X, Y))) = join(X, complement(Y)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(X, complement(join(X, Y)))
% 174.74/22.60 = { by lemma 72 R->L }
% 174.74/22.60 join(X, meet(complement(X), complement(Y)))
% 174.74/22.60 = { by lemma 55 R->L }
% 174.74/22.60 join(X, complement(join(Y, complement(complement(X)))))
% 174.74/22.60 = { by lemma 65 R->L }
% 174.74/22.60 complement(meet(complement(X), join(Y, complement(complement(X)))))
% 174.74/22.60 = { by lemma 78 R->L }
% 174.74/22.60 complement(meet(Y, join(complement(X), complement(Y))))
% 174.74/22.60 = { by lemma 57 }
% 174.74/22.60 complement(meet(Y, complement(meet(X, Y))))
% 174.74/22.60 = { by lemma 64 }
% 174.74/22.60 join(meet(X, Y), complement(Y))
% 174.74/22.60 = { by lemma 62 R->L }
% 174.74/22.60 join(meet(X, Y), complement(join(meet(X, Y), meet(complement(X), Y))))
% 174.74/22.60 = { by lemma 79 R->L }
% 174.74/22.60 join(complement(meet(complement(X), Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 174.74/22.60 = { by lemma 65 }
% 174.74/22.60 join(join(X, complement(Y)), meet(meet(complement(X), Y), meet(X, Y)))
% 174.74/22.60 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 174.74/22.60 join(X, join(complement(Y), meet(meet(complement(X), Y), meet(X, Y))))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(X, join(complement(Y), meet(meet(X, Y), meet(complement(X), Y))))
% 174.74/22.60 = { by lemma 74 R->L }
% 174.74/22.60 join(X, join(complement(Y), meet(complement(X), meet(Y, meet(X, Y)))))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(X, join(complement(Y), meet(meet(Y, meet(X, Y)), complement(X))))
% 174.74/22.60 = { by lemma 77 }
% 174.74/22.60 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(X, Y)))))))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(X, complement(meet(Y, join(X, complement(meet(Y, meet(Y, X)))))))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), Y))))))
% 174.74/22.60 = { by lemma 25 R->L }
% 174.74/22.60 join(X, complement(meet(Y, join(X, complement(meet(meet(Y, X), join(meet(Y, X), complement(join(complement(Y), X)))))))))
% 174.74/22.60 = { by lemma 66 }
% 174.74/22.60 join(X, complement(meet(Y, join(X, complement(meet(Y, X))))))
% 174.74/22.60 = { by lemma 80 }
% 174.74/22.60 join(X, complement(meet(Y, top)))
% 174.74/22.60 = { by lemma 45 }
% 174.74/22.60 join(X, complement(Y))
% 174.74/22.60
% 174.74/22.60 Lemma 82: meet(X, join(Y, complement(X))) = meet(X, Y).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(X, join(Y, complement(X)))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 meet(X, join(complement(X), Y))
% 174.74/22.60 = { by lemma 49 R->L }
% 174.74/22.60 meet(X, complement(complement(join(complement(X), Y))))
% 174.74/22.60 = { by lemma 56 R->L }
% 174.74/22.60 complement(join(complement(X), complement(join(complement(X), Y))))
% 174.74/22.60 = { by lemma 81 }
% 174.74/22.60 complement(join(complement(X), complement(Y)))
% 174.74/22.60 = { by lemma 56 }
% 174.74/22.60 meet(X, complement(complement(Y)))
% 174.74/22.60 = { by lemma 49 }
% 174.74/22.60 meet(X, Y)
% 174.74/22.60
% 174.74/22.60 Lemma 83: meet(X, join(complement(X), Y)) = meet(X, Y).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(X, join(complement(X), Y))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 meet(X, join(Y, complement(X)))
% 174.74/22.60 = { by lemma 82 }
% 174.74/22.60 meet(X, Y)
% 174.74/22.60
% 174.74/22.60 Lemma 84: composition(complement(converse(X)), converse(Y)) = converse(composition(Y, complement(X))).
% 174.74/22.60 Proof:
% 174.74/22.60 composition(complement(converse(X)), converse(Y))
% 174.74/22.60 = { by lemma 63 R->L }
% 174.74/22.60 composition(converse(complement(X)), converse(Y))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 174.74/22.60 converse(composition(Y, complement(X)))
% 174.74/22.60
% 174.74/22.60 Lemma 85: converse(join(composition(X, Y), composition(X, converse(Z)))) = converse(composition(X, join(Y, converse(Z)))).
% 174.74/22.60 Proof:
% 174.74/22.60 converse(join(composition(X, Y), composition(X, converse(Z))))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 converse(join(composition(X, converse(Z)), composition(X, Y)))
% 174.74/22.60 = { by axiom 6 (converse_additivity_9) }
% 174.74/22.60 join(converse(composition(X, converse(Z))), converse(composition(X, Y)))
% 174.74/22.60 = { by lemma 76 }
% 174.74/22.60 join(composition(Z, converse(X)), converse(composition(X, Y)))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) }
% 174.74/22.60 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 174.74/22.60 = { by axiom 12 (composition_distributivity_7) R->L }
% 174.74/22.60 composition(join(converse(Y), Z), converse(X))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.60 composition(join(Z, converse(Y)), converse(X))
% 174.74/22.60 = { by lemma 33 R->L }
% 174.74/22.60 composition(converse(join(Y, converse(Z))), converse(X))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 174.74/22.60 converse(composition(X, join(Y, converse(Z))))
% 174.74/22.60
% 174.74/22.60 Lemma 86: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(composition(X, Y), composition(X, Z))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.60 join(composition(X, Y), composition(X, converse(converse(Z))))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.60 converse(converse(join(composition(X, Y), composition(X, converse(converse(Z))))))
% 174.74/22.60 = { by lemma 85 }
% 174.74/22.60 converse(converse(composition(X, join(Y, converse(converse(Z))))))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.60 composition(X, join(Y, converse(converse(Z))))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.60 composition(X, join(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 87: join(converse(composition(X, Y)), composition(Z, converse(X))) = composition(join(Z, converse(Y)), converse(X)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(converse(composition(X, Y)), composition(Z, converse(X)))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) }
% 174.74/22.60 join(composition(converse(Y), converse(X)), composition(Z, converse(X)))
% 174.74/22.60 = { by axiom 12 (composition_distributivity_7) R->L }
% 174.74/22.60 composition(join(converse(Y), Z), converse(X))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 composition(join(Z, converse(Y)), converse(X))
% 174.74/22.60
% 174.74/22.60 Lemma 88: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 174.74/22.60 Proof:
% 174.74/22.60 join(composition(X, Z), composition(X, Y))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.60 join(composition(X, Z), composition(converse(converse(X)), Y))
% 174.74/22.60 = { by lemma 18 R->L }
% 174.74/22.60 join(composition(X, Z), converse(composition(converse(Y), converse(X))))
% 174.74/22.60 = { by lemma 34 R->L }
% 174.74/22.60 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 174.74/22.60 = { by lemma 87 }
% 174.74/22.60 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 174.74/22.60 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.60 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 174.74/22.60 = { by lemma 76 }
% 174.74/22.60 composition(X, converse(join(converse(Z), converse(Y))))
% 174.74/22.60 = { by lemma 33 }
% 174.74/22.60 composition(X, join(Y, converse(converse(Z))))
% 174.74/22.60 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.60 composition(X, join(Y, Z))
% 174.74/22.60
% 174.74/22.60 Lemma 89: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(complement(X), join(Y, complement(Z)))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 meet(join(Y, complement(Z)), complement(X))
% 174.74/22.60 = { by lemma 55 R->L }
% 174.74/22.60 complement(join(X, complement(join(Y, complement(Z)))))
% 174.74/22.60 = { by lemma 55 }
% 174.74/22.60 complement(join(X, meet(Z, complement(Y))))
% 174.74/22.60
% 174.74/22.60 Lemma 90: meet(join(X, complement(Y)), complement(Z)) = complement(join(Z, meet(Y, complement(X)))).
% 174.74/22.60 Proof:
% 174.74/22.60 meet(join(X, complement(Y)), complement(Z))
% 174.74/22.60 = { by lemma 40 }
% 174.74/22.60 meet(complement(Z), join(X, complement(Y)))
% 174.74/22.60 = { by lemma 89 }
% 174.74/22.60 complement(join(Z, meet(Y, complement(X))))
% 174.74/22.60
% 174.74/22.60 Lemma 91: composition(converse(X), complement(composition(X, top))) = zero.
% 174.74/22.60 Proof:
% 174.74/22.60 composition(converse(X), complement(composition(X, top)))
% 174.74/22.60 = { by lemma 39 R->L }
% 174.74/22.60 join(zero, composition(converse(X), complement(composition(X, top))))
% 174.74/22.60 = { by lemma 16 R->L }
% 174.74/22.60 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 174.74/22.60 = { by lemma 21 }
% 174.74/22.60 complement(top)
% 174.74/22.60 = { by lemma 16 }
% 174.74/22.60 zero
% 174.74/22.60
% 174.74/22.60 Lemma 92: composition(converse(X), composition(converse(Y), Z)) = converse(composition(composition(converse(Z), Y), X)).
% 174.74/22.60 Proof:
% 174.74/22.60 composition(converse(X), composition(converse(Y), Z))
% 174.74/22.60 = { by lemma 18 R->L }
% 174.74/22.60 composition(converse(X), converse(composition(converse(Z), Y)))
% 174.74/22.60 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 174.74/22.60 converse(composition(composition(converse(Z), Y), X))
% 174.74/22.60
% 174.74/22.60 Lemma 93: join(X, composition(composition(converse(sk1), sk1), X)) = X.
% 174.74/22.60 Proof:
% 174.74/22.60 join(X, composition(composition(converse(sk1), sk1), X))
% 174.74/22.60 = { by lemma 51 R->L }
% 174.74/22.61 composition(join(join(composition(converse(sk1), sk1), one), composition(converse(sk1), sk1)), X)
% 174.74/22.61 = { by axiom 11 (goals_17) }
% 174.74/22.61 composition(join(one, composition(converse(sk1), sk1)), X)
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.61 composition(join(composition(converse(sk1), sk1), one), X)
% 174.74/22.61 = { by lemma 20 }
% 174.74/22.61 X
% 174.74/22.61
% 174.74/22.61 Lemma 94: join(join(meet(composition(X, Y), Z), composition(X, meet(Y, W))), composition(X, Y)) = composition(X, Y).
% 174.74/22.61 Proof:
% 174.74/22.61 join(join(meet(composition(X, Y), Z), composition(X, meet(Y, W))), composition(X, Y))
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.61 join(composition(X, Y), join(meet(composition(X, Y), Z), composition(X, meet(Y, W))))
% 174.74/22.61 = { by lemma 69 }
% 174.74/22.61 join(meet(composition(X, Y), Z), join(composition(X, meet(Y, W)), composition(X, Y)))
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.61 join(meet(composition(X, Y), Z), join(composition(X, Y), composition(X, meet(Y, W))))
% 174.74/22.61 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.61 join(join(meet(composition(X, Y), Z), composition(X, Y)), composition(X, meet(Y, W)))
% 174.74/22.61 = { by lemma 70 }
% 174.74/22.61 join(composition(X, Y), composition(X, meet(Y, W)))
% 174.74/22.61 = { by lemma 88 }
% 174.74/22.61 composition(X, join(meet(Y, W), Y))
% 174.74/22.61 = { by lemma 70 }
% 174.74/22.61 composition(X, Y)
% 174.74/22.61
% 174.74/22.61 Lemma 95: meet(composition(X, meet(Y, Z)), composition(X, Y)) = composition(X, meet(Y, Z)).
% 174.74/22.61 Proof:
% 174.74/22.61 meet(composition(X, meet(Y, Z)), composition(X, Y))
% 174.74/22.61 = { by lemma 94 R->L }
% 174.74/22.61 meet(composition(X, meet(Y, Z)), join(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))), composition(X, Y)))
% 174.74/22.61 = { by lemma 45 R->L }
% 174.74/22.61 meet(composition(X, meet(Y, Z)), join(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))), meet(composition(X, Y), top)))
% 174.74/22.61 = { by lemma 41 R->L }
% 174.74/22.61 meet(composition(X, meet(Y, Z)), join(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))), complement(join(zero, complement(composition(X, Y))))))
% 174.74/22.61 = { by lemma 64 R->L }
% 174.74/22.61 meet(composition(X, meet(Y, Z)), complement(meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))))))
% 174.74/22.61 = { by lemma 56 R->L }
% 174.74/22.61 complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))))))
% 174.74/22.61 = { by lemma 30 R->L }
% 174.74/22.61 join(zero, complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 43 R->L }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), zero), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 16 R->L }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(top)), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 48 R->L }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(join(top, meet(composition(X, Y), W)))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), top))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by axiom 4 (def_top_12) }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), join(composition(X, meet(Y, Z)), complement(composition(X, meet(Y, Z))))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 69 R->L }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(join(complement(composition(X, meet(Y, Z))), join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), complement(join(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))), complement(composition(X, meet(Y, Z)))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 55 }
% 174.74/22.61 join(meet(join(zero, complement(composition(X, Y))), meet(composition(X, meet(Y, Z)), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 73 R->L }
% 174.74/22.61 join(meet(complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))), meet(composition(X, meet(Y, Z)), join(zero, complement(composition(X, Y))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 74 R->L }
% 174.74/22.61 join(meet(composition(X, meet(Y, Z)), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z)))))), complement(join(complement(composition(X, meet(Y, Z))), meet(join(zero, complement(composition(X, Y))), complement(join(meet(composition(X, Y), W), composition(X, meet(Y, Z))))))))
% 174.74/22.61 = { by lemma 25 }
% 174.74/22.61 composition(X, meet(Y, Z))
% 174.74/22.61
% 174.74/22.61 Lemma 96: join(complement(composition(sk1, X)), join(Y, composition(sk1, complement(X)))) = join(Y, complement(composition(sk1, X))).
% 174.74/22.61 Proof:
% 174.74/22.61 join(complement(composition(sk1, X)), join(Y, composition(sk1, complement(X))))
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.61 join(complement(composition(sk1, X)), join(composition(sk1, complement(X)), Y))
% 174.74/22.61 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.61 join(join(complement(composition(sk1, X)), composition(sk1, complement(X))), Y)
% 174.74/22.61 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(converse(composition(sk1, complement(X))))), Y)
% 174.74/22.61 = { by lemma 84 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(complement(converse(X)), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 39 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(join(zero, composition(complement(converse(X)), converse(sk1))))), Y)
% 174.74/22.61 = { by lemma 47 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(join(converse(zero), composition(complement(converse(X)), converse(sk1))))), Y)
% 174.74/22.61 = { by lemma 91 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(join(converse(composition(converse(converse(sk1)), complement(composition(converse(sk1), top)))), composition(complement(converse(X)), converse(sk1))))), Y)
% 174.74/22.61 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(join(converse(composition(sk1, complement(composition(converse(sk1), top)))), composition(complement(converse(X)), converse(sk1))))), Y)
% 174.74/22.61 = { by lemma 87 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), converse(complement(composition(converse(sk1), top)))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 63 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(converse(composition(converse(sk1), top)))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 18 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(converse(top), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 38 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(top, sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by axiom 4 (def_top_12) }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(join(composition(top, converse(sk1)), complement(composition(top, converse(sk1)))), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 38 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(join(composition(top, converse(sk1)), complement(composition(converse(top), converse(sk1)))), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(join(composition(top, converse(sk1)), complement(converse(composition(sk1, top)))), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 63 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(join(composition(top, converse(sk1)), converse(complement(composition(sk1, top)))), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 33 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(converse(join(complement(composition(sk1, top)), converse(composition(top, converse(sk1))))), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 18 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(converse(composition(converse(sk1), join(complement(composition(sk1, top)), converse(composition(top, converse(sk1)))))))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 85 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(converse(join(composition(converse(sk1), complement(composition(sk1, top))), composition(converse(sk1), converse(composition(top, converse(sk1)))))))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 91 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(converse(join(zero, composition(converse(sk1), converse(composition(top, converse(sk1)))))))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 39 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(converse(composition(converse(sk1), converse(composition(top, converse(sk1))))))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 76 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(composition(top, converse(sk1)), converse(converse(sk1))))), converse(sk1)))), Y)
% 174.74/22.61 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(composition(top, converse(sk1)), sk1))), converse(sk1)))), Y)
% 174.74/22.61 = { by axiom 9 (composition_associativity_5) R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(converse(X)), complement(composition(top, composition(converse(sk1), sk1)))), converse(sk1)))), Y)
% 174.74/22.61 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(join(complement(composition(top, composition(converse(sk1), sk1))), complement(converse(X))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 57 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(top, composition(converse(sk1), sk1)), converse(X))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 40 }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), composition(top, composition(converse(sk1), sk1)))), converse(sk1)))), Y)
% 174.74/22.61 = { by lemma 83 R->L }
% 174.74/22.61 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), composition(top, composition(converse(sk1), sk1))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 38 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), composition(converse(top), composition(converse(sk1), sk1))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 92 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), converse(composition(composition(converse(sk1), sk1), top))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 34 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(converse(complement(converse(X))), composition(composition(converse(sk1), sk1), top))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 4 (def_top_12) }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(converse(complement(converse(X))), composition(composition(converse(sk1), sk1), join(converse(complement(converse(X))), complement(converse(complement(converse(X)))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(converse(complement(converse(X))), composition(composition(converse(sk1), sk1), join(complement(converse(complement(converse(X)))), converse(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(composition(composition(converse(sk1), sk1), join(complement(converse(complement(converse(X)))), converse(complement(converse(X))))), converse(complement(converse(X))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 86 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(join(composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))), composition(composition(converse(sk1), sk1), converse(complement(converse(X))))), converse(complement(converse(X))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))), join(composition(composition(converse(sk1), sk1), converse(complement(converse(X)))), converse(complement(converse(X)))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))), join(converse(complement(converse(X))), composition(composition(converse(sk1), sk1), converse(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 93 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))), converse(complement(converse(X))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), converse(join(converse(complement(converse(X))), composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 34 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), converse(composition(composition(converse(sk1), sk1), complement(converse(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 63 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), converse(composition(composition(converse(sk1), sk1), converse(complement(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 18 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), converse(composition(converse(composition(converse(sk1), sk1)), converse(complement(complement(converse(X))))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), converse(converse(composition(complement(complement(converse(X))), composition(converse(sk1), sk1))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), join(complement(converse(X)), composition(complement(complement(converse(X))), composition(converse(sk1), sk1))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 83 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), composition(complement(complement(converse(X))), composition(converse(sk1), sk1)))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 49 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(converse(X), composition(converse(X), composition(converse(sk1), sk1)))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 40 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), converse(X))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), converse(converse(converse(X))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 93 R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), converse(join(converse(converse(X)), composition(composition(converse(sk1), sk1), converse(converse(X))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 34 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), join(converse(X), converse(composition(composition(converse(sk1), sk1), converse(converse(X))))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 76 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), join(converse(X), composition(converse(X), converse(composition(converse(sk1), sk1)))))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 18 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), join(converse(X), composition(converse(X), composition(converse(sk1), sk1))))), converse(sk1)))), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(meet(composition(converse(X), composition(converse(sk1), sk1)), join(composition(converse(X), composition(converse(sk1), sk1)), converse(X)))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 71 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(composition(converse(X), composition(converse(sk1), sk1))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 92 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(composition(complement(converse(composition(composition(converse(sk1), sk1), X))), converse(sk1)))), Y)
% 174.74/22.62 = { by lemma 84 }
% 174.74/22.62 join(join(complement(composition(sk1, X)), converse(converse(composition(sk1, complement(composition(composition(converse(sk1), sk1), X)))))), Y)
% 174.74/22.62 = { by axiom 1 (converse_idempotence_8) }
% 174.74/22.62 join(join(complement(composition(sk1, X)), composition(sk1, complement(composition(composition(converse(sk1), sk1), X)))), Y)
% 174.74/22.62 = { by axiom 1 (converse_idempotence_8) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), composition(converse(converse(sk1)), complement(composition(composition(converse(sk1), sk1), X)))), Y)
% 174.74/22.62 = { by axiom 9 (composition_associativity_5) R->L }
% 174.74/22.62 join(join(complement(composition(sk1, X)), composition(converse(converse(sk1)), complement(composition(converse(sk1), composition(sk1, X))))), Y)
% 174.74/22.62 = { by lemma 21 }
% 174.74/22.62 join(complement(composition(sk1, X)), Y)
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.62 join(Y, complement(composition(sk1, X)))
% 174.74/22.62
% 174.74/22.62 Lemma 97: join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))) = meet(composition(X, Y), composition(X, Z)).
% 174.74/22.62 Proof:
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z)))
% 174.74/22.62 = { by lemma 95 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), composition(X, Y)))
% 174.74/22.62 = { by lemma 40 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), composition(X, meet(Y, Z))))
% 174.74/22.62 = { by lemma 71 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), join(composition(X, meet(Y, Z)), composition(X, Z)))))
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), join(composition(X, Z), composition(X, meet(Y, Z))))))
% 174.74/22.62 = { by lemma 88 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, join(meet(Y, Z), Z)))))
% 174.74/22.62 = { by lemma 45 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, meet(join(meet(Y, Z), Z), top)))))
% 174.74/22.62 = { by lemma 41 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(join(meet(Y, Z), Z))))))))
% 174.74/22.62 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(join(Z, meet(Y, Z)))))))))
% 174.74/22.62 = { by lemma 72 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, meet(complement(Z), complement(meet(Y, Z)))))))))
% 174.74/22.62 = { by lemma 39 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(zero, meet(complement(Z), complement(meet(Y, Z))))))))))
% 174.74/22.62 = { by lemma 16 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(complement(top), meet(complement(Z), complement(meet(Y, Z))))))))))
% 174.74/22.62 = { by lemma 80 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(complement(join(Z, complement(meet(Y, Z)))), meet(complement(Z), complement(meet(Y, Z))))))))))
% 174.74/22.62 = { by lemma 55 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, join(meet(meet(Y, Z), complement(Z)), meet(complement(Z), complement(meet(Y, Z))))))))))
% 174.74/22.62 = { by lemma 61 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, complement(join(zero, complement(Z)))))))
% 174.74/22.62 = { by lemma 41 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, meet(Z, top)))))
% 174.74/22.62 = { by lemma 45 }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), meet(composition(X, meet(Y, Z)), composition(X, Z))))
% 174.74/22.62 = { by lemma 75 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, meet(Y, Z)), meet(composition(X, Y), composition(X, Z))))
% 174.74/22.62 = { by lemma 40 R->L }
% 174.74/22.62 join(meet(composition(X, Y), composition(X, Z)), meet(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))))
% 174.74/22.62 = { by lemma 67 }
% 174.74/22.62 meet(composition(X, Y), composition(X, Z))
% 174.74/22.62
% 174.74/22.62 Lemma 98: join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), meet(composition(X, Y), complement(composition(X, Z)))) = composition(X, Y).
% 174.74/22.62 Proof:
% 174.74/22.62 join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), meet(composition(X, Y), complement(composition(X, Z))))
% 174.74/22.62 = { by lemma 39 R->L }
% 174.74/22.62 join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(zero, meet(composition(X, Y), complement(composition(X, Z)))))
% 174.74/22.62 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.62 join(join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), zero), meet(composition(X, Y), complement(composition(X, Z))))
% 174.74/22.62 = { by lemma 29 }
% 174.74/22.63 join(join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z)))), meet(composition(X, Y), complement(composition(X, Z))))
% 174.74/22.63 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 174.74/22.63 join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), meet(composition(X, Y), complement(composition(X, Z)))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(meet(composition(X, Y), complement(composition(X, Z))), join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z)))))
% 174.74/22.63 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.63 join(join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), meet(composition(X, Y), complement(composition(X, Z)))), join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))))
% 174.74/22.63 = { by lemma 69 }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(composition(X, meet(Y, Z)), join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), meet(composition(X, Y), complement(composition(X, Z))))))
% 174.74/22.63 = { by lemma 68 R->L }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(composition(X, meet(Y, Z)), meet(composition(X, Y), complement(composition(X, Z))))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, meet(Y, Z)))))
% 174.74/22.63 = { by lemma 97 }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(meet(composition(X, Y), composition(X, Z)), join(meet(composition(X, Y), complement(composition(X, Z))), composition(X, meet(Y, Z)))))
% 174.74/22.63 = { by axiom 7 (maddux2_join_associativity_2) }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(join(meet(composition(X, Y), composition(X, Z)), meet(composition(X, Y), complement(composition(X, Z)))), composition(X, meet(Y, Z))))
% 174.74/22.63 = { by lemma 44 }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(composition(X, Y), composition(X, meet(Y, Z))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 join(meet(composition(X, Y), composition(X, Z)), join(composition(X, meet(Y, Z)), composition(X, Y)))
% 174.74/22.63 = { by lemma 69 R->L }
% 174.74/22.63 join(composition(X, Y), join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 join(join(meet(composition(X, Y), composition(X, Z)), composition(X, meet(Y, Z))), composition(X, Y))
% 174.74/22.63 = { by lemma 94 }
% 174.74/22.63 composition(X, Y)
% 174.74/22.63
% 174.74/22.63 Goal 1 (goals_18): tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))) = tuple(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))).
% 174.74/22.63 Proof:
% 174.74/22.63 tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 tuple(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 71 R->L }
% 174.74/22.63 tuple(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), composition(sk1, sk2))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 94 }
% 174.74/22.63 tuple(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), composition(sk1, sk2)), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 40 }
% 174.74/22.63 tuple(meet(composition(sk1, sk2), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 82 R->L }
% 174.74/22.63 tuple(meet(composition(sk1, sk2), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 64 R->L }
% 174.74/22.63 tuple(meet(composition(sk1, sk2), complement(meet(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 56 R->L }
% 174.74/22.63 tuple(complement(join(complement(composition(sk1, sk2)), meet(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 89 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 67 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(composition(sk1, sk2), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 90 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 97 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 71 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(composition(sk1, sk2), meet(composition(sk1, sk3), join(composition(sk1, sk3), complement(composition(sk1, sk2))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 74 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), meet(composition(sk1, sk2), composition(sk1, sk3))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 40 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), join(composition(sk1, sk3), complement(composition(sk1, sk2)))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 97 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(complement(composition(sk1, sk2)), composition(sk1, sk3))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), join(composition(sk1, sk3), complement(composition(sk1, sk2)))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 40 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 64 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 97 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, sk2), composition(sk1, sk3))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 25 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), complement(composition(sk1, sk2))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, sk2)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 55 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(composition(sk1, sk2), complement(meet(composition(sk1, sk2), composition(sk1, sk3))))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, sk2)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 58 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(top), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, sk2)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 16 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, sk2)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 30 }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(join(complement(meet(composition(sk1, sk2), composition(sk1, sk3))), complement(composition(sk1, sk2))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, sk2))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 74 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(meet(composition(sk1, sk2), composition(sk1, sk3)), meet(composition(sk1, sk2), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 97 R->L }
% 174.74/22.63 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.63 = { by lemma 40 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk2))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 98 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 40 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 55 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 72 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 31 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), zero), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by axiom 5 (def_zero_13) }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 55 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 75 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 40 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 89 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(join(meet(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 62 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(complement(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 72 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 55 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 40 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 64 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), join(join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 98 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), composition(sk1, sk2)), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 40 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(meet(composition(sk1, sk2), join(composition(sk1, sk3), complement(composition(sk1, sk2)))), meet(join(composition(sk1, sk3), complement(composition(sk1, sk2))), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 61 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(composition(sk1, sk3), complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 96 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, complement(sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 86 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, join(sk3, complement(sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 61 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, join(sk3, complement(sk2))), meet(join(sk3, complement(sk2)), complement(sk2)))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 82 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, sk3), meet(join(sk3, complement(sk2)), complement(sk2)))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 90 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, sk3), complement(join(sk2, meet(sk2, complement(sk3)))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 67 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, join(meet(sk2, sk3), complement(sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 86 R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), join(composition(sk1, meet(sk2, sk3)), composition(sk1, complement(sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 96 }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(composition(sk1, meet(sk2, sk3)), complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.64 tuple(meet(complement(complement(composition(sk1, sk2))), join(complement(composition(sk1, sk2)), composition(sk1, meet(sk2, sk3)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 40 }
% 174.74/22.64 tuple(meet(join(complement(composition(sk1, sk2)), composition(sk1, meet(sk2, sk3))), complement(complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 55 R->L }
% 174.74/22.64 tuple(complement(join(complement(composition(sk1, sk2)), complement(join(complement(composition(sk1, sk2)), composition(sk1, meet(sk2, sk3)))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.64 = { by lemma 79 R->L }
% 174.74/22.64 tuple(complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(composition(sk1, meet(sk2, sk3)), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 56 }
% 174.74/22.65 tuple(meet(composition(sk1, meet(sk2, sk3)), complement(meet(composition(sk1, meet(sk2, sk3)), complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 57 R->L }
% 174.74/22.65 tuple(meet(composition(sk1, meet(sk2, sk3)), join(complement(composition(sk1, meet(sk2, sk3))), complement(complement(composition(sk1, sk2))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 64 R->L }
% 174.74/22.65 tuple(meet(composition(sk1, meet(sk2, sk3)), complement(meet(complement(composition(sk1, sk2)), complement(complement(composition(sk1, meet(sk2, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 56 R->L }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(complement(composition(sk1, sk2)), complement(complement(composition(sk1, meet(sk2, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 49 R->L }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, meet(sk2, sk3))), meet(complement(composition(sk1, sk2)), complement(complement(complement(complement(composition(sk1, meet(sk2, sk3))))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 56 R->L }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, meet(sk2, sk3))), complement(join(complement(complement(composition(sk1, sk2))), complement(complement(complement(composition(sk1, meet(sk2, sk3))))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 65 R->L }
% 174.74/22.65 tuple(complement(complement(meet(complement(complement(composition(sk1, meet(sk2, sk3)))), join(complement(complement(composition(sk1, sk2))), complement(complement(complement(composition(sk1, meet(sk2, sk3))))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 78 }
% 174.74/22.65 tuple(complement(complement(meet(complement(complement(composition(sk1, sk2))), join(complement(complement(composition(sk1, meet(sk2, sk3)))), complement(complement(complement(composition(sk1, sk2)))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 174.74/22.65 tuple(complement(complement(meet(complement(complement(composition(sk1, sk2))), join(complement(complement(complement(composition(sk1, sk2)))), complement(complement(composition(sk1, meet(sk2, sk3)))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 89 }
% 174.74/22.65 tuple(complement(complement(complement(join(complement(composition(sk1, sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(complement(complement(complement(composition(sk1, sk2)))))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 49 }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(complement(complement(complement(composition(sk1, sk2)))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 49 }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, sk2)), meet(complement(composition(sk1, meet(sk2, sk3))), complement(complement(composition(sk1, sk2)))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 55 R->L }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, sk2)), complement(join(complement(composition(sk1, sk2)), complement(complement(composition(sk1, meet(sk2, sk3)))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 81 }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, sk2)), complement(complement(complement(composition(sk1, meet(sk2, sk3))))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 49 }
% 174.74/22.65 tuple(complement(join(complement(composition(sk1, sk2)), complement(composition(sk1, meet(sk2, sk3))))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 55 }
% 174.74/22.65 tuple(meet(composition(sk1, meet(sk2, sk3)), complement(complement(composition(sk1, sk2)))), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 49 }
% 174.74/22.65 tuple(meet(composition(sk1, meet(sk2, sk3)), composition(sk1, sk2)), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 95 }
% 174.74/22.65 tuple(composition(sk1, meet(sk2, sk3)), join(meet(composition(sk1, sk2), composition(sk1, sk3)), composition(sk1, meet(sk2, sk3))))
% 174.74/22.65 = { by lemma 97 }
% 174.74/22.65 tuple(composition(sk1, meet(sk2, sk3)), meet(composition(sk1, sk2), composition(sk1, sk3)))
% 174.74/22.65 % SZS output end Proof
% 174.74/22.65
% 174.74/22.65 RESULT: Unsatisfiable (the axioms are contradictory).
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