TSTP Solution File: REL033-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL033-4 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n020.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 13:44:19 EDT 2023
% Result : Unsatisfiable 235.12s 30.71s
% Output : Proof 236.43s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL033-4 : TPTP v8.1.2. Released v4.0.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.14/0.34 % Computer : n020.cluster.edu
% 0.14/0.34 % Model : x86_64 x86_64
% 0.14/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.34 % Memory : 8042.1875MB
% 0.14/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.34 % CPULimit : 300
% 0.14/0.34 % WCLimit : 300
% 0.14/0.34 % DateTime : Fri Aug 25 18:59:45 EDT 2023
% 0.14/0.35 % CPUTime :
% 235.12/30.71 Command-line arguments: --flatten
% 235.12/30.71
% 235.12/30.71 % SZS status Unsatisfiable
% 235.12/30.71
% 235.81/30.86 % SZS output start Proof
% 235.81/30.86 Take the following subset of the input axioms:
% 235.81/30.87 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 235.81/30.87 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 235.81/30.87 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 235.81/30.87 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 235.81/30.87 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 235.81/30.87 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 235.81/30.87 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 235.81/30.87 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 235.81/30.87 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 235.81/30.87 fof(goals_17, negated_conjecture, composition(sk1, top)=sk1).
% 235.81/30.87 fof(goals_18, negated_conjecture, join(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3)))!=meet(sk1, composition(sk2, sk3)) | join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3))!=composition(meet(sk1, sk2), sk3)).
% 235.81/30.87 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 235.81/30.87 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 235.81/30.87 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 235.81/30.87 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 235.81/30.87 fof(modular_law_1_15, axiom, ![A2, B2, C2]: join(meet(composition(A2, B2), C2), meet(composition(A2, meet(B2, composition(converse(A2), C2))), C2))=meet(composition(A2, meet(B2, composition(converse(A2), C2))), C2)).
% 235.81/30.87 fof(modular_law_2_16, axiom, ![A2, B2, C2]: join(meet(composition(A2, B2), C2), meet(composition(meet(A2, composition(C2, converse(B2))), B2), C2))=meet(composition(meet(A2, composition(C2, converse(B2))), B2), C2)).
% 235.81/30.87
% 235.81/30.87 Now clausify the problem and encode Horn clauses using encoding 3 of
% 235.81/30.87 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 235.81/30.87 We repeatedly replace C & s=t => u=v by the two clauses:
% 235.81/30.87 fresh(y, y, x1...xn) = u
% 235.81/30.87 C => fresh(s, t, x1...xn) = v
% 235.81/30.87 where fresh is a fresh function symbol and x1..xn are the free
% 235.81/30.87 variables of u and v.
% 235.81/30.87 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 235.81/30.87 input problem has no model of domain size 1).
% 235.81/30.87
% 235.81/30.87 The encoding turns the above axioms into the following unit equations and goals:
% 235.81/30.87
% 235.81/30.87 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 235.81/30.87 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 235.81/30.87 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 235.81/30.87 Axiom 4 (goals_17): composition(sk1, top) = sk1.
% 235.81/30.87 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 235.81/30.87 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 235.81/30.87 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 235.81/30.87 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 235.81/30.87 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 235.81/30.87 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 235.81/30.87 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 235.81/30.87 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 235.81/30.87 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 235.81/30.87 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 235.81/30.87 Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 235.81/30.87 Axiom 16 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 235.81/30.87
% 235.81/30.87 Lemma 17: complement(top) = zero.
% 235.81/30.87 Proof:
% 235.81/30.87 complement(top)
% 235.81/30.87 = { by axiom 5 (def_top_12) }
% 235.81/30.87 complement(join(complement(X), complement(complement(X))))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 235.81/30.87 meet(X, complement(X))
% 235.81/30.87 = { by axiom 6 (def_zero_13) R->L }
% 235.81/30.87 zero
% 235.81/30.87
% 235.81/30.87 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, join(Y, complement(X)))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(X, join(complement(X), Y))
% 235.81/30.87 = { by axiom 8 (maddux2_join_associativity_2) }
% 235.81/30.87 join(join(X, complement(X)), Y)
% 235.81/30.87 = { by axiom 5 (def_top_12) R->L }
% 235.81/30.87 join(top, Y)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) }
% 235.81/30.87 join(Y, top)
% 235.81/30.87
% 235.81/30.87 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 235.81/30.87 Proof:
% 235.81/30.87 converse(composition(converse(X), Y))
% 235.81/30.87 = { by axiom 9 (converse_multiplicativity_10) }
% 235.81/30.87 composition(converse(Y), converse(converse(X)))
% 235.81/30.87 = { by axiom 1 (converse_idempotence_8) }
% 235.81/30.87 composition(converse(Y), X)
% 235.81/30.87
% 235.81/30.87 Lemma 20: composition(converse(one), X) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 composition(converse(one), X)
% 235.81/30.87 = { by lemma 19 R->L }
% 235.81/30.87 converse(composition(converse(X), one))
% 235.81/30.87 = { by axiom 3 (composition_identity_6) }
% 235.81/30.87 converse(converse(X))
% 235.81/30.87 = { by axiom 1 (converse_idempotence_8) }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 21: composition(one, X) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 composition(one, X)
% 235.81/30.87 = { by lemma 20 R->L }
% 235.81/30.87 composition(converse(one), composition(one, X))
% 235.81/30.87 = { by axiom 10 (composition_associativity_5) }
% 235.81/30.87 composition(composition(converse(one), one), X)
% 235.81/30.87 = { by axiom 3 (composition_identity_6) }
% 235.81/30.87 composition(converse(one), X)
% 235.81/30.87 = { by lemma 20 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 22: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 235.81/30.87 Proof:
% 235.81/30.87 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 235.81/30.87 = { by axiom 13 (converse_cancellativity_11) }
% 235.81/30.87 complement(X)
% 235.81/30.87
% 235.81/30.87 Lemma 23: join(complement(X), complement(X)) = complement(X).
% 235.81/30.87 Proof:
% 235.81/30.87 join(complement(X), complement(X))
% 235.81/30.87 = { by lemma 20 R->L }
% 235.81/30.87 join(complement(X), composition(converse(one), complement(X)))
% 235.81/30.87 = { by lemma 21 R->L }
% 235.81/30.87 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 235.81/30.87 = { by lemma 22 }
% 235.81/30.87 complement(X)
% 235.81/30.87
% 235.81/30.87 Lemma 24: join(top, complement(X)) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 join(top, complement(X))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(complement(X), top)
% 235.81/30.87 = { by lemma 18 R->L }
% 235.81/30.87 join(X, join(complement(X), complement(X)))
% 235.81/30.87 = { by lemma 23 }
% 235.81/30.87 join(X, complement(X))
% 235.81/30.87 = { by axiom 5 (def_top_12) R->L }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 25: join(Y, top) = join(X, top).
% 235.81/30.87 Proof:
% 235.81/30.87 join(Y, top)
% 235.81/30.87 = { by lemma 24 R->L }
% 235.81/30.87 join(Y, join(top, complement(Y)))
% 235.81/30.87 = { by lemma 18 }
% 235.81/30.87 join(top, top)
% 235.81/30.87 = { by lemma 18 R->L }
% 235.81/30.87 join(X, join(top, complement(X)))
% 235.81/30.87 = { by lemma 24 }
% 235.81/30.87 join(X, top)
% 235.81/30.87
% 235.81/30.87 Lemma 26: join(X, top) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, top)
% 235.81/30.87 = { by lemma 25 }
% 235.81/30.87 join(zero, top)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(top, zero)
% 235.81/30.87 = { by lemma 17 R->L }
% 235.81/30.87 join(top, complement(top))
% 235.81/30.87 = { by axiom 5 (def_top_12) R->L }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 27: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 235.81/30.87 Proof:
% 235.81/30.87 converse(join(X, converse(Y)))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 converse(join(converse(Y), X))
% 235.81/30.87 = { by axiom 7 (converse_additivity_9) }
% 235.81/30.87 join(converse(converse(Y)), converse(X))
% 235.81/30.87 = { by axiom 1 (converse_idempotence_8) }
% 235.81/30.87 join(Y, converse(X))
% 235.81/30.87
% 235.81/30.87 Lemma 28: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 235.81/30.87 Proof:
% 235.81/30.87 converse(join(converse(X), Y))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 converse(join(Y, converse(X)))
% 235.81/30.87 = { by lemma 27 }
% 235.81/30.87 join(X, converse(Y))
% 235.81/30.87
% 235.81/30.87 Lemma 29: join(X, join(complement(X), Y)) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, join(complement(X), Y))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(X, join(Y, complement(X)))
% 235.81/30.87 = { by lemma 18 }
% 235.81/30.87 join(Y, top)
% 235.81/30.87 = { by lemma 25 R->L }
% 235.81/30.87 join(Z, top)
% 235.81/30.87 = { by lemma 26 }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 30: join(X, converse(top)) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, converse(top))
% 235.81/30.87 = { by axiom 5 (def_top_12) }
% 235.81/30.87 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 235.81/30.87 = { by lemma 28 }
% 235.81/30.87 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 235.81/30.87 = { by lemma 29 }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 31: converse(top) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 converse(top)
% 235.81/30.87 = { by lemma 26 R->L }
% 235.81/30.87 converse(join(X, top))
% 235.81/30.87 = { by axiom 7 (converse_additivity_9) }
% 235.81/30.87 join(converse(X), converse(top))
% 235.81/30.87 = { by lemma 30 }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 32: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(meet(X, Y), complement(join(complement(X), Y)))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.87 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 235.81/30.87 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 33: join(zero, meet(X, X)) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(zero, meet(X, X))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.87 join(zero, complement(join(complement(X), complement(X))))
% 235.81/30.87 = { by axiom 6 (def_zero_13) }
% 235.81/30.87 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 235.81/30.87 = { by lemma 32 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 34: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 235.81/30.87 Proof:
% 235.81/30.87 join(zero, join(X, complement(complement(Y))))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(zero, join(complement(complement(Y)), X))
% 235.81/30.87 = { by lemma 23 R->L }
% 235.81/30.87 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 235.81/30.87 join(zero, join(meet(Y, Y), X))
% 235.81/30.87 = { by axiom 8 (maddux2_join_associativity_2) }
% 235.81/30.87 join(join(zero, meet(Y, Y)), X)
% 235.81/30.87 = { by lemma 33 }
% 235.81/30.87 join(Y, X)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) }
% 235.81/30.87 join(X, Y)
% 235.81/30.87
% 235.81/30.87 Lemma 35: join(zero, complement(complement(X))) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(zero, complement(complement(X)))
% 235.81/30.87 = { by axiom 6 (def_zero_13) }
% 235.81/30.87 join(meet(X, complement(X)), complement(complement(X)))
% 235.81/30.87 = { by lemma 23 R->L }
% 235.81/30.87 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 235.81/30.87 = { by lemma 32 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 36: join(X, zero) = join(X, X).
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, zero)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(zero, X)
% 235.81/30.87 = { by lemma 35 R->L }
% 235.81/30.87 join(zero, join(zero, complement(complement(X))))
% 235.81/30.87 = { by lemma 23 R->L }
% 235.81/30.87 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 235.81/30.87 = { by lemma 34 }
% 235.81/30.87 join(zero, join(complement(complement(X)), X))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) }
% 235.81/30.87 join(zero, join(X, complement(complement(X))))
% 235.81/30.87 = { by lemma 34 }
% 235.81/30.87 join(X, X)
% 235.81/30.87
% 235.81/30.87 Lemma 37: join(zero, complement(X)) = complement(X).
% 235.81/30.87 Proof:
% 235.81/30.87 join(zero, complement(X))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(complement(X), zero)
% 235.81/30.87 = { by lemma 36 }
% 235.81/30.87 join(complement(X), complement(X))
% 235.81/30.87 = { by lemma 23 }
% 235.81/30.87 complement(X)
% 235.81/30.87
% 235.81/30.87 Lemma 38: join(X, zero) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, zero)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(zero, X)
% 235.81/30.87 = { by lemma 34 R->L }
% 235.81/30.87 join(zero, join(zero, complement(complement(X))))
% 235.81/30.87 = { by lemma 37 }
% 235.81/30.87 join(zero, complement(complement(X)))
% 235.81/30.87 = { by lemma 35 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 39: join(zero, X) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(zero, X)
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(X, zero)
% 235.81/30.87 = { by lemma 38 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 40: meet(Y, X) = meet(X, Y).
% 235.81/30.87 Proof:
% 235.81/30.87 meet(Y, X)
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.87 complement(join(complement(Y), complement(X)))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 complement(join(complement(X), complement(Y)))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 235.81/30.87 meet(X, Y)
% 235.81/30.87
% 235.81/30.87 Lemma 41: complement(join(zero, complement(X))) = meet(X, top).
% 235.81/30.87 Proof:
% 235.81/30.87 complement(join(zero, complement(X)))
% 235.81/30.87 = { by lemma 17 R->L }
% 235.81/30.87 complement(join(complement(top), complement(X)))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 235.81/30.87 meet(top, X)
% 235.81/30.87 = { by lemma 40 R->L }
% 235.81/30.87 meet(X, top)
% 235.81/30.87
% 235.81/30.87 Lemma 42: join(X, complement(zero)) = top.
% 235.81/30.87 Proof:
% 235.81/30.87 join(X, complement(zero))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(complement(zero), X)
% 235.81/30.87 = { by lemma 34 R->L }
% 235.81/30.87 join(zero, join(complement(zero), complement(complement(X))))
% 235.81/30.87 = { by lemma 29 }
% 235.81/30.87 top
% 235.81/30.87
% 235.81/30.87 Lemma 43: meet(X, zero) = zero.
% 235.81/30.87 Proof:
% 235.81/30.87 meet(X, zero)
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.87 complement(join(complement(X), complement(zero)))
% 235.81/30.87 = { by lemma 42 }
% 235.81/30.87 complement(top)
% 235.81/30.87 = { by lemma 17 }
% 235.81/30.87 zero
% 235.81/30.87
% 235.81/30.87 Lemma 44: join(meet(X, Y), meet(X, complement(Y))) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 join(meet(X, Y), meet(X, complement(Y)))
% 235.81/30.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.87 join(meet(X, complement(Y)), meet(X, Y))
% 235.81/30.87 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.87 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 235.81/30.87 = { by lemma 32 }
% 235.81/30.87 X
% 235.81/30.87
% 235.81/30.87 Lemma 45: meet(X, top) = X.
% 235.81/30.87 Proof:
% 235.81/30.87 meet(X, top)
% 235.81/30.87 = { by lemma 41 R->L }
% 235.81/30.87 complement(join(zero, complement(X)))
% 235.81/30.88 = { by lemma 37 R->L }
% 235.81/30.88 join(zero, complement(join(zero, complement(X))))
% 235.81/30.88 = { by lemma 41 }
% 235.81/30.88 join(zero, meet(X, top))
% 235.81/30.88 = { by lemma 42 R->L }
% 235.81/30.88 join(zero, meet(X, join(complement(zero), complement(zero))))
% 235.81/30.88 = { by lemma 23 }
% 235.81/30.88 join(zero, meet(X, complement(zero)))
% 235.81/30.88 = { by lemma 43 R->L }
% 235.81/30.88 join(meet(X, zero), meet(X, complement(zero)))
% 235.81/30.88 = { by lemma 44 }
% 235.81/30.88 X
% 235.81/30.88
% 235.81/30.88 Lemma 46: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 235.81/30.88 Proof:
% 235.81/30.88 join(meet(X, Y), meet(X, Y))
% 235.81/30.88 = { by lemma 40 }
% 235.81/30.88 join(meet(Y, X), meet(X, Y))
% 235.81/30.88 = { by lemma 40 }
% 235.81/30.88 join(meet(Y, X), meet(Y, X))
% 235.81/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.88 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 235.81/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 235.81/30.88 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 235.81/30.88 = { by lemma 23 }
% 235.81/30.88 complement(join(complement(Y), complement(X)))
% 235.81/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 235.81/30.88 meet(Y, X)
% 235.81/30.88 = { by lemma 40 R->L }
% 235.81/30.88 meet(X, Y)
% 235.81/30.88
% 235.81/30.88 Lemma 47: converse(zero) = zero.
% 235.81/30.88 Proof:
% 235.81/30.88 converse(zero)
% 235.81/30.88 = { by lemma 39 R->L }
% 235.81/30.88 join(zero, converse(zero))
% 235.81/30.88 = { by lemma 28 R->L }
% 235.81/30.88 converse(join(converse(zero), zero))
% 235.81/30.88 = { by lemma 36 }
% 235.81/30.88 converse(join(converse(zero), converse(zero)))
% 235.81/30.88 = { by lemma 27 }
% 235.81/30.88 join(zero, converse(converse(zero)))
% 235.81/30.88 = { by axiom 1 (converse_idempotence_8) }
% 235.81/30.88 join(zero, zero)
% 235.81/30.88 = { by lemma 45 R->L }
% 235.81/30.88 join(zero, meet(zero, top))
% 235.81/30.88 = { by lemma 45 R->L }
% 235.81/30.88 join(meet(zero, top), meet(zero, top))
% 235.81/30.88 = { by lemma 46 }
% 235.81/30.88 meet(zero, top)
% 235.81/30.88 = { by lemma 45 }
% 235.81/30.88 zero
% 235.81/30.88
% 235.81/30.88 Lemma 48: join(top, X) = top.
% 235.81/30.88 Proof:
% 235.81/30.88 join(top, X)
% 235.81/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.88 join(X, top)
% 235.81/30.88 = { by lemma 25 R->L }
% 235.81/30.88 join(Y, top)
% 235.81/30.88 = { by lemma 26 }
% 235.81/30.88 top
% 235.81/30.88
% 235.81/30.88 Lemma 49: complement(complement(X)) = X.
% 235.81/30.88 Proof:
% 235.81/30.88 complement(complement(X))
% 235.81/30.88 = { by lemma 37 R->L }
% 235.81/30.88 join(zero, complement(complement(X)))
% 235.81/30.88 = { by lemma 35 }
% 235.81/30.88 X
% 235.81/30.88
% 235.81/30.88 Lemma 50: meet(zero, X) = zero.
% 235.81/30.88 Proof:
% 235.81/30.88 meet(zero, X)
% 235.81/30.88 = { by lemma 40 }
% 235.81/30.88 meet(X, zero)
% 235.81/30.88 = { by lemma 43 }
% 235.81/30.88 zero
% 235.81/30.88
% 235.81/30.88 Lemma 51: composition(top, zero) = zero.
% 235.81/30.88 Proof:
% 235.81/30.88 composition(top, zero)
% 235.81/30.88 = { by lemma 31 R->L }
% 235.81/30.88 composition(converse(top), zero)
% 235.81/30.88 = { by lemma 39 R->L }
% 235.81/30.88 join(zero, composition(converse(top), zero))
% 235.81/30.88 = { by lemma 17 R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), zero))
% 235.81/30.88 = { by lemma 17 R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(top)))
% 235.81/30.88 = { by lemma 48 R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 235.81/30.88 = { by lemma 31 R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 235.81/30.88 = { by lemma 21 R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(join(composition(one, top), composition(converse(top), top)))))
% 235.81/30.88 = { by axiom 12 (composition_distributivity_7) R->L }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(composition(join(one, converse(top)), top))))
% 235.81/30.88 = { by lemma 30 }
% 235.81/30.88 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 235.81/30.88 = { by lemma 22 }
% 235.81/30.88 complement(top)
% 235.81/30.88 = { by lemma 17 }
% 235.81/30.88 zero
% 235.81/30.88
% 235.81/30.88 Lemma 52: composition(X, zero) = zero.
% 235.81/30.88 Proof:
% 235.81/30.88 composition(X, zero)
% 235.81/30.88 = { by lemma 39 R->L }
% 235.81/30.88 join(zero, composition(X, zero))
% 235.81/30.88 = { by lemma 51 R->L }
% 235.81/30.88 join(composition(top, zero), composition(X, zero))
% 235.81/30.88 = { by axiom 12 (composition_distributivity_7) R->L }
% 235.81/30.88 composition(join(top, X), zero)
% 235.81/30.88 = { by lemma 48 }
% 235.81/30.88 composition(top, zero)
% 235.81/30.88 = { by lemma 51 }
% 235.81/30.88 zero
% 235.81/30.88
% 235.81/30.88 Lemma 53: composition(zero, X) = zero.
% 235.81/30.88 Proof:
% 235.81/30.88 composition(zero, X)
% 235.81/30.88 = { by lemma 47 R->L }
% 235.81/30.88 composition(converse(zero), X)
% 235.81/30.88 = { by lemma 19 R->L }
% 235.81/30.88 converse(composition(converse(X), zero))
% 235.81/30.88 = { by lemma 52 }
% 235.81/30.88 converse(zero)
% 235.81/30.88 = { by lemma 47 }
% 235.81/30.88 zero
% 235.81/30.88
% 235.81/30.88 Lemma 54: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 235.81/30.88 Proof:
% 235.81/30.88 join(Z, join(X, Y))
% 235.81/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 235.81/30.88 join(join(X, Y), Z)
% 235.81/30.88 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 236.43/30.88 join(X, join(Y, Z))
% 236.43/30.88
% 236.43/30.88 Lemma 55: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 236.43/30.88 Proof:
% 236.43/30.88 meet(X, join(complement(Y), complement(Z)))
% 236.43/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.88 meet(X, join(complement(Z), complement(Y)))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 meet(join(complement(Z), complement(Y)), X)
% 236.43/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.88 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 236.43/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 236.43/30.88 complement(join(meet(Z, Y), complement(X)))
% 236.43/30.88 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.88 complement(join(complement(X), meet(Z, Y)))
% 236.43/30.88 = { by lemma 40 R->L }
% 236.43/30.88 complement(join(complement(X), meet(Y, Z)))
% 236.43/30.88
% 236.43/30.88 Lemma 56: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 236.43/30.88 Proof:
% 236.43/30.88 join(complement(X), complement(Y))
% 236.43/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.88 join(complement(Y), complement(X))
% 236.43/30.88 = { by lemma 33 R->L }
% 236.43/30.88 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 236.43/30.88 = { by lemma 55 }
% 236.43/30.88 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 236.43/30.88 = { by lemma 37 }
% 236.43/30.88 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 236.43/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 236.43/30.88 complement(join(meet(Y, X), meet(Y, X)))
% 236.43/30.88 = { by lemma 46 }
% 236.43/30.88 complement(meet(Y, X))
% 236.43/30.88 = { by lemma 40 R->L }
% 236.43/30.88 complement(meet(X, Y))
% 236.43/30.88
% 236.43/30.88 Lemma 57: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 236.43/30.88 Proof:
% 236.43/30.88 complement(join(X, complement(Y)))
% 236.43/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.88 complement(join(complement(Y), X))
% 236.43/30.88 = { by lemma 45 R->L }
% 236.43/30.88 complement(join(complement(Y), meet(X, top)))
% 236.43/30.88 = { by lemma 40 R->L }
% 236.43/30.88 complement(join(complement(Y), meet(top, X)))
% 236.43/30.88 = { by lemma 55 R->L }
% 236.43/30.88 meet(Y, join(complement(top), complement(X)))
% 236.43/30.88 = { by lemma 17 }
% 236.43/30.88 meet(Y, join(zero, complement(X)))
% 236.43/30.88 = { by lemma 37 }
% 236.43/30.88 meet(Y, complement(X))
% 236.43/30.88
% 236.43/30.88 Lemma 58: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 236.43/30.88 Proof:
% 236.43/30.88 complement(join(complement(X), Y))
% 236.43/30.88 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.88 complement(join(Y, complement(X)))
% 236.43/30.88 = { by lemma 57 }
% 236.43/30.88 meet(X, complement(Y))
% 236.43/30.88
% 236.43/30.88 Lemma 59: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 236.43/30.88 Proof:
% 236.43/30.88 complement(meet(X, complement(Y)))
% 236.43/30.88 = { by lemma 39 R->L }
% 236.43/30.88 complement(join(zero, meet(X, complement(Y))))
% 236.43/30.88 = { by lemma 57 R->L }
% 236.43/30.88 complement(join(zero, complement(join(Y, complement(X)))))
% 236.43/30.88 = { by lemma 41 }
% 236.43/30.88 meet(join(Y, complement(X)), top)
% 236.43/30.88 = { by lemma 45 }
% 236.43/30.88 join(Y, complement(X))
% 236.43/30.88
% 236.43/30.88 Lemma 60: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 236.43/30.88 Proof:
% 236.43/30.88 complement(meet(complement(X), Y))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 complement(meet(Y, complement(X)))
% 236.43/30.88 = { by lemma 59 }
% 236.43/30.88 join(X, complement(Y))
% 236.43/30.88
% 236.43/30.88 Lemma 61: join(X, complement(meet(X, Y))) = top.
% 236.43/30.88 Proof:
% 236.43/30.88 join(X, complement(meet(X, Y)))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 join(X, complement(meet(Y, X)))
% 236.43/30.88 = { by lemma 56 R->L }
% 236.43/30.88 join(X, join(complement(Y), complement(X)))
% 236.43/30.88 = { by lemma 18 }
% 236.43/30.88 join(complement(Y), top)
% 236.43/30.88 = { by lemma 26 }
% 236.43/30.88 top
% 236.43/30.88
% 236.43/30.88 Lemma 62: meet(X, join(X, complement(Y))) = X.
% 236.43/30.88 Proof:
% 236.43/30.88 meet(X, join(X, complement(Y)))
% 236.43/30.88 = { by lemma 59 R->L }
% 236.43/30.88 meet(X, complement(meet(Y, complement(X))))
% 236.43/30.88 = { by lemma 58 R->L }
% 236.43/30.88 complement(join(complement(X), meet(Y, complement(X))))
% 236.43/30.88 = { by lemma 37 R->L }
% 236.43/30.88 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 236.43/30.88 = { by lemma 17 R->L }
% 236.43/30.88 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 236.43/30.88 = { by lemma 61 R->L }
% 236.43/30.88 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 236.43/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 236.43/30.88 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 236.43/30.88 = { by lemma 40 R->L }
% 236.43/30.88 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 236.43/30.88 = { by lemma 32 }
% 236.43/30.88 X
% 236.43/30.88
% 236.43/30.88 Lemma 63: meet(X, join(X, Y)) = X.
% 236.43/30.88 Proof:
% 236.43/30.88 meet(X, join(X, Y))
% 236.43/30.88 = { by lemma 45 R->L }
% 236.43/30.88 meet(X, join(X, meet(Y, top)))
% 236.43/30.88 = { by lemma 41 R->L }
% 236.43/30.88 meet(X, join(X, complement(join(zero, complement(Y)))))
% 236.43/30.88 = { by lemma 62 }
% 236.43/30.88 X
% 236.43/30.88
% 236.43/30.88 Lemma 64: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 236.43/30.88 Proof:
% 236.43/30.88 meet(complement(X), complement(Y))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 meet(complement(Y), complement(X))
% 236.43/30.88 = { by lemma 37 R->L }
% 236.43/30.88 meet(join(zero, complement(Y)), complement(X))
% 236.43/30.88 = { by lemma 57 R->L }
% 236.43/30.88 complement(join(X, complement(join(zero, complement(Y)))))
% 236.43/30.88 = { by lemma 41 }
% 236.43/30.88 complement(join(X, meet(Y, top)))
% 236.43/30.88 = { by lemma 45 }
% 236.43/30.88 complement(join(X, Y))
% 236.43/30.88
% 236.43/30.88 Lemma 65: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 236.43/30.88 Proof:
% 236.43/30.88 meet(complement(Z), meet(Y, X))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 meet(complement(Z), meet(X, Y))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.88 meet(meet(X, Y), complement(Z))
% 236.43/30.88 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.88 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 236.43/30.88 = { by lemma 64 }
% 236.43/30.88 complement(join(join(complement(X), complement(Y)), Z))
% 236.43/30.88 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 236.43/30.88 complement(join(complement(X), join(complement(Y), Z)))
% 236.43/30.88 = { by lemma 58 }
% 236.43/30.88 meet(X, complement(join(complement(Y), Z)))
% 236.43/30.88 = { by lemma 58 }
% 236.43/30.88 meet(X, meet(Y, complement(Z)))
% 236.43/30.88
% 236.43/30.88 Lemma 66: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 236.43/30.88 Proof:
% 236.43/30.88 meet(Y, meet(Z, X))
% 236.43/30.88 = { by lemma 45 R->L }
% 236.43/30.88 meet(meet(Y, top), meet(Z, X))
% 236.43/30.88 = { by lemma 41 R->L }
% 236.43/30.88 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 236.43/30.88 = { by lemma 65 }
% 236.43/30.88 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 236.43/30.88 = { by lemma 41 }
% 236.43/30.88 meet(X, meet(Z, meet(Y, top)))
% 236.43/30.88 = { by lemma 45 }
% 236.43/30.88 meet(X, meet(Z, Y))
% 236.43/30.88 = { by lemma 40 R->L }
% 236.43/30.88 meet(X, meet(Y, Z))
% 236.43/30.88
% 236.43/30.88 Lemma 67: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 236.43/30.88 Proof:
% 236.43/30.88 meet(Y, meet(X, Z))
% 236.43/30.88 = { by lemma 40 }
% 236.43/30.89 meet(Y, meet(Z, X))
% 236.43/30.89 = { by lemma 66 }
% 236.43/30.89 meet(X, meet(Y, Z))
% 236.43/30.89
% 236.43/30.89 Lemma 68: meet(meet(X, Y), Z) = meet(X, meet(Y, Z)).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(meet(X, Y), Z)
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 meet(Z, meet(X, Y))
% 236.43/30.89 = { by lemma 66 R->L }
% 236.43/30.89 meet(X, meet(Y, Z))
% 236.43/30.89
% 236.43/30.89 Lemma 69: join(X, complement(meet(Y, X))) = top.
% 236.43/30.89 Proof:
% 236.43/30.89 join(X, complement(meet(Y, X)))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 join(X, complement(meet(X, Y)))
% 236.43/30.89 = { by lemma 61 }
% 236.43/30.89 top
% 236.43/30.89
% 236.43/30.89 Lemma 70: join(complement(X), meet(Y, complement(Z))) = complement(meet(X, join(Z, complement(Y)))).
% 236.43/30.89 Proof:
% 236.43/30.89 join(complement(X), meet(Y, complement(Z)))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.89 join(meet(Y, complement(Z)), complement(X))
% 236.43/30.89 = { by lemma 57 R->L }
% 236.43/30.89 join(complement(join(Z, complement(Y))), complement(X))
% 236.43/30.89 = { by lemma 56 }
% 236.43/30.89 complement(meet(join(Z, complement(Y)), X))
% 236.43/30.89 = { by lemma 40 R->L }
% 236.43/30.89 complement(meet(X, join(Z, complement(Y))))
% 236.43/30.89
% 236.43/30.89 Lemma 71: complement(join(X, join(Y, complement(Z)))) = meet(Z, complement(join(X, Y))).
% 236.43/30.89 Proof:
% 236.43/30.89 complement(join(X, join(Y, complement(Z))))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.89 complement(join(join(Y, complement(Z)), X))
% 236.43/30.89 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 236.43/30.89 complement(join(Y, join(complement(Z), X)))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.89 complement(join(Y, join(X, complement(Z))))
% 236.43/30.89 = { by axiom 8 (maddux2_join_associativity_2) }
% 236.43/30.89 complement(join(join(Y, X), complement(Z)))
% 236.43/30.89 = { by lemma 57 }
% 236.43/30.89 meet(Z, complement(join(Y, X)))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.89 meet(Z, complement(join(X, Y)))
% 236.43/30.89
% 236.43/30.89 Lemma 72: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 236.43/30.89 Proof:
% 236.43/30.89 join(meet(X, Y), meet(Y, complement(X)))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 join(meet(Y, X), meet(Y, complement(X)))
% 236.43/30.89 = { by lemma 44 }
% 236.43/30.89 Y
% 236.43/30.89
% 236.43/30.89 Lemma 73: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 236.43/30.89 Proof:
% 236.43/30.89 join(meet(X, Y), meet(complement(X), Y))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 join(meet(X, Y), meet(Y, complement(X)))
% 236.43/30.89 = { by lemma 72 }
% 236.43/30.89 Y
% 236.43/30.89
% 236.43/30.89 Lemma 74: complement(meet(Y, join(X, complement(Y)))) = complement(meet(X, join(Y, complement(X)))).
% 236.43/30.89 Proof:
% 236.43/30.89 complement(meet(Y, join(X, complement(Y))))
% 236.43/30.89 = { by lemma 70 R->L }
% 236.43/30.89 join(complement(Y), meet(Y, complement(X)))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.89 join(meet(Y, complement(X)), complement(Y))
% 236.43/30.89 = { by lemma 73 R->L }
% 236.43/30.89 join(meet(Y, complement(X)), join(meet(X, complement(Y)), meet(complement(X), complement(Y))))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.89 join(meet(Y, complement(X)), join(meet(complement(X), complement(Y)), meet(X, complement(Y))))
% 236.43/30.89 = { by axiom 8 (maddux2_join_associativity_2) }
% 236.43/30.89 join(join(meet(Y, complement(X)), meet(complement(X), complement(Y))), meet(X, complement(Y)))
% 236.43/30.89 = { by lemma 72 }
% 236.43/30.89 join(complement(X), meet(X, complement(Y)))
% 236.43/30.89 = { by lemma 70 }
% 236.43/30.89 complement(meet(X, join(Y, complement(X))))
% 236.43/30.89
% 236.43/30.89 Lemma 75: meet(complement(X), join(Y, complement(Z))) = complement(join(X, meet(Z, complement(Y)))).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(complement(X), join(Y, complement(Z)))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 meet(join(Y, complement(Z)), complement(X))
% 236.43/30.89 = { by lemma 57 R->L }
% 236.43/30.89 complement(join(X, complement(join(Y, complement(Z)))))
% 236.43/30.89 = { by lemma 57 }
% 236.43/30.89 complement(join(X, meet(Z, complement(Y))))
% 236.43/30.89
% 236.43/30.89 Lemma 76: meet(composition(sk1, top), meet(X, Y)) = meet(Y, meet(sk1, X)).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(composition(sk1, top), meet(X, Y))
% 236.43/30.89 = { by lemma 66 }
% 236.43/30.89 meet(Y, meet(composition(sk1, top), X))
% 236.43/30.89 = { by axiom 4 (goals_17) }
% 236.43/30.89 meet(Y, meet(sk1, X))
% 236.43/30.89
% 236.43/30.89 Lemma 77: meet(composition(sk1, top), meet(X, Y)) = meet(X, meet(sk1, Y)).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(composition(sk1, top), meet(X, Y))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 meet(composition(sk1, top), meet(Y, X))
% 236.43/30.89 = { by lemma 76 }
% 236.43/30.89 meet(X, meet(sk1, Y))
% 236.43/30.89
% 236.43/30.89 Lemma 78: meet(meet(sk1, X), complement(composition(sk1, top))) = zero.
% 236.43/30.89 Proof:
% 236.43/30.89 meet(meet(sk1, X), complement(composition(sk1, top)))
% 236.43/30.89 = { by lemma 57 R->L }
% 236.43/30.89 complement(join(composition(sk1, top), complement(meet(sk1, X))))
% 236.43/30.89 = { by axiom 4 (goals_17) }
% 236.43/30.89 complement(join(sk1, complement(meet(sk1, X))))
% 236.43/30.89 = { by lemma 61 }
% 236.43/30.89 complement(top)
% 236.43/30.89 = { by lemma 17 }
% 236.43/30.89 zero
% 236.43/30.89
% 236.43/30.89 Lemma 79: meet(composition(meet(X, Y), Z), composition(Y, Z)) = composition(meet(X, Y), Z).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(Y, Z))
% 236.43/30.89 = { by lemma 45 R->L }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(meet(Y, top), Z))
% 236.43/30.89 = { by lemma 41 R->L }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, complement(Y))), Z))
% 236.43/30.89 = { by lemma 72 R->L }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, join(meet(meet(X, Y), complement(Y)), meet(complement(Y), complement(meet(X, Y)))))), Z))
% 236.43/30.89 = { by lemma 57 R->L }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, join(complement(join(Y, complement(meet(X, Y)))), meet(complement(Y), complement(meet(X, Y)))))), Z))
% 236.43/30.89 = { by lemma 69 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, join(complement(top), meet(complement(Y), complement(meet(X, Y)))))), Z))
% 236.43/30.89 = { by lemma 17 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, join(zero, meet(complement(Y), complement(meet(X, Y)))))), Z))
% 236.43/30.89 = { by lemma 39 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, meet(complement(Y), complement(meet(X, Y))))), Z))
% 236.43/30.89 = { by lemma 64 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, complement(join(Y, meet(X, Y))))), Z))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(complement(join(zero, complement(join(meet(X, Y), Y)))), Z))
% 236.43/30.89 = { by lemma 41 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(meet(join(meet(X, Y), Y), top), Z))
% 236.43/30.89 = { by lemma 45 }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(join(meet(X, Y), Y), Z))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.89 meet(composition(meet(X, Y), Z), composition(join(Y, meet(X, Y)), Z))
% 236.43/30.89 = { by axiom 12 (composition_distributivity_7) }
% 236.43/30.89 meet(composition(meet(X, Y), Z), join(composition(Y, Z), composition(meet(X, Y), Z)))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.89 meet(composition(meet(X, Y), Z), join(composition(meet(X, Y), Z), composition(Y, Z)))
% 236.43/30.89 = { by lemma 63 }
% 236.43/30.89 composition(meet(X, Y), Z)
% 236.43/30.89
% 236.43/30.89 Lemma 80: composition(converse(complement(composition(sk1, top))), composition(sk1, top)) = zero.
% 236.43/30.89 Proof:
% 236.43/30.89 composition(converse(complement(composition(sk1, top))), composition(sk1, top))
% 236.43/30.89 = { by lemma 19 R->L }
% 236.43/30.89 converse(composition(converse(composition(sk1, top)), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 38 R->L }
% 236.43/30.89 converse(join(composition(converse(composition(sk1, top)), complement(composition(sk1, top))), zero))
% 236.43/30.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.89 converse(join(zero, composition(converse(composition(sk1, top)), complement(composition(sk1, top)))))
% 236.43/30.89 = { by lemma 17 R->L }
% 236.43/30.89 converse(join(complement(top), composition(converse(composition(sk1, top)), complement(composition(sk1, top)))))
% 236.43/30.89 = { by axiom 4 (goals_17) R->L }
% 236.43/30.89 converse(join(complement(top), composition(converse(composition(sk1, top)), complement(composition(composition(sk1, top), top)))))
% 236.43/30.89 = { by lemma 22 }
% 236.43/30.89 converse(complement(top))
% 236.43/30.89 = { by lemma 17 }
% 236.43/30.89 converse(zero)
% 236.43/30.89 = { by lemma 47 }
% 236.43/30.89 zero
% 236.43/30.89
% 236.43/30.89 Lemma 81: join(meet(sk1, X), meet(composition(sk1, top), complement(X))) = composition(sk1, top).
% 236.43/30.89 Proof:
% 236.43/30.89 join(meet(sk1, X), meet(composition(sk1, top), complement(X)))
% 236.43/30.89 = { by axiom 4 (goals_17) }
% 236.43/30.89 join(meet(sk1, X), meet(sk1, complement(X)))
% 236.43/30.89 = { by lemma 44 }
% 236.43/30.89 sk1
% 236.43/30.89 = { by axiom 4 (goals_17) R->L }
% 236.43/30.89 composition(sk1, top)
% 236.43/30.89
% 236.43/30.89 Lemma 82: meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), X)) = composition(complement(composition(sk1, top)), X).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), X))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top)))
% 236.43/30.89 = { by lemma 39 R->L }
% 236.43/30.89 join(zero, meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 50 R->L }
% 236.43/30.89 join(meet(zero, composition(sk1, top)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 52 R->L }
% 236.43/30.89 join(meet(composition(complement(composition(sk1, top)), zero), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 43 R->L }
% 236.43/30.89 join(meet(composition(complement(composition(sk1, top)), meet(X, zero)), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 80 R->L }
% 236.43/30.89 join(meet(composition(complement(composition(sk1, top)), meet(X, composition(converse(complement(composition(sk1, top))), composition(sk1, top)))), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by axiom 15 (modular_law_1_15) R->L }
% 236.43/30.89 join(join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), meet(X, composition(converse(complement(composition(sk1, top))), composition(sk1, top)))), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 80 }
% 236.43/30.89 join(join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), meet(X, zero)), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 43 }
% 236.43/30.89 join(join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), zero), composition(sk1, top))), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 52 }
% 236.43/30.89 join(join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), meet(zero, composition(sk1, top))), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 50 }
% 236.43/30.89 join(join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), zero), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 38 }
% 236.43/30.89 join(meet(composition(complement(composition(sk1, top)), X), composition(sk1, top)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 40 R->L }
% 236.43/30.89 join(meet(composition(sk1, top), composition(complement(composition(sk1, top)), X)), meet(composition(complement(composition(sk1, top)), X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 72 }
% 236.43/30.89 composition(complement(composition(sk1, top)), X)
% 236.43/30.89
% 236.43/30.89 Lemma 83: meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y)) = zero.
% 236.43/30.89 Proof:
% 236.43/30.89 meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y))
% 236.43/30.89 = { by lemma 82 R->L }
% 236.43/30.89 meet(meet(sk1, X), meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), Y)))
% 236.43/30.89 = { by lemma 67 }
% 236.43/30.89 meet(complement(composition(sk1, top)), meet(meet(sk1, X), composition(complement(composition(sk1, top)), Y)))
% 236.43/30.89 = { by lemma 65 }
% 236.43/30.89 meet(composition(complement(composition(sk1, top)), Y), meet(meet(sk1, X), complement(composition(sk1, top))))
% 236.43/30.89 = { by lemma 78 }
% 236.43/30.89 meet(composition(complement(composition(sk1, top)), Y), zero)
% 236.43/30.89 = { by lemma 43 }
% 236.43/30.89 zero
% 236.43/30.89
% 236.43/30.89 Lemma 84: meet(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)) = composition(meet(sk1, X), Y).
% 236.43/30.89 Proof:
% 236.43/30.89 meet(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))
% 236.43/30.89 = { by lemma 40 }
% 236.43/30.89 meet(composition(meet(sk1, X), Y), meet(sk1, composition(X, Y)))
% 236.43/30.89 = { by lemma 77 R->L }
% 236.43/30.89 meet(composition(sk1, top), meet(composition(meet(sk1, X), Y), composition(X, Y)))
% 236.43/30.89 = { by lemma 79 }
% 236.43/30.89 meet(composition(sk1, top), composition(meet(sk1, X), Y))
% 236.43/30.89 = { by lemma 40 R->L }
% 236.43/30.89 meet(composition(meet(sk1, X), Y), composition(sk1, top))
% 236.43/30.89 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.89 complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top))))
% 236.43/30.90 = { by lemma 37 R->L }
% 236.43/30.90 join(zero, complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 50 R->L }
% 236.43/30.90 join(meet(zero, complement(composition(sk1, top))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 53 R->L }
% 236.43/30.90 join(meet(composition(zero, Y), complement(composition(sk1, top))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 83 R->L }
% 236.43/30.90 join(meet(composition(meet(meet(sk1, X), composition(complement(composition(sk1, top)), converse(Y))), Y), complement(composition(sk1, top))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by axiom 16 (modular_law_2_16) R->L }
% 236.43/30.90 join(join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(composition(meet(meet(sk1, X), composition(complement(composition(sk1, top)), converse(Y))), Y), complement(composition(sk1, top)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 83 }
% 236.43/30.90 join(join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(composition(zero, Y), complement(composition(sk1, top)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 53 }
% 236.43/30.90 join(join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 50 }
% 236.43/30.90 join(join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), zero), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 38 }
% 236.43/30.90 join(meet(composition(meet(sk1, X), Y), complement(composition(sk1, top))), complement(join(complement(composition(meet(sk1, X), Y)), complement(composition(sk1, top)))))
% 236.43/30.90 = { by lemma 32 }
% 236.43/30.90 composition(meet(sk1, X), Y)
% 236.43/30.90
% 236.43/30.90 Lemma 85: join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)) = meet(sk1, composition(X, Y)).
% 236.43/30.90 Proof:
% 236.43/30.90 join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))
% 236.43/30.90 = { by lemma 44 R->L }
% 236.43/30.90 join(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), composition(meet(sk1, X), Y)), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 40 }
% 236.43/30.90 join(meet(composition(meet(sk1, X), Y), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.90 join(complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 37 R->L }
% 236.43/30.90 join(join(zero, complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 17 R->L }
% 236.43/30.90 join(join(complement(top), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 26 R->L }
% 236.43/30.90 join(join(complement(join(meet(sk1, composition(X, Y)), top)), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by axiom 5 (def_top_12) }
% 236.43/30.90 join(join(complement(join(meet(sk1, composition(X, Y)), join(composition(meet(sk1, X), Y), complement(composition(meet(sk1, X), Y))))), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 54 R->L }
% 236.43/30.90 join(join(complement(join(complement(composition(meet(sk1, X), Y)), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.90 join(join(complement(join(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 57 }
% 236.43/30.90 join(join(meet(composition(meet(sk1, X), Y), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))), complement(join(complement(composition(meet(sk1, X), Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 32 }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.90 = { by lemma 63 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), meet(sk1, composition(X, Y)))))
% 236.43/30.90 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))))))
% 236.43/30.90 = { by lemma 40 }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), meet(complement(composition(meet(sk1, X), Y)), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))
% 236.43/30.90 = { by axiom 4 (goals_17) R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), meet(complement(composition(meet(composition(sk1, top), X), Y)), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))
% 236.43/30.90 = { by axiom 4 (goals_17) R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), meet(complement(composition(meet(composition(sk1, top), X), Y)), join(meet(sk1, composition(X, Y)), composition(meet(composition(sk1, top), X), Y))))))
% 236.43/30.90 = { by lemma 40 }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), meet(join(meet(sk1, composition(X, Y)), composition(meet(composition(sk1, top), X), Y)), complement(composition(meet(composition(sk1, top), X), Y))))))
% 236.43/30.90 = { by lemma 57 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), complement(join(composition(meet(composition(sk1, top), X), Y), complement(join(meet(sk1, composition(X, Y)), composition(meet(composition(sk1, top), X), Y))))))))
% 236.43/30.90 = { by lemma 64 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), join(meet(sk1, composition(X, Y)), complement(join(composition(meet(composition(sk1, top), X), Y), meet(complement(meet(sk1, composition(X, Y))), complement(composition(meet(composition(sk1, top), X), Y))))))))
% 236.43/30.90 = { by lemma 60 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(meet(complement(meet(sk1, composition(X, Y))), join(composition(meet(composition(sk1, top), X), Y), meet(complement(meet(sk1, composition(X, Y))), complement(composition(meet(composition(sk1, top), X), Y))))))))
% 236.43/30.90 = { by lemma 37 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(meet(complement(meet(sk1, composition(X, Y))), join(composition(meet(composition(sk1, top), X), Y), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))))))
% 236.43/30.90 = { by lemma 45 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(meet(complement(meet(sk1, composition(X, Y))), join(meet(composition(meet(composition(sk1, top), X), Y), top), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))))))
% 236.43/30.90 = { by lemma 41 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))))))
% 236.43/30.90 = { by lemma 38 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(join(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))), zero))))
% 236.43/30.90 = { by lemma 17 R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(join(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))), complement(top)))))
% 236.43/30.90 = { by axiom 5 (def_top_12) }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(join(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))), complement(join(join(complement(complement(meet(sk1, composition(X, Y)))), complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y))))), complement(join(complement(complement(meet(sk1, composition(X, Y)))), complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y))))))))))))
% 236.43/30.90 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 236.43/30.90 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(join(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))), complement(join(join(complement(complement(meet(sk1, composition(X, Y)))), complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y))))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y))))))))))
% 236.43/30.91 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(join(meet(complement(meet(sk1, composition(X, Y))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))), complement(join(complement(complement(meet(sk1, composition(X, Y)))), join(complement(join(zero, complement(composition(meet(composition(sk1, top), X), Y)))), meet(complement(meet(sk1, composition(X, Y))), join(zero, complement(composition(meet(composition(sk1, top), X), Y)))))))))))
% 236.43/30.91 = { by lemma 32 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), complement(complement(meet(sk1, composition(X, Y))))))
% 236.43/30.91 = { by lemma 49 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(composition(meet(sk1, X), Y))), meet(sk1, composition(X, Y))))
% 236.43/30.91 = { by lemma 68 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), meet(complement(composition(meet(sk1, X), Y)), meet(sk1, composition(X, Y)))))
% 236.43/30.91 = { by lemma 40 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), meet(meet(sk1, composition(X, Y)), complement(composition(meet(sk1, X), Y)))))
% 236.43/30.91 = { by lemma 66 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), meet(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), meet(sk1, composition(X, Y)))))
% 236.43/30.91 = { by lemma 40 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), meet(meet(sk1, composition(X, Y)), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))
% 236.43/30.91 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))))))
% 236.43/30.91 = { by lemma 37 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(zero, complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 17 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(complement(top), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 26 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(complement(join(composition(meet(sk1, X), Y), top)), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 18 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(complement(join(meet(sk1, composition(X, Y)), join(composition(meet(sk1, X), Y), complement(meet(sk1, composition(X, Y)))))), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 54 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(complement(join(complement(meet(sk1, composition(X, Y))), join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(complement(join(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(meet(sk1, composition(X, Y))))), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 57 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), join(meet(meet(sk1, composition(X, Y)), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)))), complement(join(complement(meet(sk1, composition(X, Y))), complement(join(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y))))))))
% 236.43/30.91 = { by lemma 32 }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(complement(composition(meet(sk1, X), Y)), meet(sk1, composition(X, Y))))
% 236.43/30.91 = { by lemma 40 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), meet(meet(sk1, composition(X, Y)), complement(composition(meet(sk1, X), Y))))
% 236.43/30.91 = { by lemma 58 R->L }
% 236.43/30.91 join(composition(meet(sk1, X), Y), complement(join(complement(meet(sk1, composition(X, Y))), composition(meet(sk1, X), Y))))
% 236.43/30.91 = { by lemma 84 R->L }
% 236.43/30.91 join(meet(meet(sk1, composition(X, Y)), composition(meet(sk1, X), Y)), complement(join(complement(meet(sk1, composition(X, Y))), composition(meet(sk1, X), Y))))
% 236.43/30.91 = { by lemma 32 }
% 236.43/30.91 meet(sk1, composition(X, Y))
% 236.43/30.91
% 236.43/30.91 Lemma 86: meet(composition(sk1, top), join(X, composition(complement(composition(sk1, top)), Y))) = meet(X, composition(sk1, top)).
% 236.43/30.91 Proof:
% 236.43/30.91 meet(composition(sk1, top), join(X, composition(complement(composition(sk1, top)), Y)))
% 236.43/30.91 = { by lemma 49 R->L }
% 236.43/30.91 meet(complement(complement(composition(sk1, top))), join(X, composition(complement(composition(sk1, top)), Y)))
% 236.43/30.91 = { by lemma 82 R->L }
% 236.43/30.91 meet(complement(complement(composition(sk1, top))), join(X, meet(complement(composition(sk1, top)), composition(complement(composition(sk1, top)), Y))))
% 236.43/30.91 = { by lemma 40 }
% 236.43/30.91 meet(complement(complement(composition(sk1, top))), join(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))
% 236.43/30.91 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.91 meet(complement(complement(composition(sk1, top))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X))
% 236.43/30.91 = { by lemma 40 }
% 236.43/30.91 meet(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(complement(composition(sk1, top))))
% 236.43/30.91 = { by lemma 72 R->L }
% 236.43/30.91 meet(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))))))
% 236.43/30.91 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 236.43/30.91 meet(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))
% 236.43/30.91 = { by lemma 71 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)))))
% 236.43/30.91 = { by lemma 64 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), complement(X)))))
% 236.43/30.91 = { by lemma 57 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(join(X, complement(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))))))
% 236.43/30.91 = { by lemma 60 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), join(X, complement(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))))))
% 236.43/30.91 = { by lemma 74 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(X, join(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), complement(X))))))
% 236.43/30.91 = { by lemma 56 }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(X, complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X))))))
% 236.43/30.91 = { by lemma 59 }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(X))))
% 236.43/30.91 = { by lemma 73 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(join(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X))))))
% 236.43/30.91 = { by lemma 64 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), meet(complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)), complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X))))))
% 236.43/30.91 = { by lemma 57 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), complement(join(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), complement(complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X))))))))
% 236.43/30.91 = { by lemma 60 R->L }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)), join(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), complement(complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X))))))))
% 236.43/30.91 = { by lemma 74 }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), join(complement(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)), complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)))))))
% 236.43/30.91 = { by lemma 56 }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), complement(meet(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), complement(meet(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)))))))
% 236.43/30.91 = { by lemma 59 }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)), complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)))))
% 236.43/30.91 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.91 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)), meet(meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)))))
% 236.43/30.91 = { by lemma 40 R->L }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(complement(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X)), meet(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)))))
% 236.43/30.92 = { by lemma 60 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(X)), meet(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)))))
% 236.43/30.92 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), join(complement(X), meet(meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X), meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X))))))
% 236.43/30.92 = { by lemma 68 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), join(complement(X), meet(complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), meet(X, meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)))))))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), join(complement(X), meet(meet(X, meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)), complement(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))))))
% 236.43/30.92 = { by lemma 70 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, meet(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), X)))))))))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, meet(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))))))))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(meet(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), X))))))))
% 236.43/30.92 = { by lemma 32 R->L }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(meet(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), join(meet(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))), complement(join(complement(X), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))))))))))
% 236.43/30.92 = { by lemma 62 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top)))))))))))
% 236.43/30.92 = { by lemma 69 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(meet(X, top)))))
% 236.43/30.92 = { by lemma 45 }
% 236.43/30.92 complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), complement(X))))
% 236.43/30.92 = { by lemma 71 }
% 236.43/30.92 meet(X, complement(join(meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))), meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))))))
% 236.43/30.92 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.92 meet(X, complement(join(meet(composition(complement(composition(sk1, top)), Y), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(complement(composition(sk1, top)), Y))))))
% 236.43/30.92 = { by lemma 72 }
% 236.43/30.92 meet(X, complement(complement(composition(sk1, top))))
% 236.43/30.92 = { by lemma 49 }
% 236.43/30.92 meet(X, composition(sk1, top))
% 236.43/30.92
% 236.43/30.92 Goal 1 (goals_18): tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3)))) = tuple(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3))).
% 236.43/30.92 Proof:
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) R->L }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, top), composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) R->L }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(composition(sk1, top), composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) R->L }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(composition(sk1, top), sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(composition(sk1, top), composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) R->L }
% 236.43/30.92 tuple(join(meet(composition(sk1, top), composition(sk2, sk3)), composition(meet(composition(sk1, top), sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(composition(sk1, top), composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) }
% 236.43/30.92 tuple(join(meet(composition(sk1, top), composition(sk2, sk3)), composition(meet(composition(sk1, top), sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(sk1, composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(composition(sk1, top), sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(sk1, composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 4 (goals_17) }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(composition(sk1, top), sk2), sk3), meet(sk1, composition(sk2, sk3))))
% 236.43/30.92 = { by axiom 2 (maddux1_join_commutativity_1) }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(composition(sk1, top), sk2), sk3)))
% 236.43/30.92 = { by axiom 4 (goals_17) }
% 236.43/30.92 tuple(join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 85 }
% 236.43/30.92 tuple(meet(sk1, composition(sk2, sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by axiom 4 (goals_17) R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(sk2, sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 tuple(meet(composition(sk2, sk3), composition(sk1, top)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 86 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), join(composition(sk2, sk3), composition(complement(composition(sk1, top)), sk3))), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by axiom 12 (composition_distributivity_7) R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(sk2, complement(composition(sk1, top))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 72 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(meet(composition(sk1, top), join(sk2, complement(composition(sk1, top)))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(meet(join(sk2, complement(composition(sk1, top))), composition(sk1, top)), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 81 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(meet(join(sk2, complement(composition(sk1, top))), join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2)))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 59 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2)))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 40 }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(meet(join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2))), complement(meet(composition(sk1, top), complement(sk2)))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 57 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(complement(join(meet(composition(sk1, top), complement(sk2)), complement(join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2)))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 64 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2)))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 73 R->L }
% 236.43/30.92 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(complement(meet(sk1, sk2)), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2)))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.92 = { by lemma 75 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(complement(meet(sk1, sk2)), meet(complement(meet(composition(sk1, top), complement(sk2))), join(meet(composition(sk1, top), complement(sk2)), complement(complement(meet(sk1, sk2))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 67 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(complement(meet(composition(sk1, top), complement(sk2))), meet(complement(meet(sk1, sk2)), join(meet(composition(sk1, top), complement(sk2)), complement(complement(meet(sk1, sk2))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 65 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(join(meet(composition(sk1, top), complement(sk2)), complement(complement(meet(sk1, sk2)))), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2)))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 57 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(join(meet(composition(sk1, top), complement(sk2)), complement(complement(meet(sk1, sk2)))), complement(join(meet(composition(sk1, top), complement(sk2)), complement(complement(meet(sk1, sk2))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by axiom 6 (def_zero_13) R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), zero), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 38 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), meet(complement(meet(sk1, sk2)), complement(meet(composition(sk1, top), complement(sk2))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 64 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(meet(sk1, sk2), complement(join(meet(composition(sk1, top), complement(sk2)), complement(join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2))))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 57 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(meet(sk1, sk2), meet(join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2))), complement(meet(composition(sk1, top), complement(sk2))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 40 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(meet(sk1, sk2), meet(complement(meet(composition(sk1, top), complement(sk2))), join(meet(sk1, sk2), meet(composition(sk1, top), complement(sk2))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 81 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(meet(sk1, sk2), meet(complement(meet(composition(sk1, top), complement(sk2))), composition(sk1, top))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 67 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), meet(meet(sk1, sk2), composition(sk1, top))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 37 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), join(zero, complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top)))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 78 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), join(meet(meet(sk1, sk2), complement(composition(sk1, top))), complement(join(complement(meet(sk1, sk2)), complement(composition(sk1, top)))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 32 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(complement(meet(composition(sk1, top), complement(sk2))), meet(sk1, sk2)), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 59 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(join(sk2, complement(composition(sk1, top))), meet(sk1, sk2)), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 76 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(composition(sk1, top), meet(sk2, join(sk2, complement(composition(sk1, top))))), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 63 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(composition(sk1, top), sk2), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by axiom 4 (goals_17) }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), meet(join(sk2, complement(composition(sk1, top))), complement(composition(sk1, top)))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 40 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), meet(complement(composition(sk1, top)), join(sk2, complement(composition(sk1, top))))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 75 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), complement(join(composition(sk1, top), meet(composition(sk1, top), complement(sk2))))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), complement(join(composition(sk1, top), complement(join(complement(composition(sk1, top)), complement(complement(sk2))))))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 60 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), complement(complement(meet(complement(composition(sk1, top)), join(complement(composition(sk1, top)), complement(complement(sk2))))))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 62 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), complement(complement(complement(composition(sk1, top))))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 49 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(join(meet(sk1, sk2), complement(composition(sk1, top))), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by axiom 12 (composition_distributivity_7) }
% 236.43/30.93 tuple(meet(composition(sk1, top), join(composition(meet(sk1, sk2), sk3), composition(complement(composition(sk1, top)), sk3))), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 86 }
% 236.43/30.93 tuple(meet(composition(meet(sk1, sk2), sk3), composition(sk1, top)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 40 }
% 236.43/30.93 tuple(meet(composition(sk1, top), composition(meet(sk1, sk2), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 79 R->L }
% 236.43/30.93 tuple(meet(composition(sk1, top), meet(composition(meet(sk1, sk2), sk3), composition(sk2, sk3))), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 77 }
% 236.43/30.93 tuple(meet(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3))), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 40 R->L }
% 236.43/30.93 tuple(meet(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 84 }
% 236.43/30.93 tuple(composition(meet(sk1, sk2), sk3), join(meet(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 236.43/30.93 = { by lemma 85 }
% 236.43/30.93 tuple(composition(meet(sk1, sk2), sk3), meet(sk1, composition(sk2, sk3)))
% 236.43/30.93 % SZS output end Proof
% 236.43/30.93
% 236.43/30.93 RESULT: Unsatisfiable (the axioms are contradictory).
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