TSTP Solution File: REL029-4 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL029-4 : 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 : n031.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 13:44:13 EDT 2023
% Result : Unsatisfiable 55.09s 7.48s
% Output : Proof 56.68s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL029-4 : 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
% 0.14/0.35 % Computer : n031.cluster.edu
% 0.14/0.35 % Model : x86_64 x86_64
% 0.14/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.14/0.35 % Memory : 8042.1875MB
% 0.14/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.14/0.35 % CPULimit : 300
% 0.14/0.35 % WCLimit : 300
% 0.14/0.35 % DateTime : Fri Aug 25 19:56:52 EDT 2023
% 0.14/0.35 % CPUTime :
% 55.09/7.48 Command-line arguments: --lhs-weight 9 --flip-ordering --complete-subsets --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 55.09/7.48
% 55.09/7.48 % SZS status Unsatisfiable
% 55.09/7.48
% 55.95/7.62 % SZS output start Proof
% 55.95/7.62 Take the following subset of the input axioms:
% 55.95/7.62 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 55.95/7.62 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 55.95/7.62 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 55.95/7.62 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 55.95/7.62 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 55.95/7.62 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 55.95/7.62 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 55.95/7.62 fof(dedekind_law_14, axiom, ![A2, B2, C2]: join(meet(composition(A2, B2), C2), composition(meet(A2, composition(C2, converse(B2))), meet(B2, composition(converse(A2), C2))))=composition(meet(A2, composition(C2, converse(B2))), meet(B2, composition(converse(A2), C2)))).
% 55.95/7.62 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 55.95/7.62 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 55.95/7.62 fof(goals_17, negated_conjecture, join(sk1, one)=one).
% 55.95/7.62 fof(goals_18, negated_conjecture, join(sk2, one)=one).
% 55.95/7.62 fof(goals_19, negated_conjecture, join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))!=meet(composition(sk1, sk3), composition(sk2, sk3)) | join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3))!=composition(meet(sk1, sk2), sk3)).
% 55.95/7.62 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 55.95/7.62 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 55.95/7.62 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 55.95/7.62 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 55.95/7.62
% 55.95/7.62 Now clausify the problem and encode Horn clauses using encoding 3 of
% 55.95/7.62 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 55.95/7.62 We repeatedly replace C & s=t => u=v by the two clauses:
% 55.95/7.62 fresh(y, y, x1...xn) = u
% 55.95/7.62 C => fresh(s, t, x1...xn) = v
% 55.95/7.62 where fresh is a fresh function symbol and x1..xn are the free
% 55.95/7.62 variables of u and v.
% 55.95/7.62 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 55.95/7.62 input problem has no model of domain size 1).
% 55.95/7.62
% 55.95/7.62 The encoding turns the above axioms into the following unit equations and goals:
% 55.95/7.62
% 55.95/7.62 Axiom 1 (composition_identity_6): composition(X, one) = X.
% 55.95/7.62 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 55.95/7.62 Axiom 3 (goals_17): join(sk1, one) = one.
% 55.95/7.62 Axiom 4 (goals_18): join(sk2, one) = one.
% 55.95/7.62 Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
% 55.95/7.62 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 55.95/7.62 Axiom 7 (def_top_12): top = join(X, complement(X)).
% 55.95/7.62 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 55.95/7.62 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 55.95/7.62 Axiom 10 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 55.95/7.62 Axiom 11 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 55.95/7.62 Axiom 12 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 55.95/7.62 Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 55.95/7.62 Axiom 14 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 55.95/7.62 Axiom 15 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 55.95/7.62 Axiom 16 (dedekind_law_14): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
% 55.95/7.62
% 55.95/7.62 Lemma 17: complement(top) = zero.
% 55.95/7.62 Proof:
% 55.95/7.62 complement(top)
% 55.95/7.62 = { by axiom 7 (def_top_12) }
% 55.95/7.62 complement(join(complement(X), complement(complement(X))))
% 55.95/7.62 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.95/7.62 meet(X, complement(X))
% 55.95/7.62 = { by axiom 6 (def_zero_13) R->L }
% 55.95/7.62 zero
% 55.95/7.62
% 55.95/7.62 Lemma 18: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 55.95/7.62 Proof:
% 55.95/7.62 join(meet(X, Y), complement(join(complement(X), Y)))
% 55.95/7.62 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.62 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 55.95/7.62 = { by axiom 15 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 55.95/7.62 X
% 55.95/7.62
% 55.95/7.62 Lemma 19: join(zero, meet(X, X)) = X.
% 55.95/7.62 Proof:
% 55.95/7.62 join(zero, meet(X, X))
% 55.95/7.62 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.62 join(zero, complement(join(complement(X), complement(X))))
% 55.95/7.62 = { by axiom 6 (def_zero_13) }
% 55.95/7.62 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 55.95/7.62 = { by lemma 18 }
% 55.95/7.62 X
% 55.95/7.62
% 55.95/7.62 Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 55.95/7.62 Proof:
% 55.95/7.62 converse(composition(converse(X), Y))
% 55.95/7.62 = { by axiom 8 (converse_multiplicativity_10) }
% 55.95/7.62 composition(converse(Y), converse(converse(X)))
% 55.95/7.62 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.62 composition(converse(Y), X)
% 55.95/7.62
% 55.95/7.62 Lemma 21: composition(converse(one), X) = X.
% 55.95/7.62 Proof:
% 55.95/7.62 composition(converse(one), X)
% 55.95/7.62 = { by lemma 20 R->L }
% 55.95/7.62 converse(composition(converse(X), one))
% 55.95/7.63 = { by axiom 1 (composition_identity_6) }
% 55.95/7.63 converse(converse(X))
% 55.95/7.63 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 22: composition(one, X) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 composition(one, X)
% 55.95/7.63 = { by lemma 21 R->L }
% 55.95/7.63 composition(converse(one), composition(one, X))
% 55.95/7.63 = { by axiom 9 (composition_associativity_5) }
% 55.95/7.63 composition(composition(converse(one), one), X)
% 55.95/7.63 = { by axiom 1 (composition_identity_6) }
% 55.95/7.63 composition(converse(one), X)
% 55.95/7.63 = { by lemma 21 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 55.95/7.63 = { by axiom 14 (converse_cancellativity_11) }
% 55.95/7.63 complement(X)
% 55.95/7.63
% 55.95/7.63 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(X), complement(X))
% 55.95/7.63 = { by lemma 21 R->L }
% 55.95/7.63 join(complement(X), composition(converse(one), complement(X)))
% 55.95/7.63 = { by lemma 22 R->L }
% 55.95/7.63 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 55.95/7.63 = { by lemma 23 }
% 55.95/7.63 complement(X)
% 55.95/7.63
% 55.95/7.63 Lemma 25: join(zero, zero) = zero.
% 55.95/7.63 Proof:
% 55.95/7.63 join(zero, zero)
% 55.95/7.63 = { by lemma 17 R->L }
% 55.95/7.63 join(zero, complement(top))
% 55.95/7.63 = { by lemma 17 R->L }
% 55.95/7.63 join(complement(top), complement(top))
% 55.95/7.63 = { by lemma 24 }
% 55.95/7.63 complement(top)
% 55.95/7.63 = { by lemma 17 }
% 55.95/7.63 zero
% 55.95/7.63
% 55.95/7.63 Lemma 26: join(zero, join(zero, X)) = join(X, zero).
% 55.95/7.63 Proof:
% 55.95/7.63 join(zero, join(zero, X))
% 55.95/7.63 = { by axiom 11 (maddux2_join_associativity_2) }
% 55.95/7.63 join(join(zero, zero), X)
% 55.95/7.63 = { by lemma 25 }
% 55.95/7.63 join(zero, X)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(X, zero)
% 55.95/7.63
% 55.95/7.63 Lemma 27: join(X, zero) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, zero)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(zero, X)
% 55.95/7.63 = { by lemma 19 R->L }
% 55.95/7.63 join(zero, join(zero, meet(X, X)))
% 55.95/7.63 = { by lemma 26 }
% 55.95/7.63 join(meet(X, X), zero)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(zero, meet(X, X))
% 55.95/7.63 = { by lemma 19 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 28: join(zero, X) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 join(zero, X)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(X, zero)
% 55.95/7.63 = { by lemma 27 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 29: complement(zero) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 complement(zero)
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 join(zero, complement(zero))
% 55.95/7.63 = { by axiom 7 (def_top_12) R->L }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 30: converse(one) = one.
% 55.95/7.63 Proof:
% 55.95/7.63 converse(one)
% 55.95/7.63 = { by axiom 1 (composition_identity_6) R->L }
% 55.95/7.63 composition(converse(one), one)
% 55.95/7.63 = { by lemma 21 }
% 55.95/7.63 one
% 55.95/7.63
% 55.95/7.63 Lemma 31: join(X, join(Y, complement(X))) = join(Y, top).
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, join(Y, complement(X)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(X, join(complement(X), Y))
% 55.95/7.63 = { by axiom 11 (maddux2_join_associativity_2) }
% 55.95/7.63 join(join(X, complement(X)), Y)
% 55.95/7.63 = { by axiom 7 (def_top_12) R->L }
% 55.95/7.63 join(top, Y)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(Y, top)
% 55.95/7.63
% 55.95/7.63 Lemma 32: join(top, complement(X)) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 join(top, complement(X))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(complement(X), top)
% 55.95/7.63 = { by lemma 31 R->L }
% 55.95/7.63 join(X, join(complement(X), complement(X)))
% 55.95/7.63 = { by lemma 24 }
% 55.95/7.63 join(X, complement(X))
% 55.95/7.63 = { by axiom 7 (def_top_12) R->L }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 33: join(Y, top) = join(X, top).
% 55.95/7.63 Proof:
% 55.95/7.63 join(Y, top)
% 55.95/7.63 = { by lemma 32 R->L }
% 55.95/7.63 join(Y, join(top, complement(Y)))
% 55.95/7.63 = { by lemma 31 }
% 55.95/7.63 join(top, top)
% 55.95/7.63 = { by lemma 31 R->L }
% 55.95/7.63 join(X, join(top, complement(X)))
% 55.95/7.63 = { by lemma 32 }
% 55.95/7.63 join(X, top)
% 55.95/7.63
% 55.95/7.63 Lemma 34: join(sk1, join(one, X)) = join(X, one).
% 55.95/7.63 Proof:
% 55.95/7.63 join(sk1, join(one, X))
% 55.95/7.63 = { by axiom 11 (maddux2_join_associativity_2) }
% 55.95/7.63 join(join(sk1, one), X)
% 55.95/7.63 = { by axiom 3 (goals_17) }
% 55.95/7.63 join(one, X)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(X, one)
% 55.95/7.63
% 55.95/7.63 Lemma 35: join(X, top) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, top)
% 55.95/7.63 = { by lemma 33 }
% 55.95/7.63 join(sk1, top)
% 55.95/7.63 = { by axiom 7 (def_top_12) }
% 55.95/7.63 join(sk1, join(one, complement(one)))
% 55.95/7.63 = { by lemma 34 }
% 55.95/7.63 join(complement(one), one)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(one, complement(one))
% 55.95/7.63 = { by axiom 7 (def_top_12) R->L }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 36: join(top, X) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 join(top, X)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(X, top)
% 55.95/7.63 = { by lemma 33 R->L }
% 55.95/7.63 join(Y, top)
% 55.95/7.63 = { by lemma 35 }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 37: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 55.95/7.63 Proof:
% 55.95/7.63 converse(join(X, converse(Y)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 converse(join(converse(Y), X))
% 55.95/7.63 = { by axiom 10 (converse_additivity_9) }
% 55.95/7.63 join(converse(converse(Y)), converse(X))
% 55.95/7.63 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.63 join(Y, converse(X))
% 55.95/7.63
% 55.95/7.63 Lemma 38: converse(top) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 converse(top)
% 55.95/7.63 = { by lemma 36 R->L }
% 55.95/7.63 converse(join(top, converse(top)))
% 55.95/7.63 = { by lemma 37 }
% 55.95/7.63 join(top, converse(top))
% 55.95/7.63 = { by lemma 36 }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 39: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 converse(join(converse(X), Y))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 converse(join(Y, converse(X)))
% 55.95/7.63 = { by lemma 37 }
% 55.95/7.63 join(X, converse(Y))
% 55.95/7.63
% 55.95/7.63 Lemma 40: converse(zero) = zero.
% 55.95/7.63 Proof:
% 55.95/7.63 converse(zero)
% 55.95/7.63 = { by lemma 27 R->L }
% 55.95/7.63 join(converse(zero), zero)
% 55.95/7.63 = { by lemma 26 R->L }
% 55.95/7.63 join(zero, join(zero, converse(zero)))
% 55.95/7.63 = { by lemma 39 R->L }
% 55.95/7.63 join(zero, converse(join(converse(zero), zero)))
% 55.95/7.63 = { by lemma 27 }
% 55.95/7.63 join(zero, converse(converse(zero)))
% 55.95/7.63 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.63 join(zero, zero)
% 55.95/7.63 = { by lemma 25 }
% 55.95/7.63 zero
% 55.95/7.63
% 55.95/7.63 Lemma 41: complement(complement(X)) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 complement(complement(X))
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 join(zero, complement(complement(X)))
% 55.95/7.63 = { by lemma 24 R->L }
% 55.95/7.63 join(zero, complement(join(complement(X), complement(X))))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.95/7.63 join(zero, meet(X, X))
% 55.95/7.63 = { by lemma 19 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 42: meet(Y, X) = meet(X, Y).
% 55.95/7.63 Proof:
% 55.95/7.63 meet(Y, X)
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.63 complement(join(complement(Y), complement(X)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 complement(join(complement(X), complement(Y)))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.95/7.63 meet(X, Y)
% 55.95/7.63
% 55.95/7.63 Lemma 43: join(meet(X, Y), meet(X, complement(Y))) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 join(meet(X, Y), meet(X, complement(Y)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(meet(X, complement(Y)), meet(X, Y))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.63 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 55.95/7.63 = { by lemma 18 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 44: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 55.95/7.63 Proof:
% 55.95/7.63 join(meet(X, Y), meet(Y, complement(X)))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(meet(Y, X), meet(Y, complement(X)))
% 55.95/7.63 = { by lemma 43 }
% 55.95/7.63 Y
% 55.95/7.63
% 55.95/7.63 Lemma 45: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 55.95/7.63 Proof:
% 55.95/7.63 join(meet(X, Y), meet(complement(X), Y))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(meet(X, Y), meet(Y, complement(X)))
% 55.95/7.63 = { by lemma 44 }
% 55.95/7.63 Y
% 55.95/7.63
% 55.95/7.63 Lemma 46: complement(join(zero, complement(X))) = meet(X, top).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(join(zero, complement(X)))
% 55.95/7.63 = { by lemma 17 R->L }
% 55.95/7.63 complement(join(complement(top), complement(X)))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.95/7.63 meet(top, X)
% 55.95/7.63 = { by lemma 42 R->L }
% 55.95/7.63 meet(X, top)
% 55.95/7.63
% 55.95/7.63 Lemma 47: meet(X, top) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 meet(X, top)
% 55.95/7.63 = { by lemma 46 R->L }
% 55.95/7.63 complement(join(zero, complement(X)))
% 55.95/7.63 = { by lemma 28 }
% 55.95/7.63 complement(complement(X))
% 55.95/7.63 = { by lemma 41 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 48: meet(top, X) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 meet(top, X)
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 meet(X, top)
% 55.95/7.63 = { by lemma 47 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 49: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(join(complement(X), meet(Y, Z)))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 complement(join(complement(X), meet(Z, Y)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 complement(join(meet(Z, Y), complement(X)))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.63 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 55.95/7.63 meet(join(complement(Z), complement(Y)), X)
% 55.95/7.63 = { by lemma 42 R->L }
% 55.95/7.63 meet(X, join(complement(Z), complement(Y)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 meet(X, join(complement(Y), complement(Z)))
% 55.95/7.63
% 55.95/7.63 Lemma 50: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(X), complement(Y))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 join(complement(Y), complement(X))
% 55.95/7.63 = { by lemma 48 R->L }
% 55.95/7.63 meet(top, join(complement(Y), complement(X)))
% 55.95/7.63 = { by lemma 49 R->L }
% 55.95/7.63 complement(join(complement(top), meet(Y, X)))
% 55.95/7.63 = { by lemma 17 }
% 55.95/7.63 complement(join(zero, meet(Y, X)))
% 55.95/7.63 = { by lemma 28 }
% 55.95/7.63 complement(meet(Y, X))
% 55.95/7.63 = { by lemma 42 R->L }
% 55.95/7.63 complement(meet(X, Y))
% 55.95/7.63
% 55.95/7.63 Lemma 51: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(meet(X, complement(Y)))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 complement(meet(complement(Y), X))
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 complement(meet(join(zero, complement(Y)), X))
% 55.95/7.63 = { by lemma 50 R->L }
% 55.95/7.63 join(complement(join(zero, complement(Y))), complement(X))
% 55.95/7.63 = { by lemma 46 }
% 55.95/7.63 join(meet(Y, top), complement(X))
% 55.95/7.63 = { by lemma 47 }
% 55.95/7.63 join(Y, complement(X))
% 55.95/7.63
% 55.95/7.63 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(join(X, complement(Y)))
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 complement(join(zero, join(X, complement(Y))))
% 55.95/7.63 = { by lemma 51 R->L }
% 55.95/7.63 complement(join(zero, complement(meet(Y, complement(X)))))
% 55.95/7.63 = { by lemma 46 }
% 55.95/7.63 meet(meet(Y, complement(X)), top)
% 55.95/7.63 = { by lemma 47 }
% 55.95/7.63 meet(Y, complement(X))
% 55.95/7.63
% 55.95/7.63 Lemma 53: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(join(complement(X), Y))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.63 complement(join(Y, complement(X)))
% 55.95/7.63 = { by lemma 52 }
% 55.95/7.63 meet(X, complement(Y))
% 55.95/7.63
% 55.95/7.63 Lemma 54: join(X, converse(complement(converse(X)))) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, converse(complement(converse(X))))
% 55.95/7.63 = { by lemma 39 R->L }
% 55.95/7.63 converse(join(converse(X), complement(converse(X))))
% 55.95/7.63 = { by axiom 7 (def_top_12) R->L }
% 55.95/7.63 converse(top)
% 55.95/7.63 = { by lemma 38 }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 55: join(X, composition(Y, X)) = composition(join(Y, one), X).
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, composition(Y, X))
% 55.95/7.63 = { by lemma 22 R->L }
% 55.95/7.63 join(composition(one, X), composition(Y, X))
% 55.95/7.63 = { by axiom 13 (composition_distributivity_7) R->L }
% 55.95/7.63 composition(join(one, Y), X)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 composition(join(Y, one), X)
% 55.95/7.63
% 55.95/7.63 Lemma 56: composition(top, zero) = zero.
% 55.95/7.63 Proof:
% 55.95/7.63 composition(top, zero)
% 55.95/7.63 = { by lemma 17 R->L }
% 55.95/7.63 composition(top, complement(top))
% 55.95/7.63 = { by lemma 36 R->L }
% 55.95/7.63 composition(join(top, one), complement(top))
% 55.95/7.63 = { by lemma 38 R->L }
% 55.95/7.63 composition(join(converse(top), one), complement(top))
% 55.95/7.63 = { by lemma 55 R->L }
% 55.95/7.63 join(complement(top), composition(converse(top), complement(top)))
% 55.95/7.63 = { by lemma 36 R->L }
% 55.95/7.63 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 55.95/7.63 = { by lemma 55 }
% 55.95/7.63 join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
% 55.95/7.63 = { by lemma 36 }
% 55.95/7.63 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 55.95/7.63 = { by lemma 23 }
% 55.95/7.63 complement(top)
% 55.95/7.63 = { by lemma 17 }
% 55.95/7.63 zero
% 55.95/7.63
% 55.95/7.63 Lemma 57: composition(X, zero) = zero.
% 55.95/7.63 Proof:
% 55.95/7.63 composition(X, zero)
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 join(zero, composition(X, zero))
% 55.95/7.63 = { by lemma 56 R->L }
% 55.95/7.63 join(composition(top, zero), composition(X, zero))
% 55.95/7.63 = { by axiom 13 (composition_distributivity_7) R->L }
% 55.95/7.63 composition(join(top, X), zero)
% 55.95/7.63 = { by lemma 36 }
% 55.95/7.63 composition(top, zero)
% 55.95/7.63 = { by lemma 56 }
% 55.95/7.63 zero
% 55.95/7.63
% 55.95/7.63 Lemma 58: composition(zero, X) = zero.
% 55.95/7.63 Proof:
% 55.95/7.63 composition(zero, X)
% 55.95/7.63 = { by lemma 40 R->L }
% 55.95/7.63 composition(converse(zero), X)
% 55.95/7.63 = { by lemma 20 R->L }
% 55.95/7.63 converse(composition(converse(X), zero))
% 55.95/7.63 = { by lemma 57 }
% 55.95/7.63 converse(zero)
% 55.95/7.63 = { by lemma 40 }
% 55.95/7.63 zero
% 55.95/7.63
% 55.95/7.63 Lemma 59: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 meet(complement(X), complement(Y))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 meet(complement(Y), complement(X))
% 55.95/7.63 = { by lemma 28 R->L }
% 55.95/7.63 meet(join(zero, complement(Y)), complement(X))
% 55.95/7.63 = { by lemma 52 R->L }
% 55.95/7.63 complement(join(X, complement(join(zero, complement(Y)))))
% 55.95/7.63 = { by lemma 46 }
% 55.95/7.63 complement(join(X, meet(Y, top)))
% 55.95/7.63 = { by lemma 47 }
% 55.95/7.63 complement(join(X, Y))
% 55.95/7.63
% 55.95/7.63 Lemma 60: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 complement(meet(complement(X), Y))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 complement(meet(Y, complement(X)))
% 55.95/7.63 = { by lemma 51 }
% 55.95/7.63 join(X, complement(Y))
% 55.95/7.63
% 55.95/7.63 Lemma 61: join(X, complement(meet(X, Y))) = top.
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, complement(meet(X, Y)))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(X, complement(meet(Y, X)))
% 55.95/7.63 = { by lemma 50 R->L }
% 55.95/7.63 join(X, join(complement(Y), complement(X)))
% 55.95/7.63 = { by lemma 31 }
% 55.95/7.63 join(complement(Y), top)
% 55.95/7.63 = { by lemma 35 }
% 55.95/7.63 top
% 55.95/7.63
% 55.95/7.63 Lemma 62: meet(X, join(X, complement(Y))) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 meet(X, join(X, complement(Y)))
% 55.95/7.63 = { by lemma 27 R->L }
% 55.95/7.63 join(meet(X, join(X, complement(Y))), zero)
% 55.95/7.63 = { by lemma 17 R->L }
% 55.95/7.63 join(meet(X, join(X, complement(Y))), complement(top))
% 55.95/7.63 = { by lemma 60 R->L }
% 55.95/7.63 join(meet(X, complement(meet(complement(X), Y))), complement(top))
% 55.95/7.63 = { by lemma 61 R->L }
% 55.95/7.63 join(meet(X, complement(meet(complement(X), Y))), complement(join(complement(X), complement(meet(complement(X), Y)))))
% 55.95/7.63 = { by lemma 18 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 63: join(X, meet(X, Y)) = X.
% 55.95/7.63 Proof:
% 55.95/7.63 join(X, meet(X, Y))
% 55.95/7.63 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.63 join(X, complement(join(complement(X), complement(Y))))
% 55.95/7.63 = { by lemma 60 R->L }
% 55.95/7.63 complement(meet(complement(X), join(complement(X), complement(Y))))
% 55.95/7.63 = { by lemma 62 }
% 55.95/7.63 complement(complement(X))
% 55.95/7.63 = { by lemma 41 }
% 55.95/7.63 X
% 55.95/7.63
% 55.95/7.63 Lemma 64: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(X), meet(X, Y))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(complement(X), meet(Y, X))
% 55.95/7.63 = { by lemma 63 R->L }
% 55.95/7.63 join(join(complement(X), meet(complement(X), Y)), meet(Y, X))
% 55.95/7.63 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 55.95/7.63 join(complement(X), join(meet(complement(X), Y), meet(Y, X)))
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(complement(X), join(meet(Y, X), meet(complement(X), Y)))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(complement(X), join(meet(Y, X), meet(Y, complement(X))))
% 55.95/7.63 = { by lemma 43 }
% 55.95/7.63 join(complement(X), Y)
% 55.95/7.63 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.63 join(Y, complement(X))
% 55.95/7.63
% 55.95/7.63 Lemma 65: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
% 55.95/7.63 Proof:
% 55.95/7.63 meet(X, complement(meet(X, Y)))
% 55.95/7.63 = { by lemma 50 R->L }
% 55.95/7.63 meet(X, join(complement(X), complement(Y)))
% 55.95/7.63 = { by lemma 49 R->L }
% 55.95/7.63 complement(join(complement(X), meet(X, Y)))
% 55.95/7.63 = { by lemma 64 }
% 55.95/7.63 complement(join(Y, complement(X)))
% 55.95/7.63 = { by lemma 52 }
% 55.95/7.63 meet(X, complement(Y))
% 55.95/7.63
% 55.95/7.63 Lemma 66: join(complement(one), converse(complement(one))) = complement(one).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(one), converse(complement(one)))
% 55.95/7.63 = { by lemma 47 R->L }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), top)))
% 55.95/7.63 = { by lemma 54 R->L }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), join(one, converse(complement(converse(one)))))))
% 55.95/7.63 = { by lemma 30 }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), join(one, converse(complement(one))))))
% 55.95/7.63 = { by lemma 42 }
% 55.95/7.63 join(complement(one), converse(meet(join(one, converse(complement(one))), complement(one))))
% 55.95/7.63 = { by lemma 52 R->L }
% 55.95/7.63 join(complement(one), converse(complement(join(one, complement(join(one, converse(complement(one))))))))
% 55.95/7.63 = { by lemma 59 R->L }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), complement(complement(join(one, converse(complement(one))))))))
% 55.95/7.63 = { by lemma 59 R->L }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), complement(meet(complement(one), complement(converse(complement(one))))))))
% 55.95/7.63 = { by lemma 65 }
% 55.95/7.63 join(complement(one), converse(meet(complement(one), complement(complement(converse(complement(one)))))))
% 55.95/7.63 = { by lemma 59 }
% 55.95/7.63 join(complement(one), converse(complement(join(one, complement(converse(complement(one)))))))
% 55.95/7.63 = { by lemma 52 }
% 55.95/7.63 join(complement(one), converse(meet(converse(complement(one)), complement(one))))
% 55.95/7.63 = { by lemma 39 R->L }
% 55.95/7.63 converse(join(converse(complement(one)), meet(converse(complement(one)), complement(one))))
% 55.95/7.63 = { by lemma 63 }
% 55.95/7.63 converse(converse(complement(one)))
% 55.95/7.63 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.63 complement(one)
% 55.95/7.63
% 55.95/7.63 Lemma 67: converse(complement(one)) = complement(one).
% 55.95/7.63 Proof:
% 55.95/7.63 converse(complement(one))
% 55.95/7.63 = { by lemma 66 R->L }
% 55.95/7.63 converse(join(complement(one), converse(complement(one))))
% 55.95/7.63 = { by lemma 37 }
% 55.95/7.63 join(complement(one), converse(complement(one)))
% 55.95/7.63 = { by lemma 66 }
% 55.95/7.63 complement(one)
% 55.95/7.63
% 55.95/7.63 Lemma 68: join(complement(one), composition(converse(X), complement(X))) = complement(one).
% 55.95/7.63 Proof:
% 55.95/7.63 join(complement(one), composition(converse(X), complement(X)))
% 55.95/7.63 = { by axiom 1 (composition_identity_6) R->L }
% 55.95/7.63 join(complement(one), composition(converse(X), complement(composition(X, one))))
% 55.95/7.63 = { by lemma 23 }
% 55.95/7.63 complement(one)
% 55.95/7.64
% 55.95/7.64 Lemma 69: converse(join(X, composition(converse(Y), Z))) = join(converse(X), composition(converse(Z), Y)).
% 55.95/7.64 Proof:
% 55.95/7.64 converse(join(X, composition(converse(Y), Z)))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.64 converse(join(composition(converse(Y), Z), X))
% 55.95/7.64 = { by axiom 10 (converse_additivity_9) }
% 55.95/7.64 join(converse(composition(converse(Y), Z)), converse(X))
% 55.95/7.64 = { by lemma 20 }
% 55.95/7.64 join(composition(converse(Z), Y), converse(X))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.64 join(converse(X), composition(converse(Z), Y))
% 55.95/7.64
% 55.95/7.64 Lemma 70: meet(X, complement(join(Y, X))) = zero.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, complement(join(Y, X)))
% 55.95/7.64 = { by lemma 59 R->L }
% 55.95/7.64 meet(X, meet(complement(Y), complement(X)))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(X, meet(complement(X), complement(Y)))
% 55.95/7.64 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.64 complement(join(complement(X), complement(meet(complement(X), complement(Y)))))
% 55.95/7.64 = { by lemma 61 }
% 55.95/7.64 complement(top)
% 55.95/7.64 = { by lemma 17 }
% 55.95/7.64 zero
% 55.95/7.64
% 55.95/7.64 Lemma 71: meet(one, composition(converse(complement(X)), X)) = zero.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(one, composition(converse(complement(X)), X))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(composition(converse(complement(X)), X), one)
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(complement(one)))
% 55.95/7.64 = { by lemma 67 R->L }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(converse(complement(one))))
% 55.95/7.64 = { by lemma 68 R->L }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(converse(converse(X))), complement(converse(converse(X))))))))
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(converse(join(complement(one), composition(converse(X), complement(converse(converse(X))))))))
% 55.95/7.64 = { by lemma 69 }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(join(converse(complement(one)), composition(converse(complement(converse(converse(X)))), X))))
% 55.95/7.64 = { by lemma 67 }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(converse(converse(X)))), X))))
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.64 meet(composition(converse(complement(X)), X), complement(join(complement(one), composition(converse(complement(X)), X))))
% 55.95/7.64 = { by lemma 70 }
% 55.95/7.64 zero
% 55.95/7.64
% 55.95/7.64 Lemma 72: converse(complement(converse(X))) = complement(X).
% 55.95/7.64 Proof:
% 55.95/7.64 converse(complement(converse(X)))
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 converse(complement(converse(complement(complement(X)))))
% 55.95/7.64 = { by lemma 45 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), converse(complement(converse(complement(complement(X)))))))
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), meet(complement(X), complement(complement(converse(complement(converse(complement(complement(X)))))))))
% 55.95/7.64 = { by lemma 53 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X)))))))))
% 55.95/7.64 = { by lemma 48 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(top, join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 55.95/7.64 = { by lemma 54 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))), join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))))))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), converse(complement(converse(complement(complement(X)))))))))
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(converse(complement(converse(complement(complement(X))))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 55.95/7.64 = { by lemma 51 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))), join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(meet(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))), complement(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X))))))))
% 55.95/7.64 = { by lemma 52 R->L }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), complement(join(complement(complement(X)), complement(complement(converse(complement(converse(complement(complement(X)))))))))))))
% 55.95/7.64 = { by lemma 52 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(join(meet(converse(complement(converse(complement(complement(X))))), complement(complement(complement(X)))), meet(complement(converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))))))
% 55.95/7.64 = { by lemma 45 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(complement(complement(X))))))
% 55.95/7.64 = { by lemma 41 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(complement(complement(X))))
% 55.95/7.64 = { by lemma 41 }
% 55.95/7.64 join(meet(X, converse(complement(converse(complement(complement(X)))))), complement(X))
% 55.95/7.64 = { by lemma 41 }
% 55.95/7.64 join(meet(X, converse(complement(converse(X)))), complement(X))
% 55.95/7.64 = { by lemma 27 R->L }
% 55.95/7.64 join(join(meet(X, converse(complement(converse(X)))), zero), complement(X))
% 55.95/7.64 = { by lemma 58 R->L }
% 55.95/7.64 join(join(meet(X, converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 55.95/7.64 = { by lemma 22 R->L }
% 55.95/7.64 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(zero, meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 55.95/7.64 = { by lemma 71 R->L }
% 55.95/7.64 join(join(meet(composition(one, X), converse(complement(converse(X)))), composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X))))))), complement(X))
% 55.95/7.64 = { by axiom 16 (dedekind_law_14) }
% 55.95/7.64 join(composition(meet(one, composition(converse(complement(converse(X))), converse(X))), meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 55.95/7.64 = { by lemma 71 }
% 55.95/7.64 join(composition(zero, meet(X, composition(converse(one), converse(complement(converse(X)))))), complement(X))
% 55.95/7.64 = { by lemma 58 }
% 55.95/7.64 join(zero, complement(X))
% 55.95/7.64 = { by lemma 28 }
% 55.95/7.64 complement(X)
% 55.95/7.64
% 55.95/7.64 Lemma 73: complement(converse(X)) = converse(complement(X)).
% 55.95/7.64 Proof:
% 55.95/7.64 complement(converse(X))
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) R->L }
% 55.95/7.64 converse(converse(complement(converse(X))))
% 55.95/7.64 = { by lemma 72 }
% 55.95/7.64 converse(complement(X))
% 55.95/7.64
% 55.95/7.64 Lemma 74: join(complement(X), meet(Y, X)) = join(Y, complement(X)).
% 55.95/7.64 Proof:
% 55.95/7.64 join(complement(X), meet(Y, X))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 join(complement(X), meet(X, Y))
% 55.95/7.64 = { by lemma 64 }
% 55.95/7.64 join(Y, complement(X))
% 55.95/7.64
% 55.95/7.64 Lemma 75: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 55.95/7.64 Proof:
% 55.95/7.64 meet(Y, meet(X, Z))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(Y, meet(Z, X))
% 55.95/7.64 = { by lemma 47 R->L }
% 55.95/7.64 meet(meet(Y, meet(Z, X)), top)
% 55.95/7.64 = { by lemma 46 R->L }
% 55.95/7.64 complement(join(zero, complement(meet(Y, meet(Z, X)))))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 complement(join(zero, complement(meet(Y, meet(X, Z)))))
% 55.95/7.64 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.64 complement(join(zero, complement(meet(Y, complement(join(complement(X), complement(Z)))))))
% 55.95/7.64 = { by lemma 51 }
% 55.95/7.64 complement(join(zero, join(join(complement(X), complement(Z)), complement(Y))))
% 55.95/7.64 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 55.95/7.64 complement(join(zero, join(complement(X), join(complement(Z), complement(Y)))))
% 55.95/7.64 = { by lemma 50 }
% 55.95/7.64 complement(join(zero, join(complement(X), complement(meet(Z, Y)))))
% 55.95/7.64 = { by lemma 50 }
% 55.95/7.64 complement(join(zero, complement(meet(X, meet(Z, Y)))))
% 55.95/7.64 = { by lemma 42 R->L }
% 55.95/7.64 complement(join(zero, complement(meet(X, meet(Y, Z)))))
% 55.95/7.64 = { by lemma 46 }
% 55.95/7.64 meet(meet(X, meet(Y, Z)), top)
% 55.95/7.64 = { by lemma 47 }
% 55.95/7.64 meet(X, meet(Y, Z))
% 55.95/7.64
% 55.95/7.64 Lemma 76: meet(X, join(X, Y)) = X.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, join(X, Y))
% 55.95/7.64 = { by lemma 47 R->L }
% 55.95/7.64 meet(X, join(X, meet(Y, top)))
% 55.95/7.64 = { by lemma 46 R->L }
% 55.95/7.64 meet(X, join(X, complement(join(zero, complement(Y)))))
% 55.95/7.64 = { by lemma 62 }
% 55.95/7.64 X
% 55.95/7.64
% 55.95/7.64 Lemma 77: meet(X, meet(Y, composition(top, X))) = meet(X, Y).
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, meet(Y, composition(top, X)))
% 55.95/7.64 = { by lemma 75 }
% 55.95/7.64 meet(Y, meet(X, composition(top, X)))
% 55.95/7.64 = { by lemma 36 R->L }
% 55.95/7.64 meet(Y, meet(X, composition(join(top, one), X)))
% 55.95/7.64 = { by lemma 55 R->L }
% 55.95/7.64 meet(Y, meet(X, join(X, composition(top, X))))
% 55.95/7.64 = { by lemma 76 }
% 55.95/7.64 meet(Y, X)
% 55.95/7.64 = { by lemma 42 R->L }
% 55.95/7.64 meet(X, Y)
% 55.95/7.64
% 55.95/7.64 Lemma 78: meet(X, join(complement(X), Y)) = meet(X, Y).
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, join(complement(X), Y))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.64 meet(X, join(Y, complement(X)))
% 55.95/7.64 = { by lemma 51 R->L }
% 55.95/7.64 meet(X, complement(meet(X, complement(Y))))
% 55.95/7.64 = { by lemma 65 }
% 55.95/7.64 meet(X, complement(complement(Y)))
% 55.95/7.64 = { by lemma 41 }
% 55.95/7.64 meet(X, Y)
% 55.95/7.64
% 55.95/7.64 Lemma 79: converse(join(X, one)) = join(one, converse(X)).
% 55.95/7.64 Proof:
% 55.95/7.64 converse(join(X, one))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.64 converse(join(one, X))
% 55.95/7.64 = { by axiom 10 (converse_additivity_9) }
% 55.95/7.64 join(converse(one), converse(X))
% 55.95/7.64 = { by lemma 30 }
% 55.95/7.64 join(one, converse(X))
% 55.95/7.64
% 55.95/7.64 Lemma 80: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
% 55.95/7.64 Proof:
% 55.95/7.64 converse(composition(X, converse(Y)))
% 55.95/7.64 = { by axiom 8 (converse_multiplicativity_10) }
% 55.95/7.64 composition(converse(converse(Y)), converse(X))
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.64 composition(Y, converse(X))
% 55.95/7.64
% 55.95/7.64 Lemma 81: join(composition(X, join(Z, one)), Y) = join(X, join(Y, composition(X, Z))).
% 55.95/7.64 Proof:
% 55.95/7.64 join(composition(X, join(Z, one)), Y)
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.64 join(composition(X, join(one, Z)), Y)
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) R->L }
% 55.95/7.64 join(composition(X, join(one, converse(converse(Z)))), Y)
% 55.95/7.64 = { by lemma 79 R->L }
% 55.95/7.64 join(composition(X, converse(join(converse(Z), one))), Y)
% 55.95/7.64 = { by lemma 80 R->L }
% 55.95/7.64 join(converse(composition(join(converse(Z), one), converse(X))), Y)
% 55.95/7.64 = { by lemma 55 R->L }
% 55.95/7.64 join(converse(join(converse(X), composition(converse(Z), converse(X)))), Y)
% 55.95/7.64 = { by lemma 39 }
% 55.95/7.64 join(join(X, converse(composition(converse(Z), converse(X)))), Y)
% 55.95/7.64 = { by lemma 80 }
% 55.95/7.64 join(join(X, composition(X, converse(converse(Z)))), Y)
% 55.95/7.64 = { by axiom 5 (converse_idempotence_8) }
% 55.95/7.64 join(join(X, composition(X, Z)), Y)
% 55.95/7.64 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 55.95/7.64 join(X, join(composition(X, Z), Y))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.64 join(X, join(Y, composition(X, Z)))
% 55.95/7.64
% 55.95/7.64 Lemma 82: meet(sk1, meet(one, X)) = meet(X, sk1).
% 55.95/7.64 Proof:
% 55.95/7.64 meet(sk1, meet(one, X))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(sk1, meet(X, one))
% 55.95/7.64 = { by lemma 75 }
% 55.95/7.64 meet(X, meet(sk1, one))
% 55.95/7.64 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.64 meet(X, complement(join(complement(sk1), complement(one))))
% 55.95/7.64 = { by lemma 28 R->L }
% 55.95/7.64 meet(X, join(zero, complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 17 R->L }
% 55.95/7.64 meet(X, join(complement(top), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 35 R->L }
% 55.95/7.64 meet(X, join(complement(join(one, top)), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 31 R->L }
% 55.95/7.64 meet(X, join(complement(join(sk1, join(one, complement(sk1)))), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 34 }
% 55.95/7.64 meet(X, join(complement(join(complement(sk1), one)), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 55.95/7.64 meet(X, join(complement(join(one, complement(sk1))), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 52 }
% 55.95/7.64 meet(X, join(meet(sk1, complement(one)), complement(join(complement(sk1), complement(one)))))
% 55.95/7.64 = { by lemma 18 }
% 55.95/7.64 meet(X, sk1)
% 55.95/7.64
% 55.95/7.64 Lemma 83: meet(one, composition(converse(X), complement(X))) = zero.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(one, composition(converse(X), complement(X)))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(composition(converse(X), complement(X)), one)
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 meet(composition(converse(X), complement(X)), complement(complement(one)))
% 55.95/7.64 = { by lemma 68 R->L }
% 55.95/7.64 meet(composition(converse(X), complement(X)), complement(join(complement(one), composition(converse(X), complement(X)))))
% 55.95/7.64 = { by lemma 70 }
% 55.95/7.64 zero
% 55.95/7.64
% 55.95/7.64 Lemma 84: meet(X, zero) = zero.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, zero)
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(zero, X)
% 55.95/7.64 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 55.95/7.64 complement(join(complement(zero), complement(X)))
% 55.95/7.64 = { by lemma 29 }
% 55.95/7.64 complement(join(top, complement(X)))
% 55.95/7.64 = { by lemma 32 }
% 55.95/7.64 complement(top)
% 55.95/7.64 = { by lemma 17 }
% 55.95/7.64 zero
% 55.95/7.64
% 55.95/7.64 Lemma 85: meet(sk1, composition(converse(X), complement(X))) = zero.
% 55.95/7.64 Proof:
% 55.95/7.64 meet(sk1, composition(converse(X), complement(X)))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(composition(converse(X), complement(X)), sk1)
% 55.95/7.64 = { by lemma 82 R->L }
% 55.95/7.64 meet(sk1, meet(one, composition(converse(X), complement(X))))
% 55.95/7.64 = { by lemma 83 }
% 55.95/7.64 meet(sk1, zero)
% 55.95/7.64 = { by lemma 84 }
% 55.95/7.64 zero
% 55.95/7.64
% 55.95/7.64 Lemma 86: meet(X, composition(top, sk1)) = composition(X, sk1).
% 55.95/7.64 Proof:
% 55.95/7.64 meet(X, composition(top, sk1))
% 55.95/7.64 = { by lemma 78 R->L }
% 55.95/7.64 meet(X, join(complement(X), composition(top, sk1)))
% 55.95/7.64 = { by axiom 7 (def_top_12) }
% 55.95/7.64 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), sk1)))
% 55.95/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 55.95/7.64 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), sk1)))
% 55.95/7.64 = { by axiom 13 (composition_distributivity_7) }
% 55.95/7.64 meet(X, join(complement(X), join(composition(complement(complement(X)), sk1), composition(complement(X), sk1))))
% 55.95/7.64 = { by lemma 81 R->L }
% 55.95/7.64 meet(X, join(composition(complement(X), join(sk1, one)), composition(complement(complement(X)), sk1)))
% 55.95/7.64 = { by axiom 3 (goals_17) }
% 55.95/7.64 meet(X, join(composition(complement(X), one), composition(complement(complement(X)), sk1)))
% 55.95/7.64 = { by axiom 1 (composition_identity_6) }
% 55.95/7.64 meet(X, join(complement(X), composition(complement(complement(X)), sk1)))
% 55.95/7.64 = { by lemma 78 }
% 55.95/7.64 meet(X, composition(complement(complement(X)), sk1))
% 55.95/7.64 = { by lemma 41 }
% 55.95/7.64 meet(X, composition(X, sk1))
% 55.95/7.64 = { by lemma 42 }
% 55.95/7.64 meet(composition(X, sk1), X)
% 55.95/7.64 = { by lemma 41 R->L }
% 55.95/7.64 meet(composition(X, sk1), complement(complement(X)))
% 55.95/7.64 = { by lemma 28 R->L }
% 55.95/7.64 join(zero, meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 57 R->L }
% 56.68/7.64 join(composition(meet(X, composition(complement(X), converse(sk1))), zero), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 85 R->L }
% 56.68/7.64 join(composition(meet(X, composition(complement(X), converse(sk1))), meet(sk1, composition(converse(X), complement(X)))), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by axiom 16 (dedekind_law_14) R->L }
% 56.68/7.64 join(join(meet(composition(X, sk1), complement(X)), composition(meet(X, composition(complement(X), converse(sk1))), meet(sk1, composition(converse(X), complement(X))))), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 85 }
% 56.68/7.64 join(join(meet(composition(X, sk1), complement(X)), composition(meet(X, composition(complement(X), converse(sk1))), zero)), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 57 }
% 56.68/7.64 join(join(meet(composition(X, sk1), complement(X)), zero), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 27 }
% 56.68/7.64 join(meet(composition(X, sk1), complement(X)), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 42 R->L }
% 56.68/7.64 join(meet(complement(X), composition(X, sk1)), meet(composition(X, sk1), complement(complement(X))))
% 56.68/7.64 = { by lemma 44 }
% 56.68/7.64 composition(X, sk1)
% 56.68/7.64
% 56.68/7.64 Lemma 87: join(sk1, converse(sk1)) = converse(sk1).
% 56.68/7.64 Proof:
% 56.68/7.64 join(sk1, converse(sk1))
% 56.68/7.64 = { by lemma 41 R->L }
% 56.68/7.64 join(sk1, converse(complement(complement(sk1))))
% 56.68/7.64 = { by lemma 73 R->L }
% 56.68/7.64 join(sk1, complement(converse(complement(sk1))))
% 56.68/7.64 = { by lemma 74 R->L }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(sk1, converse(complement(sk1))))
% 56.68/7.64 = { by lemma 77 R->L }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(sk1, meet(converse(complement(sk1)), composition(top, sk1))))
% 56.68/7.64 = { by lemma 86 }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(sk1, composition(converse(complement(sk1)), sk1)))
% 56.68/7.64 = { by lemma 42 }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(composition(converse(complement(sk1)), sk1), sk1))
% 56.68/7.64 = { by lemma 82 R->L }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(sk1, meet(one, composition(converse(complement(sk1)), sk1))))
% 56.68/7.64 = { by lemma 71 }
% 56.68/7.64 join(complement(converse(complement(sk1))), meet(sk1, zero))
% 56.68/7.64 = { by lemma 84 }
% 56.68/7.64 join(complement(converse(complement(sk1))), zero)
% 56.68/7.64 = { by lemma 27 }
% 56.68/7.64 complement(converse(complement(sk1)))
% 56.68/7.64 = { by lemma 73 }
% 56.68/7.64 converse(complement(complement(sk1)))
% 56.68/7.64 = { by lemma 41 }
% 56.68/7.64 converse(sk1)
% 56.68/7.64
% 56.68/7.64 Lemma 88: converse(sk1) = sk1.
% 56.68/7.64 Proof:
% 56.68/7.64 converse(sk1)
% 56.68/7.64 = { by lemma 87 R->L }
% 56.68/7.64 join(sk1, converse(sk1))
% 56.68/7.64 = { by lemma 37 R->L }
% 56.68/7.64 converse(join(sk1, converse(sk1)))
% 56.68/7.64 = { by lemma 87 }
% 56.68/7.64 converse(converse(sk1))
% 56.68/7.64 = { by axiom 5 (converse_idempotence_8) }
% 56.68/7.64 sk1
% 56.68/7.64
% 56.68/7.64 Lemma 89: join(one, converse(sk1)) = one.
% 56.68/7.64 Proof:
% 56.68/7.64 join(one, converse(sk1))
% 56.68/7.64 = { by lemma 79 R->L }
% 56.68/7.64 converse(join(sk1, one))
% 56.68/7.64 = { by axiom 3 (goals_17) }
% 56.68/7.64 converse(one)
% 56.68/7.64 = { by lemma 30 }
% 56.68/7.64 one
% 56.68/7.64
% 56.68/7.64 Lemma 90: join(one, converse(sk2)) = one.
% 56.68/7.64 Proof:
% 56.68/7.64 join(one, converse(sk2))
% 56.68/7.64 = { by lemma 79 R->L }
% 56.68/7.64 converse(join(sk2, one))
% 56.68/7.64 = { by axiom 4 (goals_18) }
% 56.68/7.64 converse(one)
% 56.68/7.64 = { by lemma 30 }
% 56.68/7.64 one
% 56.68/7.64
% 56.68/7.64 Lemma 91: meet(X, join(Y, X)) = X.
% 56.68/7.64 Proof:
% 56.68/7.64 meet(X, join(Y, X))
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.64 meet(X, join(X, Y))
% 56.68/7.64 = { by lemma 76 }
% 56.68/7.64 X
% 56.68/7.64
% 56.68/7.64 Lemma 92: meet(one, converse(sk2)) = converse(sk2).
% 56.68/7.64 Proof:
% 56.68/7.64 meet(one, converse(sk2))
% 56.68/7.64 = { by lemma 42 }
% 56.68/7.64 meet(converse(sk2), one)
% 56.68/7.64 = { by lemma 90 R->L }
% 56.68/7.64 meet(converse(sk2), join(one, converse(sk2)))
% 56.68/7.64 = { by lemma 91 }
% 56.68/7.64 converse(sk2)
% 56.68/7.64
% 56.68/7.64 Lemma 93: join(X, join(Y, meet(X, Z))) = join(Y, X).
% 56.68/7.64 Proof:
% 56.68/7.64 join(X, join(Y, meet(X, Z)))
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.64 join(X, join(meet(X, Z), Y))
% 56.68/7.64 = { by axiom 11 (maddux2_join_associativity_2) }
% 56.68/7.64 join(join(X, meet(X, Z)), Y)
% 56.68/7.64 = { by lemma 63 }
% 56.68/7.64 join(X, Y)
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.64 join(Y, X)
% 56.68/7.64
% 56.68/7.64 Lemma 94: join(converse(X), join(Y, converse(Z))) = join(Y, converse(join(X, Z))).
% 56.68/7.64 Proof:
% 56.68/7.64 join(converse(X), join(Y, converse(Z)))
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.64 join(converse(X), join(converse(Z), Y))
% 56.68/7.64 = { by axiom 11 (maddux2_join_associativity_2) }
% 56.68/7.64 join(join(converse(X), converse(Z)), Y)
% 56.68/7.64 = { by axiom 10 (converse_additivity_9) R->L }
% 56.68/7.64 join(converse(join(X, Z)), Y)
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.64 join(Y, converse(join(X, Z)))
% 56.68/7.64
% 56.68/7.64 Lemma 95: meet(one, converse(meet(sk2, X))) = converse(meet(X, sk2)).
% 56.68/7.64 Proof:
% 56.68/7.64 meet(one, converse(meet(sk2, X)))
% 56.68/7.64 = { by lemma 42 }
% 56.68/7.64 meet(one, converse(meet(X, sk2)))
% 56.68/7.64 = { by lemma 42 }
% 56.68/7.64 meet(converse(meet(X, sk2)), one)
% 56.68/7.64 = { by lemma 90 R->L }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(one, converse(sk2)))
% 56.68/7.64 = { by lemma 63 R->L }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(one, converse(join(sk2, meet(sk2, X)))))
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(one, converse(join(meet(sk2, X), sk2))))
% 56.68/7.64 = { by lemma 94 R->L }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(converse(meet(sk2, X)), join(one, converse(sk2))))
% 56.68/7.64 = { by lemma 90 }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(converse(meet(sk2, X)), one))
% 56.68/7.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(one, converse(meet(sk2, X))))
% 56.68/7.64 = { by lemma 42 R->L }
% 56.68/7.64 meet(converse(meet(X, sk2)), join(one, converse(meet(X, sk2))))
% 56.68/7.64 = { by lemma 91 }
% 56.68/7.64 converse(meet(X, sk2))
% 56.68/7.65
% 56.68/7.65 Lemma 96: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
% 56.68/7.65 Proof:
% 56.68/7.65 converse(meet(X, converse(Y)))
% 56.68/7.65 = { by lemma 41 R->L }
% 56.68/7.65 converse(complement(complement(meet(X, converse(Y)))))
% 56.68/7.65 = { by lemma 50 R->L }
% 56.68/7.65 converse(complement(join(complement(X), complement(converse(Y)))))
% 56.68/7.65 = { by lemma 73 }
% 56.68/7.65 converse(complement(join(complement(X), converse(complement(Y)))))
% 56.68/7.65 = { by lemma 37 R->L }
% 56.68/7.65 converse(complement(converse(join(complement(Y), converse(complement(X))))))
% 56.68/7.65 = { by lemma 72 }
% 56.68/7.65 complement(join(complement(Y), converse(complement(X))))
% 56.68/7.65 = { by lemma 53 }
% 56.68/7.65 meet(Y, complement(converse(complement(X))))
% 56.68/7.65 = { by lemma 73 }
% 56.68/7.65 meet(Y, converse(complement(complement(X))))
% 56.68/7.65 = { by lemma 41 }
% 56.68/7.65 meet(Y, converse(X))
% 56.68/7.65
% 56.68/7.65 Lemma 97: meet(Y, composition(X, sk1)) = meet(X, composition(Y, sk1)).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(Y, composition(X, sk1))
% 56.68/7.65 = { by lemma 86 R->L }
% 56.68/7.65 meet(Y, meet(X, composition(top, sk1)))
% 56.68/7.65 = { by lemma 75 R->L }
% 56.68/7.65 meet(X, meet(Y, composition(top, sk1)))
% 56.68/7.65 = { by lemma 86 }
% 56.68/7.65 meet(X, composition(Y, sk1))
% 56.68/7.65
% 56.68/7.65 Lemma 98: meet(composition(top, sk1), X) = composition(X, sk1).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(composition(top, sk1), X)
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(X, composition(top, sk1))
% 56.68/7.65 = { by lemma 86 }
% 56.68/7.65 composition(X, sk1)
% 56.68/7.65
% 56.68/7.65 Lemma 99: converse(meet(sk1, sk2)) = composition(converse(sk2), sk1).
% 56.68/7.65 Proof:
% 56.68/7.65 converse(meet(sk1, sk2))
% 56.68/7.65 = { by lemma 95 R->L }
% 56.68/7.65 meet(one, converse(meet(sk2, sk1)))
% 56.68/7.65 = { by lemma 88 R->L }
% 56.68/7.65 meet(one, converse(meet(sk2, converse(sk1))))
% 56.68/7.65 = { by lemma 96 }
% 56.68/7.65 meet(one, meet(sk1, converse(sk2)))
% 56.68/7.65 = { by lemma 75 }
% 56.68/7.65 meet(sk1, meet(one, converse(sk2)))
% 56.68/7.65 = { by lemma 92 }
% 56.68/7.65 meet(sk1, converse(sk2))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(converse(sk2), sk1)
% 56.68/7.65 = { by lemma 22 R->L }
% 56.68/7.65 meet(converse(sk2), composition(one, sk1))
% 56.68/7.65 = { by lemma 97 R->L }
% 56.68/7.65 meet(one, composition(converse(sk2), sk1))
% 56.68/7.65 = { by lemma 98 R->L }
% 56.68/7.65 meet(one, meet(composition(top, sk1), converse(sk2)))
% 56.68/7.65 = { by lemma 75 }
% 56.68/7.65 meet(composition(top, sk1), meet(one, converse(sk2)))
% 56.68/7.65 = { by lemma 98 }
% 56.68/7.65 composition(meet(one, converse(sk2)), sk1)
% 56.68/7.65 = { by lemma 92 }
% 56.68/7.65 composition(converse(sk2), sk1)
% 56.68/7.65
% 56.68/7.65 Lemma 100: join(composition(join(Z, one), X), Y) = join(X, join(Y, composition(Z, X))).
% 56.68/7.65 Proof:
% 56.68/7.65 join(composition(join(Z, one), X), Y)
% 56.68/7.65 = { by lemma 55 R->L }
% 56.68/7.65 join(join(X, composition(Z, X)), Y)
% 56.68/7.65 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 56.68/7.65 join(X, join(composition(Z, X), Y))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.65 join(X, join(Y, composition(Z, X)))
% 56.68/7.65
% 56.68/7.65 Lemma 101: composition(meet(sk1, X), sk1) = meet(sk1, X).
% 56.68/7.65 Proof:
% 56.68/7.65 composition(meet(sk1, X), sk1)
% 56.68/7.65 = { by lemma 86 R->L }
% 56.68/7.65 meet(meet(sk1, X), composition(top, sk1))
% 56.68/7.65 = { by lemma 36 R->L }
% 56.68/7.65 meet(meet(sk1, X), composition(join(top, one), sk1))
% 56.68/7.65 = { by lemma 55 R->L }
% 56.68/7.65 meet(meet(sk1, X), join(sk1, composition(top, sk1)))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.65 meet(meet(sk1, X), join(composition(top, sk1), sk1))
% 56.68/7.65 = { by lemma 93 R->L }
% 56.68/7.65 meet(meet(sk1, X), join(sk1, join(composition(top, sk1), meet(sk1, X))))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.65 meet(meet(sk1, X), join(sk1, join(meet(sk1, X), composition(top, sk1))))
% 56.68/7.65 = { by lemma 100 R->L }
% 56.68/7.65 meet(meet(sk1, X), join(composition(join(top, one), sk1), meet(sk1, X)))
% 56.68/7.65 = { by lemma 91 }
% 56.68/7.65 meet(sk1, X)
% 56.68/7.65
% 56.68/7.65 Lemma 102: meet(meet(sk1, X), converse(meet(sk1, sk2))) = meet(meet(sk1, X), converse(sk2)).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(meet(sk1, X), converse(meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 99 }
% 56.68/7.65 meet(meet(sk1, X), composition(converse(sk2), sk1))
% 56.68/7.65 = { by lemma 97 }
% 56.68/7.65 meet(converse(sk2), composition(meet(sk1, X), sk1))
% 56.68/7.65 = { by lemma 101 }
% 56.68/7.65 meet(converse(sk2), meet(sk1, X))
% 56.68/7.65 = { by lemma 42 R->L }
% 56.68/7.65 meet(meet(sk1, X), converse(sk2))
% 56.68/7.65
% 56.68/7.65 Lemma 103: converse(meet(converse(X), Y)) = meet(X, converse(Y)).
% 56.68/7.65 Proof:
% 56.68/7.65 converse(meet(converse(X), Y))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 converse(meet(Y, converse(X)))
% 56.68/7.65 = { by lemma 96 }
% 56.68/7.65 meet(X, converse(Y))
% 56.68/7.65
% 56.68/7.65 Lemma 104: meet(one, converse(meet(meet(sk1, X), Y))) = converse(meet(Y, meet(sk1, X))).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(one, converse(meet(meet(sk1, X), Y)))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(one, converse(meet(Y, meet(sk1, X))))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), one)
% 56.68/7.65 = { by lemma 89 R->L }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(one, converse(sk1)))
% 56.68/7.65 = { by lemma 63 R->L }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(one, converse(join(sk1, meet(sk1, meet(Y, X))))))
% 56.68/7.65 = { by lemma 75 }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(one, converse(join(sk1, meet(Y, meet(sk1, X))))))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(one, converse(join(meet(Y, meet(sk1, X)), sk1))))
% 56.68/7.65 = { by lemma 94 R->L }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(converse(meet(Y, meet(sk1, X))), join(one, converse(sk1))))
% 56.68/7.65 = { by lemma 89 }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(converse(meet(Y, meet(sk1, X))), one))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.65 meet(converse(meet(Y, meet(sk1, X))), join(one, converse(meet(Y, meet(sk1, X)))))
% 56.68/7.65 = { by lemma 91 }
% 56.68/7.65 converse(meet(Y, meet(sk1, X)))
% 56.68/7.65
% 56.68/7.65 Lemma 105: meet(meet(sk1, sk2), converse(sk2)) = meet(sk2, converse(meet(sk1, sk2))).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(meet(sk1, sk2), converse(sk2))
% 56.68/7.65 = { by lemma 102 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), converse(meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 103 R->L }
% 56.68/7.65 converse(meet(converse(meet(sk1, sk2)), meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 104 R->L }
% 56.68/7.65 meet(one, converse(meet(meet(sk1, sk2), converse(meet(sk1, sk2)))))
% 56.68/7.65 = { by lemma 102 }
% 56.68/7.65 meet(one, converse(meet(meet(sk1, sk2), converse(sk2))))
% 56.68/7.65 = { by lemma 104 }
% 56.68/7.65 converse(meet(converse(sk2), meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 103 }
% 56.68/7.65 meet(sk2, converse(meet(sk1, sk2)))
% 56.68/7.65
% 56.68/7.65 Lemma 106: meet(Z, meet(X, Y)) = meet(X, meet(Y, Z)).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(Z, meet(X, Y))
% 56.68/7.65 = { by lemma 75 R->L }
% 56.68/7.65 meet(X, meet(Z, Y))
% 56.68/7.65 = { by lemma 42 R->L }
% 56.68/7.65 meet(X, meet(Y, Z))
% 56.68/7.65
% 56.68/7.65 Lemma 107: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(X, complement(meet(Y, X)))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(X, complement(meet(X, Y)))
% 56.68/7.65 = { by lemma 65 }
% 56.68/7.65 meet(X, complement(Y))
% 56.68/7.65
% 56.68/7.65 Lemma 108: meet(sk2, meet(one, X)) = meet(X, sk2).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(sk2, meet(one, X))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(sk2, meet(X, one))
% 56.68/7.65 = { by lemma 75 }
% 56.68/7.65 meet(X, meet(sk2, one))
% 56.68/7.65 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 56.68/7.65 meet(X, complement(join(complement(sk2), complement(one))))
% 56.68/7.65 = { by lemma 28 R->L }
% 56.68/7.65 meet(X, join(zero, complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by lemma 17 R->L }
% 56.68/7.65 meet(X, join(complement(top), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by lemma 35 R->L }
% 56.68/7.65 meet(X, join(complement(join(one, top)), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by lemma 31 R->L }
% 56.68/7.65 meet(X, join(complement(join(sk2, join(one, complement(sk2)))), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by axiom 11 (maddux2_join_associativity_2) }
% 56.68/7.65 meet(X, join(complement(join(join(sk2, one), complement(sk2))), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by axiom 4 (goals_18) }
% 56.68/7.65 meet(X, join(complement(join(one, complement(sk2))), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by lemma 52 }
% 56.68/7.65 meet(X, join(meet(sk2, complement(one)), complement(join(complement(sk2), complement(one)))))
% 56.68/7.65 = { by lemma 18 }
% 56.68/7.65 meet(X, sk2)
% 56.68/7.65
% 56.68/7.65 Lemma 109: meet(sk2, composition(converse(X), complement(X))) = zero.
% 56.68/7.65 Proof:
% 56.68/7.65 meet(sk2, composition(converse(X), complement(X)))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(composition(converse(X), complement(X)), sk2)
% 56.68/7.65 = { by lemma 108 R->L }
% 56.68/7.65 meet(sk2, meet(one, composition(converse(X), complement(X))))
% 56.68/7.65 = { by lemma 83 }
% 56.68/7.65 meet(sk2, zero)
% 56.68/7.65 = { by lemma 84 }
% 56.68/7.65 zero
% 56.68/7.65
% 56.68/7.65 Lemma 110: meet(X, composition(X, sk2)) = composition(X, sk2).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(X, composition(X, sk2))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(composition(X, sk2), X)
% 56.68/7.65 = { by lemma 41 R->L }
% 56.68/7.65 meet(composition(X, sk2), complement(complement(X)))
% 56.68/7.65 = { by lemma 28 R->L }
% 56.68/7.65 join(zero, meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 57 R->L }
% 56.68/7.65 join(composition(meet(X, composition(complement(X), converse(sk2))), zero), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 109 R->L }
% 56.68/7.65 join(composition(meet(X, composition(complement(X), converse(sk2))), meet(sk2, composition(converse(X), complement(X)))), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by axiom 16 (dedekind_law_14) R->L }
% 56.68/7.65 join(join(meet(composition(X, sk2), complement(X)), composition(meet(X, composition(complement(X), converse(sk2))), meet(sk2, composition(converse(X), complement(X))))), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 109 }
% 56.68/7.65 join(join(meet(composition(X, sk2), complement(X)), composition(meet(X, composition(complement(X), converse(sk2))), zero)), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 57 }
% 56.68/7.65 join(join(meet(composition(X, sk2), complement(X)), zero), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 27 }
% 56.68/7.65 join(meet(composition(X, sk2), complement(X)), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 42 R->L }
% 56.68/7.65 join(meet(complement(X), composition(X, sk2)), meet(composition(X, sk2), complement(complement(X))))
% 56.68/7.65 = { by lemma 44 }
% 56.68/7.65 composition(X, sk2)
% 56.68/7.65
% 56.68/7.65 Lemma 111: meet(X, composition(top, sk2)) = composition(X, sk2).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(X, composition(top, sk2))
% 56.68/7.65 = { by lemma 78 R->L }
% 56.68/7.65 meet(X, join(complement(X), composition(top, sk2)))
% 56.68/7.65 = { by axiom 7 (def_top_12) }
% 56.68/7.65 meet(X, join(complement(X), composition(join(complement(X), complement(complement(X))), sk2)))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.65 meet(X, join(complement(X), composition(join(complement(complement(X)), complement(X)), sk2)))
% 56.68/7.65 = { by axiom 13 (composition_distributivity_7) }
% 56.68/7.65 meet(X, join(complement(X), join(composition(complement(complement(X)), sk2), composition(complement(X), sk2))))
% 56.68/7.65 = { by lemma 81 R->L }
% 56.68/7.65 meet(X, join(composition(complement(X), join(sk2, one)), composition(complement(complement(X)), sk2)))
% 56.68/7.65 = { by axiom 4 (goals_18) }
% 56.68/7.65 meet(X, join(composition(complement(X), one), composition(complement(complement(X)), sk2)))
% 56.68/7.65 = { by axiom 1 (composition_identity_6) }
% 56.68/7.65 meet(X, join(complement(X), composition(complement(complement(X)), sk2)))
% 56.68/7.65 = { by lemma 78 }
% 56.68/7.65 meet(X, composition(complement(complement(X)), sk2))
% 56.68/7.65 = { by lemma 41 }
% 56.68/7.65 meet(X, composition(X, sk2))
% 56.68/7.65 = { by lemma 110 }
% 56.68/7.65 composition(X, sk2)
% 56.68/7.65
% 56.68/7.65 Lemma 112: meet(sk2, converse(meet(sk1, sk2))) = meet(sk1, sk2).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(sk2, converse(meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 105 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), converse(sk2))
% 56.68/7.65 = { by lemma 41 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), converse(complement(complement(sk2))))
% 56.68/7.65 = { by lemma 73 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), complement(converse(complement(sk2))))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(complement(converse(complement(sk2))), meet(sk1, sk2))
% 56.68/7.65 = { by lemma 106 }
% 56.68/7.65 meet(sk1, meet(sk2, complement(converse(complement(sk2)))))
% 56.68/7.65 = { by lemma 107 R->L }
% 56.68/7.65 meet(sk1, meet(sk2, complement(meet(converse(complement(sk2)), sk2))))
% 56.68/7.65 = { by lemma 106 R->L }
% 56.68/7.65 meet(complement(meet(converse(complement(sk2)), sk2)), meet(sk1, sk2))
% 56.68/7.65 = { by lemma 42 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(converse(complement(sk2)), sk2)))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(sk2, converse(complement(sk2)))))
% 56.68/7.65 = { by lemma 77 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(sk2, meet(converse(complement(sk2)), composition(top, sk2)))))
% 56.68/7.65 = { by lemma 111 }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(sk2, composition(converse(complement(sk2)), sk2))))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(composition(converse(complement(sk2)), sk2), sk2)))
% 56.68/7.65 = { by lemma 108 R->L }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(sk2, meet(one, composition(converse(complement(sk2)), sk2)))))
% 56.68/7.65 = { by lemma 71 }
% 56.68/7.65 meet(meet(sk1, sk2), complement(meet(sk2, zero)))
% 56.68/7.65 = { by lemma 84 }
% 56.68/7.65 meet(meet(sk1, sk2), complement(zero))
% 56.68/7.65 = { by lemma 29 }
% 56.68/7.65 meet(meet(sk1, sk2), top)
% 56.68/7.65 = { by lemma 47 }
% 56.68/7.65 meet(sk1, sk2)
% 56.68/7.65
% 56.68/7.65 Lemma 113: converse(meet(sk1, sk2)) = meet(sk1, sk2).
% 56.68/7.65 Proof:
% 56.68/7.65 converse(meet(sk1, sk2))
% 56.68/7.65 = { by lemma 91 R->L }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), join(one, converse(meet(sk1, sk2))))
% 56.68/7.65 = { by lemma 79 R->L }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), converse(join(meet(sk1, sk2), one)))
% 56.68/7.65 = { by lemma 34 R->L }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), converse(join(sk1, join(one, meet(sk1, sk2)))))
% 56.68/7.65 = { by lemma 93 }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), converse(join(one, sk1)))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), converse(join(sk1, one)))
% 56.68/7.65 = { by axiom 3 (goals_17) }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), converse(one))
% 56.68/7.65 = { by lemma 30 }
% 56.68/7.65 meet(converse(meet(sk1, sk2)), one)
% 56.68/7.65 = { by lemma 42 R->L }
% 56.68/7.65 meet(one, converse(meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 112 R->L }
% 56.68/7.65 meet(one, converse(meet(sk2, converse(meet(sk1, sk2)))))
% 56.68/7.65 = { by lemma 95 }
% 56.68/7.65 converse(meet(converse(meet(sk1, sk2)), sk2))
% 56.68/7.65 = { by lemma 103 }
% 56.68/7.65 meet(meet(sk1, sk2), converse(sk2))
% 56.68/7.65 = { by lemma 105 }
% 56.68/7.65 meet(sk2, converse(meet(sk1, sk2)))
% 56.68/7.65 = { by lemma 112 }
% 56.68/7.65 meet(sk1, sk2)
% 56.68/7.65
% 56.68/7.65 Lemma 114: join(X, join(Y, Z)) = join(Y, join(X, Z)).
% 56.68/7.65 Proof:
% 56.68/7.65 join(X, join(Y, Z))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.65 join(join(Y, Z), X)
% 56.68/7.65 = { by axiom 11 (maddux2_join_associativity_2) R->L }
% 56.68/7.65 join(Y, join(Z, X))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.65 join(Y, join(X, Z))
% 56.68/7.65
% 56.68/7.65 Lemma 115: join(X, join(Y, composition(sk1, X))) = join(X, Y).
% 56.68/7.65 Proof:
% 56.68/7.65 join(X, join(Y, composition(sk1, X)))
% 56.68/7.65 = { by lemma 100 R->L }
% 56.68/7.65 join(composition(join(sk1, one), X), Y)
% 56.68/7.65 = { by axiom 3 (goals_17) }
% 56.68/7.65 join(composition(one, X), Y)
% 56.68/7.65 = { by lemma 22 }
% 56.68/7.65 join(X, Y)
% 56.68/7.65
% 56.68/7.65 Lemma 116: meet(X, join(Y, join(Z, X))) = X.
% 56.68/7.65 Proof:
% 56.68/7.65 meet(X, join(Y, join(Z, X)))
% 56.68/7.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.65 meet(X, join(Y, join(X, Z)))
% 56.68/7.65 = { by lemma 114 R->L }
% 56.68/7.65 meet(X, join(X, join(Y, Z)))
% 56.68/7.65 = { by lemma 76 }
% 56.68/7.65 X
% 56.68/7.65
% 56.68/7.65 Lemma 117: meet(X, composition(sk1, X)) = composition(sk1, X).
% 56.68/7.65 Proof:
% 56.68/7.65 meet(X, composition(sk1, X))
% 56.68/7.65 = { by lemma 42 }
% 56.68/7.65 meet(composition(sk1, X), X)
% 56.68/7.65 = { by lemma 27 R->L }
% 56.68/7.65 meet(composition(sk1, X), join(X, zero))
% 56.68/7.65 = { by lemma 115 R->L }
% 56.68/7.66 meet(composition(sk1, X), join(X, join(zero, composition(sk1, X))))
% 56.68/7.66 = { by lemma 116 }
% 56.68/7.66 composition(sk1, X)
% 56.68/7.66
% 56.68/7.66 Lemma 118: join(composition(X, Y), composition(Z, Y)) = composition(join(Z, X), Y).
% 56.68/7.66 Proof:
% 56.68/7.66 join(composition(X, Y), composition(Z, Y))
% 56.68/7.66 = { by axiom 13 (composition_distributivity_7) R->L }
% 56.68/7.66 composition(join(X, Z), Y)
% 56.68/7.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.66 composition(join(Z, X), Y)
% 56.68/7.66
% 56.68/7.66 Lemma 119: composition(converse(X), complement(composition(X, top))) = zero.
% 56.68/7.66 Proof:
% 56.68/7.66 composition(converse(X), complement(composition(X, top)))
% 56.68/7.66 = { by lemma 28 R->L }
% 56.68/7.66 join(zero, composition(converse(X), complement(composition(X, top))))
% 56.68/7.66 = { by lemma 17 R->L }
% 56.68/7.66 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 56.68/7.66 = { by lemma 23 }
% 56.68/7.66 complement(top)
% 56.68/7.66 = { by lemma 17 }
% 56.68/7.66 zero
% 56.68/7.66
% 56.68/7.66 Lemma 120: meet(composition(X, Y), complement(composition(X, top))) = zero.
% 56.68/7.66 Proof:
% 56.68/7.66 meet(composition(X, Y), complement(composition(X, top)))
% 56.68/7.66 = { by lemma 27 R->L }
% 56.68/7.66 join(meet(composition(X, Y), complement(composition(X, top))), zero)
% 56.68/7.66 = { by lemma 57 R->L }
% 56.68/7.66 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), zero))
% 56.68/7.66 = { by lemma 84 R->L }
% 56.68/7.66 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, zero)))
% 56.68/7.66 = { by lemma 119 R->L }
% 56.68/7.66 join(meet(composition(X, Y), complement(composition(X, top))), composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, composition(converse(X), complement(composition(X, top))))))
% 56.68/7.66 = { by axiom 16 (dedekind_law_14) }
% 56.68/7.66 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, composition(converse(X), complement(composition(X, top)))))
% 56.68/7.66 = { by lemma 119 }
% 56.68/7.66 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), meet(Y, zero))
% 56.68/7.66 = { by lemma 84 }
% 56.68/7.66 composition(meet(X, composition(complement(composition(X, top)), converse(Y))), zero)
% 56.68/7.66 = { by lemma 57 }
% 56.68/7.66 zero
% 56.68/7.66
% 56.68/7.66 Lemma 121: meet(composition(meet(X, Y), Z), composition(Y, Z)) = composition(meet(X, Y), Z).
% 56.68/7.66 Proof:
% 56.68/7.66 meet(composition(meet(X, Y), Z), composition(Y, Z))
% 56.68/7.66 = { by lemma 63 R->L }
% 56.68/7.66 meet(composition(meet(X, Y), Z), composition(join(Y, meet(Y, X)), Z))
% 56.68/7.66 = { by lemma 42 R->L }
% 56.68/7.66 meet(composition(meet(X, Y), Z), composition(join(Y, meet(X, Y)), Z))
% 56.68/7.66 = { by lemma 118 R->L }
% 56.68/7.66 meet(composition(meet(X, Y), Z), join(composition(meet(X, Y), Z), composition(Y, Z)))
% 56.68/7.66 = { by lemma 76 }
% 56.68/7.66 composition(meet(X, Y), Z)
% 56.68/7.66
% 56.68/7.66 Lemma 122: meet(composition(meet(sk1, sk2), X), composition(meet(sk1, sk2), Y)) = meet(Y, composition(meet(sk1, sk2), X)).
% 56.68/7.66 Proof:
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), composition(meet(sk1, sk2), Y))
% 56.68/7.66 = { by lemma 113 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), composition(converse(meet(sk1, sk2)), Y))
% 56.68/7.66 = { by lemma 20 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(sk1, sk2))))
% 56.68/7.66 = { by lemma 101 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), composition(meet(sk1, sk2), sk1))))
% 56.68/7.66 = { by lemma 98 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(top, sk1), meet(sk1, sk2)))))
% 56.68/7.66 = { by lemma 75 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(sk1, meet(composition(top, sk1), sk2)))))
% 56.68/7.66 = { by lemma 98 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(sk1, composition(sk2, sk1)))))
% 56.68/7.66 = { by lemma 42 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(sk2, sk1), sk1))))
% 56.68/7.66 = { by lemma 27 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(sk2, sk1), join(sk1, zero)))))
% 56.68/7.66 = { by lemma 22 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(sk2, sk1), join(composition(one, sk1), zero)))))
% 56.68/7.66 = { by axiom 4 (goals_18) R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(sk2, sk1), join(composition(join(sk2, one), sk1), zero)))))
% 56.68/7.66 = { by lemma 100 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), meet(composition(sk2, sk1), join(sk1, join(zero, composition(sk2, sk1)))))))
% 56.68/7.66 = { by lemma 116 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(converse(Y), composition(sk2, sk1))))
% 56.68/7.66 = { by axiom 9 (composition_associativity_5) }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(composition(composition(converse(Y), sk2), sk1)))
% 56.68/7.66 = { by lemma 98 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(composition(top, sk1), composition(converse(Y), sk2))))
% 56.68/7.66 = { by lemma 111 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(composition(top, sk1), meet(converse(Y), composition(top, sk2)))))
% 56.68/7.66 = { by lemma 75 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), meet(composition(top, sk1), composition(top, sk2)))))
% 56.68/7.66 = { by lemma 111 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(composition(top, sk1), sk2))))
% 56.68/7.66 = { by axiom 9 (composition_associativity_5) R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(top, composition(sk1, sk2)))))
% 56.68/7.66 = { by lemma 88 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(top, composition(converse(sk1), sk2)))))
% 56.68/7.66 = { by lemma 20 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(top, converse(composition(converse(sk2), sk1))))))
% 56.68/7.66 = { by lemma 99 R->L }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(top, converse(converse(meet(sk1, sk2)))))))
% 56.68/7.66 = { by axiom 5 (converse_idempotence_8) }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), converse(meet(converse(Y), composition(top, meet(sk1, sk2)))))
% 56.68/7.66 = { by lemma 103 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), meet(Y, converse(composition(top, meet(sk1, sk2)))))
% 56.68/7.66 = { by axiom 8 (converse_multiplicativity_10) }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), meet(Y, composition(converse(meet(sk1, sk2)), converse(top))))
% 56.68/7.66 = { by lemma 38 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), meet(Y, composition(converse(meet(sk1, sk2)), top)))
% 56.68/7.66 = { by lemma 113 }
% 56.68/7.66 meet(composition(meet(sk1, sk2), X), meet(Y, composition(meet(sk1, sk2), top)))
% 56.68/7.66 = { by lemma 75 }
% 56.68/7.66 meet(Y, meet(composition(meet(sk1, sk2), X), composition(meet(sk1, sk2), top)))
% 56.68/7.66 = { by lemma 42 }
% 56.68/7.66 meet(Y, meet(composition(meet(sk1, sk2), top), composition(meet(sk1, sk2), X)))
% 56.68/7.66 = { by lemma 27 R->L }
% 56.68/7.66 meet(Y, join(meet(composition(meet(sk1, sk2), top), composition(meet(sk1, sk2), X)), zero))
% 56.68/7.66 = { by lemma 120 R->L }
% 56.68/7.66 meet(Y, join(meet(composition(meet(sk1, sk2), top), composition(meet(sk1, sk2), X)), meet(composition(meet(sk1, sk2), X), complement(composition(meet(sk1, sk2), top)))))
% 56.68/7.66 = { by lemma 44 }
% 56.68/7.66 meet(Y, composition(meet(sk1, sk2), X))
% 56.68/7.66
% 56.68/7.66 Lemma 123: join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)) = meet(composition(X, Y), composition(Z, Y)).
% 56.68/7.66 Proof:
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))
% 56.68/7.66 = { by lemma 76 R->L }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), join(composition(meet(X, Z), Y), composition(X, Y))))
% 56.68/7.66 = { by lemma 118 }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), composition(join(X, meet(X, Z)), Y)))
% 56.68/7.66 = { by lemma 63 }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), composition(X, Y)))
% 56.68/7.66 = { by lemma 42 }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), composition(meet(X, Z), Y)))
% 56.68/7.66 = { by lemma 121 R->L }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(X, Y), meet(composition(meet(X, Z), Y), composition(Z, Y))))
% 56.68/7.66 = { by lemma 75 }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(composition(meet(X, Z), Y), meet(composition(X, Y), composition(Z, Y))))
% 56.68/7.66 = { by lemma 42 R->L }
% 56.68/7.66 join(meet(composition(X, Y), composition(Z, Y)), meet(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))
% 56.68/7.66 = { by lemma 63 }
% 56.68/7.66 meet(composition(X, Y), composition(Z, Y))
% 56.68/7.66
% 56.68/7.66 Lemma 124: meet(W, join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y))) = meet(composition(X, Y), meet(composition(Z, Y), W)).
% 56.68/7.66 Proof:
% 56.68/7.66 meet(W, join(meet(composition(X, Y), composition(Z, Y)), composition(meet(X, Z), Y)))
% 56.68/7.66 = { by lemma 123 }
% 56.68/7.66 meet(W, meet(composition(X, Y), composition(Z, Y)))
% 56.68/7.66 = { by lemma 75 R->L }
% 56.68/7.66 meet(composition(X, Y), meet(W, composition(Z, Y)))
% 56.68/7.66 = { by lemma 42 R->L }
% 56.68/7.66 meet(composition(X, Y), meet(composition(Z, Y), W))
% 56.68/7.66
% 56.68/7.66 Goal 1 (goals_19): tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))) = tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))).
% 56.68/7.66 Proof:
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3))))
% 56.68/7.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 42 }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk2, sk1), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by axiom 12 (maddux4_definiton_of_meet_4) }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(complement(join(complement(sk2), complement(sk1))), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 24 R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(join(complement(join(complement(sk2), complement(sk1))), complement(join(complement(sk2), complement(sk1)))), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(join(meet(sk2, sk1), complement(join(complement(sk2), complement(sk1)))), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(join(meet(sk2, sk1), meet(sk2, sk1)), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 42 R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(join(meet(sk2, sk1), meet(sk1, sk2)), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 42 R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(join(meet(sk1, sk2), meet(sk1, sk2)), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 118 R->L }
% 56.68/7.66 tuple(join(meet(composition(sk1, sk3), composition(sk2, sk3)), join(composition(meet(sk1, sk2), sk3), composition(meet(sk1, sk2), sk3))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 114 }
% 56.68/7.66 tuple(join(composition(meet(sk1, sk2), sk3), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 56.68/7.66 tuple(join(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), composition(meet(sk1, sk2), sk3)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 41 R->L }
% 56.68/7.66 tuple(join(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(complement(composition(meet(sk1, sk2), sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 74 R->L }
% 56.68/7.66 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(composition(meet(sk1, sk2), sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 121 R->L }
% 56.68/7.66 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(meet(sk1, sk2), sk3), composition(sk2, sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.66 = { by lemma 42 }
% 56.68/7.66 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), composition(meet(sk1, sk2), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 122 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(meet(sk1, sk2), sk3), composition(meet(sk1, sk2), composition(sk2, sk3)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 101 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(meet(sk1, sk2), sk3), composition(composition(meet(sk1, sk2), sk1), composition(sk2, sk3)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 9 (composition_associativity_5) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(meet(sk1, sk2), sk3), composition(meet(sk1, sk2), composition(sk1, composition(sk2, sk3))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 122 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(meet(sk1, sk2), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 63 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(meet(sk1, sk2), meet(meet(sk1, sk2), composition(sk1, sk2))), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 42 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(meet(sk1, sk2), meet(composition(sk1, sk2), meet(sk1, sk2))), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 106 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(meet(sk1, sk2), meet(sk1, meet(sk2, composition(sk1, sk2)))), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 117 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(meet(sk1, sk2), meet(sk1, composition(sk1, sk2))), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 110 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(meet(sk1, sk2), composition(sk1, sk2)), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), composition(join(composition(sk1, sk2), meet(sk1, sk2)), sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 118 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), join(composition(meet(sk1, sk2), sk3), composition(composition(sk1, sk2), sk3)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 9 (composition_associativity_5) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, composition(sk2, sk3)), join(composition(meet(sk1, sk2), sk3), composition(sk1, composition(sk2, sk3))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 91 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(composition(sk1, composition(sk2, sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 117 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), composition(sk1, composition(sk2, sk3)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 41 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), composition(sk1, complement(complement(composition(sk2, sk3)))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 78 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(sk1, complement(complement(composition(sk2, sk3))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 115 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), join(composition(sk1, complement(complement(composition(sk2, sk3)))), composition(sk1, complement(composition(sk2, sk3))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 5 (converse_idempotence_8) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), join(composition(sk1, complement(complement(composition(sk2, sk3)))), composition(converse(converse(sk1)), complement(composition(sk2, sk3))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 5 (converse_idempotence_8) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), join(converse(converse(composition(sk1, complement(complement(composition(sk2, sk3)))))), composition(converse(converse(sk1)), complement(composition(sk2, sk3))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 69 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), converse(join(converse(composition(sk1, complement(complement(composition(sk2, sk3))))), composition(converse(complement(composition(sk2, sk3))), converse(sk1))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 8 (converse_multiplicativity_10) }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), converse(join(composition(converse(complement(complement(composition(sk2, sk3)))), converse(sk1)), composition(converse(complement(composition(sk2, sk3))), converse(sk1))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 13 (composition_distributivity_7) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), converse(composition(join(converse(complement(complement(composition(sk2, sk3)))), converse(complement(composition(sk2, sk3)))), converse(sk1)))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 8 (converse_multiplicativity_10) }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(converse(converse(sk1)), converse(join(converse(complement(complement(composition(sk2, sk3)))), converse(complement(composition(sk2, sk3))))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 37 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(converse(converse(sk1)), join(complement(composition(sk2, sk3)), converse(converse(complement(complement(composition(sk2, sk3)))))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 5 (converse_idempotence_8) }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(sk1, join(complement(composition(sk2, sk3)), converse(converse(complement(complement(composition(sk2, sk3)))))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 5 (converse_idempotence_8) }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(sk1, join(complement(composition(sk2, sk3)), complement(complement(composition(sk2, sk3)))))))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by axiom 7 (def_top_12) R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), join(complement(composition(sk2, sk3)), composition(sk1, top)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 78 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk2, sk3), composition(sk1, top))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 42 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(meet(composition(sk1, top), composition(sk2, sk3))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 42 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(complement(meet(composition(sk1, top), composition(sk2, sk3))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 124 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(composition(sk1, sk3), meet(composition(sk2, sk3), complement(meet(composition(sk1, top), composition(sk2, sk3)))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 107 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(composition(sk1, sk3), meet(composition(sk2, sk3), complement(composition(sk1, top))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 124 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(complement(composition(sk1, top)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 42 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 123 }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(meet(composition(sk1, sk3), composition(sk2, sk3)), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 18 R->L }
% 56.68/7.67 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(meet(composition(sk1, sk3), composition(sk2, sk3)), complement(composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, sk3), composition(sk2, sk3))), complement(composition(sk1, sk3))))), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.67 = { by lemma 52 R->L }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(complement(join(composition(sk1, sk3), complement(meet(composition(sk1, sk3), composition(sk2, sk3))))), complement(join(complement(meet(composition(sk1, sk3), composition(sk2, sk3))), complement(composition(sk1, sk3))))), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 61 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(complement(top), complement(join(complement(meet(composition(sk1, sk3), composition(sk2, sk3))), complement(composition(sk1, sk3))))), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 17 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(zero, complement(join(complement(meet(composition(sk1, sk3), composition(sk2, sk3))), complement(composition(sk1, sk3))))), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 28 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(complement(join(complement(meet(composition(sk1, sk3), composition(sk2, sk3))), complement(composition(sk1, sk3)))), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by axiom 12 (maddux4_definiton_of_meet_4) R->L }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(meet(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(sk1, sk3)), complement(composition(sk1, top)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 42 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(complement(composition(sk1, top)), meet(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(sk1, sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 75 R->L }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(meet(composition(sk1, sk3), composition(sk2, sk3)), meet(complement(composition(sk1, top)), composition(sk1, sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 123 R->L }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), meet(complement(composition(sk1, top)), composition(sk1, sk3)))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 42 R->L }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), meet(composition(sk1, sk3), complement(composition(sk1, top))))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 120 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), meet(join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)), zero)), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 84 }
% 56.68/7.68 tuple(join(complement(complement(composition(meet(sk1, sk2), sk3))), zero), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 27 }
% 56.68/7.68 tuple(complement(complement(composition(meet(sk1, sk2), sk3))), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 41 }
% 56.68/7.68 tuple(composition(meet(sk1, sk2), sk3), join(meet(composition(sk1, sk3), composition(sk2, sk3)), composition(meet(sk1, sk2), sk3)))
% 56.68/7.68 = { by lemma 123 }
% 56.68/7.68 tuple(composition(meet(sk1, sk2), sk3), meet(composition(sk1, sk3), composition(sk2, sk3)))
% 56.68/7.68 % SZS output end Proof
% 56.68/7.68
% 56.68/7.68 RESULT: Unsatisfiable (the axioms are contradictory).
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