TSTP Solution File: REL030-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL030-4 : 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 : n012.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 13:44:15 EDT 2023
% Result : Unsatisfiable 32.06s 4.47s
% Output : Proof 34.08s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : REL030-4 : 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
% 0.12/0.34 % Computer : n012.cluster.edu
% 0.12/0.34 % Model : x86_64 x86_64
% 0.12/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.34 % Memory : 8042.1875MB
% 0.12/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.34 % CPULimit : 300
% 0.12/0.34 % WCLimit : 300
% 0.12/0.34 % DateTime : Fri Aug 25 20:05:11 EDT 2023
% 0.12/0.34 % CPUTime :
% 32.06/4.47 Command-line arguments: --flatten
% 32.06/4.47
% 32.06/4.47 % SZS status Unsatisfiable
% 32.06/4.47
% 33.19/4.64 % SZS output start Proof
% 33.19/4.64 Take the following subset of the input axioms:
% 33.19/4.64 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 33.19/4.64 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 33.19/4.64 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 33.19/4.64 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 33.19/4.64 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 33.19/4.64 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 33.19/4.64 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 33.19/4.64 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 33.19/4.64 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 33.19/4.64 fof(goals_17, negated_conjecture, join(sk1, one)=one).
% 33.19/4.64 fof(goals_18, negated_conjecture, join(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))!=meet(composition(sk1, sk2), complement(composition(sk1, sk3))) | join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))!=meet(composition(sk1, sk2), complement(sk3))).
% 33.19/4.64 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 33.19/4.64 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 33.19/4.64 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 33.19/4.64 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 33.19/4.64 fof(modular_law_1_15, axiom, ![A2, B2, C2]: join(meet(composition(A2, B2), C2), meet(composition(A2, meet(B2, composition(converse(A2), C2))), C2))=meet(composition(A2, meet(B2, composition(converse(A2), C2))), C2)).
% 33.19/4.64
% 33.19/4.64 Now clausify the problem and encode Horn clauses using encoding 3 of
% 33.19/4.64 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 33.19/4.64 We repeatedly replace C & s=t => u=v by the two clauses:
% 33.19/4.64 fresh(y, y, x1...xn) = u
% 33.19/4.64 C => fresh(s, t, x1...xn) = v
% 33.19/4.64 where fresh is a fresh function symbol and x1..xn are the free
% 33.19/4.64 variables of u and v.
% 33.19/4.64 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 33.19/4.64 input problem has no model of domain size 1).
% 33.19/4.64
% 33.19/4.64 The encoding turns the above axioms into the following unit equations and goals:
% 33.19/4.64
% 33.19/4.64 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 33.19/4.64 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 33.19/4.64 Axiom 3 (goals_17): join(sk1, one) = one.
% 33.19/4.64 Axiom 4 (composition_identity_6): composition(X, one) = X.
% 33.19/4.64 Axiom 5 (def_top_12): top = join(X, complement(X)).
% 33.19/4.64 Axiom 6 (def_zero_13): zero = meet(X, complement(X)).
% 33.19/4.64 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 33.19/4.64 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 33.19/4.64 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 33.19/4.64 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 33.19/4.64 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 33.19/4.64 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 33.19/4.64 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 33.19/4.64 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 33.19/4.64 Axiom 15 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
% 33.19/4.64
% 33.19/4.64 Lemma 16: complement(top) = zero.
% 33.19/4.64 Proof:
% 33.19/4.64 complement(top)
% 33.19/4.64 = { by axiom 5 (def_top_12) }
% 33.19/4.64 complement(join(complement(X), complement(complement(X))))
% 33.19/4.64 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.19/4.64 meet(X, complement(X))
% 33.19/4.64 = { by axiom 6 (def_zero_13) R->L }
% 33.19/4.64 zero
% 33.19/4.64
% 33.19/4.64 Lemma 17: join(X, join(Y, complement(X))) = join(Y, top).
% 33.19/4.64 Proof:
% 33.19/4.64 join(X, join(Y, complement(X)))
% 33.19/4.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.64 join(X, join(complement(X), Y))
% 33.19/4.64 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.19/4.64 join(join(X, complement(X)), Y)
% 33.19/4.64 = { by axiom 5 (def_top_12) R->L }
% 33.19/4.64 join(top, Y)
% 33.19/4.64 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.19/4.64 join(Y, top)
% 33.19/4.64
% 33.19/4.64 Lemma 18: composition(X, join(sk1, one)) = X.
% 33.19/4.64 Proof:
% 33.19/4.64 composition(X, join(sk1, one))
% 33.19/4.64 = { by axiom 3 (goals_17) }
% 33.19/4.64 composition(X, one)
% 33.19/4.64 = { by axiom 4 (composition_identity_6) }
% 33.19/4.64 X
% 33.19/4.64
% 33.19/4.64 Lemma 19: composition(converse(join(sk1, one)), X) = X.
% 33.19/4.64 Proof:
% 33.19/4.64 composition(converse(join(sk1, one)), X)
% 33.19/4.64 = { by axiom 1 (converse_idempotence_8) R->L }
% 33.19/4.64 composition(converse(join(sk1, one)), converse(converse(X)))
% 33.19/4.64 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 33.19/4.64 converse(composition(converse(X), join(sk1, one)))
% 33.19/4.64 = { by lemma 18 }
% 33.19/4.64 converse(converse(X))
% 33.19/4.64 = { by axiom 1 (converse_idempotence_8) }
% 33.19/4.64 X
% 33.19/4.64
% 33.19/4.64 Lemma 20: composition(join(sk1, one), X) = X.
% 33.19/4.64 Proof:
% 33.19/4.64 composition(join(sk1, one), X)
% 33.19/4.64 = { by lemma 19 R->L }
% 33.19/4.64 composition(converse(join(sk1, one)), composition(join(sk1, one), X))
% 33.19/4.64 = { by axiom 10 (composition_associativity_5) }
% 33.19/4.64 composition(composition(converse(join(sk1, one)), join(sk1, one)), X)
% 33.19/4.64 = { by lemma 18 }
% 33.19/4.64 composition(converse(join(sk1, one)), X)
% 33.19/4.64 = { by lemma 19 }
% 33.19/4.64 X
% 33.19/4.64
% 33.19/4.64 Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 33.19/4.64 Proof:
% 33.19/4.64 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 33.19/4.64 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.64 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 33.19/4.64 = { by axiom 13 (converse_cancellativity_11) }
% 33.19/4.64 complement(X)
% 33.19/4.64
% 33.19/4.64 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 33.19/4.64 Proof:
% 33.19/4.64 join(complement(X), complement(X))
% 33.19/4.64 = { by lemma 19 R->L }
% 33.19/4.64 join(complement(X), composition(converse(join(sk1, one)), complement(X)))
% 33.19/4.64 = { by lemma 20 R->L }
% 33.19/4.64 join(complement(X), composition(converse(join(sk1, one)), complement(composition(join(sk1, one), X))))
% 33.19/4.64 = { by lemma 21 }
% 33.19/4.64 complement(X)
% 33.19/4.64
% 33.19/4.64 Lemma 23: join(top, complement(X)) = top.
% 33.19/4.64 Proof:
% 33.19/4.65 join(top, complement(X))
% 33.19/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.65 join(complement(X), top)
% 33.19/4.65 = { by lemma 17 R->L }
% 33.19/4.65 join(X, join(complement(X), complement(X)))
% 33.19/4.65 = { by lemma 22 }
% 33.19/4.65 join(X, complement(X))
% 33.19/4.65 = { by axiom 5 (def_top_12) R->L }
% 33.19/4.65 top
% 33.19/4.65
% 33.19/4.65 Lemma 24: join(Y, top) = join(X, top).
% 33.19/4.65 Proof:
% 33.19/4.65 join(Y, top)
% 33.19/4.65 = { by lemma 23 R->L }
% 33.19/4.65 join(Y, join(top, complement(Y)))
% 33.19/4.65 = { by lemma 17 }
% 33.19/4.65 join(top, top)
% 33.19/4.65 = { by lemma 17 R->L }
% 33.19/4.65 join(X, join(top, complement(X)))
% 33.19/4.65 = { by lemma 23 }
% 33.19/4.65 join(X, top)
% 33.19/4.65
% 33.19/4.65 Lemma 25: join(X, top) = top.
% 33.19/4.65 Proof:
% 33.19/4.65 join(X, top)
% 33.19/4.65 = { by lemma 24 }
% 33.19/4.65 join(complement(Y), top)
% 33.19/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.65 join(top, complement(Y))
% 33.19/4.65 = { by lemma 23 }
% 33.19/4.65 top
% 33.19/4.65
% 33.19/4.65 Lemma 26: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 33.19/4.65 Proof:
% 33.19/4.65 converse(join(X, converse(Y)))
% 33.19/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.65 converse(join(converse(Y), X))
% 33.19/4.65 = { by axiom 7 (converse_additivity_9) }
% 33.19/4.65 join(converse(converse(Y)), converse(X))
% 33.19/4.65 = { by axiom 1 (converse_idempotence_8) }
% 33.19/4.65 join(Y, converse(X))
% 33.19/4.65
% 33.19/4.65 Lemma 27: join(X, join(complement(X), Y)) = top.
% 33.19/4.65 Proof:
% 33.19/4.65 join(X, join(complement(X), Y))
% 33.19/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.65 join(X, join(Y, complement(X)))
% 33.19/4.65 = { by lemma 17 }
% 33.19/4.65 join(Y, top)
% 33.19/4.65 = { by lemma 24 R->L }
% 33.19/4.65 join(Z, top)
% 33.19/4.65 = { by lemma 25 }
% 33.19/4.65 top
% 33.19/4.65
% 33.19/4.65 Lemma 28: join(X, converse(top)) = top.
% 33.19/4.65 Proof:
% 33.19/4.65 join(X, converse(top))
% 33.19/4.65 = { by axiom 5 (def_top_12) }
% 33.19/4.65 join(X, converse(join(converse(complement(X)), complement(converse(complement(X))))))
% 33.19/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.19/4.65 join(X, converse(join(complement(converse(complement(X))), converse(complement(X)))))
% 33.19/4.65 = { by lemma 26 }
% 33.19/4.65 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 33.19/4.65 = { by lemma 27 }
% 33.19/4.65 top
% 33.19/4.65
% 33.19/4.65 Lemma 29: converse(top) = top.
% 33.19/4.65 Proof:
% 33.19/4.65 converse(top)
% 33.63/4.65 = { by lemma 25 R->L }
% 33.63/4.65 converse(join(X, top))
% 33.63/4.65 = { by axiom 7 (converse_additivity_9) }
% 33.63/4.65 join(converse(X), converse(top))
% 33.63/4.65 = { by lemma 28 }
% 33.63/4.65 top
% 33.63/4.65
% 33.63/4.65 Lemma 30: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(meet(X, Y), complement(join(complement(X), Y)))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 33.63/4.65 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 31: join(zero, meet(X, X)) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(zero, meet(X, X))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 join(zero, complement(join(complement(X), complement(X))))
% 33.63/4.65 = { by axiom 6 (def_zero_13) }
% 33.63/4.65 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 33.63/4.65 = { by lemma 30 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 32: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 33.63/4.65 Proof:
% 33.63/4.65 join(zero, join(X, complement(complement(Y))))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(zero, join(complement(complement(Y)), X))
% 33.63/4.65 = { by lemma 22 R->L }
% 33.63/4.65 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.63/4.65 join(zero, join(meet(Y, Y), X))
% 33.63/4.65 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.63/4.65 join(join(zero, meet(Y, Y)), X)
% 33.63/4.65 = { by lemma 31 }
% 33.63/4.65 join(Y, X)
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.63/4.65 join(X, Y)
% 33.63/4.65
% 33.63/4.65 Lemma 33: join(zero, complement(complement(X))) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(zero, complement(complement(X)))
% 33.63/4.65 = { by axiom 6 (def_zero_13) }
% 33.63/4.65 join(meet(X, complement(X)), complement(complement(X)))
% 33.63/4.65 = { by lemma 22 R->L }
% 33.63/4.65 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 33.63/4.65 = { by lemma 30 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 34: join(zero, complement(X)) = complement(X).
% 33.63/4.65 Proof:
% 33.63/4.65 join(zero, complement(X))
% 33.63/4.65 = { by lemma 33 R->L }
% 33.63/4.65 join(zero, join(zero, complement(complement(complement(X)))))
% 33.63/4.65 = { by lemma 22 R->L }
% 33.63/4.65 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 33.63/4.65 = { by lemma 32 }
% 33.63/4.65 join(zero, join(complement(complement(complement(X))), complement(X)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.63/4.65 join(zero, join(complement(X), complement(complement(complement(X)))))
% 33.63/4.65 = { by lemma 32 }
% 33.63/4.65 join(complement(X), complement(X))
% 33.63/4.65 = { by lemma 22 }
% 33.63/4.65 complement(X)
% 33.63/4.65
% 33.63/4.65 Lemma 35: join(X, zero) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(X, zero)
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(zero, X)
% 33.63/4.65 = { by lemma 32 R->L }
% 33.63/4.65 join(zero, join(zero, complement(complement(X))))
% 33.63/4.65 = { by lemma 34 }
% 33.63/4.65 join(zero, complement(complement(X)))
% 33.63/4.65 = { by lemma 33 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 36: join(top, X) = top.
% 33.63/4.65 Proof:
% 33.63/4.65 join(top, X)
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(X, top)
% 33.63/4.65 = { by lemma 24 R->L }
% 33.63/4.65 join(Y, top)
% 33.63/4.65 = { by lemma 25 }
% 33.63/4.65 top
% 33.63/4.65
% 33.63/4.65 Lemma 37: join(zero, X) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(zero, X)
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(X, zero)
% 33.63/4.65 = { by lemma 35 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 38: complement(complement(X)) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 complement(complement(X))
% 33.63/4.65 = { by lemma 34 R->L }
% 33.63/4.65 join(zero, complement(complement(X)))
% 33.63/4.65 = { by lemma 33 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 39: meet(Y, X) = meet(X, Y).
% 33.63/4.65 Proof:
% 33.63/4.65 meet(Y, X)
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 complement(join(complement(Y), complement(X)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 complement(join(complement(X), complement(Y)))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.63/4.65 meet(X, Y)
% 33.63/4.65
% 33.63/4.65 Lemma 40: complement(join(zero, complement(X))) = meet(X, top).
% 33.63/4.65 Proof:
% 33.63/4.65 complement(join(zero, complement(X)))
% 33.63/4.65 = { by lemma 16 R->L }
% 33.63/4.65 complement(join(complement(top), complement(X)))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.63/4.65 meet(top, X)
% 33.63/4.65 = { by lemma 39 R->L }
% 33.63/4.65 meet(X, top)
% 33.63/4.65
% 33.63/4.65 Lemma 41: join(X, complement(zero)) = top.
% 33.63/4.65 Proof:
% 33.63/4.65 join(X, complement(zero))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(complement(zero), X)
% 33.63/4.65 = { by lemma 32 R->L }
% 33.63/4.65 join(zero, join(complement(zero), complement(complement(X))))
% 33.63/4.65 = { by lemma 27 }
% 33.63/4.65 top
% 33.63/4.65
% 33.63/4.65 Lemma 42: meet(X, zero) = zero.
% 33.63/4.65 Proof:
% 33.63/4.65 meet(X, zero)
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 complement(join(complement(X), complement(zero)))
% 33.63/4.65 = { by lemma 41 }
% 33.63/4.65 complement(top)
% 33.63/4.65 = { by lemma 16 }
% 33.63/4.65 zero
% 33.63/4.65
% 33.63/4.65 Lemma 43: join(meet(X, Y), meet(X, complement(Y))) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 join(meet(X, Y), meet(X, complement(Y)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(meet(X, complement(Y)), meet(X, Y))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 33.63/4.65 = { by lemma 30 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 44: meet(X, top) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 meet(X, top)
% 33.63/4.65 = { by lemma 40 R->L }
% 33.63/4.65 complement(join(zero, complement(X)))
% 33.63/4.65 = { by lemma 34 R->L }
% 33.63/4.65 join(zero, complement(join(zero, complement(X))))
% 33.63/4.65 = { by lemma 40 }
% 33.63/4.65 join(zero, meet(X, top))
% 33.63/4.65 = { by lemma 41 R->L }
% 33.63/4.65 join(zero, meet(X, join(complement(zero), complement(zero))))
% 33.63/4.65 = { by lemma 22 }
% 33.63/4.65 join(zero, meet(X, complement(zero)))
% 33.63/4.65 = { by lemma 42 R->L }
% 33.63/4.65 join(meet(X, zero), meet(X, complement(zero)))
% 33.63/4.65 = { by lemma 43 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 45: meet(top, X) = X.
% 33.63/4.65 Proof:
% 33.63/4.65 meet(top, X)
% 33.63/4.65 = { by lemma 39 }
% 33.63/4.65 meet(X, top)
% 33.63/4.65 = { by lemma 44 }
% 33.63/4.65 X
% 33.63/4.65
% 33.63/4.65 Lemma 46: meet(zero, X) = zero.
% 33.63/4.65 Proof:
% 33.63/4.65 meet(zero, X)
% 33.63/4.65 = { by lemma 39 }
% 33.63/4.65 meet(X, zero)
% 33.63/4.65 = { by lemma 42 }
% 33.63/4.65 zero
% 33.63/4.65
% 33.63/4.65 Lemma 47: composition(join(join(sk1, one), Y), X) = join(X, composition(Y, X)).
% 33.63/4.65 Proof:
% 33.63/4.65 composition(join(join(sk1, one), Y), X)
% 33.63/4.65 = { by axiom 12 (composition_distributivity_7) }
% 33.63/4.65 join(composition(join(sk1, one), X), composition(Y, X))
% 33.63/4.65 = { by lemma 20 }
% 33.63/4.65 join(X, composition(Y, X))
% 33.63/4.65
% 33.63/4.65 Lemma 48: composition(top, zero) = zero.
% 33.63/4.65 Proof:
% 33.63/4.65 composition(top, zero)
% 33.63/4.65 = { by lemma 29 R->L }
% 33.63/4.65 composition(converse(top), zero)
% 33.63/4.65 = { by lemma 37 R->L }
% 33.63/4.65 join(zero, composition(converse(top), zero))
% 33.63/4.65 = { by lemma 16 R->L }
% 33.63/4.65 join(complement(top), composition(converse(top), zero))
% 33.63/4.65 = { by lemma 16 R->L }
% 33.63/4.65 join(complement(top), composition(converse(top), complement(top)))
% 33.63/4.65 = { by lemma 36 R->L }
% 33.63/4.65 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 33.63/4.65 = { by lemma 29 R->L }
% 33.63/4.65 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 33.63/4.65 = { by lemma 47 R->L }
% 33.63/4.65 join(complement(top), composition(converse(top), complement(composition(join(join(sk1, one), converse(top)), top))))
% 33.63/4.65 = { by lemma 28 }
% 33.63/4.65 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 33.63/4.65 = { by lemma 21 }
% 33.63/4.65 complement(top)
% 33.63/4.65 = { by lemma 16 }
% 33.63/4.65 zero
% 33.63/4.65
% 33.63/4.65 Lemma 49: composition(X, zero) = zero.
% 33.63/4.65 Proof:
% 33.63/4.65 composition(X, zero)
% 33.63/4.65 = { by lemma 37 R->L }
% 33.63/4.65 join(zero, composition(X, zero))
% 33.63/4.65 = { by lemma 48 R->L }
% 33.63/4.65 join(composition(top, zero), composition(X, zero))
% 33.63/4.65 = { by axiom 12 (composition_distributivity_7) R->L }
% 33.63/4.65 composition(join(top, X), zero)
% 33.63/4.65 = { by lemma 36 }
% 33.63/4.65 composition(top, zero)
% 33.63/4.65 = { by lemma 48 }
% 33.63/4.65 zero
% 33.63/4.65
% 33.63/4.65 Lemma 50: join(complement(X), X) = top.
% 33.63/4.65 Proof:
% 33.63/4.65 join(complement(X), X)
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 join(X, complement(X))
% 33.63/4.65 = { by axiom 5 (def_top_12) R->L }
% 33.63/4.65 top
% 33.63/4.65
% 33.63/4.65 Lemma 51: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 33.63/4.65 Proof:
% 33.63/4.65 meet(X, join(complement(Y), complement(Z)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 meet(X, join(complement(Z), complement(Y)))
% 33.63/4.65 = { by lemma 39 }
% 33.63/4.65 meet(join(complement(Z), complement(Y)), X)
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.63/4.65 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 33.63/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.63/4.65 complement(join(meet(Z, Y), complement(X)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.63/4.65 complement(join(complement(X), meet(Z, Y)))
% 33.63/4.65 = { by lemma 39 R->L }
% 33.63/4.65 complement(join(complement(X), meet(Y, Z)))
% 33.63/4.65
% 33.63/4.65 Lemma 52: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 33.63/4.65 Proof:
% 33.63/4.65 complement(join(X, complement(Y)))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 complement(join(complement(Y), X))
% 33.63/4.65 = { by lemma 45 R->L }
% 33.63/4.65 complement(join(complement(Y), meet(top, X)))
% 33.63/4.65 = { by lemma 51 R->L }
% 33.63/4.65 meet(Y, join(complement(top), complement(X)))
% 33.63/4.65 = { by lemma 16 }
% 33.63/4.65 meet(Y, join(zero, complement(X)))
% 33.63/4.65 = { by lemma 34 }
% 33.63/4.65 meet(Y, complement(X))
% 33.63/4.65
% 33.63/4.65 Lemma 53: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 33.63/4.65 Proof:
% 33.63/4.65 complement(meet(X, complement(Y)))
% 33.63/4.65 = { by lemma 37 R->L }
% 33.63/4.65 complement(join(zero, meet(X, complement(Y))))
% 33.63/4.65 = { by lemma 52 R->L }
% 33.63/4.65 complement(join(zero, complement(join(Y, complement(X)))))
% 33.63/4.65 = { by lemma 40 }
% 33.63/4.65 meet(join(Y, complement(X)), top)
% 33.63/4.65 = { by lemma 44 }
% 33.63/4.65 join(Y, complement(X))
% 33.63/4.65
% 33.63/4.65 Lemma 54: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 33.63/4.65 Proof:
% 33.63/4.65 complement(meet(complement(X), Y))
% 33.63/4.65 = { by lemma 39 }
% 33.63/4.65 complement(meet(Y, complement(X)))
% 33.63/4.65 = { by lemma 53 }
% 33.63/4.65 join(X, complement(Y))
% 33.63/4.65
% 33.63/4.65 Lemma 55: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 33.63/4.65 Proof:
% 33.63/4.65 complement(join(complement(X), Y))
% 33.63/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.63/4.65 complement(join(Y, complement(X)))
% 33.63/4.65 = { by lemma 52 }
% 33.71/4.65 meet(X, complement(Y))
% 33.71/4.65
% 33.71/4.65 Lemma 56: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 33.71/4.65 Proof:
% 33.71/4.65 join(complement(X), complement(Y))
% 33.71/4.65 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.65 join(complement(Y), complement(X))
% 33.71/4.65 = { by lemma 31 R->L }
% 33.71/4.65 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 33.71/4.65 = { by lemma 51 }
% 33.71/4.65 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 33.71/4.65 = { by lemma 34 }
% 33.71/4.65 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 33.71/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.71/4.65 complement(join(meet(Y, X), meet(Y, X)))
% 33.71/4.65 = { by lemma 39 }
% 33.71/4.65 complement(join(meet(X, Y), meet(Y, X)))
% 33.71/4.65 = { by lemma 39 }
% 33.71/4.65 complement(join(meet(X, Y), meet(X, Y)))
% 33.71/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.71/4.65 complement(join(meet(X, Y), complement(join(complement(X), complement(Y)))))
% 33.71/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.71/4.65 complement(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), complement(Y)))))
% 33.71/4.65 = { by lemma 22 }
% 33.71/4.65 complement(complement(join(complement(X), complement(Y))))
% 33.71/4.65 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.71/4.65 complement(meet(X, Y))
% 33.71/4.65
% 33.71/4.65 Lemma 57: join(X, complement(meet(X, Y))) = top.
% 33.71/4.65 Proof:
% 33.71/4.65 join(X, complement(meet(X, Y)))
% 33.71/4.65 = { by lemma 39 }
% 33.71/4.65 join(X, complement(meet(Y, X)))
% 33.71/4.65 = { by lemma 56 R->L }
% 33.71/4.65 join(X, join(complement(Y), complement(X)))
% 33.71/4.65 = { by lemma 17 }
% 33.71/4.65 join(complement(Y), top)
% 33.71/4.65 = { by lemma 25 }
% 33.71/4.65 top
% 33.71/4.65
% 33.71/4.66 Lemma 58: meet(X, join(X, complement(Y))) = X.
% 33.71/4.66 Proof:
% 33.71/4.66 meet(X, join(X, complement(Y)))
% 33.71/4.66 = { by lemma 53 R->L }
% 33.71/4.66 meet(X, complement(meet(Y, complement(X))))
% 33.71/4.66 = { by lemma 55 R->L }
% 33.71/4.66 complement(join(complement(X), meet(Y, complement(X))))
% 33.71/4.66 = { by lemma 34 R->L }
% 33.71/4.66 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 33.71/4.66 = { by lemma 16 R->L }
% 33.71/4.66 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 33.71/4.66 = { by lemma 57 R->L }
% 33.71/4.66 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.71/4.66 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 33.71/4.66 = { by lemma 39 R->L }
% 33.71/4.66 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 33.71/4.66 = { by lemma 30 }
% 33.71/4.66 X
% 33.71/4.66
% 33.71/4.66 Lemma 59: join(X, meet(X, Y)) = X.
% 33.71/4.66 Proof:
% 33.71/4.66 join(X, meet(X, Y))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.71/4.66 join(X, complement(join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 54 R->L }
% 33.71/4.66 complement(meet(complement(X), join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 58 }
% 33.71/4.66 complement(complement(X))
% 33.71/4.66 = { by lemma 38 }
% 33.71/4.66 X
% 33.71/4.66
% 33.71/4.66 Lemma 60: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 33.71/4.66 Proof:
% 33.71/4.66 complement(join(complement(X), complement(Y)))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.71/4.66 meet(X, Y)
% 33.71/4.66 = { by lemma 39 R->L }
% 33.71/4.66 meet(Y, X)
% 33.71/4.66
% 33.71/4.66 Lemma 61: join(X, meet(Y, X)) = X.
% 33.71/4.66 Proof:
% 33.71/4.66 join(X, meet(Y, X))
% 33.71/4.66 = { by lemma 60 R->L }
% 33.71/4.66 join(X, complement(join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 54 R->L }
% 33.71/4.66 complement(meet(complement(X), join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 58 }
% 33.71/4.66 complement(complement(X))
% 33.71/4.66 = { by lemma 38 }
% 33.71/4.66 X
% 33.71/4.66
% 33.71/4.66 Lemma 62: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 33.71/4.66 Proof:
% 33.71/4.66 join(Y, join(X, Z))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(join(X, Z), Y)
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.66 join(X, join(Z, Y))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.66 join(X, join(Y, Z))
% 33.71/4.66
% 33.71/4.66 Lemma 63: join(composition(sk1, X), X) = X.
% 33.71/4.66 Proof:
% 33.71/4.66 join(composition(sk1, X), X)
% 33.71/4.66 = { by lemma 20 R->L }
% 33.71/4.66 join(composition(sk1, X), composition(join(sk1, one), X))
% 33.71/4.66 = { by axiom 3 (goals_17) }
% 33.71/4.66 join(composition(sk1, X), composition(one, X))
% 33.71/4.66 = { by axiom 12 (composition_distributivity_7) R->L }
% 33.71/4.66 composition(join(sk1, one), X)
% 33.71/4.66 = { by lemma 20 }
% 33.71/4.66 X
% 33.71/4.66
% 33.71/4.66 Lemma 64: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 33.71/4.66 Proof:
% 33.71/4.66 meet(complement(X), complement(Y))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 meet(complement(Y), complement(X))
% 33.71/4.66 = { by lemma 34 R->L }
% 33.71/4.66 meet(join(zero, complement(Y)), complement(X))
% 33.71/4.66 = { by lemma 52 R->L }
% 33.71/4.66 complement(join(X, complement(join(zero, complement(Y)))))
% 33.71/4.66 = { by lemma 40 }
% 33.71/4.66 complement(join(X, meet(Y, top)))
% 33.71/4.66 = { by lemma 44 }
% 33.71/4.66 complement(join(X, Y))
% 33.71/4.66
% 33.71/4.66 Lemma 65: meet(complement(X), complement(Y)) = complement(join(Y, X)).
% 33.71/4.66 Proof:
% 33.71/4.66 meet(complement(X), complement(Y))
% 33.71/4.66 = { by lemma 64 }
% 33.71/4.66 complement(join(X, Y))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.66 complement(join(Y, X))
% 33.71/4.66
% 33.71/4.66 Lemma 66: meet(complement(Z), meet(X, Y)) = meet(X, meet(Y, complement(Z))).
% 33.71/4.66 Proof:
% 33.71/4.66 meet(complement(Z), meet(X, Y))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.71/4.66 meet(complement(Z), complement(join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 65 }
% 33.71/4.66 complement(join(join(complement(X), complement(Y)), Z))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.66 complement(join(complement(X), join(complement(Y), Z)))
% 33.71/4.66 = { by lemma 55 }
% 33.71/4.66 meet(X, complement(join(complement(Y), Z)))
% 33.71/4.66 = { by lemma 55 }
% 33.71/4.66 meet(X, meet(Y, complement(Z)))
% 33.71/4.66
% 33.71/4.66 Lemma 67: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 33.71/4.66 Proof:
% 33.71/4.66 meet(complement(Z), meet(Y, X))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 meet(complement(Z), meet(X, Y))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 meet(meet(X, Y), complement(Z))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 33.71/4.66 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 33.71/4.66 = { by lemma 64 }
% 33.71/4.66 complement(join(join(complement(X), complement(Y)), Z))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.66 complement(join(complement(X), join(complement(Y), Z)))
% 33.71/4.66 = { by lemma 55 }
% 33.71/4.66 meet(X, complement(join(complement(Y), Z)))
% 33.71/4.66 = { by lemma 55 }
% 33.71/4.66 meet(X, meet(Y, complement(Z)))
% 33.71/4.66
% 33.71/4.66 Lemma 68: join(X, join(Y, composition(sk1, X))) = join(X, Y).
% 33.71/4.66 Proof:
% 33.71/4.66 join(X, join(Y, composition(sk1, X)))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(X, join(composition(sk1, X), Y))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.66 join(join(X, composition(sk1, X)), Y)
% 33.71/4.66 = { by lemma 47 R->L }
% 33.71/4.66 join(composition(join(join(sk1, one), sk1), X), Y)
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(composition(join(sk1, join(sk1, one)), X), Y)
% 33.71/4.66 = { by axiom 3 (goals_17) }
% 33.71/4.66 join(composition(join(sk1, one), X), Y)
% 33.71/4.66 = { by lemma 20 }
% 33.71/4.66 join(X, Y)
% 33.71/4.66
% 33.71/4.66 Lemma 69: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 33.71/4.66 Proof:
% 33.71/4.66 join(composition(X, Z), composition(X, Y))
% 33.71/4.66 = { by axiom 1 (converse_idempotence_8) R->L }
% 33.71/4.66 join(composition(X, Z), composition(X, converse(converse(Y))))
% 33.71/4.66 = { by axiom 1 (converse_idempotence_8) R->L }
% 33.71/4.66 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 33.71/4.66 = { by axiom 7 (converse_additivity_9) }
% 33.71/4.66 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 33.71/4.66 = { by axiom 9 (converse_multiplicativity_10) }
% 33.71/4.66 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 33.71/4.66 = { by axiom 1 (converse_idempotence_8) }
% 33.71/4.66 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.66 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 33.71/4.66 = { by axiom 9 (converse_multiplicativity_10) }
% 33.71/4.66 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 33.71/4.66 = { by axiom 12 (composition_distributivity_7) R->L }
% 33.71/4.66 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 33.71/4.66 = { by lemma 26 R->L }
% 33.71/4.66 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 33.71/4.66 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 33.71/4.66 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 33.71/4.66 = { by axiom 1 (converse_idempotence_8) }
% 33.71/4.66 composition(X, join(Z, converse(converse(Y))))
% 33.71/4.66 = { by axiom 1 (converse_idempotence_8) }
% 33.71/4.66 composition(X, join(Z, Y))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.66 composition(X, join(Y, Z))
% 33.71/4.66
% 33.71/4.66 Lemma 70: composition(converse(X), complement(composition(X, top))) = zero.
% 33.71/4.66 Proof:
% 33.71/4.66 composition(converse(X), complement(composition(X, top)))
% 33.71/4.66 = { by lemma 37 R->L }
% 33.71/4.66 join(zero, composition(converse(X), complement(composition(X, top))))
% 33.71/4.66 = { by lemma 16 R->L }
% 33.71/4.66 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 33.71/4.66 = { by lemma 21 }
% 33.71/4.66 complement(top)
% 33.71/4.66 = { by lemma 16 }
% 33.71/4.66 zero
% 33.71/4.66
% 33.71/4.66 Lemma 71: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 33.71/4.66 Proof:
% 33.71/4.66 join(meet(X, Y), meet(Y, complement(X)))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 join(meet(Y, X), meet(Y, complement(X)))
% 33.71/4.66 = { by lemma 43 }
% 33.71/4.66 Y
% 33.71/4.66
% 33.71/4.66 Lemma 72: join(meet(X, complement(Y)), meet(X, Y)) = X.
% 33.71/4.66 Proof:
% 33.71/4.66 join(meet(X, complement(Y)), meet(X, Y))
% 33.71/4.66 = { by lemma 60 R->L }
% 33.71/4.66 join(meet(X, complement(Y)), complement(join(complement(Y), complement(X))))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 33.71/4.66 = { by lemma 30 }
% 33.71/4.66 X
% 33.71/4.66
% 33.71/4.66 Lemma 73: meet(complement(X), meet(Y, complement(composition(sk1, X)))) = meet(Y, complement(X)).
% 33.71/4.66 Proof:
% 33.71/4.66 meet(complement(X), meet(Y, complement(composition(sk1, X))))
% 33.71/4.66 = { by lemma 66 }
% 33.71/4.66 meet(Y, meet(complement(composition(sk1, X)), complement(X)))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 meet(Y, meet(complement(X), complement(composition(sk1, X))))
% 33.71/4.66 = { by lemma 65 }
% 33.71/4.66 meet(Y, complement(join(composition(sk1, X), X)))
% 33.71/4.66 = { by lemma 63 }
% 33.71/4.66 meet(Y, complement(X))
% 33.71/4.66
% 33.71/4.66 Lemma 74: join(meet(X, complement(Y)), join(Z, meet(X, Y))) = join(Z, X).
% 33.71/4.66 Proof:
% 33.71/4.66 join(meet(X, complement(Y)), join(Z, meet(X, Y)))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(meet(X, complement(Y)), join(meet(X, Y), Z))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.66 join(join(meet(X, complement(Y)), meet(X, Y)), Z)
% 33.71/4.66 = { by lemma 72 }
% 33.71/4.66 join(X, Z)
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.66 join(Z, X)
% 33.71/4.66
% 33.71/4.66 Lemma 75: join(meet(X, complement(Y)), join(meet(X, Y), Z)) = join(Z, X).
% 33.71/4.66 Proof:
% 33.71/4.66 join(meet(X, complement(Y)), join(meet(X, Y), Z))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(meet(X, complement(Y)), join(Z, meet(X, Y)))
% 33.71/4.66 = { by lemma 74 }
% 33.71/4.66 join(Z, X)
% 33.71/4.66
% 33.71/4.66 Lemma 76: join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))) = meet(X, complement(composition(sk1, Y))).
% 33.71/4.66 Proof:
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y)))
% 33.71/4.66 = { by lemma 73 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), meet(X, complement(composition(sk1, Y)))))
% 33.71/4.66 = { by lemma 30 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), join(meet(meet(X, complement(composition(sk1, Y))), complement(X)), complement(join(complement(meet(X, complement(composition(sk1, Y)))), complement(X))))))
% 33.71/4.66 = { by lemma 52 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), join(complement(join(X, complement(meet(X, complement(composition(sk1, Y)))))), complement(join(complement(meet(X, complement(composition(sk1, Y)))), complement(X))))))
% 33.71/4.66 = { by lemma 57 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), join(complement(top), complement(join(complement(meet(X, complement(composition(sk1, Y)))), complement(X))))))
% 33.71/4.66 = { by lemma 16 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), join(zero, complement(join(complement(meet(X, complement(composition(sk1, Y)))), complement(X))))))
% 33.71/4.66 = { by lemma 34 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), complement(join(complement(meet(X, complement(composition(sk1, Y)))), complement(X)))))
% 33.71/4.66 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(complement(Y), meet(meet(X, complement(composition(sk1, Y))), X)))
% 33.71/4.66 = { by lemma 66 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), meet(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))))
% 33.71/4.66 = { by lemma 59 }
% 33.71/4.66 meet(X, complement(composition(sk1, Y)))
% 33.71/4.66
% 33.71/4.66 Lemma 77: join(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))), Y) = join(meet(X, complement(Y)), Y).
% 33.71/4.66 Proof:
% 33.71/4.66 join(join(meet(X, complement(composition(sk1, Y))), meet(X, complement(Y))), Y)
% 33.71/4.66 = { by lemma 76 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), Y)
% 33.71/4.66 = { by lemma 59 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, meet(Y, meet(X, composition(sk1, Y)))))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, meet(meet(X, composition(sk1, Y)), Y)))
% 33.71/4.66 = { by lemma 37 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(zero, meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 16 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(top), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 25 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), top)), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 25 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(Z, top))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 24 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(Y, top))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(top, Y))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 50 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(join(complement(composition(sk1, Y)), composition(sk1, Y)), Y))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(complement(composition(sk1, Y)), join(composition(sk1, Y), Y)))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 63 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(X), join(complement(composition(sk1, Y)), Y))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 62 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(complement(composition(sk1, Y)), join(complement(X), Y))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(complement(join(join(complement(composition(sk1, Y)), complement(X)), Y)), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 65 R->L }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(meet(complement(Y), complement(join(complement(composition(sk1, Y)), complement(X)))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 60 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(meet(complement(Y), meet(X, composition(sk1, Y))), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 39 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, join(meet(meet(X, composition(sk1, Y)), complement(Y)), meet(meet(X, composition(sk1, Y)), Y))))
% 33.71/4.66 = { by lemma 72 }
% 33.71/4.66 join(meet(X, complement(composition(sk1, Y))), join(Y, meet(X, composition(sk1, Y))))
% 33.71/4.66 = { by lemma 74 }
% 33.71/4.66 join(Y, X)
% 33.71/4.66 = { by lemma 74 R->L }
% 33.71/4.66 join(meet(X, complement(Y)), join(Y, meet(X, Y)))
% 33.71/4.66 = { by lemma 61 }
% 33.71/4.67 join(meet(X, complement(Y)), Y)
% 33.71/4.67
% 33.71/4.67 Goal 1 (goals_18): tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))) = tuple(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))).
% 33.71/4.67 Proof:
% 33.71/4.67 tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 33.71/4.67 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.67 tuple(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 72 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 38 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(complement(sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 52 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 71 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), join(meet(composition(sk1, sk3), complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))), meet(complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), complement(composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 52 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), join(complement(join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(composition(sk1, sk3)))), meet(complement(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), complement(composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 64 }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), join(complement(join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(composition(sk1, sk3)))), complement(join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 56 }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(composition(sk1, sk3))), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(complement(composition(sk1, sk3)), join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(complement(composition(sk1, sk3)), join(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), complement(sk3))), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 76 }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(complement(composition(sk1, sk3)), join(complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(sk3))), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(join(complement(composition(sk1, sk3)), complement(meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(sk3)), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 39 R->L }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(join(complement(composition(sk1, sk3)), complement(meet(complement(composition(sk1, sk3)), composition(sk1, sk2)))), complement(sk3)), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.67 = { by lemma 57 }
% 33.71/4.67 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(join(top, complement(sk3)), join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 36 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(meet(top, join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 45 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(sk3), join(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 62 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 55 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 52 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 71 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(meet(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(meet(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 53 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(meet(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3))), complement(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 55 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3)))), complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 52 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 56 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(complement(meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 39 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(complement(meet(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 56 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 64 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3)))), meet(complement(join(complement(sk3), composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 64 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(join(meet(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), complement(join(complement(sk3), composition(sk1, sk3)))), meet(complement(join(complement(sk3), composition(sk1, sk3))), complement(complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 71 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(complement(complement(join(complement(sk3), composition(sk1, sk3)))), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 38 }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), meet(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.68 = { by lemma 52 R->L }
% 33.71/4.68 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 52 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(join(complement(sk3), composition(sk1, sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 59 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), complement(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 52 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))), complement(join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 39 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 62 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 76 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 61 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(composition(sk1, sk3), meet(composition(sk1, sk2), composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 74 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, sk3), composition(sk1, sk2))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, sk2), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 68 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(sk3), composition(sk1, sk2))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 71 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(meet(complement(composition(sk1, sk2)), join(complement(sk3), composition(sk1, sk2))), meet(join(complement(sk3), composition(sk1, sk2)), complement(complement(composition(sk1, sk2)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 39 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(meet(complement(composition(sk1, sk2)), join(complement(sk3), composition(sk1, sk2))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 39 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(meet(join(complement(sk3), composition(sk1, sk2)), complement(composition(sk1, sk2))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 52 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(composition(sk1, sk2), complement(join(complement(sk3), composition(sk1, sk2))))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(composition(sk1, sk2), complement(join(composition(sk1, sk2), complement(sk3))))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(complement(join(composition(sk1, sk2), complement(sk3))), composition(sk1, sk2))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 75 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(meet(composition(sk1, sk2), complement(sk3)), join(meet(composition(sk1, sk2), sk3), complement(join(composition(sk1, sk2), complement(sk3)))))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(meet(composition(sk1, sk2), complement(sk3)), join(meet(composition(sk1, sk2), sk3), complement(join(complement(sk3), composition(sk1, sk2)))))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 39 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(meet(composition(sk1, sk2), complement(sk3)), join(meet(sk3, composition(sk1, sk2)), complement(join(complement(sk3), composition(sk1, sk2)))))), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 30 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(meet(composition(sk1, sk2), complement(sk3)), sk3)), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 77 R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), meet(complement(complement(composition(sk1, sk2))), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by lemma 38 }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), meet(composition(sk1, sk2), join(complement(sk3), composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.69 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.69 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), meet(composition(sk1, sk2), join(composition(sk1, sk2), complement(sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 44 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), meet(composition(sk1, sk2), join(composition(sk1, sk2), meet(complement(sk3), top))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 40 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), meet(composition(sk1, sk2), join(composition(sk1, sk2), complement(join(zero, complement(complement(sk3)))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 58 }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)), composition(sk1, sk2))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(composition(sk1, sk2), complement(join(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 77 }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), join(composition(sk1, sk3), composition(sk1, join(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), complement(sk3)), sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 69 }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(join(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), complement(sk3)), sk3))), sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), complement(sk3)), sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(composition(sk1, sk2), complement(join(sk3, meet(composition(sk1, sk2), complement(sk3)))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(complement(join(sk3, meet(composition(sk1, sk2), complement(sk3)))), composition(sk1, sk2))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 75 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(meet(composition(sk1, sk2), complement(sk3)), join(meet(composition(sk1, sk2), sk3), complement(join(sk3, meet(composition(sk1, sk2), complement(sk3))))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(meet(composition(sk1, sk2), complement(sk3)), join(complement(join(sk3, meet(composition(sk1, sk2), complement(sk3)))), meet(composition(sk1, sk2), sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(sk3, join(join(meet(composition(sk1, sk2), complement(sk3)), complement(join(sk3, meet(composition(sk1, sk2), complement(sk3))))), meet(composition(sk1, sk2), sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(join(sk3, join(meet(composition(sk1, sk2), complement(sk3)), complement(join(sk3, meet(composition(sk1, sk2), complement(sk3)))))), meet(composition(sk1, sk2), sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 8 (maddux2_join_associativity_2) }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(join(join(sk3, meet(composition(sk1, sk2), complement(sk3))), complement(join(sk3, meet(composition(sk1, sk2), complement(sk3))))), meet(composition(sk1, sk2), sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 5 (def_top_12) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, join(top, meet(composition(sk1, sk2), sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 36 }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(join(complement(sk3), composition(sk1, sk2)), composition(sk1, top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, sk2), composition(sk1, top))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 44 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), meet(join(composition(sk1, sk2), composition(sk1, top)), top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 40 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, complement(join(composition(sk1, sk2), composition(sk1, top)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, complement(join(composition(sk1, top), composition(sk1, sk2)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 64 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, meet(complement(composition(sk1, top)), complement(composition(sk1, sk2)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 37 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(zero, meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 46 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(meet(zero, complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 49 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(meet(composition(sk1, zero), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 33.71/4.70 = { by lemma 42 R->L }
% 33.71/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(meet(composition(sk1, meet(sk2, zero)), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.70 = { by lemma 70 R->L }
% 34.08/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(meet(composition(sk1, meet(sk2, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.70 = { by axiom 15 (modular_law_1_15) R->L }
% 34.08/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, meet(sk2, composition(converse(sk1), complement(composition(sk1, top))))), complement(composition(sk1, top)))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.70 = { by lemma 70 }
% 34.08/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, meet(sk2, zero)), complement(composition(sk1, top)))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.70 = { by lemma 42 }
% 34.08/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(composition(sk1, zero), complement(composition(sk1, top)))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.70 = { by lemma 49 }
% 34.08/4.70 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(zero, complement(composition(sk1, top)))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 46 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(join(meet(composition(sk1, sk2), complement(composition(sk1, top))), zero), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 35 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, join(meet(composition(sk1, sk2), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, sk2))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 71 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), complement(join(zero, complement(composition(sk1, top))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 40 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), meet(composition(sk1, top), top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 44 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), composition(sk1, top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 50 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), composition(sk1, join(complement(sk3), sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 69 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), join(composition(sk1, sk3), composition(sk1, complement(sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 68 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(complement(sk3), composition(sk1, sk3))), join(complement(sk3), composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 39 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), composition(sk1, sk3)), complement(join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), composition(sk1, sk3)), complement(join(composition(sk1, sk3), complement(sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 52 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), composition(sk1, sk3)), meet(sk3, complement(composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 67 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(composition(sk1, sk3)), meet(sk3, join(complement(sk3), composition(sk1, sk3))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 44 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(composition(sk1, sk3)), meet(sk3, join(complement(sk3), meet(composition(sk1, sk3), top))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 34 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(zero, complement(composition(sk1, sk3))), meet(sk3, join(complement(sk3), meet(composition(sk1, sk3), top))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 44 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(meet(join(zero, complement(composition(sk1, sk3))), top), meet(sk3, join(complement(sk3), meet(composition(sk1, sk3), top))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 40 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(complement(join(zero, complement(join(zero, complement(composition(sk1, sk3)))))), meet(sk3, join(complement(sk3), meet(composition(sk1, sk3), top))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 67 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), meet(composition(sk1, sk3), top)), meet(sk3, complement(join(zero, complement(join(zero, complement(composition(sk1, sk3))))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 40 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), meet(composition(sk1, sk3), top)), meet(sk3, meet(join(zero, complement(composition(sk1, sk3))), top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 44 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), meet(composition(sk1, sk3), top)), meet(sk3, join(zero, complement(composition(sk1, sk3)))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 39 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), meet(composition(sk1, sk3), top)), meet(join(zero, complement(composition(sk1, sk3))), sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 39 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(meet(join(zero, complement(composition(sk1, sk3))), sk3), join(complement(sk3), meet(composition(sk1, sk3), top)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 40 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(meet(join(zero, complement(composition(sk1, sk3))), sk3), join(complement(sk3), complement(join(zero, complement(composition(sk1, sk3))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 39 }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), complement(join(zero, complement(composition(sk1, sk3))))), meet(join(zero, complement(composition(sk1, sk3))), sk3))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 60 R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), meet(join(complement(sk3), complement(join(zero, complement(composition(sk1, sk3))))), complement(join(complement(sk3), complement(join(zero, complement(composition(sk1, sk3)))))))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by axiom 6 (def_zero_13) R->L }
% 34.08/4.71 tuple(join(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), zero), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 35 }
% 34.08/4.71 tuple(meet(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))), complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 76 }
% 34.08/4.71 tuple(meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 39 }
% 34.08/4.71 tuple(meet(complement(sk3), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 73 }
% 34.08/4.71 tuple(meet(composition(sk1, sk2), complement(sk3)), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(sk3))))
% 34.08/4.71 = { by lemma 76 }
% 34.08/4.71 tuple(meet(composition(sk1, sk2), complement(sk3)), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))
% 34.08/4.71 % SZS output end Proof
% 34.08/4.71
% 34.08/4.71 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------