TSTP Solution File: REL016-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL016-4 : 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 : n014.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 13:43:56 EDT 2023
% Result : Unsatisfiable 18.48s 2.74s
% Output : Proof 19.77s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.13 % Problem : REL016-4 : TPTP v8.1.2. Released v4.0.0.
% 0.13/0.14 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.35 % Computer : n014.cluster.edu
% 0.13/0.35 % Model : x86_64 x86_64
% 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.35 % Memory : 8042.1875MB
% 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.35 % CPULimit : 300
% 0.13/0.35 % WCLimit : 300
% 0.13/0.35 % DateTime : Fri Aug 25 19:37:17 EDT 2023
% 0.13/0.36 % CPUTime :
% 18.48/2.74 Command-line arguments: --flatten
% 18.48/2.74
% 18.48/2.74 % SZS status Unsatisfiable
% 18.48/2.74
% 19.26/2.83 % SZS output start Proof
% 19.26/2.83 Take the following subset of the input axioms:
% 19.26/2.83 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 19.26/2.84 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 19.26/2.84 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 19.26/2.84 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 19.26/2.84 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 19.26/2.84 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 19.26/2.84 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 19.26/2.84 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 19.26/2.84 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 19.26/2.84 fof(goals_17, negated_conjecture, join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))!=meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))) | join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))!=meet(composition(sk1, sk2), complement(composition(sk1, sk3)))).
% 19.26/2.84 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 19.26/2.84 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 19.26/2.84 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 19.26/2.84 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 19.26/2.84
% 19.26/2.84 Now clausify the problem and encode Horn clauses using encoding 3 of
% 19.26/2.84 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 19.26/2.84 We repeatedly replace C & s=t => u=v by the two clauses:
% 19.26/2.84 fresh(y, y, x1...xn) = u
% 19.26/2.84 C => fresh(s, t, x1...xn) = v
% 19.26/2.84 where fresh is a fresh function symbol and x1..xn are the free
% 19.26/2.84 variables of u and v.
% 19.26/2.84 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 19.26/2.84 input problem has no model of domain size 1).
% 19.26/2.84
% 19.26/2.84 The encoding turns the above axioms into the following unit equations and goals:
% 19.26/2.84
% 19.26/2.84 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 19.26/2.84 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 19.26/2.84 Axiom 3 (composition_identity_6): composition(X, one) = X.
% 19.26/2.84 Axiom 4 (def_top_12): top = join(X, complement(X)).
% 19.26/2.84 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 19.26/2.84 Axiom 6 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 19.26/2.84 Axiom 7 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 19.26/2.84 Axiom 8 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 19.26/2.84 Axiom 9 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 19.26/2.84 Axiom 10 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 19.26/2.84 Axiom 11 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 19.26/2.84 Axiom 12 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 19.26/2.84 Axiom 13 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 19.26/2.84
% 19.26/2.84 Lemma 14: complement(top) = zero.
% 19.26/2.84 Proof:
% 19.26/2.84 complement(top)
% 19.26/2.84 = { by axiom 4 (def_top_12) }
% 19.26/2.84 complement(join(complement(X), complement(complement(X))))
% 19.26/2.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.84 meet(X, complement(X))
% 19.26/2.84 = { by axiom 5 (def_zero_13) R->L }
% 19.26/2.84 zero
% 19.26/2.84
% 19.26/2.84 Lemma 15: join(X, join(Y, complement(X))) = join(Y, top).
% 19.26/2.84 Proof:
% 19.26/2.84 join(X, join(Y, complement(X)))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(X, join(complement(X), Y))
% 19.26/2.84 = { by axiom 7 (maddux2_join_associativity_2) }
% 19.26/2.84 join(join(X, complement(X)), Y)
% 19.26/2.84 = { by axiom 4 (def_top_12) R->L }
% 19.26/2.84 join(top, Y)
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.84 join(Y, top)
% 19.26/2.84
% 19.26/2.84 Lemma 16: composition(converse(one), X) = X.
% 19.26/2.84 Proof:
% 19.26/2.84 composition(converse(one), X)
% 19.26/2.84 = { by axiom 1 (converse_idempotence_8) R->L }
% 19.26/2.84 composition(converse(one), converse(converse(X)))
% 19.26/2.84 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 19.26/2.84 converse(composition(converse(X), one))
% 19.26/2.84 = { by axiom 3 (composition_identity_6) }
% 19.26/2.84 converse(converse(X))
% 19.26/2.84 = { by axiom 1 (converse_idempotence_8) }
% 19.26/2.84 X
% 19.26/2.84
% 19.26/2.84 Lemma 17: join(complement(X), complement(X)) = complement(X).
% 19.26/2.84 Proof:
% 19.26/2.84 join(complement(X), complement(X))
% 19.26/2.84 = { by lemma 16 R->L }
% 19.26/2.84 join(complement(X), composition(converse(one), complement(X)))
% 19.26/2.84 = { by lemma 16 R->L }
% 19.26/2.84 join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
% 19.26/2.84 = { by axiom 3 (composition_identity_6) R->L }
% 19.26/2.84 join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
% 19.26/2.84 = { by axiom 9 (composition_associativity_5) R->L }
% 19.26/2.84 join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
% 19.26/2.84 = { by lemma 16 }
% 19.26/2.84 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 19.26/2.84 = { by axiom 12 (converse_cancellativity_11) }
% 19.26/2.84 complement(X)
% 19.26/2.84
% 19.26/2.84 Lemma 18: join(top, complement(X)) = top.
% 19.26/2.84 Proof:
% 19.26/2.84 join(top, complement(X))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(complement(X), top)
% 19.26/2.84 = { by lemma 15 R->L }
% 19.26/2.84 join(X, join(complement(X), complement(X)))
% 19.26/2.84 = { by lemma 17 }
% 19.26/2.84 join(X, complement(X))
% 19.26/2.84 = { by axiom 4 (def_top_12) R->L }
% 19.26/2.84 top
% 19.26/2.84
% 19.26/2.84 Lemma 19: join(Y, top) = join(X, top).
% 19.26/2.84 Proof:
% 19.26/2.84 join(Y, top)
% 19.26/2.84 = { by lemma 18 R->L }
% 19.26/2.84 join(Y, join(top, complement(Y)))
% 19.26/2.84 = { by lemma 15 }
% 19.26/2.84 join(top, top)
% 19.26/2.84 = { by lemma 15 R->L }
% 19.26/2.84 join(X, join(top, complement(X)))
% 19.26/2.84 = { by lemma 18 }
% 19.26/2.84 join(X, top)
% 19.26/2.84
% 19.26/2.84 Lemma 20: join(X, top) = top.
% 19.26/2.84 Proof:
% 19.26/2.84 join(X, top)
% 19.26/2.84 = { by lemma 19 }
% 19.26/2.84 join(complement(Y), top)
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(top, complement(Y))
% 19.26/2.84 = { by lemma 18 }
% 19.26/2.84 top
% 19.26/2.84
% 19.26/2.84 Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 19.26/2.84 Proof:
% 19.26/2.84 join(meet(X, Y), complement(join(complement(X), Y)))
% 19.26/2.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.84 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 19.26/2.84 = { by axiom 13 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 19.26/2.84 X
% 19.26/2.84
% 19.26/2.84 Lemma 22: join(zero, meet(X, X)) = X.
% 19.26/2.84 Proof:
% 19.26/2.84 join(zero, meet(X, X))
% 19.26/2.84 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.84 join(zero, complement(join(complement(X), complement(X))))
% 19.26/2.84 = { by axiom 5 (def_zero_13) }
% 19.26/2.84 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 19.26/2.84 = { by lemma 21 }
% 19.26/2.84 X
% 19.26/2.84
% 19.26/2.84 Lemma 23: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 19.26/2.84 Proof:
% 19.26/2.84 join(zero, join(X, complement(complement(Y))))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(zero, join(complement(complement(Y)), X))
% 19.26/2.84 = { by lemma 17 R->L }
% 19.26/2.84 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 19.26/2.84 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.84 join(zero, join(meet(Y, Y), X))
% 19.26/2.84 = { by axiom 7 (maddux2_join_associativity_2) }
% 19.26/2.84 join(join(zero, meet(Y, Y)), X)
% 19.26/2.84 = { by lemma 22 }
% 19.26/2.84 join(Y, X)
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.84 join(X, Y)
% 19.26/2.84
% 19.26/2.84 Lemma 24: join(zero, complement(complement(X))) = X.
% 19.26/2.84 Proof:
% 19.26/2.84 join(zero, complement(complement(X)))
% 19.26/2.84 = { by axiom 5 (def_zero_13) }
% 19.26/2.84 join(meet(X, complement(X)), complement(complement(X)))
% 19.26/2.84 = { by lemma 17 R->L }
% 19.26/2.84 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 19.26/2.84 = { by lemma 21 }
% 19.26/2.84 X
% 19.26/2.84
% 19.26/2.84 Lemma 25: join(zero, complement(X)) = complement(X).
% 19.26/2.84 Proof:
% 19.26/2.84 join(zero, complement(X))
% 19.26/2.84 = { by lemma 24 R->L }
% 19.26/2.84 join(zero, join(zero, complement(complement(complement(X)))))
% 19.26/2.84 = { by lemma 17 R->L }
% 19.26/2.84 join(zero, join(zero, join(complement(complement(complement(X))), complement(complement(complement(X))))))
% 19.26/2.84 = { by lemma 23 }
% 19.26/2.84 join(zero, join(complement(complement(complement(X))), complement(X)))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.84 join(zero, join(complement(X), complement(complement(complement(X)))))
% 19.26/2.84 = { by lemma 23 }
% 19.26/2.84 join(complement(X), complement(X))
% 19.26/2.84 = { by lemma 17 }
% 19.26/2.84 complement(X)
% 19.26/2.84
% 19.26/2.84 Lemma 26: join(X, zero) = X.
% 19.26/2.84 Proof:
% 19.26/2.84 join(X, zero)
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(zero, X)
% 19.26/2.84 = { by lemma 23 R->L }
% 19.26/2.84 join(zero, join(zero, complement(complement(X))))
% 19.26/2.84 = { by lemma 25 }
% 19.26/2.84 join(zero, complement(complement(X)))
% 19.26/2.84 = { by lemma 24 }
% 19.26/2.84 X
% 19.26/2.84
% 19.26/2.84 Lemma 27: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 19.26/2.84 Proof:
% 19.26/2.84 join(Y, join(X, Z))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.84 join(join(X, Z), Y)
% 19.26/2.84 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 19.26/2.84 join(X, join(Z, Y))
% 19.26/2.84 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.84 join(X, join(Y, Z))
% 19.26/2.85
% 19.26/2.85 Lemma 28: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 19.26/2.85 Proof:
% 19.26/2.85 join(Z, join(X, Y))
% 19.26/2.85 = { by lemma 27 }
% 19.26/2.85 join(X, join(Z, Y))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.85 join(X, join(Y, Z))
% 19.26/2.85
% 19.26/2.85 Lemma 29: join(join(X, Y), complement(X)) = join(top, Y).
% 19.26/2.85 Proof:
% 19.26/2.85 join(join(X, Y), complement(X))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(complement(X), join(X, Y))
% 19.26/2.85 = { by lemma 28 }
% 19.26/2.85 join(X, join(Y, complement(X)))
% 19.26/2.85 = { by lemma 15 }
% 19.26/2.85 join(Y, top)
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.85 join(top, Y)
% 19.26/2.85
% 19.26/2.85 Lemma 30: join(X, join(complement(X), Y)) = top.
% 19.26/2.85 Proof:
% 19.26/2.85 join(X, join(complement(X), Y))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(X, join(Y, complement(X)))
% 19.26/2.85 = { by lemma 15 }
% 19.26/2.85 join(Y, top)
% 19.26/2.85 = { by lemma 19 R->L }
% 19.26/2.85 join(Z, top)
% 19.26/2.85 = { by lemma 20 }
% 19.26/2.85 top
% 19.26/2.85
% 19.26/2.85 Lemma 31: join(top, X) = top.
% 19.26/2.85 Proof:
% 19.26/2.85 join(top, X)
% 19.26/2.85 = { by lemma 29 R->L }
% 19.26/2.85 join(join(Y, X), complement(Y))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(complement(Y), join(Y, X))
% 19.26/2.85 = { by lemma 27 }
% 19.26/2.85 join(Y, join(complement(Y), X))
% 19.26/2.85 = { by lemma 30 }
% 19.26/2.85 top
% 19.26/2.85
% 19.26/2.85 Lemma 32: complement(complement(X)) = X.
% 19.26/2.85 Proof:
% 19.26/2.85 complement(complement(X))
% 19.26/2.85 = { by lemma 25 R->L }
% 19.26/2.85 join(zero, complement(complement(X)))
% 19.26/2.85 = { by lemma 24 }
% 19.26/2.85 X
% 19.26/2.85
% 19.26/2.85 Lemma 33: meet(Y, X) = meet(X, Y).
% 19.26/2.85 Proof:
% 19.26/2.85 meet(Y, X)
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.85 complement(join(complement(Y), complement(X)))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 complement(join(complement(X), complement(Y)))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.85 meet(X, Y)
% 19.26/2.85
% 19.26/2.85 Lemma 34: complement(join(zero, complement(X))) = meet(X, top).
% 19.26/2.85 Proof:
% 19.26/2.85 complement(join(zero, complement(X)))
% 19.26/2.85 = { by lemma 14 R->L }
% 19.26/2.85 complement(join(complement(top), complement(X)))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.85 meet(top, X)
% 19.26/2.85 = { by lemma 33 R->L }
% 19.26/2.85 meet(X, top)
% 19.26/2.85
% 19.26/2.85 Lemma 35: join(X, complement(zero)) = top.
% 19.26/2.85 Proof:
% 19.26/2.85 join(X, complement(zero))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(complement(zero), X)
% 19.26/2.85 = { by lemma 23 R->L }
% 19.26/2.85 join(zero, join(complement(zero), complement(complement(X))))
% 19.26/2.85 = { by lemma 30 }
% 19.26/2.85 top
% 19.26/2.85
% 19.26/2.85 Lemma 36: join(meet(X, Y), meet(X, complement(Y))) = X.
% 19.26/2.85 Proof:
% 19.26/2.85 join(meet(X, Y), meet(X, complement(Y)))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(meet(X, complement(Y)), meet(X, Y))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.85 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 19.26/2.85 = { by lemma 21 }
% 19.26/2.85 X
% 19.26/2.85
% 19.26/2.85 Lemma 37: meet(X, top) = X.
% 19.26/2.85 Proof:
% 19.26/2.85 meet(X, top)
% 19.26/2.85 = { by lemma 34 R->L }
% 19.26/2.85 complement(join(zero, complement(X)))
% 19.26/2.85 = { by lemma 25 R->L }
% 19.26/2.85 join(zero, complement(join(zero, complement(X))))
% 19.26/2.85 = { by lemma 34 }
% 19.26/2.85 join(zero, meet(X, top))
% 19.26/2.85 = { by lemma 35 R->L }
% 19.26/2.85 join(zero, meet(X, join(complement(zero), complement(zero))))
% 19.26/2.85 = { by lemma 17 }
% 19.26/2.85 join(zero, meet(X, complement(zero)))
% 19.26/2.85 = { by lemma 14 R->L }
% 19.26/2.85 join(complement(top), meet(X, complement(zero)))
% 19.26/2.85 = { by lemma 35 R->L }
% 19.26/2.85 join(complement(join(complement(X), complement(zero))), meet(X, complement(zero)))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.85 join(meet(X, zero), meet(X, complement(zero)))
% 19.26/2.85 = { by lemma 36 }
% 19.26/2.85 X
% 19.26/2.85
% 19.26/2.85 Lemma 38: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 19.26/2.85 Proof:
% 19.26/2.85 meet(X, join(complement(Y), complement(Z)))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 meet(X, join(complement(Z), complement(Y)))
% 19.26/2.85 = { by lemma 33 }
% 19.26/2.85 meet(join(complement(Z), complement(Y)), X)
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.85 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.85 complement(join(meet(Z, Y), complement(X)))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.85 complement(join(complement(X), meet(Z, Y)))
% 19.26/2.85 = { by lemma 33 R->L }
% 19.26/2.85 complement(join(complement(X), meet(Y, Z)))
% 19.26/2.85
% 19.26/2.85 Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 19.26/2.85 Proof:
% 19.26/2.85 complement(join(X, complement(Y)))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 complement(join(complement(Y), X))
% 19.26/2.85 = { by lemma 37 R->L }
% 19.26/2.85 complement(join(complement(Y), meet(X, top)))
% 19.26/2.85 = { by lemma 33 R->L }
% 19.26/2.85 complement(join(complement(Y), meet(top, X)))
% 19.26/2.85 = { by lemma 38 R->L }
% 19.26/2.85 meet(Y, join(complement(top), complement(X)))
% 19.26/2.85 = { by lemma 14 }
% 19.26/2.85 meet(Y, join(zero, complement(X)))
% 19.26/2.85 = { by lemma 25 }
% 19.26/2.85 meet(Y, complement(X))
% 19.26/2.85
% 19.26/2.85 Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 19.26/2.85 Proof:
% 19.26/2.85 complement(meet(X, complement(Y)))
% 19.26/2.85 = { by lemma 26 R->L }
% 19.26/2.85 complement(join(meet(X, complement(Y)), zero))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.85 complement(join(zero, meet(X, complement(Y))))
% 19.26/2.85 = { by lemma 39 R->L }
% 19.26/2.85 complement(join(zero, complement(join(Y, complement(X)))))
% 19.26/2.85 = { by lemma 34 }
% 19.26/2.85 meet(join(Y, complement(X)), top)
% 19.26/2.85 = { by lemma 37 }
% 19.26/2.85 join(Y, complement(X))
% 19.26/2.85
% 19.26/2.85 Lemma 41: complement(meet(complement(X), Y)) = join(X, complement(Y)).
% 19.26/2.85 Proof:
% 19.26/2.85 complement(meet(complement(X), Y))
% 19.26/2.85 = { by lemma 33 }
% 19.26/2.85 complement(meet(Y, complement(X)))
% 19.26/2.85 = { by lemma 40 }
% 19.26/2.85 join(X, complement(Y))
% 19.26/2.85
% 19.26/2.85 Lemma 42: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 19.26/2.85 Proof:
% 19.26/2.85 complement(join(complement(X), Y))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 complement(join(Y, complement(X)))
% 19.26/2.85 = { by lemma 39 }
% 19.26/2.85 meet(X, complement(Y))
% 19.26/2.85
% 19.26/2.85 Lemma 43: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 19.26/2.85 Proof:
% 19.26/2.85 join(meet(X, Y), meet(X, Y))
% 19.26/2.85 = { by lemma 33 }
% 19.26/2.85 join(meet(Y, X), meet(X, Y))
% 19.26/2.85 = { by lemma 33 }
% 19.26/2.85 join(meet(Y, X), meet(Y, X))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.85 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.85 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 19.26/2.85 = { by lemma 17 }
% 19.26/2.85 complement(join(complement(Y), complement(X)))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.85 meet(Y, X)
% 19.26/2.85 = { by lemma 33 R->L }
% 19.26/2.85 meet(X, Y)
% 19.26/2.85
% 19.26/2.85 Lemma 44: meet(X, join(X, complement(Y))) = X.
% 19.26/2.85 Proof:
% 19.26/2.85 meet(X, join(X, complement(Y)))
% 19.26/2.85 = { by lemma 40 R->L }
% 19.26/2.85 meet(X, complement(meet(Y, complement(X))))
% 19.26/2.85 = { by lemma 42 R->L }
% 19.26/2.85 complement(join(complement(X), meet(Y, complement(X))))
% 19.26/2.85 = { by lemma 25 R->L }
% 19.26/2.85 join(zero, complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 14 R->L }
% 19.26/2.85 join(complement(top), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 20 R->L }
% 19.26/2.85 join(complement(join(complement(Y), top)), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 15 R->L }
% 19.26/2.85 join(complement(join(complement(X), join(complement(Y), complement(complement(X))))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.85 join(complement(join(complement(X), join(complement(complement(X)), complement(Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 22 R->L }
% 19.26/2.85 join(complement(join(complement(X), join(zero, meet(join(complement(complement(X)), complement(Y)), join(complement(complement(X)), complement(Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 38 }
% 19.26/2.85 join(complement(join(complement(X), join(zero, complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y)))))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by lemma 25 }
% 19.26/2.85 join(complement(join(complement(X), complement(join(complement(join(complement(complement(X)), complement(Y))), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.85 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.86 join(complement(join(complement(X), complement(join(meet(complement(X), Y), meet(complement(X), Y))))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.86 = { by lemma 43 }
% 19.26/2.86 join(complement(join(complement(X), complement(meet(complement(X), Y)))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.86 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.86 join(meet(X, meet(complement(X), Y)), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.86 = { by lemma 33 R->L }
% 19.26/2.86 join(meet(X, meet(Y, complement(X))), complement(join(complement(X), meet(Y, complement(X)))))
% 19.26/2.86 = { by lemma 21 }
% 19.26/2.86 X
% 19.26/2.86
% 19.26/2.86 Lemma 45: join(X, meet(X, Y)) = X.
% 19.26/2.86 Proof:
% 19.26/2.86 join(X, meet(X, Y))
% 19.26/2.86 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.86 join(X, complement(join(complement(X), complement(Y))))
% 19.26/2.86 = { by lemma 41 R->L }
% 19.26/2.86 complement(meet(complement(X), join(complement(X), complement(Y))))
% 19.26/2.86 = { by lemma 44 }
% 19.26/2.86 complement(complement(X))
% 19.26/2.86 = { by lemma 32 }
% 19.26/2.86 X
% 19.26/2.86
% 19.26/2.86 Lemma 46: complement(join(complement(X), complement(Y))) = meet(Y, X).
% 19.26/2.86 Proof:
% 19.26/2.86 complement(join(complement(X), complement(Y)))
% 19.26/2.86 = { by axiom 10 (maddux4_definiton_of_meet_4) R->L }
% 19.26/2.86 meet(X, Y)
% 19.26/2.86 = { by lemma 33 R->L }
% 19.26/2.86 meet(Y, X)
% 19.26/2.86
% 19.26/2.86 Lemma 47: join(X, meet(Y, X)) = X.
% 19.26/2.86 Proof:
% 19.26/2.86 join(X, meet(Y, X))
% 19.26/2.86 = { by lemma 46 R->L }
% 19.26/2.86 join(X, complement(join(complement(X), complement(Y))))
% 19.26/2.86 = { by lemma 41 R->L }
% 19.26/2.86 complement(meet(complement(X), join(complement(X), complement(Y))))
% 19.26/2.86 = { by lemma 44 }
% 19.26/2.86 complement(complement(X))
% 19.26/2.86 = { by lemma 32 }
% 19.26/2.86 X
% 19.26/2.86
% 19.26/2.86 Lemma 48: meet(X, join(X, Y)) = X.
% 19.26/2.86 Proof:
% 19.26/2.86 meet(X, join(X, Y))
% 19.26/2.86 = { by lemma 37 R->L }
% 19.26/2.86 meet(X, join(X, meet(Y, top)))
% 19.26/2.86 = { by lemma 34 R->L }
% 19.26/2.86 meet(X, join(X, complement(join(zero, complement(Y)))))
% 19.26/2.86 = { by lemma 44 }
% 19.26/2.86 X
% 19.26/2.86
% 19.26/2.86 Lemma 49: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 19.26/2.86 Proof:
% 19.26/2.86 meet(complement(X), complement(Y))
% 19.26/2.86 = { by lemma 33 }
% 19.26/2.86 meet(complement(Y), complement(X))
% 19.26/2.86 = { by lemma 25 R->L }
% 19.26/2.86 meet(join(zero, complement(Y)), complement(X))
% 19.26/2.86 = { by lemma 39 R->L }
% 19.26/2.86 complement(join(X, complement(join(zero, complement(Y)))))
% 19.26/2.86 = { by lemma 34 }
% 19.26/2.86 complement(join(X, meet(Y, top)))
% 19.26/2.86 = { by lemma 37 }
% 19.26/2.86 complement(join(X, Y))
% 19.26/2.86
% 19.26/2.86 Lemma 50: meet(complement(X), complement(Y)) = complement(join(Y, X)).
% 19.26/2.86 Proof:
% 19.26/2.86 meet(complement(X), complement(Y))
% 19.26/2.86 = { by lemma 49 }
% 19.26/2.86 complement(join(X, Y))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 complement(join(Y, X))
% 19.26/2.86
% 19.26/2.86 Lemma 51: join(meet(X, complement(Y)), join(Z, meet(X, Y))) = join(Z, X).
% 19.26/2.86 Proof:
% 19.26/2.86 join(meet(X, complement(Y)), join(Z, meet(X, Y)))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.86 join(meet(X, complement(Y)), join(meet(X, Y), Z))
% 19.26/2.86 = { by axiom 7 (maddux2_join_associativity_2) }
% 19.26/2.86 join(join(meet(X, complement(Y)), meet(X, Y)), Z)
% 19.26/2.86 = { by lemma 46 R->L }
% 19.26/2.86 join(join(meet(X, complement(Y)), complement(join(complement(Y), complement(X)))), Z)
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.86 join(join(meet(X, complement(Y)), complement(join(complement(X), complement(Y)))), Z)
% 19.26/2.86 = { by lemma 21 }
% 19.26/2.86 join(X, Z)
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 join(Z, X)
% 19.26/2.86
% 19.26/2.86 Lemma 52: join(meet(X, complement(Y)), X) = X.
% 19.26/2.86 Proof:
% 19.26/2.86 join(meet(X, complement(Y)), X)
% 19.26/2.86 = { by lemma 45 R->L }
% 19.26/2.86 join(meet(X, complement(Y)), join(X, meet(X, Y)))
% 19.26/2.86 = { by lemma 51 }
% 19.26/2.86 join(X, X)
% 19.26/2.86 = { by lemma 37 R->L }
% 19.26/2.86 join(X, meet(X, top))
% 19.26/2.86 = { by lemma 37 R->L }
% 19.26/2.86 join(meet(X, top), meet(X, top))
% 19.26/2.86 = { by lemma 43 }
% 19.26/2.86 meet(X, top)
% 19.26/2.86 = { by lemma 37 }
% 19.26/2.86 X
% 19.26/2.86
% 19.26/2.86 Lemma 53: join(meet(X, complement(Y)), Y) = join(X, Y).
% 19.26/2.86 Proof:
% 19.26/2.86 join(meet(X, complement(Y)), Y)
% 19.26/2.86 = { by lemma 47 R->L }
% 19.26/2.86 join(meet(X, complement(Y)), join(Y, meet(X, Y)))
% 19.26/2.86 = { by lemma 51 }
% 19.26/2.86 join(Y, X)
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 join(X, Y)
% 19.26/2.86
% 19.26/2.86 Lemma 54: meet(X, complement(join(Y, X))) = zero.
% 19.26/2.86 Proof:
% 19.26/2.86 meet(X, complement(join(Y, X)))
% 19.26/2.86 = { by lemma 39 R->L }
% 19.26/2.86 complement(join(join(Y, X), complement(X)))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.86 complement(join(complement(X), join(Y, X)))
% 19.26/2.86 = { by lemma 28 }
% 19.26/2.86 complement(join(Y, join(X, complement(X))))
% 19.26/2.86 = { by axiom 4 (def_top_12) R->L }
% 19.26/2.86 complement(join(Y, top))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 complement(join(top, Y))
% 19.26/2.86 = { by lemma 31 }
% 19.26/2.86 complement(top)
% 19.26/2.86 = { by lemma 14 }
% 19.26/2.86 zero
% 19.26/2.86
% 19.26/2.86 Lemma 55: join(composition(X, Z), composition(X, Y)) = composition(X, join(Y, Z)).
% 19.26/2.86 Proof:
% 19.26/2.86 join(composition(X, Z), composition(X, Y))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) R->L }
% 19.26/2.86 join(composition(X, Z), composition(X, converse(converse(Y))))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) R->L }
% 19.26/2.86 converse(converse(join(composition(X, Z), composition(X, converse(converse(Y))))))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.86 converse(converse(join(composition(X, converse(converse(Y))), composition(X, Z))))
% 19.26/2.86 = { by axiom 6 (converse_additivity_9) }
% 19.26/2.86 converse(join(converse(composition(X, converse(converse(Y)))), converse(composition(X, Z))))
% 19.26/2.86 = { by axiom 8 (converse_multiplicativity_10) }
% 19.26/2.86 converse(join(composition(converse(converse(converse(Y))), converse(X)), converse(composition(X, Z))))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) }
% 19.26/2.86 converse(join(composition(converse(Y), converse(X)), converse(composition(X, Z))))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 converse(join(converse(composition(X, Z)), composition(converse(Y), converse(X))))
% 19.26/2.86 = { by axiom 8 (converse_multiplicativity_10) }
% 19.26/2.86 converse(join(composition(converse(Z), converse(X)), composition(converse(Y), converse(X))))
% 19.26/2.86 = { by axiom 11 (composition_distributivity_7) R->L }
% 19.26/2.86 converse(composition(join(converse(Z), converse(Y)), converse(X)))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.86 converse(composition(join(converse(Y), converse(Z)), converse(X)))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) R->L }
% 19.26/2.86 converse(composition(join(converse(converse(converse(Y))), converse(Z)), converse(X)))
% 19.26/2.86 = { by axiom 6 (converse_additivity_9) R->L }
% 19.26/2.86 converse(composition(converse(join(converse(converse(Y)), Z)), converse(X)))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 converse(composition(converse(join(Z, converse(converse(Y)))), converse(X)))
% 19.26/2.86 = { by axiom 8 (converse_multiplicativity_10) R->L }
% 19.26/2.86 converse(converse(composition(X, join(Z, converse(converse(Y))))))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) }
% 19.26/2.86 composition(X, join(Z, converse(converse(Y))))
% 19.26/2.86 = { by axiom 1 (converse_idempotence_8) }
% 19.26/2.86 composition(X, join(Z, Y))
% 19.26/2.86 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.86 composition(X, join(Y, Z))
% 19.26/2.86
% 19.26/2.86 Lemma 56: meet(meet(X, complement(Z)), complement(Y)) = meet(X, complement(join(Y, Z))).
% 19.26/2.86 Proof:
% 19.26/2.86 meet(meet(X, complement(Z)), complement(Y))
% 19.26/2.86 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.86 meet(complement(join(complement(X), complement(complement(Z)))), complement(Y))
% 19.26/2.86 = { by lemma 49 }
% 19.26/2.86 complement(join(join(complement(X), complement(complement(Z))), Y))
% 19.26/2.86 = { by axiom 7 (maddux2_join_associativity_2) R->L }
% 19.26/2.86 complement(join(complement(X), join(complement(complement(Z)), Y)))
% 19.26/2.86 = { by lemma 42 }
% 19.26/2.86 meet(X, complement(join(complement(complement(Z)), Y)))
% 19.26/2.86 = { by lemma 42 }
% 19.26/2.86 meet(X, meet(complement(Z), complement(Y)))
% 19.26/2.86 = { by lemma 50 }
% 19.26/2.86 meet(X, complement(join(Y, Z)))
% 19.26/2.86
% 19.26/2.86 Lemma 57: join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))) = meet(composition(X, Y), complement(Z)).
% 19.26/2.86 Proof:
% 19.26/2.86 join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))
% 19.26/2.86 = { by lemma 48 R->L }
% 19.26/2.86 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), composition(X, Y)))
% 19.26/2.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))))
% 19.26/2.87 = { by lemma 28 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(meet(composition(X, Y), complement(Z)), composition(X, Y))))
% 19.26/2.87 = { by lemma 52 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), composition(X, Y)))
% 19.26/2.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), meet(composition(X, meet(Y, complement(W))), complement(Z))))
% 19.26/2.87 = { by lemma 48 R->L }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), meet(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(composition(X, Y), meet(composition(X, meet(Y, complement(W))), Z))))))
% 19.26/2.87 = { by lemma 51 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), meet(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(composition(X, Y), composition(X, meet(Y, complement(W)))))))
% 19.26/2.87 = { by lemma 55 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), meet(meet(composition(X, meet(Y, complement(W))), complement(Z)), composition(X, join(meet(Y, complement(W)), Y)))))
% 19.26/2.87 = { by lemma 52 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), join(composition(X, Y), meet(meet(composition(X, meet(Y, complement(W))), complement(Z)), composition(X, Y))))
% 19.26/2.87 = { by lemma 47 }
% 19.26/2.87 meet(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), composition(X, Y))
% 19.26/2.87 = { by lemma 33 }
% 19.26/2.87 meet(composition(X, Y), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))
% 19.26/2.87 = { by lemma 32 R->L }
% 19.26/2.87 meet(composition(X, Y), complement(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))
% 19.26/2.87 = { by lemma 45 R->L }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), meet(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), join(meet(composition(X, Y), complement(Z)), Z)))))
% 19.26/2.87 = { by lemma 33 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), meet(join(meet(composition(X, Y), complement(Z)), Z), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))))
% 19.26/2.87 = { by lemma 39 R->L }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))), complement(join(meet(composition(X, Y), complement(Z)), Z)))))))
% 19.26/2.87 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(complement(join(meet(composition(X, Y), complement(Z)), Z)), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))))
% 19.26/2.87 = { by lemma 53 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(complement(join(composition(X, Y), Z)), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))))
% 19.26/2.87 = { by lemma 50 R->L }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(meet(complement(Z), complement(composition(X, Y))), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))))
% 19.26/2.87 = { by lemma 28 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(meet(composition(X, Y), complement(Z)), meet(complement(Z), complement(composition(X, Y)))))))))
% 19.26/2.87 = { by lemma 33 R->L }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), join(meet(complement(Z), composition(X, Y)), meet(complement(Z), complement(composition(X, Y)))))))))
% 19.26/2.87 = { by lemma 36 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), complement(Z))))))
% 19.26/2.87 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(join(complement(Z), meet(composition(X, meet(Y, complement(W))), complement(Z)))))))
% 19.26/2.87 = { by lemma 47 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), complement(complement(Z)))))
% 19.26/2.87 = { by lemma 32 }
% 19.26/2.87 meet(composition(X, Y), complement(join(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))), Z)))
% 19.26/2.87 = { by lemma 56 R->L }
% 19.26/2.87 meet(meet(composition(X, Y), complement(Z)), complement(complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))
% 19.26/2.87 = { by lemma 32 }
% 19.26/2.87 meet(meet(composition(X, Y), complement(Z)), join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))
% 19.26/2.87 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.87 complement(join(complement(meet(composition(X, Y), complement(Z))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))))
% 19.26/2.87 = { by lemma 25 R->L }
% 19.26/2.87 join(zero, complement(join(complement(meet(composition(X, Y), complement(Z))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))))))
% 19.26/2.87 = { by lemma 54 R->L }
% 19.26/2.87 join(meet(meet(composition(X, Y), complement(Z)), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z))))), complement(join(complement(meet(composition(X, Y), complement(Z))), complement(join(meet(composition(X, meet(Y, complement(W))), complement(Z)), meet(composition(X, Y), complement(Z)))))))
% 19.26/2.87 = { by lemma 21 }
% 19.26/2.87 meet(composition(X, Y), complement(Z))
% 19.26/2.87
% 19.26/2.87 Goal 1 (goals_17): tuple(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))) = tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))).
% 19.26/2.87 Proof:
% 19.26/2.88 tuple(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))
% 19.26/2.88 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.88 tuple(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 19.26/2.88 = { by lemma 57 }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))
% 19.26/2.88 = { by lemma 21 R->L }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))), complement(join(complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))))
% 19.26/2.88 = { by axiom 7 (maddux2_join_associativity_2) }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))), complement(join(join(complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))))
% 19.26/2.88 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))), complement(join(join(complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))), complement(join(complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))))))
% 19.26/2.88 = { by axiom 4 (def_top_12) R->L }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))), complement(top)))
% 19.26/2.88 = { by lemma 14 }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))), zero))
% 19.26/2.88 = { by lemma 26 }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(complement(join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 19.26/2.88 = { by lemma 34 }
% 19.26/2.88 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), top), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 19.26/2.89 = { by lemma 37 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(zero, complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))))))
% 19.26/2.89 = { by lemma 25 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 19.26/2.89 = { by lemma 57 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))))))
% 19.26/2.89 = { by lemma 56 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), composition(sk1, sk3)))))))
% 19.26/2.89 = { by lemma 53 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(composition(sk1, meet(sk2, complement(sk3))), composition(sk1, sk3)))))))
% 19.26/2.89 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, meet(sk2, complement(sk3)))))))))
% 19.26/2.89 = { by lemma 55 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, join(meet(sk2, complement(sk3)), sk3)))))))
% 19.26/2.89 = { by lemma 53 }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, join(sk2, sk3)))))))
% 19.26/2.89 = { by lemma 55 R->L }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(composition(sk1, sk3), composition(sk1, sk2)))))))
% 19.26/2.89 = { by axiom 2 (maddux1_join_commutativity_1) }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(composition(sk1, sk2), composition(sk1, sk3)))))))
% 19.26/2.89 = { by lemma 53 R->L }
% 19.26/2.89 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), composition(sk1, sk3)))))))
% 19.77/2.90 = { by lemma 57 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(join(join(meet(composition(sk1, meet(sk2, complement(X))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), composition(sk1, sk3)))))))
% 19.77/2.90 = { by lemma 56 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(meet(composition(sk1, meet(sk2, complement(X))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 54 }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), zero)))
% 19.77/2.90 = { by lemma 26 }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))
% 19.77/2.90 = { by lemma 33 }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))
% 19.77/2.90 = { by axiom 10 (maddux4_definiton_of_meet_4) }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3))))))))
% 19.77/2.90 = { by lemma 25 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(zero, complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 14 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(top), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 31 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(join(top, meet(composition(sk1, sk2), complement(composition(sk1, sk3))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 29 R->L }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(complement(join(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))), complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 39 }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), join(meet(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))), complement(join(complement(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3)))), complement(join(meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))), meet(composition(sk1, sk2), complement(composition(sk1, sk3)))))))))
% 19.77/2.90 = { by lemma 21 }
% 19.77/2.90 tuple(meet(composition(sk1, sk2), complement(composition(sk1, sk3))), meet(composition(sk1, meet(sk2, complement(sk3))), complement(composition(sk1, sk3))))
% 19.77/2.90 % SZS output end Proof
% 19.77/2.90
% 19.77/2.90 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------