TSTP Solution File: REL026-4 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : REL026-4 : 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 : n009.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 13:44:07 EDT 2023
% Result : Unsatisfiable 14.44s 2.19s
% Output : Proof 15.46s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.10/0.12 % Problem : REL026-4 : TPTP v8.1.2. Released v4.0.0.
% 0.12/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.13/0.34 % Computer : n009.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Aug 25 21:43:06 EDT 2023
% 0.13/0.34 % CPUTime :
% 14.44/2.19 Command-line arguments: --flatten
% 14.44/2.19
% 14.44/2.19 % SZS status Unsatisfiable
% 14.44/2.19
% 15.33/2.30 % SZS output start Proof
% 15.33/2.30 Take the following subset of the input axioms:
% 15.33/2.30 fof(composition_associativity_5, axiom, ![A, B, C]: composition(A, composition(B, C))=composition(composition(A, B), C)).
% 15.33/2.30 fof(composition_distributivity_7, axiom, ![A2, B2, C2]: composition(join(A2, B2), C2)=join(composition(A2, C2), composition(B2, C2))).
% 15.33/2.30 fof(composition_identity_6, axiom, ![A2]: composition(A2, one)=A2).
% 15.33/2.30 fof(converse_additivity_9, axiom, ![A2, B2]: converse(join(A2, B2))=join(converse(A2), converse(B2))).
% 15.33/2.30 fof(converse_cancellativity_11, axiom, ![A2, B2]: join(composition(converse(A2), complement(composition(A2, B2))), complement(B2))=complement(B2)).
% 15.33/2.30 fof(converse_idempotence_8, axiom, ![A2]: converse(converse(A2))=A2).
% 15.33/2.30 fof(converse_multiplicativity_10, axiom, ![A2, B2]: converse(composition(A2, B2))=composition(converse(B2), converse(A2))).
% 15.33/2.30 fof(def_top_12, axiom, ![A2]: top=join(A2, complement(A2))).
% 15.33/2.30 fof(def_zero_13, axiom, ![A2]: zero=meet(A2, complement(A2))).
% 15.33/2.30 fof(goals_17, negated_conjecture, join(sk1, one)=one).
% 15.33/2.30 fof(goals_18, negated_conjecture, join(composition(sk1, sk2), meet(composition(sk1, top), sk2))!=meet(composition(sk1, top), sk2) | join(meet(composition(sk1, top), sk2), composition(sk1, sk2))!=composition(sk1, sk2)).
% 15.33/2.30 fof(maddux1_join_commutativity_1, axiom, ![A2, B2]: join(A2, B2)=join(B2, A2)).
% 15.33/2.30 fof(maddux2_join_associativity_2, axiom, ![A2, B2, C2]: join(A2, join(B2, C2))=join(join(A2, B2), C2)).
% 15.33/2.30 fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A2, B2]: A2=join(complement(join(complement(A2), complement(B2))), complement(join(complement(A2), B2)))).
% 15.33/2.30 fof(maddux4_definiton_of_meet_4, axiom, ![A2, B2]: meet(A2, B2)=complement(join(complement(A2), complement(B2)))).
% 15.33/2.30 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)).
% 15.33/2.31 fof(modular_law_2_16, axiom, ![A2, B2, C2]: join(meet(composition(A2, B2), C2), meet(composition(meet(A2, composition(C2, converse(B2))), B2), C2))=meet(composition(meet(A2, composition(C2, converse(B2))), B2), C2)).
% 15.33/2.31
% 15.33/2.31 Now clausify the problem and encode Horn clauses using encoding 3 of
% 15.33/2.31 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 15.33/2.31 We repeatedly replace C & s=t => u=v by the two clauses:
% 15.33/2.31 fresh(y, y, x1...xn) = u
% 15.33/2.31 C => fresh(s, t, x1...xn) = v
% 15.33/2.31 where fresh is a fresh function symbol and x1..xn are the free
% 15.33/2.31 variables of u and v.
% 15.33/2.31 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 15.33/2.31 input problem has no model of domain size 1).
% 15.33/2.31
% 15.33/2.31 The encoding turns the above axioms into the following unit equations and goals:
% 15.33/2.31
% 15.33/2.31 Axiom 1 (converse_idempotence_8): converse(converse(X)) = X.
% 15.33/2.31 Axiom 2 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
% 15.33/2.31 Axiom 3 (goals_17): join(sk1, one) = one.
% 15.33/2.31 Axiom 4 (composition_identity_6): composition(X, one) = X.
% 15.33/2.31 Axiom 5 (def_zero_13): zero = meet(X, complement(X)).
% 15.33/2.31 Axiom 6 (def_top_12): top = join(X, complement(X)).
% 15.33/2.31 Axiom 7 (converse_additivity_9): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 15.33/2.31 Axiom 8 (maddux2_join_associativity_2): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 15.33/2.31 Axiom 9 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 15.33/2.31 Axiom 10 (composition_associativity_5): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 15.33/2.31 Axiom 11 (maddux4_definiton_of_meet_4): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 15.33/2.31 Axiom 12 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 15.33/2.31 Axiom 13 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 15.33/2.31 Axiom 14 (maddux3_a_kind_of_de_Morgan_3): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 15.33/2.31 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).
% 15.33/2.31 Axiom 16 (modular_law_2_16): join(meet(composition(X, Y), Z), meet(composition(meet(X, composition(Z, converse(Y))), Y), Z)) = meet(composition(meet(X, composition(Z, converse(Y))), Y), Z).
% 15.33/2.31
% 15.33/2.31 Lemma 17: complement(top) = zero.
% 15.33/2.31 Proof:
% 15.33/2.31 complement(top)
% 15.33/2.31 = { by axiom 6 (def_top_12) }
% 15.33/2.31 complement(join(complement(X), complement(complement(X))))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.33/2.31 meet(X, complement(X))
% 15.33/2.31 = { by axiom 5 (def_zero_13) R->L }
% 15.33/2.31 zero
% 15.33/2.31
% 15.33/2.31 Lemma 18: join(X, join(Y, complement(X))) = join(Y, top).
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, join(Y, complement(X)))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(X, join(complement(X), Y))
% 15.33/2.31 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.33/2.31 join(join(X, complement(X)), Y)
% 15.33/2.31 = { by axiom 6 (def_top_12) R->L }
% 15.33/2.31 join(top, Y)
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.33/2.31 join(Y, top)
% 15.33/2.31
% 15.33/2.31 Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
% 15.33/2.31 Proof:
% 15.33/2.31 converse(composition(converse(X), Y))
% 15.33/2.31 = { by axiom 9 (converse_multiplicativity_10) }
% 15.33/2.31 composition(converse(Y), converse(converse(X)))
% 15.33/2.31 = { by axiom 1 (converse_idempotence_8) }
% 15.33/2.31 composition(converse(Y), X)
% 15.33/2.31
% 15.33/2.31 Lemma 20: composition(X, join(sk1, one)) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 composition(X, join(sk1, one))
% 15.33/2.31 = { by axiom 3 (goals_17) }
% 15.33/2.31 composition(X, one)
% 15.33/2.31 = { by axiom 4 (composition_identity_6) }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 21: composition(converse(join(sk1, one)), X) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 composition(converse(join(sk1, one)), X)
% 15.33/2.31 = { by lemma 19 R->L }
% 15.33/2.31 converse(composition(converse(X), join(sk1, one)))
% 15.33/2.31 = { by lemma 20 }
% 15.33/2.31 converse(converse(X))
% 15.33/2.31 = { by axiom 1 (converse_idempotence_8) }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 22: composition(join(sk1, one), X) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 composition(join(sk1, one), X)
% 15.33/2.31 = { by lemma 21 R->L }
% 15.33/2.31 composition(converse(join(sk1, one)), composition(join(sk1, one), X))
% 15.33/2.31 = { by axiom 10 (composition_associativity_5) }
% 15.33/2.31 composition(composition(converse(join(sk1, one)), join(sk1, one)), X)
% 15.33/2.31 = { by lemma 20 }
% 15.33/2.31 composition(converse(join(sk1, one)), X)
% 15.33/2.31 = { by lemma 21 }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
% 15.33/2.31 Proof:
% 15.33/2.31 join(complement(X), composition(converse(Y), complement(composition(Y, X))))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(composition(converse(Y), complement(composition(Y, X))), complement(X))
% 15.33/2.31 = { by axiom 13 (converse_cancellativity_11) }
% 15.33/2.31 complement(X)
% 15.33/2.31
% 15.33/2.31 Lemma 24: join(complement(X), complement(X)) = complement(X).
% 15.33/2.31 Proof:
% 15.33/2.31 join(complement(X), complement(X))
% 15.33/2.31 = { by lemma 21 R->L }
% 15.33/2.31 join(complement(X), composition(converse(join(sk1, one)), complement(X)))
% 15.33/2.31 = { by lemma 22 R->L }
% 15.33/2.31 join(complement(X), composition(converse(join(sk1, one)), complement(composition(join(sk1, one), X))))
% 15.33/2.31 = { by lemma 23 }
% 15.33/2.31 complement(X)
% 15.33/2.31
% 15.33/2.31 Lemma 25: join(top, complement(X)) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 join(top, complement(X))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(complement(X), top)
% 15.33/2.31 = { by lemma 18 R->L }
% 15.33/2.31 join(X, join(complement(X), complement(X)))
% 15.33/2.31 = { by lemma 24 }
% 15.33/2.31 join(X, complement(X))
% 15.33/2.31 = { by axiom 6 (def_top_12) R->L }
% 15.33/2.31 top
% 15.33/2.31
% 15.33/2.31 Lemma 26: join(Y, top) = join(X, top).
% 15.33/2.31 Proof:
% 15.33/2.31 join(Y, top)
% 15.33/2.31 = { by lemma 25 R->L }
% 15.33/2.31 join(Y, join(top, complement(Y)))
% 15.33/2.31 = { by lemma 18 }
% 15.33/2.31 join(top, top)
% 15.33/2.31 = { by lemma 18 R->L }
% 15.33/2.31 join(X, join(top, complement(X)))
% 15.33/2.31 = { by lemma 25 }
% 15.33/2.31 join(X, top)
% 15.33/2.31
% 15.33/2.31 Lemma 27: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 join(meet(X, Y), complement(join(complement(X), Y)))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.31 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 15.33/2.31 = { by axiom 14 (maddux3_a_kind_of_de_Morgan_3) R->L }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 28: join(zero, meet(X, X)) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 join(zero, meet(X, X))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.31 join(zero, complement(join(complement(X), complement(X))))
% 15.33/2.31 = { by axiom 5 (def_zero_13) }
% 15.33/2.31 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.33/2.31 = { by lemma 27 }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 29: join(zero, join(X, complement(complement(Y)))) = join(X, Y).
% 15.33/2.31 Proof:
% 15.33/2.31 join(zero, join(X, complement(complement(Y))))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(zero, join(complement(complement(Y)), X))
% 15.33/2.31 = { by lemma 24 R->L }
% 15.33/2.31 join(zero, join(complement(join(complement(Y), complement(Y))), X))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.33/2.31 join(zero, join(meet(Y, Y), X))
% 15.33/2.31 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.33/2.31 join(join(zero, meet(Y, Y)), X)
% 15.33/2.31 = { by lemma 28 }
% 15.33/2.31 join(Y, X)
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.33/2.31 join(X, Y)
% 15.33/2.31
% 15.33/2.31 Lemma 30: join(zero, complement(complement(X))) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 join(zero, complement(complement(X)))
% 15.33/2.31 = { by axiom 5 (def_zero_13) }
% 15.33/2.31 join(meet(X, complement(X)), complement(complement(X)))
% 15.33/2.31 = { by lemma 24 R->L }
% 15.33/2.31 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 15.33/2.31 = { by lemma 27 }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 31: join(X, zero) = join(X, X).
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, zero)
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(zero, X)
% 15.33/2.31 = { by lemma 30 R->L }
% 15.33/2.31 join(zero, join(zero, complement(complement(X))))
% 15.33/2.31 = { by lemma 24 R->L }
% 15.33/2.31 join(zero, join(zero, join(complement(complement(X)), complement(complement(X)))))
% 15.33/2.31 = { by lemma 29 }
% 15.33/2.31 join(zero, join(complement(complement(X)), X))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.33/2.31 join(zero, join(X, complement(complement(X))))
% 15.33/2.31 = { by lemma 29 }
% 15.33/2.31 join(X, X)
% 15.33/2.31
% 15.33/2.31 Lemma 32: join(zero, complement(X)) = complement(X).
% 15.33/2.31 Proof:
% 15.33/2.31 join(zero, complement(X))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(complement(X), zero)
% 15.33/2.31 = { by lemma 31 }
% 15.33/2.31 join(complement(X), complement(X))
% 15.33/2.31 = { by lemma 24 }
% 15.33/2.31 complement(X)
% 15.33/2.31
% 15.33/2.31 Lemma 33: join(X, zero) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, zero)
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(zero, X)
% 15.33/2.31 = { by lemma 29 R->L }
% 15.33/2.31 join(zero, join(zero, complement(complement(X))))
% 15.33/2.31 = { by lemma 32 }
% 15.33/2.31 join(zero, complement(complement(X)))
% 15.33/2.31 = { by lemma 30 }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 34: join(zero, X) = X.
% 15.33/2.31 Proof:
% 15.33/2.31 join(zero, X)
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(X, zero)
% 15.33/2.31 = { by lemma 33 }
% 15.33/2.31 X
% 15.33/2.31
% 15.33/2.31 Lemma 35: join(X, top) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, top)
% 15.33/2.31 = { by lemma 26 }
% 15.33/2.31 join(zero, top)
% 15.33/2.31 = { by lemma 34 }
% 15.33/2.31 top
% 15.33/2.31
% 15.33/2.31 Lemma 36: converse(join(X, converse(Y))) = join(Y, converse(X)).
% 15.33/2.31 Proof:
% 15.33/2.31 converse(join(X, converse(Y)))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 converse(join(converse(Y), X))
% 15.33/2.31 = { by axiom 7 (converse_additivity_9) }
% 15.33/2.31 join(converse(converse(Y)), converse(X))
% 15.33/2.31 = { by axiom 1 (converse_idempotence_8) }
% 15.33/2.31 join(Y, converse(X))
% 15.33/2.31
% 15.33/2.31 Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
% 15.33/2.31 Proof:
% 15.33/2.31 converse(join(converse(X), Y))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 converse(join(Y, converse(X)))
% 15.33/2.31 = { by lemma 36 }
% 15.33/2.31 join(X, converse(Y))
% 15.33/2.31
% 15.33/2.31 Lemma 38: join(X, converse(complement(converse(X)))) = converse(top).
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, converse(complement(converse(X))))
% 15.33/2.31 = { by lemma 37 R->L }
% 15.33/2.31 converse(join(converse(X), complement(converse(X))))
% 15.33/2.31 = { by axiom 6 (def_top_12) R->L }
% 15.33/2.31 converse(top)
% 15.33/2.31
% 15.33/2.31 Lemma 39: join(X, join(complement(X), Y)) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, join(complement(X), Y))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(X, join(Y, complement(X)))
% 15.33/2.31 = { by lemma 18 }
% 15.33/2.31 join(Y, top)
% 15.33/2.31 = { by lemma 26 R->L }
% 15.33/2.31 join(Z, top)
% 15.33/2.31 = { by lemma 35 }
% 15.33/2.31 top
% 15.33/2.31
% 15.33/2.31 Lemma 40: join(X, converse(top)) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, converse(top))
% 15.33/2.31 = { by lemma 38 R->L }
% 15.33/2.31 join(X, join(complement(X), converse(complement(converse(complement(X))))))
% 15.33/2.31 = { by lemma 39 }
% 15.33/2.31 top
% 15.33/2.31
% 15.33/2.31 Lemma 41: converse(top) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 converse(top)
% 15.33/2.31 = { by lemma 35 R->L }
% 15.33/2.31 converse(join(X, top))
% 15.33/2.31 = { by axiom 7 (converse_additivity_9) }
% 15.33/2.31 join(converse(X), converse(top))
% 15.33/2.31 = { by lemma 40 }
% 15.33/2.31 top
% 15.33/2.31
% 15.33/2.31 Lemma 42: meet(Y, X) = meet(X, Y).
% 15.33/2.31 Proof:
% 15.33/2.31 meet(Y, X)
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.31 complement(join(complement(Y), complement(X)))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 complement(join(complement(X), complement(Y)))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.33/2.31 meet(X, Y)
% 15.33/2.31
% 15.33/2.31 Lemma 43: complement(join(zero, complement(X))) = meet(X, top).
% 15.33/2.31 Proof:
% 15.33/2.31 complement(join(zero, complement(X)))
% 15.33/2.31 = { by lemma 17 R->L }
% 15.33/2.31 complement(join(complement(top), complement(X)))
% 15.33/2.31 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.33/2.31 meet(top, X)
% 15.33/2.31 = { by lemma 42 R->L }
% 15.33/2.31 meet(X, top)
% 15.33/2.31
% 15.33/2.31 Lemma 44: join(X, complement(zero)) = top.
% 15.33/2.31 Proof:
% 15.33/2.31 join(X, complement(zero))
% 15.33/2.31 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.31 join(complement(zero), X)
% 15.33/2.31 = { by lemma 29 R->L }
% 15.33/2.32 join(zero, join(complement(zero), complement(complement(X))))
% 15.33/2.32 = { by lemma 39 }
% 15.33/2.32 top
% 15.33/2.32
% 15.33/2.32 Lemma 45: meet(X, zero) = zero.
% 15.33/2.32 Proof:
% 15.33/2.32 meet(X, zero)
% 15.33/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.32 complement(join(complement(X), complement(zero)))
% 15.33/2.32 = { by lemma 44 }
% 15.33/2.32 complement(top)
% 15.33/2.32 = { by lemma 17 }
% 15.33/2.32 zero
% 15.33/2.32
% 15.33/2.32 Lemma 46: join(meet(X, Y), meet(X, complement(Y))) = X.
% 15.33/2.32 Proof:
% 15.33/2.32 join(meet(X, Y), meet(X, complement(Y)))
% 15.33/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.32 join(meet(X, complement(Y)), meet(X, Y))
% 15.33/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.32 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 15.33/2.32 = { by lemma 27 }
% 15.33/2.32 X
% 15.33/2.32
% 15.33/2.32 Lemma 47: meet(X, top) = X.
% 15.33/2.32 Proof:
% 15.33/2.32 meet(X, top)
% 15.33/2.32 = { by lemma 43 R->L }
% 15.33/2.32 complement(join(zero, complement(X)))
% 15.33/2.32 = { by lemma 32 R->L }
% 15.33/2.32 join(zero, complement(join(zero, complement(X))))
% 15.33/2.32 = { by lemma 43 }
% 15.33/2.32 join(zero, meet(X, top))
% 15.33/2.32 = { by lemma 44 R->L }
% 15.33/2.32 join(zero, meet(X, join(complement(zero), complement(zero))))
% 15.33/2.32 = { by lemma 24 }
% 15.33/2.32 join(zero, meet(X, complement(zero)))
% 15.33/2.32 = { by lemma 45 R->L }
% 15.33/2.32 join(meet(X, zero), meet(X, complement(zero)))
% 15.33/2.32 = { by lemma 46 }
% 15.33/2.32 X
% 15.33/2.32
% 15.33/2.32 Lemma 48: join(meet(X, Y), meet(X, Y)) = meet(X, Y).
% 15.33/2.32 Proof:
% 15.33/2.32 join(meet(X, Y), meet(X, Y))
% 15.33/2.32 = { by lemma 42 }
% 15.33/2.32 join(meet(Y, X), meet(X, Y))
% 15.33/2.32 = { by lemma 42 }
% 15.33/2.32 join(meet(Y, X), meet(Y, X))
% 15.33/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.32 join(meet(Y, X), complement(join(complement(Y), complement(X))))
% 15.33/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.33/2.32 join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X))))
% 15.33/2.32 = { by lemma 24 }
% 15.33/2.32 complement(join(complement(Y), complement(X)))
% 15.33/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.33/2.32 meet(Y, X)
% 15.33/2.32 = { by lemma 42 R->L }
% 15.33/2.32 meet(X, Y)
% 15.33/2.32
% 15.33/2.32 Lemma 49: converse(zero) = zero.
% 15.33/2.32 Proof:
% 15.33/2.32 converse(zero)
% 15.33/2.32 = { by lemma 34 R->L }
% 15.33/2.32 join(zero, converse(zero))
% 15.33/2.32 = { by lemma 37 R->L }
% 15.33/2.32 converse(join(converse(zero), zero))
% 15.33/2.32 = { by lemma 31 }
% 15.33/2.32 converse(join(converse(zero), converse(zero)))
% 15.33/2.32 = { by lemma 36 }
% 15.33/2.32 join(zero, converse(converse(zero)))
% 15.33/2.32 = { by axiom 1 (converse_idempotence_8) }
% 15.33/2.32 join(zero, zero)
% 15.33/2.32 = { by lemma 47 R->L }
% 15.33/2.32 join(zero, meet(zero, top))
% 15.33/2.32 = { by lemma 47 R->L }
% 15.33/2.32 join(meet(zero, top), meet(zero, top))
% 15.33/2.32 = { by lemma 48 }
% 15.33/2.32 meet(zero, top)
% 15.33/2.32 = { by lemma 47 }
% 15.33/2.32 zero
% 15.33/2.32
% 15.33/2.32 Lemma 50: join(top, X) = top.
% 15.33/2.32 Proof:
% 15.33/2.32 join(top, X)
% 15.33/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.33/2.32 join(X, top)
% 15.33/2.32 = { by lemma 26 R->L }
% 15.33/2.32 join(Y, top)
% 15.33/2.32 = { by lemma 35 }
% 15.33/2.32 top
% 15.33/2.32
% 15.33/2.32 Lemma 51: complement(complement(X)) = X.
% 15.33/2.32 Proof:
% 15.33/2.32 complement(complement(X))
% 15.33/2.32 = { by lemma 32 R->L }
% 15.33/2.32 join(zero, complement(complement(X)))
% 15.33/2.32 = { by lemma 30 }
% 15.33/2.32 X
% 15.33/2.32
% 15.33/2.32 Lemma 52: meet(zero, X) = zero.
% 15.33/2.32 Proof:
% 15.33/2.32 meet(zero, X)
% 15.33/2.32 = { by lemma 42 }
% 15.33/2.32 meet(X, zero)
% 15.33/2.32 = { by lemma 45 }
% 15.33/2.32 zero
% 15.33/2.32
% 15.33/2.32 Lemma 53: composition(join(join(sk1, one), Y), X) = join(X, composition(Y, X)).
% 15.33/2.32 Proof:
% 15.33/2.32 composition(join(join(sk1, one), Y), X)
% 15.33/2.32 = { by axiom 12 (composition_distributivity_7) }
% 15.33/2.32 join(composition(join(sk1, one), X), composition(Y, X))
% 15.33/2.32 = { by lemma 22 }
% 15.33/2.32 join(X, composition(Y, X))
% 15.33/2.32
% 15.33/2.32 Lemma 54: composition(top, zero) = zero.
% 15.33/2.32 Proof:
% 15.33/2.32 composition(top, zero)
% 15.33/2.32 = { by lemma 41 R->L }
% 15.33/2.32 composition(converse(top), zero)
% 15.33/2.32 = { by lemma 34 R->L }
% 15.46/2.32 join(zero, composition(converse(top), zero))
% 15.46/2.32 = { by lemma 17 R->L }
% 15.46/2.32 join(complement(top), composition(converse(top), zero))
% 15.46/2.32 = { by lemma 17 R->L }
% 15.46/2.32 join(complement(top), composition(converse(top), complement(top)))
% 15.46/2.32 = { by lemma 50 R->L }
% 15.46/2.32 join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
% 15.46/2.32 = { by lemma 41 R->L }
% 15.46/2.32 join(complement(top), composition(converse(top), complement(join(top, composition(converse(top), top)))))
% 15.46/2.32 = { by lemma 53 R->L }
% 15.46/2.32 join(complement(top), composition(converse(top), complement(composition(join(join(sk1, one), converse(top)), top))))
% 15.46/2.32 = { by lemma 40 }
% 15.46/2.32 join(complement(top), composition(converse(top), complement(composition(top, top))))
% 15.46/2.32 = { by lemma 23 }
% 15.46/2.32 complement(top)
% 15.46/2.32 = { by lemma 17 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 55: composition(X, zero) = zero.
% 15.46/2.32 Proof:
% 15.46/2.32 composition(X, zero)
% 15.46/2.32 = { by lemma 34 R->L }
% 15.46/2.32 join(zero, composition(X, zero))
% 15.46/2.32 = { by lemma 54 R->L }
% 15.46/2.32 join(composition(top, zero), composition(X, zero))
% 15.46/2.32 = { by axiom 12 (composition_distributivity_7) R->L }
% 15.46/2.32 composition(join(top, X), zero)
% 15.46/2.32 = { by lemma 50 }
% 15.46/2.32 composition(top, zero)
% 15.46/2.32 = { by lemma 54 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 56: composition(converse(X), complement(composition(X, top))) = zero.
% 15.46/2.32 Proof:
% 15.46/2.32 composition(converse(X), complement(composition(X, top)))
% 15.46/2.32 = { by lemma 34 R->L }
% 15.46/2.32 join(zero, composition(converse(X), complement(composition(X, top))))
% 15.46/2.32 = { by lemma 17 R->L }
% 15.46/2.32 join(complement(top), composition(converse(X), complement(composition(X, top))))
% 15.46/2.32 = { by lemma 23 }
% 15.46/2.32 complement(top)
% 15.46/2.32 = { by lemma 17 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 57: composition(converse(X), composition(complement(composition(X, top)), Y)) = zero.
% 15.46/2.32 Proof:
% 15.46/2.32 composition(converse(X), composition(complement(composition(X, top)), Y))
% 15.46/2.32 = { by lemma 19 R->L }
% 15.46/2.32 converse(composition(converse(composition(complement(composition(X, top)), Y)), X))
% 15.46/2.32 = { by axiom 9 (converse_multiplicativity_10) }
% 15.46/2.32 converse(composition(composition(converse(Y), converse(complement(composition(X, top)))), X))
% 15.46/2.32 = { by axiom 10 (composition_associativity_5) R->L }
% 15.46/2.32 converse(composition(converse(Y), composition(converse(complement(composition(X, top))), X)))
% 15.46/2.32 = { by lemma 19 R->L }
% 15.46/2.32 converse(composition(converse(Y), converse(composition(converse(X), complement(composition(X, top))))))
% 15.46/2.32 = { by lemma 56 }
% 15.46/2.32 converse(composition(converse(Y), converse(zero)))
% 15.46/2.32 = { by lemma 49 }
% 15.46/2.32 converse(composition(converse(Y), zero))
% 15.46/2.32 = { by lemma 55 }
% 15.46/2.32 converse(zero)
% 15.46/2.32 = { by lemma 49 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 58: composition(zero, X) = zero.
% 15.46/2.32 Proof:
% 15.46/2.32 composition(zero, X)
% 15.46/2.32 = { by lemma 57 R->L }
% 15.46/2.32 composition(composition(converse(Y), composition(complement(composition(Y, top)), Z)), X)
% 15.46/2.32 = { by axiom 10 (composition_associativity_5) R->L }
% 15.46/2.32 composition(converse(Y), composition(composition(complement(composition(Y, top)), Z), X))
% 15.46/2.32 = { by axiom 10 (composition_associativity_5) R->L }
% 15.46/2.32 composition(converse(Y), composition(complement(composition(Y, top)), composition(Z, X)))
% 15.46/2.32 = { by lemma 57 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 59: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
% 15.46/2.32 Proof:
% 15.46/2.32 meet(X, join(complement(Y), complement(Z)))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.32 meet(X, join(complement(Z), complement(Y)))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 meet(join(complement(Z), complement(Y)), X)
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.46/2.32 complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.32 complement(join(meet(Z, Y), complement(X)))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.32 complement(join(complement(X), meet(Z, Y)))
% 15.46/2.32 = { by lemma 42 R->L }
% 15.46/2.32 complement(join(complement(X), meet(Y, Z)))
% 15.46/2.32
% 15.46/2.32 Lemma 60: complement(join(X, complement(Y))) = meet(Y, complement(X)).
% 15.46/2.32 Proof:
% 15.46/2.32 complement(join(X, complement(Y)))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.32 complement(join(complement(Y), X))
% 15.46/2.32 = { by lemma 47 R->L }
% 15.46/2.32 complement(join(complement(Y), meet(X, top)))
% 15.46/2.32 = { by lemma 42 R->L }
% 15.46/2.32 complement(join(complement(Y), meet(top, X)))
% 15.46/2.32 = { by lemma 59 R->L }
% 15.46/2.32 meet(Y, join(complement(top), complement(X)))
% 15.46/2.32 = { by lemma 17 }
% 15.46/2.32 meet(Y, join(zero, complement(X)))
% 15.46/2.32 = { by lemma 32 }
% 15.46/2.32 meet(Y, complement(X))
% 15.46/2.32
% 15.46/2.32 Lemma 61: complement(meet(X, complement(Y))) = join(Y, complement(X)).
% 15.46/2.32 Proof:
% 15.46/2.32 complement(meet(X, complement(Y)))
% 15.46/2.32 = { by lemma 34 R->L }
% 15.46/2.32 complement(join(zero, meet(X, complement(Y))))
% 15.46/2.32 = { by lemma 60 R->L }
% 15.46/2.32 complement(join(zero, complement(join(Y, complement(X)))))
% 15.46/2.32 = { by lemma 43 }
% 15.46/2.32 meet(join(Y, complement(X)), top)
% 15.46/2.32 = { by lemma 47 }
% 15.46/2.32 join(Y, complement(X))
% 15.46/2.32
% 15.46/2.32 Lemma 62: complement(join(complement(X), Y)) = meet(X, complement(Y)).
% 15.46/2.32 Proof:
% 15.46/2.32 complement(join(complement(X), Y))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.32 complement(join(Y, complement(X)))
% 15.46/2.32 = { by lemma 60 }
% 15.46/2.32 meet(X, complement(Y))
% 15.46/2.32
% 15.46/2.32 Lemma 63: join(complement(X), complement(Y)) = complement(meet(X, Y)).
% 15.46/2.32 Proof:
% 15.46/2.32 join(complement(X), complement(Y))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.32 join(complement(Y), complement(X))
% 15.46/2.32 = { by lemma 28 R->L }
% 15.46/2.32 join(zero, meet(join(complement(Y), complement(X)), join(complement(Y), complement(X))))
% 15.46/2.32 = { by lemma 59 }
% 15.46/2.32 join(zero, complement(join(complement(join(complement(Y), complement(X))), meet(Y, X))))
% 15.46/2.32 = { by lemma 32 }
% 15.46/2.32 complement(join(complement(join(complement(Y), complement(X))), meet(Y, X)))
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.32 complement(join(meet(Y, X), meet(Y, X)))
% 15.46/2.32 = { by lemma 48 }
% 15.46/2.32 complement(meet(Y, X))
% 15.46/2.32 = { by lemma 42 R->L }
% 15.46/2.32 complement(meet(X, Y))
% 15.46/2.32
% 15.46/2.32 Lemma 64: join(X, complement(meet(X, Y))) = top.
% 15.46/2.32 Proof:
% 15.46/2.32 join(X, complement(meet(X, Y)))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 join(X, complement(meet(Y, X)))
% 15.46/2.32 = { by lemma 63 R->L }
% 15.46/2.32 join(X, join(complement(Y), complement(X)))
% 15.46/2.32 = { by lemma 18 }
% 15.46/2.32 join(complement(Y), top)
% 15.46/2.32 = { by lemma 35 }
% 15.46/2.32 top
% 15.46/2.32
% 15.46/2.32 Lemma 65: meet(X, meet(Y, complement(X))) = zero.
% 15.46/2.32 Proof:
% 15.46/2.32 meet(X, meet(Y, complement(X)))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 meet(X, meet(complement(X), Y))
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.46/2.32 complement(join(complement(X), complement(meet(complement(X), Y))))
% 15.46/2.32 = { by lemma 64 }
% 15.46/2.32 complement(top)
% 15.46/2.32 = { by lemma 17 }
% 15.46/2.32 zero
% 15.46/2.32
% 15.46/2.32 Lemma 66: join(X, meet(X, Y)) = X.
% 15.46/2.32 Proof:
% 15.46/2.32 join(X, meet(X, Y))
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.46/2.32 join(X, complement(join(complement(X), complement(Y))))
% 15.46/2.32 = { by lemma 61 R->L }
% 15.46/2.32 complement(meet(join(complement(X), complement(Y)), complement(X)))
% 15.46/2.32 = { by lemma 42 R->L }
% 15.46/2.32 complement(meet(complement(X), join(complement(X), complement(Y))))
% 15.46/2.32 = { by lemma 61 R->L }
% 15.46/2.32 complement(meet(complement(X), complement(meet(Y, complement(complement(X))))))
% 15.46/2.32 = { by lemma 62 R->L }
% 15.46/2.32 complement(complement(join(complement(complement(X)), meet(Y, complement(complement(X))))))
% 15.46/2.32 = { by lemma 32 R->L }
% 15.46/2.32 complement(join(zero, complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 15.46/2.32 = { by lemma 65 R->L }
% 15.46/2.32 complement(join(meet(complement(X), meet(Y, complement(complement(X)))), complement(join(complement(complement(X)), meet(Y, complement(complement(X)))))))
% 15.46/2.32 = { by lemma 27 }
% 15.46/2.32 complement(complement(X))
% 15.46/2.32 = { by lemma 51 }
% 15.46/2.32 X
% 15.46/2.32
% 15.46/2.32 Lemma 67: join(Y, join(X, Z)) = join(X, join(Y, Z)).
% 15.46/2.32 Proof:
% 15.46/2.32 join(Y, join(X, Z))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.32 join(join(X, Z), Y)
% 15.46/2.32 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.32 join(X, join(Z, Y))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.32 join(X, join(Y, Z))
% 15.46/2.32
% 15.46/2.32 Lemma 68: join(Z, join(X, Y)) = join(X, join(Y, Z)).
% 15.46/2.32 Proof:
% 15.46/2.32 join(Z, join(X, Y))
% 15.46/2.32 = { by lemma 67 }
% 15.46/2.32 join(X, join(Z, Y))
% 15.46/2.32 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.32 join(X, join(Y, Z))
% 15.46/2.32
% 15.46/2.32 Lemma 69: join(composition(sk1, X), X) = X.
% 15.46/2.32 Proof:
% 15.46/2.32 join(composition(sk1, X), X)
% 15.46/2.32 = { by lemma 22 R->L }
% 15.46/2.32 join(composition(sk1, X), composition(join(sk1, one), X))
% 15.46/2.32 = { by axiom 3 (goals_17) }
% 15.46/2.32 join(composition(sk1, X), composition(one, X))
% 15.46/2.32 = { by axiom 12 (composition_distributivity_7) R->L }
% 15.46/2.32 composition(join(sk1, one), X)
% 15.46/2.32 = { by lemma 22 }
% 15.46/2.32 X
% 15.46/2.32
% 15.46/2.32 Lemma 70: meet(complement(X), complement(Y)) = complement(join(X, Y)).
% 15.46/2.32 Proof:
% 15.46/2.32 meet(complement(X), complement(Y))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 meet(complement(Y), complement(X))
% 15.46/2.32 = { by lemma 32 R->L }
% 15.46/2.32 meet(join(zero, complement(Y)), complement(X))
% 15.46/2.32 = { by lemma 60 R->L }
% 15.46/2.32 complement(join(X, complement(join(zero, complement(Y)))))
% 15.46/2.32 = { by lemma 43 }
% 15.46/2.32 complement(join(X, meet(Y, top)))
% 15.46/2.32 = { by lemma 47 }
% 15.46/2.32 complement(join(X, Y))
% 15.46/2.32
% 15.46/2.32 Lemma 71: meet(complement(Z), meet(Y, X)) = meet(X, meet(Y, complement(Z))).
% 15.46/2.32 Proof:
% 15.46/2.32 meet(complement(Z), meet(Y, X))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 meet(complement(Z), meet(X, Y))
% 15.46/2.32 = { by lemma 42 }
% 15.46/2.32 meet(meet(X, Y), complement(Z))
% 15.46/2.32 = { by axiom 11 (maddux4_definiton_of_meet_4) }
% 15.46/2.32 meet(complement(join(complement(X), complement(Y))), complement(Z))
% 15.46/2.32 = { by lemma 70 }
% 15.46/2.32 complement(join(join(complement(X), complement(Y)), Z))
% 15.46/2.32 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.32 complement(join(complement(X), join(complement(Y), Z)))
% 15.46/2.32 = { by lemma 62 }
% 15.46/2.32 meet(X, complement(join(complement(Y), Z)))
% 15.46/2.32 = { by lemma 62 }
% 15.46/2.33 meet(X, meet(Y, complement(Z)))
% 15.46/2.33
% 15.46/2.33 Lemma 72: meet(Y, meet(Z, X)) = meet(X, meet(Y, Z)).
% 15.46/2.33 Proof:
% 15.46/2.33 meet(Y, meet(Z, X))
% 15.46/2.33 = { by lemma 47 R->L }
% 15.46/2.33 meet(meet(Y, top), meet(Z, X))
% 15.46/2.33 = { by lemma 43 R->L }
% 15.46/2.33 meet(complement(join(zero, complement(Y))), meet(Z, X))
% 15.46/2.33 = { by lemma 71 }
% 15.46/2.33 meet(X, meet(Z, complement(join(zero, complement(Y)))))
% 15.46/2.33 = { by lemma 43 }
% 15.46/2.33 meet(X, meet(Z, meet(Y, top)))
% 15.46/2.33 = { by lemma 47 }
% 15.46/2.33 meet(X, meet(Z, Y))
% 15.46/2.33 = { by lemma 42 R->L }
% 15.46/2.33 meet(X, meet(Y, Z))
% 15.46/2.33
% 15.46/2.33 Lemma 73: meet(Y, meet(X, Z)) = meet(X, meet(Y, Z)).
% 15.46/2.33 Proof:
% 15.46/2.33 meet(Y, meet(X, Z))
% 15.46/2.33 = { by lemma 72 R->L }
% 15.46/2.33 meet(X, meet(Z, Y))
% 15.46/2.33 = { by lemma 42 R->L }
% 15.46/2.33 meet(X, meet(Y, Z))
% 15.46/2.33
% 15.46/2.33 Lemma 74: meet(meet(X, Y), complement(Y)) = zero.
% 15.46/2.33 Proof:
% 15.46/2.33 meet(meet(X, Y), complement(Y))
% 15.46/2.33 = { by lemma 60 R->L }
% 15.46/2.33 complement(join(Y, complement(meet(X, Y))))
% 15.46/2.33 = { by lemma 42 }
% 15.46/2.33 complement(join(Y, complement(meet(Y, X))))
% 15.46/2.33 = { by lemma 64 }
% 15.46/2.33 complement(top)
% 15.46/2.33 = { by lemma 17 }
% 15.46/2.33 zero
% 15.46/2.33
% 15.46/2.33 Lemma 75: join(join(X, composition(sk1, Y)), Y) = join(X, Y).
% 15.46/2.33 Proof:
% 15.46/2.33 join(join(X, composition(sk1, Y)), Y)
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.33 join(Y, join(X, composition(sk1, Y)))
% 15.46/2.33 = { by lemma 68 }
% 15.46/2.33 join(X, join(composition(sk1, Y), Y))
% 15.46/2.33 = { by lemma 69 }
% 15.46/2.33 join(X, Y)
% 15.46/2.33
% 15.46/2.33 Lemma 76: join(meet(X, Y), meet(Y, complement(X))) = Y.
% 15.46/2.33 Proof:
% 15.46/2.33 join(meet(X, Y), meet(Y, complement(X)))
% 15.46/2.33 = { by lemma 42 }
% 15.46/2.33 join(meet(Y, X), meet(Y, complement(X)))
% 15.46/2.33 = { by lemma 46 }
% 15.46/2.33 Y
% 15.46/2.33
% 15.46/2.33 Lemma 77: join(meet(X, Y), meet(complement(X), Y)) = Y.
% 15.46/2.33 Proof:
% 15.46/2.33 join(meet(X, Y), meet(complement(X), Y))
% 15.46/2.33 = { by lemma 42 }
% 15.46/2.33 join(meet(X, Y), meet(Y, complement(X)))
% 15.46/2.33 = { by lemma 76 }
% 15.46/2.33 Y
% 15.46/2.33
% 15.46/2.33 Lemma 78: meet(composition(X, Y), complement(composition(X, top))) = zero.
% 15.46/2.33 Proof:
% 15.46/2.33 meet(composition(X, Y), complement(composition(X, top)))
% 15.46/2.33 = { by lemma 33 R->L }
% 15.46/2.33 join(meet(composition(X, Y), complement(composition(X, top))), zero)
% 15.46/2.33 = { by lemma 52 R->L }
% 15.46/2.33 join(meet(composition(X, Y), complement(composition(X, top))), meet(zero, complement(composition(X, top))))
% 15.46/2.33 = { by lemma 55 R->L }
% 15.46/2.33 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, zero), complement(composition(X, top))))
% 15.46/2.33 = { by lemma 45 R->L }
% 15.46/2.33 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, zero)), complement(composition(X, top))))
% 15.46/2.33 = { by lemma 56 R->L }
% 15.46/2.33 join(meet(composition(X, Y), complement(composition(X, top))), meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top))))
% 15.46/2.33 = { by axiom 15 (modular_law_1_15) }
% 15.46/2.33 meet(composition(X, meet(Y, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top)))
% 15.46/2.33 = { by lemma 56 }
% 15.46/2.33 meet(composition(X, meet(Y, zero)), complement(composition(X, top)))
% 15.46/2.33 = { by lemma 45 }
% 15.46/2.33 meet(composition(X, zero), complement(composition(X, top)))
% 15.46/2.33 = { by lemma 55 }
% 15.46/2.33 meet(zero, complement(composition(X, top)))
% 15.46/2.33 = { by lemma 52 }
% 15.46/2.33 zero
% 15.46/2.33
% 15.46/2.33 Lemma 79: join(meet(composition(sk1, top), X), composition(sk1, X)) = meet(composition(sk1, top), X).
% 15.46/2.33 Proof:
% 15.46/2.33 join(meet(composition(sk1, top), X), composition(sk1, X))
% 15.46/2.33 = { by lemma 27 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), join(meet(composition(sk1, X), complement(composition(sk1, top))), complement(join(complement(composition(sk1, X)), complement(composition(sk1, top))))))
% 15.46/2.33 = { by lemma 78 }
% 15.46/2.33 join(meet(composition(sk1, top), X), join(zero, complement(join(complement(composition(sk1, X)), complement(composition(sk1, top))))))
% 15.46/2.33 = { by lemma 32 }
% 15.46/2.33 join(meet(composition(sk1, top), X), complement(join(complement(composition(sk1, X)), complement(composition(sk1, top)))))
% 15.46/2.33 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, X), composition(sk1, top)))
% 15.46/2.33 = { by lemma 42 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), composition(sk1, X)))
% 15.46/2.33 = { by lemma 27 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(meet(composition(sk1, X), complement(X)), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by lemma 60 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(complement(join(X, complement(composition(sk1, X)))), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by lemma 69 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(complement(join(join(composition(sk1, X), X), complement(composition(sk1, X)))), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(complement(join(composition(sk1, X), join(X, complement(composition(sk1, X))))), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(complement(join(composition(sk1, X), join(complement(composition(sk1, X)), X))), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by lemma 39 }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(complement(top), complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by lemma 17 }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), join(zero, complement(join(complement(composition(sk1, X)), complement(X))))))
% 15.46/2.33 = { by lemma 32 }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), complement(join(complement(composition(sk1, X)), complement(X)))))
% 15.46/2.33 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, top), meet(composition(sk1, X), X)))
% 15.46/2.33 = { by lemma 73 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(composition(sk1, X), meet(composition(sk1, top), X)))
% 15.46/2.33 = { by lemma 42 R->L }
% 15.46/2.33 join(meet(composition(sk1, top), X), meet(meet(composition(sk1, top), X), composition(sk1, X)))
% 15.46/2.33 = { by lemma 66 }
% 15.46/2.33 meet(composition(sk1, top), X)
% 15.46/2.33
% 15.46/2.33 Lemma 80: meet(join(sk1, one), composition(converse(complement(X)), X)) = zero.
% 15.46/2.33 Proof:
% 15.46/2.33 meet(join(sk1, one), composition(converse(complement(X)), X))
% 15.46/2.33 = { by lemma 42 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), join(sk1, one))
% 15.46/2.33 = { by lemma 51 R->L }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(complement(join(sk1, one))))
% 15.46/2.33 = { by lemma 23 R->L }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), complement(composition(join(zero, complement(X)), join(sk1, one)))))))
% 15.46/2.33 = { by lemma 20 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), complement(join(zero, complement(X)))))))
% 15.46/2.33 = { by lemma 43 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(join(zero, complement(X))), meet(X, top)))))
% 15.46/2.33 = { by lemma 32 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(complement(X)), meet(X, top)))))
% 15.46/2.33 = { by lemma 47 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), complement(join(complement(join(sk1, one)), composition(converse(complement(X)), X))))
% 15.46/2.33 = { by lemma 62 }
% 15.46/2.33 meet(composition(converse(complement(X)), X), meet(join(sk1, one), complement(composition(converse(complement(X)), X))))
% 15.46/2.33 = { by lemma 65 }
% 15.46/2.33 zero
% 15.46/2.33
% 15.46/2.33 Lemma 81: join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)) = join(composition(sk1, top), complement(X)).
% 15.46/2.33 Proof:
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X))
% 15.46/2.33 = { by lemma 66 R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), meet(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), composition(sk1, top)))
% 15.46/2.33 = { by lemma 42 R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), meet(composition(sk1, top), join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X))))
% 15.46/2.33 = { by lemma 79 }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), meet(composition(sk1, top), join(meet(composition(sk1, top), X), complement(X))))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))))
% 15.46/2.33 = { by lemma 33 R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), join(meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))), zero))
% 15.46/2.33 = { by lemma 17 R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), join(meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))), complement(top)))
% 15.46/2.33 = { by axiom 6 (def_top_12) }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), join(meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))), complement(join(join(complement(composition(sk1, top)), complement(X)), complement(join(complement(composition(sk1, top)), complement(X)))))))
% 15.46/2.33 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), join(meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))), complement(join(join(complement(composition(sk1, top)), complement(X)), meet(composition(sk1, top), X)))))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), join(meet(composition(sk1, top), join(complement(X), meet(composition(sk1, top), X))), complement(join(complement(composition(sk1, top)), join(complement(X), meet(composition(sk1, top), X))))))
% 15.46/2.33 = { by lemma 27 }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), complement(X)), composition(sk1, top))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(complement(X), composition(sk1, top)))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(composition(sk1, top), complement(X)))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), composition(sk1, top)), complement(X))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.33 join(join(composition(sk1, top), join(meet(composition(sk1, top), X), composition(sk1, X))), complement(X))
% 15.46/2.33 = { by lemma 67 }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), join(composition(sk1, top), composition(sk1, X))), complement(X))
% 15.46/2.33 = { by lemma 47 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), meet(join(composition(sk1, top), composition(sk1, X)), top)), complement(X))
% 15.46/2.33 = { by lemma 43 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), complement(join(zero, complement(join(composition(sk1, top), composition(sk1, X)))))), complement(X))
% 15.46/2.33 = { by lemma 70 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), complement(join(zero, meet(complement(composition(sk1, top)), complement(composition(sk1, X)))))), complement(X))
% 15.46/2.33 = { by lemma 34 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), complement(join(zero, join(zero, meet(complement(composition(sk1, top)), complement(composition(sk1, X))))))), complement(X))
% 15.46/2.33 = { by lemma 78 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), complement(join(zero, join(meet(composition(sk1, X), complement(composition(sk1, top))), meet(complement(composition(sk1, top)), complement(composition(sk1, X))))))), complement(X))
% 15.46/2.33 = { by lemma 76 }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), complement(join(zero, complement(composition(sk1, top))))), complement(X))
% 15.46/2.33 = { by lemma 43 }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), meet(composition(sk1, top), top)), complement(X))
% 15.46/2.33 = { by lemma 47 }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, top)), complement(X))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.33 join(join(composition(sk1, top), meet(composition(sk1, top), X)), complement(X))
% 15.46/2.33 = { by lemma 66 }
% 15.46/2.33 join(composition(sk1, top), complement(X))
% 15.46/2.33
% 15.46/2.33 Lemma 82: join(join(meet(composition(sk1, top), X), composition(sk1, X)), meet(X, complement(composition(sk1, top)))) = X.
% 15.46/2.33 Proof:
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), meet(X, complement(composition(sk1, top))))
% 15.46/2.33 = { by lemma 34 R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(zero, meet(X, complement(composition(sk1, top)))))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), zero), meet(X, complement(composition(sk1, top))))
% 15.46/2.33 = { by lemma 31 }
% 15.46/2.33 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(meet(composition(sk1, top), X), composition(sk1, X))), meet(X, complement(composition(sk1, top))))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(join(meet(composition(sk1, top), X), composition(sk1, X)), meet(X, complement(composition(sk1, top)))))
% 15.46/2.33 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.33 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(meet(X, complement(composition(sk1, top))), join(meet(composition(sk1, top), X), composition(sk1, X))))
% 15.46/2.33 = { by lemma 79 }
% 15.46/2.33 join(meet(composition(sk1, top), X), join(meet(X, complement(composition(sk1, top))), join(meet(composition(sk1, top), X), composition(sk1, X))))
% 15.46/2.33 = { by lemma 67 }
% 15.46/2.33 join(meet(X, complement(composition(sk1, top))), join(meet(composition(sk1, top), X), join(meet(composition(sk1, top), X), composition(sk1, X))))
% 15.46/2.33 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.46/2.33 join(join(meet(X, complement(composition(sk1, top))), meet(composition(sk1, top), X)), join(meet(composition(sk1, top), X), composition(sk1, X)))
% 15.46/2.33 = { by lemma 68 }
% 15.46/2.33 join(meet(composition(sk1, top), X), join(composition(sk1, X), join(meet(X, complement(composition(sk1, top))), meet(composition(sk1, top), X))))
% 15.46/2.33 = { by lemma 67 R->L }
% 15.46/2.34 join(meet(composition(sk1, top), X), join(meet(X, complement(composition(sk1, top))), join(composition(sk1, X), meet(composition(sk1, top), X))))
% 15.46/2.34 = { by lemma 79 R->L }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(meet(X, complement(composition(sk1, top))), join(composition(sk1, X), meet(composition(sk1, top), X))))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(join(composition(sk1, X), meet(composition(sk1, top), X)), meet(X, complement(composition(sk1, top)))))
% 15.46/2.34 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(composition(sk1, X), join(meet(composition(sk1, top), X), meet(X, complement(composition(sk1, top))))))
% 15.46/2.34 = { by lemma 76 }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(composition(sk1, X), X))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), composition(sk1, X)), join(X, composition(sk1, X)))
% 15.46/2.34 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.46/2.34 join(join(join(meet(composition(sk1, top), X), composition(sk1, X)), X), composition(sk1, X))
% 15.46/2.34 = { by lemma 75 }
% 15.46/2.34 join(join(meet(composition(sk1, top), X), X), composition(sk1, X))
% 15.46/2.34 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.34 join(meet(composition(sk1, top), X), join(X, composition(sk1, X)))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.34 join(meet(composition(sk1, top), X), join(composition(sk1, X), X))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.34 join(join(composition(sk1, X), X), meet(composition(sk1, top), X))
% 15.46/2.34 = { by axiom 8 (maddux2_join_associativity_2) R->L }
% 15.46/2.34 join(composition(sk1, X), join(X, meet(composition(sk1, top), X)))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.34 join(composition(sk1, X), join(meet(composition(sk1, top), X), X))
% 15.46/2.34 = { by lemma 75 R->L }
% 15.46/2.34 join(composition(sk1, X), join(join(meet(composition(sk1, top), X), composition(sk1, X)), X))
% 15.46/2.34 = { by lemma 47 R->L }
% 15.46/2.34 join(composition(sk1, X), meet(join(join(meet(composition(sk1, top), X), composition(sk1, X)), X), top))
% 15.46/2.34 = { by lemma 43 R->L }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, complement(join(join(meet(composition(sk1, top), X), composition(sk1, X)), X)))))
% 15.46/2.34 = { by lemma 75 }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, complement(join(meet(composition(sk1, top), X), X)))))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, complement(join(X, meet(composition(sk1, top), X))))))
% 15.46/2.34 = { by lemma 70 R->L }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, meet(complement(X), complement(meet(composition(sk1, top), X))))))
% 15.46/2.34 = { by lemma 34 R->L }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, join(zero, meet(complement(X), complement(meet(composition(sk1, top), X)))))))
% 15.46/2.34 = { by lemma 74 R->L }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, join(meet(meet(composition(sk1, top), X), complement(X)), meet(complement(X), complement(meet(composition(sk1, top), X)))))))
% 15.46/2.34 = { by lemma 76 }
% 15.46/2.34 join(composition(sk1, X), complement(join(zero, complement(X))))
% 15.46/2.34 = { by lemma 43 }
% 15.46/2.34 join(composition(sk1, X), meet(X, top))
% 15.46/2.34 = { by lemma 47 }
% 15.46/2.34 join(composition(sk1, X), X)
% 15.46/2.34 = { by lemma 69 }
% 15.46/2.34 X
% 15.46/2.34
% 15.46/2.34 Goal 1 (goals_18): tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(composition(sk1, sk2), meet(composition(sk1, top), sk2))) = tuple(composition(sk1, sk2), meet(composition(sk1, top), sk2)).
% 15.46/2.34 Proof:
% 15.46/2.34 tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(composition(sk1, sk2), meet(composition(sk1, top), sk2)))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.34 tuple(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 51 R->L }
% 15.46/2.34 tuple(complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 79 }
% 15.46/2.34 tuple(complement(complement(meet(composition(sk1, top), sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 27 R->L }
% 15.46/2.34 tuple(complement(complement(join(meet(meet(composition(sk1, top), sk2), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 72 }
% 15.46/2.34 tuple(complement(complement(join(meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), meet(meet(composition(sk1, top), sk2), sk2)), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 71 }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, meet(meet(composition(sk1, top), sk2), complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 60 R->L }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, complement(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(meet(composition(sk1, top), sk2))))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, complement(join(complement(meet(composition(sk1, top), sk2)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 68 }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, complement(join(meet(composition(sk1, top), sk2), join(composition(sk1, sk2), complement(meet(composition(sk1, top), sk2)))))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 18 }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, complement(join(composition(sk1, sk2), top))), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 35 }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, complement(top)), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 17 }
% 15.46/2.34 tuple(complement(complement(join(meet(sk2, zero), complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 45 }
% 15.46/2.34 tuple(complement(complement(join(zero, complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 32 }
% 15.46/2.34 tuple(complement(complement(complement(join(complement(meet(composition(sk1, top), sk2)), meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 62 }
% 15.46/2.34 tuple(complement(complement(meet(meet(composition(sk1, top), sk2), complement(meet(sk2, complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 61 }
% 15.46/2.34 tuple(complement(complement(meet(meet(composition(sk1, top), sk2), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 79 R->L }
% 15.46/2.34 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 42 }
% 15.46/2.34 tuple(complement(complement(meet(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 81 }
% 15.46/2.34 tuple(complement(complement(meet(join(composition(sk1, top), complement(sk2)), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 61 R->L }
% 15.46/2.34 tuple(complement(complement(meet(complement(meet(sk2, complement(composition(sk1, top)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 42 }
% 15.46/2.34 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 79 }
% 15.46/2.34 tuple(complement(complement(meet(meet(composition(sk1, top), sk2), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 27 R->L }
% 15.46/2.34 tuple(complement(complement(meet(join(meet(meet(composition(sk1, top), sk2), complement(sk2)), complement(join(complement(meet(composition(sk1, top), sk2)), complement(sk2)))), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 74 }
% 15.46/2.34 tuple(complement(complement(meet(join(zero, complement(join(complement(meet(composition(sk1, top), sk2)), complement(sk2)))), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 32 }
% 15.46/2.34 tuple(complement(complement(meet(complement(join(complement(meet(composition(sk1, top), sk2)), complement(sk2))), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.34 tuple(complement(complement(meet(meet(meet(composition(sk1, top), sk2), sk2), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 42 }
% 15.46/2.34 tuple(complement(complement(meet(complement(meet(sk2, complement(composition(sk1, top)))), meet(meet(composition(sk1, top), sk2), sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 72 R->L }
% 15.46/2.34 tuple(complement(complement(meet(meet(composition(sk1, top), sk2), meet(sk2, complement(meet(sk2, complement(composition(sk1, top)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.34 = { by lemma 79 R->L }
% 15.46/2.34 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(meet(sk2, complement(composition(sk1, top)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 42 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(complement(meet(sk2, complement(composition(sk1, top)))), sk2)))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 82 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(complement(meet(sk2, complement(composition(sk1, top)))), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 42 }
% 15.46/2.35 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top)))), complement(meet(sk2, complement(composition(sk1, top)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 60 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), complement(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 70 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 33 R->L }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), zero))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by axiom 5 (def_zero_13) }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), complement(join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 60 }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 71 R->L }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(complement(meet(sk2, complement(composition(sk1, top)))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 73 R->L }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), meet(complement(meet(sk2, complement(composition(sk1, top)))), join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 42 }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), meet(join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))))), complement(meet(sk2, complement(composition(sk1, top))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 60 R->L }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(join(meet(sk2, complement(composition(sk1, top))), complement(join(meet(sk2, complement(composition(sk1, top))), complement(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 60 }
% 15.46/2.35 tuple(complement(complement(join(meet(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top)))))))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 77 }
% 15.46/2.35 tuple(complement(complement(complement(join(meet(sk2, complement(composition(sk1, top))), meet(complement(join(meet(composition(sk1, top), sk2), composition(sk1, sk2))), complement(meet(sk2, complement(composition(sk1, top))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 70 }
% 15.46/2.35 tuple(complement(complement(complement(join(meet(sk2, complement(composition(sk1, top))), complement(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 60 }
% 15.46/2.35 tuple(complement(complement(meet(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top)))), complement(meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 42 R->L }
% 15.46/2.35 tuple(complement(complement(meet(complement(meet(sk2, complement(composition(sk1, top)))), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 61 }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, top), complement(sk2)), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 81 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2)), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), meet(sk2, complement(composition(sk1, top))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 82 }
% 15.46/2.35 tuple(complement(complement(meet(join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2)), sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 42 R->L }
% 15.46/2.35 tuple(complement(complement(meet(sk2, join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 69 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(join(meet(composition(sk1, top), sk2), composition(sk1, sk2)), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 81 }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(composition(sk1, top), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), composition(sk1, top))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 41 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), composition(sk1, converse(top)))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 38 R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), composition(sk1, join(sk2, converse(complement(converse(sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by axiom 1 (converse_idempotence_8) R->L }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(converse(composition(sk1, join(sk2, converse(complement(converse(sk2))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by axiom 9 (converse_multiplicativity_10) }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(composition(converse(join(sk2, converse(complement(converse(sk2))))), converse(sk1))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by lemma 36 }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(composition(join(complement(converse(sk2)), converse(sk2)), converse(sk1))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.35 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.35 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(composition(join(converse(sk2), complement(converse(sk2))), converse(sk1))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 12 (composition_distributivity_7) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(join(composition(converse(sk2), converse(sk1)), composition(complement(converse(sk2)), converse(sk1)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(join(converse(composition(sk1, sk2)), composition(complement(converse(sk2)), converse(sk1)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(join(composition(complement(converse(sk2)), converse(sk1)), converse(composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 1 (converse_idempotence_8) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(join(composition(converse(converse(complement(converse(sk2)))), converse(sk1)), converse(composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 9 (converse_multiplicativity_10) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(join(converse(composition(sk1, converse(complement(converse(sk2))))), converse(composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 7 (converse_additivity_9) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(converse(join(composition(sk1, converse(complement(converse(sk2)))), composition(sk1, sk2)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), converse(converse(join(composition(sk1, sk2), composition(sk1, converse(complement(converse(sk2))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 1 (converse_idempotence_8) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(complement(converse(sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 32 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(zero, complement(converse(sk2)))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 27 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))), complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 62 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(complement(join(complement(join(zero, complement(converse(sk2)))), converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))), complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 38 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(complement(converse(top)), complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 41 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(complement(top), complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 17 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(zero, complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 32 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(complement(join(complement(join(zero, complement(converse(sk2)))), complement(converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 11 (maddux4_definiton_of_meet_4) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 33 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), zero)))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 52 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 58 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(composition(zero, complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 80 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(composition(meet(join(sk1, one), composition(converse(complement(converse(complement(join(zero, complement(converse(sk2))))))), converse(complement(join(zero, complement(converse(sk2))))))), complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 16 (modular_law_2_16) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), join(meet(composition(join(sk1, one), complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(composition(meet(join(sk1, one), composition(converse(complement(converse(complement(join(zero, complement(converse(sk2))))))), converse(complement(join(zero, complement(converse(sk2))))))), complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 80 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), join(meet(composition(join(sk1, one), complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(composition(zero, complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 22 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), join(meet(complement(join(zero, complement(converse(sk2)))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(composition(zero, complement(join(zero, complement(converse(sk2))))), converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 58 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), join(meet(complement(join(zero, complement(converse(sk2)))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(zero, converse(complement(converse(complement(join(zero, complement(converse(sk2))))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 52 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), join(meet(complement(join(zero, complement(converse(sk2)))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), zero))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 33 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(join(meet(join(zero, complement(converse(sk2))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))), meet(complement(join(zero, complement(converse(sk2)))), converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 77 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, converse(converse(complement(converse(complement(join(zero, complement(converse(sk2)))))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 1 (converse_idempotence_8) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, complement(converse(complement(join(zero, complement(converse(sk2)))))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 43 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, complement(converse(meet(converse(sk2), top))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 47 }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, complement(converse(converse(sk2))))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 1 (converse_idempotence_8) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, sk2), composition(sk1, complement(sk2))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), join(composition(sk1, complement(sk2)), composition(sk1, sk2)))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by axiom 8 (maddux2_join_associativity_2) }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(join(complement(sk2), composition(sk1, complement(sk2))), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.36 = { by lemma 53 R->L }
% 15.46/2.36 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(composition(join(join(sk1, one), sk1), complement(sk2)), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.37 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(composition(join(sk1, join(sk1, one)), complement(sk2)), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by axiom 3 (goals_17) }
% 15.46/2.37 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(composition(join(sk1, one), complement(sk2)), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 22 }
% 15.46/2.37 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(complement(sk2), composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.37 tuple(complement(complement(meet(join(composition(sk1, sk2), sk2), join(composition(sk1, sk2), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by axiom 2 (maddux1_join_commutativity_1) R->L }
% 15.46/2.37 tuple(complement(complement(meet(join(sk2, composition(sk1, sk2)), join(composition(sk1, sk2), complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 51 R->L }
% 15.46/2.37 tuple(complement(complement(meet(join(sk2, composition(sk1, sk2)), join(composition(sk1, sk2), complement(complement(complement(sk2))))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 42 }
% 15.46/2.37 tuple(complement(complement(meet(join(composition(sk1, sk2), complement(complement(complement(sk2)))), join(sk2, composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 63 R->L }
% 15.46/2.37 tuple(complement(join(complement(join(composition(sk1, sk2), complement(complement(complement(sk2))))), complement(join(sk2, composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 60 }
% 15.46/2.37 tuple(complement(join(meet(complement(complement(sk2)), complement(composition(sk1, sk2))), complement(join(sk2, composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by axiom 2 (maddux1_join_commutativity_1) }
% 15.46/2.37 tuple(complement(join(complement(join(sk2, composition(sk1, sk2))), meet(complement(complement(sk2)), complement(composition(sk1, sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 42 R->L }
% 15.46/2.37 tuple(complement(join(complement(join(sk2, composition(sk1, sk2))), meet(complement(composition(sk1, sk2)), complement(complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 70 R->L }
% 15.46/2.37 tuple(complement(join(meet(complement(sk2), complement(composition(sk1, sk2))), meet(complement(composition(sk1, sk2)), complement(complement(sk2))))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 76 }
% 15.46/2.37 tuple(complement(complement(composition(sk1, sk2))), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 51 }
% 15.46/2.37 tuple(composition(sk1, sk2), join(meet(composition(sk1, top), sk2), composition(sk1, sk2)))
% 15.46/2.37 = { by lemma 79 }
% 15.46/2.37 tuple(composition(sk1, sk2), meet(composition(sk1, top), sk2))
% 15.46/2.37 % SZS output end Proof
% 15.46/2.37
% 15.46/2.37 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------