0.07/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.07/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n023.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:51:37 EDT 2019
0.12/0.33	% CPUTime    : 
22.17/22.37	% SZS status Unsatisfiable
22.17/22.37	
22.21/22.37	% SZS output start Proof
22.21/22.37	Take the following subset of the input axioms:
40.77/40.95	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
40.77/40.95	  fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
40.77/40.95	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
40.77/40.95	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
40.77/40.95	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
40.77/40.95	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
40.77/40.95	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
40.77/40.95	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
40.77/40.95	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
40.77/40.95	  fof(goals_14, negated_conjecture, join(composition(sk1, converse(sk2)), sk3)=sk3).
40.77/40.95	  fof(goals_15, negated_conjecture, join(composition(complement(sk3), sk2), complement(sk1))!=complement(sk1)).
40.77/40.95	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
40.77/40.95	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
40.77/40.95	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
40.77/40.95	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
40.77/40.95	
40.77/40.95	Now clausify the problem and encode Horn clauses using encoding 3 of
40.77/40.95	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
40.77/40.95	We repeatedly replace C & s=t => u=v by the two clauses:
40.77/40.95	  fresh(y, y, x1...xn) = u
40.77/40.95	  C => fresh(s, t, x1...xn) = v
40.77/40.95	where fresh is a fresh function symbol and x1..xn are the free
40.77/40.95	variables of u and v.
40.77/40.95	A predicate p(X) is encoded as p(X)=true (this is sound, because the
40.77/40.95	input problem has no model of domain size 1).
40.77/40.95	
40.77/40.95	The encoding turns the above axioms into the following unit equations and goals:
40.77/40.95	
40.77/40.95	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
40.77/40.95	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
40.77/40.95	Axiom 3 (composition_identity_6): X = composition(X, one).
40.77/40.95	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
40.77/40.95	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
40.77/40.95	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
40.77/40.95	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
40.77/40.95	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
40.77/40.95	Axiom 9 (def_top_12): join(X, complement(X)) = top.
40.77/40.95	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
40.77/40.95	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
40.77/40.95	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
40.77/40.95	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
40.77/40.96	Axiom 14 (goals_14): join(composition(sk1, converse(sk2)), sk3) = sk3.
40.77/40.96	
40.77/40.96	Lemma 15: meet(X, Y) = meet(Y, X).
40.77/40.96	Proof:
40.77/40.96	  meet(X, Y)
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  complement(join(complement(X), complement(Y)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  complement(join(complement(Y), complement(X)))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  meet(Y, X)
40.77/40.96	
40.77/40.96	Lemma 16: complement(top) = zero.
40.77/40.96	Proof:
40.77/40.96	  complement(top)
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  complement(join(complement(?), complement(complement(?))))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  meet(?, complement(?))
40.77/40.96	= { by axiom 8 (def_zero_13) }
40.77/40.96	  zero
40.77/40.96	
40.77/40.96	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
40.77/40.96	Proof:
40.77/40.96	  complement(join(zero, complement(X)))
40.77/40.96	= { by lemma 16 }
40.77/40.96	  complement(join(complement(top), complement(X)))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  meet(top, X)
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(X, top)
40.77/40.96	
40.77/40.96	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
40.77/40.96	Proof:
40.77/40.96	  converse(composition(converse(X), Y))
40.77/40.96	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.96	  composition(converse(Y), converse(converse(X)))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  composition(converse(Y), X)
40.77/40.96	
40.77/40.96	Lemma 19: composition(converse(one), X) = X.
40.77/40.96	Proof:
40.77/40.96	  composition(converse(one), X)
40.77/40.96	= { by lemma 18 }
40.77/40.96	  converse(composition(converse(X), one))
40.77/40.96	= { by axiom 3 (composition_identity_6) }
40.77/40.96	  converse(converse(X))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 20: composition(one, X) = X.
40.77/40.96	Proof:
40.77/40.96	  composition(one, X)
40.77/40.96	= { by lemma 19 }
40.77/40.96	  composition(converse(one), composition(one, X))
40.77/40.96	= { by axiom 12 (composition_associativity_5) }
40.77/40.96	  composition(composition(converse(one), one), X)
40.77/40.96	= { by axiom 3 (composition_identity_6) }
40.77/40.96	  composition(converse(one), X)
40.77/40.96	= { by lemma 19 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 21: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
40.77/40.96	Proof:
40.77/40.96	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
40.77/40.96	= { by axiom 7 (converse_cancellativity_11) }
40.77/40.96	  complement(Y)
40.77/40.96	
40.77/40.96	Lemma 22: join(complement(X), complement(X)) = complement(X).
40.77/40.96	Proof:
40.77/40.96	  join(complement(X), complement(X))
40.77/40.96	= { by lemma 19 }
40.77/40.96	  join(complement(X), composition(converse(one), complement(X)))
40.77/40.96	= { by lemma 20 }
40.77/40.96	  join(complement(X), composition(converse(one), complement(composition(one, X))))
40.77/40.96	= { by lemma 21 }
40.77/40.96	  complement(X)
40.77/40.96	
40.77/40.96	Lemma 23: meet(X, X) = complement(complement(X)).
40.77/40.96	Proof:
40.77/40.96	  meet(X, X)
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  complement(join(complement(X), complement(X)))
40.77/40.96	= { by lemma 22 }
40.77/40.96	  complement(complement(X))
40.77/40.96	
40.77/40.96	Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
40.77/40.96	Proof:
40.77/40.96	  join(meet(X, Y), complement(join(complement(X), Y)))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
40.77/40.96	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 25: join(meet(X, Y), meet(X, complement(Y))) = X.
40.77/40.96	Proof:
40.77/40.96	  join(meet(X, Y), meet(X, complement(Y)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(X, complement(Y)), meet(X, Y))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
40.77/40.96	= { by lemma 24 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 26: join(zero, complement(complement(X))) = X.
40.77/40.96	Proof:
40.77/40.96	  join(zero, complement(complement(X)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(complement(complement(X)), zero)
40.77/40.96	= { by lemma 23 }
40.77/40.96	  join(meet(X, X), zero)
40.77/40.96	= { by axiom 8 (def_zero_13) }
40.77/40.96	  join(meet(X, X), meet(X, complement(X)))
40.77/40.96	= { by lemma 25 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 27: join(X, join(Y, Z)) = join(Z, join(X, Y)).
40.77/40.96	Proof:
40.77/40.96	  join(X, join(Y, Z))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  join(join(X, Y), Z)
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(Z, join(X, Y))
40.77/40.96	
40.77/40.96	Lemma 28: join(X, join(complement(X), Y)) = join(Y, top).
40.77/40.96	Proof:
40.77/40.96	  join(X, join(complement(X), Y))
40.77/40.96	= { by lemma 27 }
40.77/40.96	  join(complement(X), join(Y, X))
40.77/40.96	= { by lemma 27 }
40.77/40.96	  join(Y, join(X, complement(X)))
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  join(Y, top)
40.77/40.96	
40.77/40.96	Lemma 29: join(X, join(Y, complement(X))) = join(Y, top).
40.77/40.96	Proof:
40.77/40.96	  join(X, join(Y, complement(X)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(X, join(complement(X), Y))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  join(join(X, complement(X)), Y)
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  join(top, Y)
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(Y, top)
40.77/40.96	
40.77/40.96	Lemma 30: join(X, top) = top.
40.77/40.96	Proof:
40.77/40.96	  join(X, top)
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  join(X, join(complement(X), complement(complement(X))))
40.77/40.96	= { by lemma 28 }
40.77/40.96	  join(complement(complement(X)), top)
40.77/40.96	= { by lemma 29 }
40.77/40.96	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
40.77/40.96	= { by lemma 22 }
40.77/40.96	  join(complement(X), complement(complement(X)))
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  top
40.77/40.96	
40.77/40.96	Lemma 31: join(zero, meet(X, top)) = X.
40.77/40.96	Proof:
40.77/40.96	  join(zero, meet(X, top))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(X, top), zero)
40.77/40.96	= { by lemma 16 }
40.77/40.96	  join(meet(X, top), complement(top))
40.77/40.96	= { by lemma 30 }
40.77/40.96	  join(meet(X, top), complement(join(complement(X), top)))
40.77/40.96	= { by lemma 24 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 32: join(zero, complement(X)) = complement(X).
40.77/40.96	Proof:
40.77/40.96	  join(zero, complement(X))
40.77/40.96	= { by lemma 26 }
40.77/40.96	  join(zero, complement(join(zero, complement(complement(X)))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  join(zero, meet(complement(X), top))
40.77/40.96	= { by lemma 31 }
40.77/40.96	  complement(X)
40.77/40.96	
40.77/40.96	Lemma 33: complement(complement(X)) = X.
40.77/40.96	Proof:
40.77/40.96	  complement(complement(X))
40.77/40.96	= { by lemma 32 }
40.77/40.96	  join(zero, complement(complement(X)))
40.77/40.96	= { by lemma 26 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 34: join(meet(Y, X), meet(X, complement(Y))) = X.
40.77/40.96	Proof:
40.77/40.96	  join(meet(Y, X), meet(X, complement(Y)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(X, Y), meet(X, complement(Y)))
40.77/40.96	= { by lemma 25 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 35: join(X, zero) = X.
40.77/40.96	Proof:
40.77/40.96	  join(X, zero)
40.77/40.96	= { by lemma 33 }
40.77/40.96	  join(complement(complement(X)), zero)
40.77/40.96	= { by lemma 23 }
40.77/40.96	  join(meet(X, X), zero)
40.77/40.96	= { by axiom 8 (def_zero_13) }
40.77/40.96	  join(meet(X, X), meet(X, complement(X)))
40.77/40.96	= { by lemma 34 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 36: join(zero, X) = X.
40.77/40.96	Proof:
40.77/40.96	  join(zero, X)
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(X, zero)
40.77/40.96	= { by lemma 35 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 37: meet(X, top) = X.
40.77/40.96	Proof:
40.77/40.96	  meet(X, top)
40.77/40.96	= { by lemma 17 }
40.77/40.96	  complement(join(zero, complement(X)))
40.77/40.96	= { by lemma 32 }
40.77/40.96	  join(zero, complement(join(zero, complement(X))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  join(zero, meet(X, top))
40.77/40.96	= { by lemma 31 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 38: meet(top, X) = X.
40.77/40.96	Proof:
40.77/40.96	  meet(top, X)
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(X, top)
40.77/40.96	= { by lemma 37 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 39: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
40.77/40.96	Proof:
40.77/40.96	  meet(X, join(complement(Y), complement(Z)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  meet(X, join(complement(Z), complement(Y)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(join(complement(Z), complement(Y)), X)
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  complement(join(meet(Z, Y), complement(X)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  complement(join(complement(X), meet(Z, Y)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  complement(join(complement(X), meet(Y, Z)))
40.77/40.96	
40.77/40.96	Lemma 40: complement(join(X, complement(Y))) = meet(Y, complement(X)).
40.77/40.96	Proof:
40.77/40.96	  complement(join(X, complement(Y)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  complement(join(complement(Y), X))
40.77/40.96	= { by lemma 38 }
40.77/40.96	  complement(join(complement(Y), meet(top, X)))
40.77/40.96	= { by lemma 39 }
40.77/40.96	  meet(Y, join(complement(top), complement(X)))
40.77/40.96	= { by lemma 16 }
40.77/40.96	  meet(Y, join(zero, complement(X)))
40.77/40.96	= { by lemma 32 }
40.77/40.96	  meet(Y, complement(X))
40.77/40.96	
40.77/40.96	Lemma 41: complement(join(complement(X), Y)) = meet(X, complement(Y)).
40.77/40.96	Proof:
40.77/40.96	  complement(join(complement(X), Y))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  complement(join(Y, complement(X)))
40.77/40.96	= { by lemma 40 }
40.77/40.96	  meet(X, complement(Y))
40.77/40.96	
40.77/40.96	Lemma 42: meet(complement(X), complement(Y)) = complement(join(X, Y)).
40.77/40.96	Proof:
40.77/40.96	  meet(complement(X), complement(Y))
40.77/40.96	= { by lemma 32 }
40.77/40.96	  meet(join(zero, complement(X)), complement(Y))
40.77/40.96	= { by lemma 40 }
40.77/40.96	  complement(join(Y, complement(join(zero, complement(X)))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  complement(join(Y, meet(X, top)))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  complement(join(Y, X))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  complement(join(X, Y))
40.77/40.96	
40.77/40.96	Lemma 43: meet(X, meet(Y, complement(Z))) = meet(complement(Z), meet(X, Y)).
40.77/40.96	Proof:
40.77/40.96	  meet(X, meet(Y, complement(Z)))
40.77/40.96	= { by lemma 41 }
40.77/40.96	  meet(X, complement(join(complement(Y), Z)))
40.77/40.96	= { by lemma 41 }
40.77/40.96	  complement(join(complement(X), join(complement(Y), Z)))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  complement(join(join(complement(X), complement(Y)), Z))
40.77/40.96	= { by lemma 42 }
40.77/40.96	  meet(complement(join(complement(X), complement(Y))), complement(Z))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  meet(meet(X, Y), complement(Z))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(complement(Z), meet(X, Y))
40.77/40.96	
40.77/40.96	Lemma 44: meet(meet(X, Z), Y) = meet(X, meet(Y, Z)).
40.77/40.96	Proof:
40.77/40.96	  meet(meet(X, Z), Y)
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(Y, meet(X, Z))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  meet(meet(Y, top), meet(X, Z))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  meet(complement(join(zero, complement(Y))), meet(X, Z))
40.77/40.96	= { by lemma 43 }
40.77/40.96	  meet(X, meet(Z, complement(join(zero, complement(Y)))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  meet(X, meet(Z, meet(Y, top)))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  meet(X, meet(Z, Y))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(X, meet(Y, Z))
40.77/40.96	
40.77/40.96	Lemma 45: meet(converse(X), converse(join(X, Y))) = converse(X).
40.77/40.96	Proof:
40.77/40.96	  meet(converse(X), converse(join(X, Y)))
40.77/40.96	= { by lemma 35 }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), zero)
40.77/40.96	= { by lemma 16 }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(top))
40.77/40.96	= { by lemma 30 }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(converse(Y), top)))
40.77/40.96	= { by axiom 9 (def_top_12) }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(converse(Y), join(converse(X), complement(converse(X))))))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(join(converse(Y), converse(X)), complement(converse(X)))))
40.77/40.96	= { by axiom 11 (converse_additivity_9) }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(converse(join(Y, X)), complement(converse(X)))))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(complement(converse(X)), converse(join(Y, X)))))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(converse(X), converse(join(X, Y))), complement(join(complement(converse(X)), converse(join(X, Y)))))
40.77/40.96	= { by lemma 24 }
40.77/40.96	  converse(X)
40.77/40.96	
40.77/40.96	Lemma 46: meet(X, join(X, Y)) = X.
40.77/40.96	Proof:
40.77/40.96	  meet(X, join(X, Y))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  meet(converse(converse(X)), join(X, Y))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  meet(converse(converse(X)), converse(converse(join(X, Y))))
40.77/40.96	= { by axiom 11 (converse_additivity_9) }
40.77/40.96	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
40.77/40.96	= { by lemma 45 }
40.77/40.96	  converse(converse(X))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 47: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
40.77/40.96	Proof:
40.77/40.96	  meet(complement(X), join(X, Y))
40.77/40.96	= { by lemma 33 }
40.77/40.96	  meet(complement(X), join(X, complement(complement(Y))))
40.77/40.96	= { by lemma 25 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(X), join(X, complement(complement(Y)))), complement(Y)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
40.77/40.96	= { by lemma 41 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), complement(join(complement(join(X, complement(complement(Y)))), X))))
40.77/40.96	= { by lemma 41 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(complement(complement(Y)), join(complement(join(X, complement(complement(Y)))), X))))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(join(complement(complement(Y)), complement(join(X, complement(complement(Y))))), X)))
40.77/40.96	= { by lemma 42 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(join(complement(complement(Y)), complement(join(X, complement(complement(Y)))))), complement(X)))
40.77/40.96	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(Y), join(X, complement(complement(Y)))), complement(X)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(complement(Y), join(X, complement(complement(Y))))))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(join(X, complement(complement(Y))), complement(Y))))
40.77/40.96	= { by lemma 43 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
40.77/40.96	= { by lemma 40 }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
40.77/40.96	= { by axiom 8 (def_zero_13) }
40.77/40.96	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
40.77/40.96	= { by lemma 35 }
40.77/40.96	  meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
40.77/40.96	= { by lemma 44 }
40.77/40.96	  meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
40.77/40.96	= { by lemma 33 }
40.77/40.96	  meet(complement(X), meet(Y, join(X, Y)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  meet(complement(X), meet(Y, join(Y, X)))
40.77/40.96	= { by lemma 46 }
40.77/40.96	  meet(complement(X), Y)
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(Y, complement(X))
40.77/40.96	
40.77/40.96	Lemma 48: meet(X, join(Y, complement(X))) = meet(X, Y).
40.77/40.96	Proof:
40.77/40.96	  meet(X, join(Y, complement(X)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  meet(X, join(complement(X), Y))
40.77/40.96	= { by lemma 36 }
40.77/40.96	  meet(X, join(zero, join(complement(X), Y)))
40.77/40.96	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.96	  meet(X, join(join(zero, complement(X)), Y))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  meet(X, meet(join(join(zero, complement(X)), Y), top))
40.77/40.96	= { by lemma 44 }
40.77/40.96	  meet(meet(X, top), join(join(zero, complement(X)), Y))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
40.77/40.96	= { by lemma 47 }
40.77/40.96	  meet(Y, complement(join(zero, complement(X))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  meet(Y, meet(X, top))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  meet(Y, X)
40.77/40.96	= { by lemma 15 }
40.77/40.96	  meet(X, Y)
40.77/40.96	
40.77/40.96	Lemma 49: complement(meet(Y, complement(X))) = join(X, complement(Y)).
40.77/40.96	Proof:
40.77/40.96	  complement(meet(Y, complement(X)))
40.77/40.96	= { by lemma 36 }
40.77/40.96	  complement(join(zero, meet(Y, complement(X))))
40.77/40.96	= { by lemma 40 }
40.77/40.96	  complement(join(zero, complement(join(X, complement(Y)))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  meet(join(X, complement(Y)), top)
40.77/40.96	= { by lemma 37 }
40.77/40.96	  join(X, complement(Y))
40.77/40.96	
40.77/40.96	Lemma 50: complement(meet(complement(X), Y)) = join(X, complement(Y)).
40.77/40.96	Proof:
40.77/40.96	  complement(meet(complement(X), Y))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  complement(meet(Y, complement(X)))
40.77/40.96	= { by lemma 49 }
40.77/40.96	  join(X, complement(Y))
40.77/40.96	
40.77/40.96	Lemma 51: join(meet(Y, X), meet(complement(Y), X)) = X.
40.77/40.96	Proof:
40.77/40.96	  join(meet(Y, X), meet(complement(Y), X))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(meet(Y, X), meet(X, complement(Y)))
40.77/40.96	= { by lemma 34 }
40.77/40.96	  X
40.77/40.96	
40.77/40.96	Lemma 52: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
40.77/40.96	Proof:
40.77/40.96	  converse(composition(X, converse(Y)))
40.77/40.96	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.96	  composition(converse(converse(Y)), converse(X))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  composition(Y, converse(X))
40.77/40.96	
40.77/40.96	Lemma 53: converse(join(X, converse(Y))) = join(Y, converse(X)).
40.77/40.96	Proof:
40.77/40.96	  converse(join(X, converse(Y)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  converse(join(converse(Y), X))
40.77/40.96	= { by axiom 11 (converse_additivity_9) }
40.77/40.96	  join(converse(converse(Y)), converse(X))
40.77/40.96	= { by axiom 5 (converse_idempotence_8) }
40.77/40.96	  join(Y, converse(X))
40.77/40.96	
40.77/40.96	Lemma 54: converse(join(converse(X), Y)) = join(X, converse(Y)).
40.77/40.96	Proof:
40.77/40.96	  converse(join(converse(X), Y))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  converse(join(Y, converse(X)))
40.77/40.96	= { by lemma 53 }
40.77/40.96	  join(X, converse(Y))
40.77/40.96	
40.77/40.96	Lemma 55: join(X, converse(top)) = converse(top).
40.77/40.96	Proof:
40.77/40.96	  join(X, converse(top))
40.77/40.96	= { by lemma 54 }
40.77/40.96	  converse(join(converse(X), top))
40.77/40.96	= { by lemma 30 }
40.77/40.96	  converse(top)
40.77/40.96	
40.77/40.96	Lemma 56: converse(top) = top.
40.77/40.96	Proof:
40.77/40.96	  converse(top)
40.77/40.96	= { by lemma 55 }
40.77/40.96	  join(?, converse(top))
40.77/40.96	= { by lemma 55 }
40.77/40.96	  join(?, join(complement(?), converse(top)))
40.77/40.96	= { by lemma 28 }
40.77/40.96	  join(converse(top), top)
40.77/40.96	= { by lemma 30 }
40.77/40.96	  top
40.77/40.96	
40.77/40.96	Lemma 57: converse(composition(X, top)) = composition(top, converse(X)).
40.77/40.96	Proof:
40.77/40.96	  converse(composition(X, top))
40.77/40.96	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.96	  composition(converse(top), converse(X))
40.77/40.96	= { by lemma 56 }
40.77/40.96	  composition(top, converse(X))
40.77/40.96	
40.77/40.96	Lemma 58: join(complement(join(X, Y)), meet(Y, complement(X))) = complement(X).
40.77/40.96	Proof:
40.77/40.96	  join(complement(join(X, Y)), meet(Y, complement(X)))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(complement(join(Y, X)), meet(Y, complement(X)))
40.77/40.96	= { by lemma 15 }
40.77/40.96	  join(complement(join(Y, X)), meet(complement(X), Y))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(complement(X), Y), complement(join(Y, X)))
40.77/40.96	= { by lemma 32 }
40.77/40.96	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
40.77/40.96	= { by lemma 37 }
40.77/40.96	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
40.77/40.96	= { by lemma 17 }
40.77/40.96	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
40.77/40.96	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.96	  join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y)))
40.77/40.96	= { by lemma 24 }
40.77/40.96	  join(zero, complement(X))
40.77/40.96	= { by lemma 32 }
40.77/40.97	  complement(X)
40.77/40.97	
40.77/40.97	Lemma 59: meet(join(X, Y), join(X, complement(Y))) = X.
40.77/40.97	Proof:
40.77/40.97	  meet(join(X, Y), join(X, complement(Y)))
40.77/40.97	= { by lemma 49 }
40.77/40.97	  meet(join(X, Y), complement(meet(Y, complement(X))))
40.77/40.97	= { by lemma 41 }
40.77/40.97	  complement(join(complement(join(X, Y)), meet(Y, complement(X))))
40.77/40.97	= { by lemma 58 }
40.77/40.97	  complement(complement(X))
40.77/40.97	= { by lemma 33 }
40.77/40.97	  X
40.77/40.97	
40.77/40.97	Lemma 60: meet(join(Y, X), join(X, complement(Y))) = X.
40.77/40.97	Proof:
40.77/40.97	  meet(join(Y, X), join(X, complement(Y)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  meet(join(X, Y), join(X, complement(Y)))
40.77/40.97	= { by lemma 59 }
40.77/40.97	  X
40.77/40.97	
40.77/40.97	Lemma 61: join(X, converse(complement(converse(X)))) = top.
40.77/40.97	Proof:
40.77/40.97	  join(X, converse(complement(converse(X))))
40.77/40.97	= { by lemma 54 }
40.77/40.97	  converse(join(converse(X), complement(converse(X))))
40.77/40.97	= { by axiom 9 (def_top_12) }
40.77/40.97	  converse(top)
40.77/40.97	= { by lemma 56 }
40.77/40.97	  top
40.77/40.97	
40.77/40.97	Lemma 62: converse(complement(converse(complement(X)))) = X.
40.77/40.97	Proof:
40.77/40.97	  converse(complement(converse(complement(X))))
40.77/40.97	= { by lemma 60 }
40.77/40.97	  meet(join(X, converse(complement(converse(complement(X))))), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  meet(join(X, converse(complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by lemma 54 }
40.77/40.97	  meet(converse(join(converse(X), complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by lemma 38 }
40.77/40.97	  meet(converse(meet(top, join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by lemma 61 }
40.77/40.97	  meet(converse(meet(join(converse(X), converse(complement(converse(converse(X))))), join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by lemma 59 }
40.77/40.97	  meet(converse(converse(X)), join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  meet(X, join(converse(complement(converse(complement(X)))), complement(X)))
40.77/40.97	= { by lemma 48 }
40.77/40.97	  meet(X, converse(complement(converse(complement(X)))))
40.77/40.97	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.97	  complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))
40.77/40.97	= { by lemma 32 }
40.77/40.97	  join(zero, complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))
40.77/40.97	= { by lemma 16 }
40.77/40.97	  join(complement(top), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))
40.77/40.97	= { by lemma 61 }
40.77/40.97	  join(complement(join(complement(X), converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))
40.77/40.97	= { by lemma 41 }
40.77/40.97	  join(meet(X, complement(converse(complement(converse(complement(X)))))), complement(join(complement(X), complement(converse(complement(converse(complement(X))))))))
40.77/40.97	= { by lemma 24 }
40.77/40.97	  X
40.77/40.97	
40.77/40.97	Lemma 63: converse(complement(X)) = complement(converse(X)).
40.77/40.97	Proof:
40.77/40.97	  converse(complement(X))
40.77/40.97	= { by lemma 32 }
40.77/40.97	  converse(join(zero, complement(X)))
40.77/40.97	= { by lemma 62 }
40.77/40.97	  converse(converse(complement(converse(complement(join(zero, complement(X)))))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  complement(converse(complement(join(zero, complement(X)))))
40.77/40.97	= { by lemma 17 }
40.77/40.97	  complement(converse(meet(X, top)))
40.77/40.97	= { by lemma 37 }
40.77/40.97	  complement(converse(X))
40.77/40.97	
40.77/40.97	Lemma 64: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
40.77/40.97	Proof:
40.77/40.97	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
40.77/40.97	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.97	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
40.77/40.97	= { by axiom 13 (composition_distributivity_7) }
40.77/40.97	  composition(join(converse(X), Y), converse(Z))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  composition(join(Y, converse(X)), converse(Z))
40.77/40.97	
40.77/40.97	Lemma 65: composition(converse(X), complement(composition(X, top))) = zero.
40.77/40.97	Proof:
40.77/40.97	  composition(converse(X), complement(composition(X, top)))
40.77/40.97	= { by lemma 36 }
40.77/40.97	  join(zero, composition(converse(X), complement(composition(X, top))))
40.77/40.97	= { by lemma 16 }
40.77/40.97	  join(complement(top), composition(converse(X), complement(composition(X, top))))
40.77/40.97	= { by lemma 21 }
40.77/40.97	  complement(top)
40.77/40.97	= { by lemma 16 }
40.77/40.97	  zero
40.77/40.97	
40.77/40.97	Lemma 66: composition(X, converse(join(complement(composition(top, X)), Y))) = composition(X, converse(Y)).
40.77/40.97	Proof:
40.77/40.97	  composition(X, converse(join(complement(composition(top, X)), Y)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  composition(X, converse(join(Y, complement(composition(top, X)))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  composition(X, converse(join(Y, complement(composition(top, converse(converse(X)))))))
40.77/40.97	= { by lemma 52 }
40.77/40.97	  converse(composition(join(Y, complement(composition(top, converse(converse(X))))), converse(X)))
40.77/40.97	= { by lemma 57 }
40.77/40.97	  converse(composition(join(Y, complement(converse(composition(converse(X), top)))), converse(X)))
40.77/40.97	= { by lemma 63 }
40.77/40.97	  converse(composition(join(Y, converse(complement(composition(converse(X), top)))), converse(X)))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  converse(composition(join(Y, converse(complement(composition(converse(X), top)))), converse(converse(converse(X)))))
40.77/40.97	= { by lemma 64 }
40.77/40.97	  converse(join(composition(Y, converse(converse(converse(X)))), converse(composition(converse(converse(X)), complement(composition(converse(X), top))))))
40.77/40.97	= { by lemma 65 }
40.77/40.97	  converse(join(composition(Y, converse(converse(converse(X)))), converse(zero)))
40.77/40.97	= { by lemma 54 }
40.77/40.97	  converse(converse(join(converse(composition(Y, converse(converse(converse(X))))), zero)))
40.77/40.97	= { by lemma 35 }
40.77/40.97	  converse(converse(converse(composition(Y, converse(converse(converse(X)))))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  converse(composition(Y, converse(converse(converse(X)))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  converse(composition(Y, converse(X)))
40.77/40.97	= { by lemma 52 }
40.77/40.97	  composition(X, converse(Y))
40.77/40.97	
40.77/40.97	Lemma 67: join(X, meet(Y, complement(X))) = join(X, Y).
40.77/40.97	Proof:
40.77/40.97	  join(X, meet(Y, complement(X)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  join(meet(Y, complement(X)), X)
40.77/40.97	= { by lemma 47 }
40.77/40.97	  join(meet(complement(X), join(X, Y)), X)
40.77/40.97	= { by lemma 46 }
40.77/40.97	  join(meet(complement(X), join(X, Y)), meet(X, join(X, Y)))
40.77/40.97	= { by lemma 33 }
40.77/40.97	  join(meet(complement(X), join(X, Y)), meet(complement(complement(X)), join(X, Y)))
40.77/40.97	= { by lemma 51 }
40.77/40.97	  join(X, Y)
40.77/40.97	
40.77/40.97	Lemma 68: composition(join(X, converse(Y)), complement(composition(Y, top))) = composition(X, complement(composition(Y, top))).
40.77/40.97	Proof:
40.77/40.97	  composition(join(X, converse(Y)), complement(composition(Y, top)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  composition(join(converse(Y), X), complement(composition(Y, top)))
40.77/40.97	= { by axiom 13 (composition_distributivity_7) }
40.77/40.97	  join(composition(converse(Y), complement(composition(Y, top))), composition(X, complement(composition(Y, top))))
40.77/40.97	= { by lemma 65 }
40.77/40.97	  join(zero, composition(X, complement(composition(Y, top))))
40.77/40.97	= { by lemma 36 }
40.77/40.97	  composition(X, complement(composition(Y, top)))
40.77/40.97	
40.77/40.97	Lemma 69: converse(join(X, composition(converse(Z), Y))) = join(converse(X), composition(converse(Y), Z)).
40.77/40.97	Proof:
40.77/40.97	  converse(join(X, composition(converse(Z), Y)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  converse(join(composition(converse(Z), Y), X))
40.77/40.97	= { by axiom 11 (converse_additivity_9) }
40.77/40.97	  join(converse(composition(converse(Z), Y)), converse(X))
40.77/40.97	= { by lemma 18 }
40.77/40.97	  join(composition(converse(Y), Z), converse(X))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  join(converse(X), composition(converse(Y), Z))
40.77/40.97	
40.77/40.97	Lemma 70: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
40.77/40.97	Proof:
40.77/40.97	  meet(join(X, complement(Y)), complement(meet(X, Y)))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  meet(join(complement(Y), X), complement(meet(X, Y)))
40.77/40.97	= { by lemma 15 }
40.77/40.97	  meet(join(complement(Y), X), complement(meet(Y, X)))
40.77/40.97	= { by lemma 40 }
40.77/40.97	  complement(join(meet(Y, X), complement(join(complement(Y), X))))
40.77/40.97	= { by lemma 24 }
40.77/40.97	  complement(Y)
40.77/40.97	
40.77/40.97	Lemma 71: meet(converse(X), converse(meet(X, Y))) = converse(meet(X, Y)).
40.77/40.97	Proof:
40.77/40.97	  meet(converse(X), converse(meet(X, Y)))
40.77/40.97	= { by lemma 15 }
40.77/40.97	  meet(converse(meet(X, Y)), converse(X))
40.77/40.97	= { by lemma 24 }
40.77/40.97	  meet(converse(meet(X, Y)), converse(join(meet(X, Y), complement(join(complement(X), Y)))))
40.77/40.97	= { by lemma 45 }
40.77/40.97	  converse(meet(X, Y))
40.77/40.97	
40.77/40.97	Lemma 72: join(zero, meet(X, X)) = X.
40.77/40.97	Proof:
40.77/40.97	  join(zero, meet(X, X))
40.77/40.97	= { by axiom 8 (def_zero_13) }
40.77/40.97	  join(meet(X, complement(X)), meet(X, X))
40.77/40.97	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.97	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
40.77/40.97	= { by lemma 24 }
40.77/40.97	  X
40.77/40.97	
40.77/40.97	Lemma 73: join(X, X) = X.
40.77/40.97	Proof:
40.77/40.97	  join(X, X)
40.77/40.97	= { by lemma 72 }
40.77/40.97	  join(join(zero, meet(X, X)), X)
40.77/40.97	= { by lemma 33 }
40.77/40.97	  join(join(zero, meet(X, X)), complement(complement(X)))
40.77/40.97	= { by axiom 6 (maddux2_join_associativity_2) }
40.77/40.97	  join(zero, join(meet(X, X), complement(complement(X))))
40.77/40.97	= { by lemma 23 }
40.77/40.97	  join(zero, join(complement(complement(X)), complement(complement(X))))
40.77/40.97	= { by lemma 22 }
40.77/40.97	  join(zero, complement(complement(X)))
40.77/40.97	= { by lemma 26 }
40.77/40.97	  X
40.77/40.97	
40.77/40.97	Lemma 74: join(complement(X), complement(Y)) = complement(meet(X, Y)).
40.77/40.97	Proof:
40.77/40.97	  join(complement(X), complement(Y))
40.77/40.97	= { by lemma 72 }
40.77/40.97	  join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y))))
40.77/40.97	= { by lemma 39 }
40.77/40.97	  join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
40.77/40.97	= { by lemma 32 }
40.77/40.97	  complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))
40.77/40.97	= { by axiom 2 (maddux4_definiton_of_meet_4) }
40.77/40.97	  complement(join(meet(X, Y), meet(X, Y)))
40.77/40.97	= { by lemma 73 }
40.77/40.97	  complement(meet(X, Y))
40.77/40.97	
40.77/40.97	Lemma 75: join(X, converse(meet(Y, complement(converse(X))))) = join(X, converse(Y)).
40.77/40.97	Proof:
40.77/40.97	  join(X, converse(meet(Y, complement(converse(X)))))
40.77/40.97	= { by lemma 54 }
40.77/40.97	  converse(join(converse(X), meet(Y, complement(converse(X)))))
40.77/40.97	= { by lemma 67 }
40.77/40.97	  converse(join(converse(X), Y))
40.77/40.97	= { by lemma 54 }
40.77/40.97	  join(X, converse(Y))
40.77/40.97	
40.77/40.97	Lemma 76: meet(X, join(complement(X), Y)) = meet(X, Y).
40.77/40.97	Proof:
40.77/40.97	  meet(X, join(complement(X), Y))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  meet(X, join(Y, complement(X)))
40.77/40.97	= { by lemma 48 }
40.77/40.97	  meet(X, Y)
40.77/40.97	
40.77/40.97	Lemma 77: composition(converse(X), join(complement(composition(X, top)), Y)) = composition(converse(X), Y).
40.77/40.97	Proof:
40.77/40.97	  composition(converse(X), join(complement(composition(X, top)), Y))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  composition(converse(X), join(Y, complement(composition(X, top))))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  composition(converse(X), join(complement(composition(X, top)), Y))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  converse(converse(composition(converse(X), join(complement(composition(X, top)), Y))))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  converse(converse(composition(converse(X), join(Y, complement(composition(X, top))))))
40.77/40.97	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.97	  converse(composition(converse(join(Y, complement(composition(X, top)))), converse(converse(X))))
40.77/40.97	= { by axiom 11 (converse_additivity_9) }
40.77/40.97	  converse(composition(join(converse(Y), converse(complement(composition(X, top)))), converse(converse(X))))
40.77/40.97	= { by lemma 64 }
40.77/40.97	  converse(join(composition(converse(Y), converse(converse(X))), converse(composition(converse(X), complement(composition(X, top))))))
40.77/40.97	= { by axiom 10 (converse_multiplicativity_10) }
40.77/40.97	  converse(join(converse(composition(converse(X), Y)), converse(composition(converse(X), complement(composition(X, top))))))
40.77/40.97	= { by axiom 11 (converse_additivity_9) }
40.77/40.97	  converse(converse(join(composition(converse(X), Y), composition(converse(X), complement(composition(X, top))))))
40.77/40.97	= { by axiom 1 (maddux1_join_commutativity_1) }
40.77/40.97	  converse(converse(join(composition(converse(X), complement(composition(X, top))), composition(converse(X), Y))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  join(composition(converse(X), complement(composition(X, top))), composition(converse(X), Y))
40.77/40.97	= { by lemma 65 }
40.77/40.97	  join(zero, composition(converse(X), Y))
40.77/40.97	= { by lemma 36 }
40.77/40.97	  composition(converse(X), Y)
40.77/40.97	
40.77/40.97	Lemma 78: join(complement(one), composition(X, complement(converse(X)))) = complement(one).
40.77/40.97	Proof:
40.77/40.97	  join(complement(one), composition(X, complement(converse(X))))
40.77/40.97	= { by axiom 5 (converse_idempotence_8) }
40.77/40.97	  join(complement(one), composition(converse(converse(X)), complement(converse(X))))
40.77/40.97	= { by axiom 3 (composition_identity_6) }
40.77/40.97	  join(complement(one), composition(converse(converse(X)), complement(composition(converse(X), one))))
40.77/40.97	= { by lemma 21 }
40.77/40.97	  complement(one)
40.77/40.97	
40.77/40.97	Lemma 79: converse(complement(converse(X))) = complement(X).
40.77/40.97	Proof:
40.77/40.97	  converse(complement(converse(X)))
40.77/40.97	= { by lemma 37 }
40.77/40.97	  converse(complement(converse(meet(X, top))))
40.77/40.97	= { by lemma 17 }
40.77/40.97	  converse(complement(converse(complement(join(zero, complement(X))))))
40.77/40.97	= { by lemma 62 }
40.77/40.97	  join(zero, complement(X))
40.77/40.97	= { by lemma 32 }
59.00/59.18	  complement(X)
59.00/59.18	
59.00/59.18	Goal 1 (goals_15): join(composition(complement(sk3), sk2), complement(sk1)) = complement(sk1).
59.00/59.18	Proof:
59.00/59.18	  join(composition(complement(sk3), sk2), complement(sk1))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), composition(complement(sk3), sk2))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(composition(complement(sk3), sk2), complement(sk1))
59.00/59.18	= { by lemma 51 }
59.00/59.18	  join(meet(sk1, join(composition(complement(sk3), sk2), complement(sk1))), meet(complement(sk1), join(composition(complement(sk3), sk2), complement(sk1))))
59.00/59.18	= { by lemma 48 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), meet(complement(sk1), join(composition(complement(sk3), sk2), complement(sk1))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), meet(join(composition(complement(sk3), sk2), complement(sk1)), complement(sk1)))
59.00/59.18	= { by lemma 40 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(join(sk1, complement(join(composition(complement(sk3), sk2), complement(sk1))))))
59.00/59.18	= { by lemma 40 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(join(sk1, meet(sk1, complement(composition(complement(sk3), sk2))))))
59.00/59.18	= { by lemma 41 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(join(sk1, complement(join(complement(sk1), composition(complement(sk3), sk2))))))
59.00/59.18	= { by lemma 50 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(complement(meet(complement(sk1), join(complement(sk1), composition(complement(sk3), sk2))))))
59.00/59.18	= { by lemma 46 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(complement(complement(sk1))))
59.00/59.18	= { by lemma 33 }
59.00/59.18	  join(meet(sk1, composition(complement(sk3), sk2)), complement(sk1))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(sk3), sk2)))
59.00/59.18	= { by lemma 20 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(one, composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 33 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(complement(one)), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 78 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk3, complement(converse(sk3))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 14 (goals_14) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(join(composition(sk1, converse(sk2)), sk3), complement(converse(sk3))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(join(sk3, composition(sk1, converse(sk2))), complement(converse(sk3))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 13 (composition_distributivity_7) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), join(composition(sk3, complement(converse(sk3))), composition(composition(sk1, converse(sk2)), complement(converse(sk3)))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 27 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(composition(composition(sk1, converse(sk2)), complement(converse(sk3))), join(complement(one), composition(sk3, complement(converse(sk3)))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 78 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(composition(composition(sk1, converse(sk2)), complement(converse(sk3))), complement(one))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(composition(sk1, converse(sk2)), complement(converse(sk3))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 12 (composition_associativity_5) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), complement(converse(sk3)))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 77 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), join(complement(composition(sk2, top)), complement(converse(sk3))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 67 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), join(complement(composition(sk2, top)), meet(complement(converse(sk3)), complement(complement(composition(sk2, top))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 77 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), meet(complement(converse(sk3)), complement(complement(composition(sk2, top)))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 33 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), meet(complement(converse(sk3)), composition(sk2, top)))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 18 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, converse(composition(converse(meet(complement(converse(sk3)), composition(sk2, top))), sk2))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 10 (converse_multiplicativity_10) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(complement(converse(sk3)), composition(sk2, top)))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(composition(sk2, top), complement(converse(sk3))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 45 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(meet(composition(sk2, top), complement(converse(sk3)))), converse(join(meet(composition(sk2, top), complement(converse(sk3))), meet(converse(sk3), composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(meet(composition(sk2, top), complement(converse(sk3)))), converse(join(meet(converse(sk3), composition(sk2, top)), meet(composition(sk2, top), complement(converse(sk3))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 34 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(meet(composition(sk2, top), complement(converse(sk3)))), converse(composition(sk2, top)))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), converse(meet(composition(sk2, top), complement(converse(sk3)))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 76 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), join(complement(converse(composition(sk2, top))), converse(meet(composition(sk2, top), complement(converse(sk3))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 75 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), join(complement(converse(composition(sk2, top))), converse(meet(meet(composition(sk2, top), complement(converse(sk3))), complement(converse(complement(converse(composition(sk2, top)))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 76 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), converse(meet(meet(composition(sk2, top), complement(converse(sk3))), complement(converse(complement(converse(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 5 (converse_idempotence_8) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(converse(converse(composition(sk2, top)))), converse(meet(meet(composition(sk2, top), complement(converse(sk3))), complement(converse(complement(converse(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 63 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(converse(converse(composition(sk2, top)))), converse(meet(meet(composition(sk2, top), complement(converse(sk3))), complement(complement(converse(converse(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 33 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(converse(converse(composition(sk2, top)))), converse(meet(meet(composition(sk2, top), complement(converse(sk3))), converse(converse(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(converse(converse(composition(sk2, top)))), converse(meet(converse(converse(composition(sk2, top))), meet(composition(sk2, top), complement(converse(sk3))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 71 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(converse(converse(composition(sk2, top))), meet(composition(sk2, top), complement(converse(sk3)))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(meet(composition(sk2, top), complement(converse(sk3))), converse(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 5 (converse_idempotence_8) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(meet(composition(sk2, top), complement(converse(sk3))), composition(sk2, top)))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 15 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(composition(sk2, top), meet(composition(sk2, top), complement(converse(sk3)))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 40 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(converse(meet(composition(sk2, top), complement(join(converse(sk3), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 33 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(complement(converse(meet(composition(sk2, top), complement(join(converse(sk3), complement(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 5 (converse_idempotence_8) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 58 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 27 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 41 }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.18	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.18	= { by lemma 44 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 43 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 15 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(complement(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 42 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 6 (maddux2_join_associativity_2) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 41 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 41 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 42 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 6 (maddux2_join_associativity_2) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 27 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 73 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 15 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 67 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 40 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 15 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 24 }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), join(complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.19	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.19	= { by lemma 40 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))), complement(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 74 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(meet(join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(meet(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(join(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))), complement(meet(meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))), join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 70 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), complement(join(complement(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 41 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), meet(complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 42 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))), converse(join(converse(sk3), complement(composition(sk2, top)))))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 75 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 35 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), zero))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), zero))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 16 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 56 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(converse(top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 30 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(converse(join(complement(composition(sk2, top)), top)))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 29 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(converse(join(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), join(complement(composition(sk2, top)), complement(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 74 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(converse(join(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(converse(join(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), complement(meet(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 54 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), converse(complement(meet(converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), join(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), complement(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), converse(complement(meet(composition(sk2, top), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 24 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(join(converse(sk3), complement(composition(sk2, top)))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 2 (maddux4_definiton_of_meet_4) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), complement(join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 22 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), complement(complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 33 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top)))), converse(join(converse(sk3), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 53 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(join(composition(sk2, top), converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(join(composition(sk2, top), converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 42 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(meet(complement(composition(sk2, top)), complement(converse(converse(join(converse(sk3), complement(composition(sk2, top))))))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 75 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 40 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), complement(converse(join(converse(sk3), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 48 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(converse(composition(sk2, top)), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 33 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(complement(converse(composition(sk2, top)))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(converse(complement(composition(sk2, top)))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 70 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(converse(meet(join(converse(sk3), complement(composition(sk2, top))), complement(meet(converse(sk3), composition(sk2, top)))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 71 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(meet(join(converse(sk3), complement(composition(sk2, top))), complement(meet(converse(sk3), composition(sk2, top))))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 70 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), converse(complement(composition(sk2, top))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(meet(converse(join(converse(sk3), complement(composition(sk2, top)))), complement(converse(composition(sk2, top))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(meet(complement(converse(composition(sk2, top))), converse(join(converse(sk3), complement(composition(sk2, top)))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 50 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(join(converse(composition(sk2, top)), complement(converse(join(converse(sk3), complement(composition(sk2, top)))))), join(complement(converse(join(converse(sk3), complement(composition(sk2, top))))), complement(converse(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 60 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(converse(join(converse(sk3), complement(composition(sk2, top)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 20 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(converse(join(converse(sk3), composition(one, complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 68 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(converse(join(converse(sk3), composition(join(one, converse(sk2)), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 53 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(converse(join(converse(sk3), composition(converse(join(sk2, converse(one))), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 69 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), join(sk2, converse(one)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 3 (composition_identity_6) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), join(sk2, composition(converse(one), one)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 19 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), join(sk2, one))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), join(one, sk2))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 5 (converse_idempotence_8) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), join(one, converse(converse(sk2))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 53 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), converse(join(converse(sk2), converse(one))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 10 (converse_multiplicativity_10) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), converse(composition(join(converse(sk2), converse(one)), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), converse(composition(join(converse(one), converse(sk2)), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 68 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), converse(composition(converse(one), complement(composition(sk2, top))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 18 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(converse(complement(composition(sk2, top))), one)))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 63 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(complement(converse(composition(sk2, top))), one)))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 57 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), composition(complement(composition(top, converse(sk2))), one)))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 3 (composition_identity_6) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(join(converse(converse(sk3)), complement(composition(top, converse(sk2)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 40 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(composition(top, converse(sk2)), complement(converse(converse(sk3))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(converse(converse(sk3))), composition(top, converse(sk2)))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 33 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(meet(complement(converse(converse(sk3))), complement(complement(composition(top, converse(sk2)))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 66 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(join(complement(composition(top, converse(sk2))), meet(complement(converse(converse(sk3))), complement(complement(composition(top, converse(sk2))))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 67 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(join(complement(composition(top, converse(sk2))), complement(converse(converse(sk3))))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 66 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, composition(converse(sk2), converse(complement(converse(converse(sk3)))))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 10 (converse_multiplicativity_10) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, converse(composition(complement(converse(converse(sk3))), sk2))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by axiom 5 (converse_idempotence_8) }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(complement(join(complement(one), composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 41 }
59.00/59.20	  join(complement(sk1), meet(sk1, composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2))))
59.00/59.20	= { by lemma 15 }
59.00/59.20	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), sk1))
59.00/59.20	= { by lemma 33 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(complement(sk1))))
59.00/59.21	= { by lemma 79 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(converse(complement(converse(sk1))))))
59.00/59.21	= { by lemma 21 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(converse(join(complement(converse(sk1)), composition(converse(converse(converse(composition(complement(sk3), sk2)))), complement(composition(converse(converse(composition(complement(sk3), sk2))), converse(sk1)))))))))
59.00/59.21	= { by axiom 5 (converse_idempotence_8) }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(converse(join(complement(converse(sk1)), composition(converse(composition(complement(sk3), sk2)), complement(composition(converse(converse(composition(complement(sk3), sk2))), converse(sk1)))))))))
59.00/59.21	= { by axiom 10 (converse_multiplicativity_10) }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(converse(join(complement(converse(sk1)), composition(converse(composition(complement(sk3), sk2)), complement(converse(composition(sk1, converse(composition(complement(sk3), sk2)))))))))))
59.00/59.21	= { by lemma 69 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(join(converse(complement(converse(sk1))), composition(converse(complement(converse(composition(sk1, converse(composition(complement(sk3), sk2)))))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by lemma 79 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(join(complement(sk1), composition(converse(complement(converse(composition(sk1, converse(composition(complement(sk3), sk2)))))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by lemma 79 }
59.00/59.21	  join(complement(sk1), meet(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)), complement(join(complement(sk1), composition(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by lemma 40 }
59.00/59.21	  join(complement(sk1), complement(join(join(complement(sk1), composition(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), composition(complement(sk3), sk2))), complement(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by axiom 6 (maddux2_join_associativity_2) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(composition(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), composition(complement(sk3), sk2)), complement(composition(meet(one, complement(composition(sk1, converse(composition(complement(sk3), sk2))))), composition(complement(sk3), sk2)))))))
59.00/59.21	= { by lemma 15 }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(composition(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), composition(complement(sk3), sk2)), complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2)))))))
59.00/59.21	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2))), composition(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by lemma 24 }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2))), composition(join(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one))), composition(complement(sk3), sk2))))))
59.00/59.21	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2))), composition(join(complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one)), meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one)), composition(complement(sk3), sk2))))))
59.00/59.21	= { by axiom 1 (maddux1_join_commutativity_1) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(composition(join(complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one)), meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one)), composition(complement(sk3), sk2)), complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2)))))))
59.00/59.21	= { by axiom 13 (composition_distributivity_7) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(join(composition(complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one)), composition(complement(sk3), sk2)), composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2))), complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2)))))))
59.00/59.21	= { by axiom 6 (maddux2_join_associativity_2) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(composition(complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one)), composition(complement(sk3), sk2)), join(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2)), complement(composition(meet(complement(composition(sk1, converse(composition(complement(sk3), sk2)))), one), composition(complement(sk3), sk2))))))))
59.00/59.21	= { by axiom 9 (def_top_12) }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), join(composition(complement(join(complement(complement(composition(sk1, converse(composition(complement(sk3), sk2))))), one)), composition(complement(sk3), sk2)), top))))
59.00/59.21	= { by lemma 30 }
59.00/59.21	  join(complement(sk1), complement(join(complement(sk1), top)))
59.00/59.21	= { by lemma 30 }
59.00/59.21	  join(complement(sk1), complement(top))
59.00/59.21	= { by lemma 16 }
59.00/59.21	  join(complement(sk1), zero)
59.00/59.21	= { by lemma 35 }
59.00/59.21	  complement(sk1)
59.00/59.21	% SZS output end Proof
59.00/59.21	
59.00/59.21	RESULT: Unsatisfiable (the axioms are contradictory).
59.00/59.21	EOF