0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n006.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 11:21:09 EDT 2019
0.12/0.33	% CPUTime    : 
0.88/1.04	% SZS status Unsatisfiable
0.88/1.04	
0.88/1.04	% SZS output start Proof
0.88/1.04	Take the following subset of the input axioms:
0.88/1.05	  fof(composition_associativity_5, axiom, ![A, C, B]: composition(composition(A, B), C)=composition(A, composition(B, C))).
0.88/1.05	  fof(composition_distributivity_7, axiom, ![A, C, B]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
0.88/1.05	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
0.88/1.05	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
0.88/1.05	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
0.88/1.05	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
0.88/1.05	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
0.88/1.05	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
0.88/1.05	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
0.88/1.05	  fof(goals_17, negated_conjecture, ![A]: zero=meet(composition(sk1, A), composition(sk1, complement(A)))).
0.88/1.05	  fof(goals_18, negated_conjecture, one!=join(composition(converse(sk1), sk1), one)).
0.88/1.05	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
0.88/1.05	  fof(maddux2_join_associativity_2, axiom, ![A, C, B]: join(join(A, B), C)=join(A, join(B, C))).
0.88/1.05	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
0.88/1.05	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
0.88/1.05	  fof(modular_law_2_16, axiom, ![A, C, B]: meet(composition(meet(A, composition(C, converse(B))), B), C)=join(meet(composition(A, B), C), meet(composition(meet(A, composition(C, converse(B))), B), C))).
0.88/1.05	
0.88/1.05	Now clausify the problem and encode Horn clauses using encoding 3 of
0.88/1.05	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.88/1.05	We repeatedly replace C & s=t => u=v by the two clauses:
0.88/1.05	  fresh(y, y, x1...xn) = u
0.88/1.05	  C => fresh(s, t, x1...xn) = v
0.88/1.05	where fresh is a fresh function symbol and x1..xn are the free
0.88/1.05	variables of u and v.
0.88/1.05	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.88/1.05	input problem has no model of domain size 1).
0.88/1.05	
0.88/1.05	The encoding turns the above axioms into the following unit equations and goals:
0.88/1.05	
0.88/1.05	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
0.88/1.05	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
0.88/1.05	Axiom 3 (composition_identity_6): X = composition(X, one).
0.88/1.05	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
0.88/1.05	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
0.88/1.05	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
0.88/1.05	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
0.88/1.05	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
0.88/1.05	Axiom 9 (def_top_12): join(X, complement(X)) = top.
0.88/1.05	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
0.88/1.05	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
0.88/1.05	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
0.88/1.05	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
0.88/1.05	Axiom 14 (modular_law_2_16): meet(composition(meet(X, composition(Y, converse(Z))), Z), Y) = join(meet(composition(X, Z), Y), meet(composition(meet(X, composition(Y, converse(Z))), Z), Y)).
0.92/1.08	Axiom 15 (goals_17): zero = meet(composition(sk1, X), composition(sk1, complement(X))).
0.92/1.08	
0.92/1.08	Lemma 16: meet(X, Y) = meet(Y, X).
0.92/1.08	Proof:
0.92/1.08	  meet(X, Y)
0.92/1.08	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.08	  complement(join(complement(X), complement(Y)))
0.92/1.08	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.08	  complement(join(complement(Y), complement(X)))
0.92/1.08	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.08	  meet(Y, X)
0.92/1.08	
0.92/1.08	Lemma 17: complement(top) = zero.
0.92/1.08	Proof:
0.92/1.08	  complement(top)
0.92/1.08	= { by axiom 9 (def_top_12) }
0.92/1.08	  complement(join(complement(?), complement(complement(?))))
0.92/1.08	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.08	  meet(?, complement(?))
0.92/1.08	= { by axiom 8 (def_zero_13) }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 18: complement(join(zero, complement(X))) = meet(X, top).
0.92/1.09	Proof:
0.92/1.09	  complement(join(zero, complement(X)))
0.92/1.09	= { by lemma 17 }
0.92/1.09	  complement(join(complement(top), complement(X)))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  meet(top, X)
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(X, top)
0.92/1.09	
0.92/1.09	Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
0.92/1.09	Proof:
0.92/1.09	  converse(composition(converse(X), Y))
0.92/1.09	= { by axiom 10 (converse_multiplicativity_10) }
0.92/1.09	  composition(converse(Y), converse(converse(X)))
0.92/1.09	= { by axiom 5 (converse_idempotence_8) }
0.92/1.09	  composition(converse(Y), X)
0.92/1.09	
0.92/1.09	Lemma 20: composition(converse(one), X) = X.
0.92/1.09	Proof:
0.92/1.09	  composition(converse(one), X)
0.92/1.09	= { by lemma 19 }
0.92/1.09	  converse(composition(converse(X), one))
0.92/1.09	= { by axiom 3 (composition_identity_6) }
0.92/1.09	  converse(converse(X))
0.92/1.09	= { by axiom 5 (converse_idempotence_8) }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 21: composition(one, X) = X.
0.92/1.09	Proof:
0.92/1.09	  composition(one, X)
0.92/1.09	= { by lemma 20 }
0.92/1.09	  composition(converse(one), composition(one, X))
0.92/1.09	= { by axiom 12 (composition_associativity_5) }
0.92/1.09	  composition(composition(converse(one), one), X)
0.92/1.09	= { by axiom 3 (composition_identity_6) }
0.92/1.09	  composition(converse(one), X)
0.92/1.09	= { by lemma 20 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 22: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
0.92/1.09	Proof:
0.92/1.09	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
0.92/1.09	= { by axiom 7 (converse_cancellativity_11) }
0.92/1.09	  complement(Y)
0.92/1.09	
0.92/1.09	Lemma 23: join(complement(X), complement(X)) = complement(X).
0.92/1.09	Proof:
0.92/1.09	  join(complement(X), complement(X))
0.92/1.09	= { by lemma 20 }
0.92/1.09	  join(complement(X), composition(converse(one), complement(X)))
0.92/1.09	= { by lemma 21 }
0.92/1.09	  join(complement(X), composition(converse(one), complement(composition(one, X))))
0.92/1.09	= { by lemma 22 }
0.92/1.09	  complement(X)
0.92/1.09	
0.92/1.09	Lemma 24: join(X, join(Y, Z)) = join(Z, join(X, Y)).
0.92/1.09	Proof:
0.92/1.09	  join(X, join(Y, Z))
0.92/1.09	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.09	  join(join(X, Y), Z)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(Z, join(X, Y))
0.92/1.09	
0.92/1.09	Lemma 25: join(X, join(complement(X), Y)) = join(Y, top).
0.92/1.09	Proof:
0.92/1.09	  join(X, join(complement(X), Y))
0.92/1.09	= { by lemma 24 }
0.92/1.09	  join(complement(X), join(Y, X))
0.92/1.09	= { by lemma 24 }
0.92/1.09	  join(Y, join(X, complement(X)))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(Y, top)
0.92/1.09	
0.92/1.09	Lemma 26: join(top, complement(X)) = top.
0.92/1.09	Proof:
0.92/1.09	  join(top, complement(X))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(complement(X), top)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(top, complement(X))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(join(X, complement(X)), complement(X))
0.92/1.09	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.09	  join(X, join(complement(X), complement(X)))
0.92/1.09	= { by lemma 23 }
0.92/1.09	  join(X, complement(X))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  top
0.92/1.09	
0.92/1.09	Lemma 27: join(X, top) = top.
0.92/1.09	Proof:
0.92/1.09	  join(X, top)
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(X, join(complement(X), complement(complement(X))))
0.92/1.09	= { by lemma 25 }
0.92/1.09	  join(complement(complement(X)), top)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(top, complement(complement(X)))
0.92/1.09	= { by lemma 26 }
0.92/1.09	  top
0.92/1.09	
0.92/1.09	Lemma 28: join(meet(X, Y), complement(join(complement(X), Y))) = X.
0.92/1.09	Proof:
0.92/1.09	  join(meet(X, Y), complement(join(complement(X), Y)))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
0.92/1.09	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 29: join(zero, meet(X, top)) = X.
0.92/1.09	Proof:
0.92/1.09	  join(zero, meet(X, top))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(meet(X, top), zero)
0.92/1.09	= { by lemma 17 }
0.92/1.09	  join(meet(X, top), complement(top))
0.92/1.09	= { by lemma 27 }
0.92/1.09	  join(meet(X, top), complement(join(complement(X), top)))
0.92/1.09	= { by lemma 28 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 30: join(meet(X, Y), meet(X, complement(Y))) = X.
0.92/1.09	Proof:
0.92/1.09	  join(meet(X, Y), meet(X, complement(Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(meet(X, complement(Y)), meet(X, Y))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
0.92/1.09	= { by lemma 28 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 31: meet(X, zero) = zero.
0.92/1.09	Proof:
0.92/1.09	  meet(X, zero)
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(zero, X)
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  complement(join(complement(zero), complement(X)))
0.92/1.09	= { by lemma 17 }
0.92/1.09	  complement(join(complement(complement(top)), complement(X)))
0.92/1.09	= { by lemma 23 }
0.92/1.09	  complement(join(complement(join(complement(top), complement(top))), complement(X)))
0.92/1.09	= { by lemma 17 }
0.92/1.09	  complement(join(complement(join(zero, complement(top))), complement(X)))
0.92/1.09	= { by lemma 18 }
0.92/1.09	  complement(join(meet(top, top), complement(X)))
0.92/1.09	= { by lemma 29 }
0.92/1.09	  complement(join(join(zero, meet(meet(top, top), top)), complement(X)))
0.92/1.09	= { by axiom 8 (def_zero_13) }
0.92/1.09	  complement(join(join(meet(top, complement(top)), meet(meet(top, top), top)), complement(X)))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  complement(join(join(meet(top, complement(top)), meet(top, meet(top, top))), complement(X)))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  complement(join(join(meet(top, complement(top)), meet(top, complement(join(complement(top), complement(top))))), complement(X)))
0.92/1.09	= { by lemma 23 }
0.92/1.09	  complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(X)))
0.92/1.09	= { by lemma 30 }
0.92/1.09	  complement(join(top, complement(X)))
0.92/1.09	= { by lemma 26 }
0.92/1.09	  complement(top)
0.92/1.09	= { by lemma 17 }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 32: join(zero, meet(X, X)) = X.
0.92/1.09	Proof:
0.92/1.09	  join(zero, meet(X, X))
0.92/1.09	= { by axiom 8 (def_zero_13) }
0.92/1.09	  join(meet(X, complement(X)), meet(X, X))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
0.92/1.09	= { by lemma 28 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 33: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
0.92/1.09	Proof:
0.92/1.09	  join(X, meet(Y, Y))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(meet(Y, Y), X)
0.92/1.09	= { by lemma 32 }
0.92/1.09	  join(meet(Y, Y), join(zero, meet(X, X)))
0.92/1.09	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.09	  join(join(meet(Y, Y), zero), meet(X, X))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(join(zero, meet(Y, Y)), meet(X, X))
0.92/1.09	= { by lemma 32 }
0.92/1.09	  join(Y, meet(X, X))
0.92/1.09	
0.92/1.09	Lemma 34: join(X, zero) = X.
0.92/1.09	Proof:
0.92/1.09	  join(X, zero)
0.92/1.09	= { by lemma 31 }
0.92/1.09	  join(X, meet(zero, zero))
0.92/1.09	= { by lemma 33 }
0.92/1.09	  join(zero, meet(X, X))
0.92/1.09	= { by lemma 32 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 35: join(zero, X) = X.
0.92/1.09	Proof:
0.92/1.09	  join(zero, X)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(X, zero)
0.92/1.09	= { by lemma 34 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 36: meet(X, X) = X.
0.92/1.09	Proof:
0.92/1.09	  meet(X, X)
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(zero, meet(X, X))
0.92/1.09	= { by lemma 32 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 37: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
0.92/1.09	Proof:
0.92/1.09	  complement(join(complement(X), meet(Y, Z)))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  complement(join(complement(X), meet(Z, Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  complement(join(meet(Z, Y), complement(X)))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
0.92/1.09	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.09	  meet(join(complement(Z), complement(Y)), X)
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(X, join(complement(Z), complement(Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  meet(X, join(complement(Y), complement(Z)))
0.92/1.09	
0.92/1.09	Lemma 38: meet(X, top) = X.
0.92/1.09	Proof:
0.92/1.09	  meet(X, top)
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(zero, meet(X, top))
0.92/1.09	= { by lemma 29 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 39: meet(top, X) = X.
0.92/1.09	Proof:
0.92/1.09	  meet(top, X)
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(X, top)
0.92/1.09	= { by lemma 38 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 40: join(complement(X), complement(Y)) = complement(meet(X, Y)).
0.92/1.09	Proof:
0.92/1.09	  join(complement(X), complement(Y))
0.92/1.09	= { by lemma 39 }
0.92/1.09	  meet(top, join(complement(X), complement(Y)))
0.92/1.09	= { by lemma 37 }
0.92/1.09	  complement(join(complement(top), meet(X, Y)))
0.92/1.09	= { by lemma 17 }
0.92/1.09	  complement(join(zero, meet(X, Y)))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  complement(meet(X, Y))
0.92/1.09	
0.92/1.09	Lemma 41: complement(join(Y, complement(X))) = meet(X, complement(Y)).
0.92/1.09	Proof:
0.92/1.09	  complement(join(Y, complement(X)))
0.92/1.09	= { by lemma 36 }
0.92/1.09	  complement(join(Y, meet(complement(X), complement(X))))
0.92/1.09	= { by lemma 33 }
0.92/1.09	  complement(join(complement(X), meet(Y, Y)))
0.92/1.09	= { by lemma 37 }
0.92/1.09	  meet(X, join(complement(Y), complement(Y)))
0.92/1.09	= { by lemma 40 }
0.92/1.09	  meet(X, complement(meet(Y, Y)))
0.92/1.09	= { by lemma 36 }
0.92/1.09	  meet(X, complement(Y))
0.92/1.09	
0.92/1.09	Lemma 42: complement(meet(X, complement(Y))) = join(Y, complement(X)).
0.92/1.09	Proof:
0.92/1.09	  complement(meet(X, complement(Y)))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  complement(meet(complement(Y), X))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  complement(meet(join(zero, complement(Y)), X))
0.92/1.09	= { by lemma 40 }
0.92/1.09	  join(complement(join(zero, complement(Y))), complement(X))
0.92/1.09	= { by lemma 18 }
0.92/1.09	  join(meet(Y, top), complement(X))
0.92/1.09	= { by lemma 38 }
0.92/1.09	  join(Y, complement(X))
0.92/1.09	
0.92/1.09	Lemma 43: join(complement(converse(X)), converse(join(X, Y))) = top.
0.92/1.09	Proof:
0.92/1.09	  join(complement(converse(X)), converse(join(X, Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(complement(converse(X)), converse(join(Y, X)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(converse(join(Y, X)), complement(converse(X)))
0.92/1.09	= { by axiom 11 (converse_additivity_9) }
0.92/1.09	  join(join(converse(Y), converse(X)), complement(converse(X)))
0.92/1.09	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.09	  join(converse(Y), join(converse(X), complement(converse(X))))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(converse(Y), top)
0.92/1.09	= { by lemma 27 }
0.92/1.09	  top
0.92/1.09	
0.92/1.09	Lemma 44: complement(complement(X)) = X.
0.92/1.09	Proof:
0.92/1.09	  complement(complement(X))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  complement(join(zero, complement(X)))
0.92/1.09	= { by lemma 18 }
0.92/1.09	  meet(X, top)
0.92/1.09	= { by lemma 38 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 45: join(meet(Y, complement(X)), complement(join(X, Y))) = complement(X).
0.92/1.09	Proof:
0.92/1.09	  join(meet(Y, complement(X)), complement(join(X, Y)))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  join(meet(complement(X), Y), complement(join(X, Y)))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(meet(join(zero, complement(X)), Y), complement(join(X, Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
0.92/1.09	= { by lemma 38 }
0.92/1.09	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
0.92/1.09	= { by lemma 18 }
0.92/1.09	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y)))
0.92/1.09	= { by lemma 28 }
0.92/1.09	  join(zero, complement(X))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  complement(X)
0.92/1.09	
0.92/1.09	Lemma 46: meet(zero, X) = zero.
0.92/1.09	Proof:
0.92/1.09	  meet(zero, X)
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(X, zero)
0.92/1.09	= { by lemma 31 }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 47: converse(join(X, converse(Y))) = join(Y, converse(X)).
0.92/1.09	Proof:
0.92/1.09	  converse(join(X, converse(Y)))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  converse(join(converse(Y), X))
0.92/1.09	= { by axiom 11 (converse_additivity_9) }
0.92/1.09	  join(converse(converse(Y)), converse(X))
0.92/1.09	= { by axiom 5 (converse_idempotence_8) }
0.92/1.09	  join(Y, converse(X))
0.92/1.09	
0.92/1.09	Lemma 48: converse(join(converse(X), Y)) = join(X, converse(Y)).
0.92/1.09	Proof:
0.92/1.09	  converse(join(converse(X), Y))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  converse(join(Y, converse(X)))
0.92/1.09	= { by lemma 47 }
0.92/1.09	  join(X, converse(Y))
0.92/1.09	
0.92/1.09	Lemma 49: converse(zero) = zero.
0.92/1.09	Proof:
0.92/1.09	  converse(zero)
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(zero, converse(zero))
0.92/1.09	= { by lemma 48 }
0.92/1.09	  converse(join(converse(zero), zero))
0.92/1.09	= { by lemma 34 }
0.92/1.09	  converse(converse(zero))
0.92/1.09	= { by axiom 5 (converse_idempotence_8) }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 50: join(top, X) = top.
0.92/1.09	Proof:
0.92/1.09	  join(top, X)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(X, top)
0.92/1.09	= { by lemma 27 }
0.92/1.09	  top
0.92/1.09	
0.92/1.09	Lemma 51: join(X, converse(top)) = converse(top).
0.92/1.09	Proof:
0.92/1.09	  join(X, converse(top))
0.92/1.09	= { by lemma 48 }
0.92/1.09	  converse(join(converse(X), top))
0.92/1.09	= { by lemma 27 }
0.92/1.09	  converse(top)
0.92/1.09	
0.92/1.09	Lemma 52: converse(top) = top.
0.92/1.09	Proof:
0.92/1.09	  converse(top)
0.92/1.09	= { by lemma 51 }
0.92/1.09	  join(?, converse(top))
0.92/1.09	= { by lemma 51 }
0.92/1.09	  join(?, join(complement(?), converse(top)))
0.92/1.09	= { by lemma 25 }
0.92/1.09	  join(converse(top), top)
0.92/1.09	= { by lemma 27 }
0.92/1.09	  top
0.92/1.09	
0.92/1.09	Lemma 53: join(Y, composition(X, Y)) = composition(join(X, one), Y).
0.92/1.09	Proof:
0.92/1.09	  join(Y, composition(X, Y))
0.92/1.09	= { by lemma 21 }
0.92/1.09	  join(composition(one, Y), composition(X, Y))
0.92/1.09	= { by axiom 13 (composition_distributivity_7) }
0.92/1.09	  composition(join(one, X), Y)
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  composition(join(X, one), Y)
0.92/1.09	
0.92/1.09	Lemma 54: composition(top, zero) = zero.
0.92/1.09	Proof:
0.92/1.09	  composition(top, zero)
0.92/1.09	= { by lemma 50 }
0.92/1.09	  composition(join(top, one), zero)
0.92/1.09	= { by lemma 52 }
0.92/1.09	  composition(join(converse(top), one), zero)
0.92/1.09	= { by lemma 17 }
0.92/1.09	  composition(join(converse(top), one), complement(top))
0.92/1.09	= { by lemma 53 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(top)))
0.92/1.09	= { by lemma 27 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(composition(top, top), top))))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(join(top, complement(top)), composition(top, top)))))
0.92/1.09	= { by lemma 17 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(join(top, zero), composition(top, top)))))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(join(zero, top), composition(top, top)))))
0.92/1.09	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(zero, join(top, composition(top, top))))))
0.92/1.09	= { by lemma 53 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(join(zero, composition(join(top, one), top)))))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
0.92/1.09	= { by lemma 50 }
0.92/1.09	  join(complement(top), composition(converse(top), complement(composition(top, top))))
0.92/1.09	= { by lemma 22 }
0.92/1.09	  complement(top)
0.92/1.09	= { by lemma 17 }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 55: composition(zero, X) = zero.
0.92/1.09	Proof:
0.92/1.09	  composition(zero, X)
0.92/1.09	= { by lemma 49 }
0.92/1.09	  composition(converse(zero), X)
0.92/1.09	= { by lemma 19 }
0.92/1.09	  converse(composition(converse(X), zero))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  converse(join(zero, composition(converse(X), zero)))
0.92/1.09	= { by lemma 54 }
0.92/1.09	  converse(join(composition(top, zero), composition(converse(X), zero)))
0.92/1.09	= { by axiom 13 (composition_distributivity_7) }
0.92/1.09	  converse(composition(join(top, converse(X)), zero))
0.92/1.09	= { by lemma 50 }
0.92/1.09	  converse(composition(top, zero))
0.92/1.09	= { by lemma 54 }
0.92/1.09	  converse(zero)
0.92/1.09	= { by lemma 49 }
0.92/1.09	  zero
0.92/1.09	
0.92/1.09	Lemma 56: join(complement(one), converse(complement(one))) = complement(one).
0.92/1.09	Proof:
0.92/1.09	  join(complement(one), converse(complement(one)))
0.92/1.09	= { by axiom 3 (composition_identity_6) }
0.92/1.09	  join(complement(one), composition(converse(complement(one)), one))
0.92/1.09	= { by lemma 35 }
0.92/1.09	  join(complement(one), composition(converse(join(zero, complement(one))), one))
0.92/1.09	= { by lemma 38 }
0.92/1.09	  join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
0.92/1.09	= { by lemma 18 }
0.92/1.09	  join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
0.92/1.09	= { by axiom 3 (composition_identity_6) }
0.92/1.09	  join(complement(one), composition(converse(join(zero, complement(one))), complement(composition(join(zero, complement(one)), one))))
0.92/1.09	= { by lemma 22 }
0.92/1.09	  complement(one)
0.92/1.09	
0.92/1.09	Lemma 57: join(meet(Y, X), meet(X, complement(Y))) = X.
0.92/1.09	Proof:
0.92/1.09	  join(meet(Y, X), meet(X, complement(Y)))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  join(meet(X, Y), meet(X, complement(Y)))
0.92/1.09	= { by lemma 30 }
0.92/1.09	  X
0.92/1.09	
0.92/1.09	Lemma 58: meet(complement(X), converse(complement(converse(X)))) = complement(X).
0.92/1.09	Proof:
0.92/1.09	  meet(complement(X), converse(complement(converse(X))))
0.92/1.09	= { by lemma 16 }
0.92/1.09	  meet(converse(complement(converse(X))), complement(X))
0.92/1.09	= { by lemma 34 }
0.92/1.09	  join(meet(converse(complement(converse(X))), complement(X)), zero)
0.92/1.09	= { by lemma 17 }
0.92/1.09	  join(meet(converse(complement(converse(X))), complement(X)), complement(top))
0.92/1.09	= { by lemma 52 }
0.92/1.09	  join(meet(converse(complement(converse(X))), complement(X)), complement(converse(top)))
0.92/1.09	= { by axiom 9 (def_top_12) }
0.92/1.09	  join(meet(converse(complement(converse(X))), complement(X)), complement(converse(join(converse(X), complement(converse(X))))))
0.92/1.09	= { by lemma 48 }
0.92/1.09	  join(meet(converse(complement(converse(X))), complement(X)), complement(join(X, converse(complement(converse(X))))))
0.92/1.09	= { by lemma 45 }
0.92/1.09	  complement(X)
0.92/1.09	
0.92/1.09	Lemma 59: join(converse(X), complement(converse(meet(X, Y)))) = top.
0.92/1.09	Proof:
0.92/1.09	  join(converse(X), complement(converse(meet(X, Y))))
0.92/1.09	= { by axiom 1 (maddux1_join_commutativity_1) }
0.92/1.09	  join(complement(converse(meet(X, Y))), converse(X))
0.92/1.09	= { by lemma 28 }
0.92/1.09	  join(complement(converse(meet(X, Y))), converse(join(meet(X, Y), complement(join(complement(X), Y)))))
0.92/1.09	= { by lemma 43 }
0.92/1.12	  top
0.92/1.12	
0.92/1.12	Lemma 60: meet(converse(sk1), composition(complement(one), converse(sk1))) = zero.
0.92/1.12	Proof:
0.92/1.12	  meet(converse(sk1), composition(complement(one), converse(sk1)))
0.92/1.12	= { by lemma 44 }
0.92/1.12	  meet(converse(complement(complement(sk1))), composition(complement(one), converse(sk1)))
0.92/1.12	= { by lemma 56 }
0.92/1.12	  meet(converse(complement(complement(sk1))), composition(join(complement(one), converse(complement(one))), converse(sk1)))
0.92/1.12	= { by lemma 47 }
0.92/1.12	  meet(converse(complement(complement(sk1))), composition(converse(join(complement(one), converse(complement(one)))), converse(sk1)))
0.92/1.12	= { by axiom 10 (converse_multiplicativity_10) }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(composition(sk1, join(complement(one), converse(complement(one))))))
0.92/1.12	= { by lemma 56 }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(composition(sk1, complement(one))))
0.92/1.12	= { by lemma 57 }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(join(meet(sk1, composition(sk1, complement(one))), meet(composition(sk1, complement(one)), complement(sk1)))))
0.92/1.12	= { by axiom 3 (composition_identity_6) }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(join(meet(composition(sk1, one), composition(sk1, complement(one))), meet(composition(sk1, complement(one)), complement(sk1)))))
0.92/1.12	= { by axiom 15 (goals_17) }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(join(zero, meet(composition(sk1, complement(one)), complement(sk1)))))
0.92/1.12	= { by lemma 35 }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(meet(composition(sk1, complement(one)), complement(sk1))))
0.92/1.12	= { by lemma 16 }
0.92/1.12	  meet(converse(complement(complement(sk1))), converse(meet(complement(sk1), composition(sk1, complement(one)))))
0.92/1.12	= { by lemma 16 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(complement(complement(sk1))))
0.92/1.12	= { by lemma 58 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(meet(complement(complement(sk1)), converse(complement(converse(complement(sk1)))))))
0.92/1.12	= { by lemma 35 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(zero, meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by axiom 8 (def_zero_13) }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), complement(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 40 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), join(complement(converse(converse(complement(sk1)))), complement(converse(complement(converse(complement(sk1))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by axiom 5 (converse_idempotence_8) }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), join(converse(converse(complement(converse(converse(complement(sk1)))))), complement(converse(complement(converse(complement(sk1))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 58 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), join(converse(converse(complement(converse(converse(complement(sk1)))))), complement(converse(meet(complement(converse(complement(sk1))), converse(complement(converse(converse(complement(sk1)))))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 16 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), join(converse(converse(complement(converse(converse(complement(sk1)))))), complement(converse(meet(converse(complement(converse(converse(complement(sk1))))), complement(converse(complement(sk1)))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 59 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), top), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 16 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(top, meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1)))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 38 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(top, meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1)))))), top), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 18 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, complement(meet(top, meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 40 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, join(complement(top), complement(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 40 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, join(complement(top), join(complement(converse(converse(complement(sk1)))), complement(converse(complement(converse(complement(sk1))))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by axiom 6 (maddux2_join_associativity_2) }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, join(join(complement(top), complement(converse(converse(complement(sk1))))), complement(converse(complement(converse(complement(sk1)))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 42 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, complement(meet(converse(complement(converse(complement(sk1)))), complement(join(complement(top), complement(converse(converse(complement(sk1)))))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by axiom 2 (maddux4_definiton_of_meet_4) }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(complement(join(zero, complement(meet(converse(complement(converse(complement(sk1)))), meet(top, converse(converse(complement(sk1)))))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 18 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(meet(converse(complement(converse(complement(sk1)))), meet(top, converse(converse(complement(sk1))))), top), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 38 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(converse(complement(converse(complement(sk1)))), meet(top, converse(converse(complement(sk1))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 39 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(converse(complement(converse(complement(sk1)))), converse(converse(complement(sk1)))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by lemma 16 }
0.92/1.12	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(converse(converse(complement(sk1))), converse(complement(converse(complement(sk1))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.12	= { by axiom 5 (converse_idempotence_8) }
0.92/1.13	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(complement(sk1), converse(complement(converse(complement(sk1))))), meet(complement(complement(sk1)), converse(complement(converse(complement(sk1))))))))
0.92/1.13	= { by lemma 16 }
0.92/1.13	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(join(meet(complement(sk1), converse(complement(converse(complement(sk1))))), meet(converse(complement(converse(complement(sk1)))), complement(complement(sk1))))))
0.92/1.13	= { by lemma 57 }
0.92/1.13	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), converse(converse(complement(converse(complement(sk1))))))
0.92/1.13	= { by axiom 5 (converse_idempotence_8) }
0.92/1.13	  meet(converse(meet(complement(sk1), composition(sk1, complement(one)))), complement(converse(complement(sk1))))
0.92/1.13	= { by lemma 41 }
0.92/1.13	  complement(join(converse(complement(sk1)), complement(converse(meet(complement(sk1), composition(sk1, complement(one)))))))
0.92/1.13	= { by lemma 59 }
0.92/1.13	  complement(top)
0.92/1.13	= { by lemma 17 }
0.98/1.14	  zero
0.98/1.14	
0.98/1.14	Goal 1 (goals_18): one = join(composition(converse(sk1), sk1), one).
0.98/1.14	Proof:
0.98/1.14	  one
0.98/1.14	= { by lemma 34 }
0.98/1.14	  join(one, zero)
0.98/1.14	= { by lemma 46 }
0.98/1.14	  join(one, meet(zero, complement(one)))
0.98/1.14	= { by lemma 55 }
0.98/1.14	  join(one, meet(composition(zero, sk1), complement(one)))
0.98/1.14	= { by lemma 60 }
0.98/1.14	  join(one, meet(composition(meet(converse(sk1), composition(complement(one), converse(sk1))), sk1), complement(one)))
0.98/1.14	= { by axiom 14 (modular_law_2_16) }
0.98/1.14	  join(one, join(meet(composition(converse(sk1), sk1), complement(one)), meet(composition(meet(converse(sk1), composition(complement(one), converse(sk1))), sk1), complement(one))))
0.98/1.14	= { by lemma 60 }
0.98/1.14	  join(one, join(meet(composition(converse(sk1), sk1), complement(one)), meet(composition(zero, sk1), complement(one))))
0.98/1.14	= { by lemma 55 }
0.98/1.14	  join(one, join(meet(composition(converse(sk1), sk1), complement(one)), meet(zero, complement(one))))
0.98/1.14	= { by lemma 46 }
0.98/1.14	  join(one, join(meet(composition(converse(sk1), sk1), complement(one)), zero))
0.98/1.14	= { by lemma 34 }
0.98/1.14	  join(one, meet(composition(converse(sk1), sk1), complement(one)))
0.98/1.14	= { by lemma 41 }
0.98/1.14	  join(one, complement(join(one, complement(composition(converse(sk1), sk1)))))
0.98/1.14	= { by axiom 1 (maddux1_join_commutativity_1) }
0.98/1.14	  join(complement(join(one, complement(composition(converse(sk1), sk1)))), one)
0.98/1.14	= { by lemma 44 }
0.98/1.14	  join(complement(join(one, complement(complement(complement(composition(converse(sk1), sk1)))))), one)
0.98/1.14	= { by axiom 5 (converse_idempotence_8) }
0.98/1.14	  join(complement(join(one, complement(converse(converse(complement(complement(composition(converse(sk1), sk1)))))))), one)
0.98/1.14	= { by lemma 28 }
0.98/1.14	  join(complement(join(one, complement(join(meet(converse(converse(complement(complement(composition(converse(sk1), sk1))))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one)))), complement(join(complement(converse(converse(complement(complement(composition(converse(sk1), sk1)))))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one))))))))), one)
0.98/1.14	= { by lemma 43 }
0.98/1.14	  join(complement(join(one, complement(join(meet(converse(converse(complement(complement(composition(converse(sk1), sk1))))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one)))), complement(top))))), one)
0.98/1.14	= { by lemma 17 }
0.98/1.14	  join(complement(join(one, complement(join(meet(converse(converse(complement(complement(composition(converse(sk1), sk1))))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one)))), zero)))), one)
0.98/1.14	= { by lemma 34 }
0.98/1.14	  join(complement(join(one, complement(meet(converse(converse(complement(complement(composition(converse(sk1), sk1))))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one))))))), one)
0.98/1.14	= { by axiom 5 (converse_idempotence_8) }
0.98/1.14	  join(complement(join(one, complement(meet(complement(complement(composition(converse(sk1), sk1))), converse(join(converse(complement(complement(composition(converse(sk1), sk1)))), converse(one))))))), one)
0.98/1.14	= { by axiom 11 (converse_additivity_9) }
0.98/1.14	  join(complement(join(one, complement(meet(complement(complement(composition(converse(sk1), sk1))), converse(converse(join(complement(complement(composition(converse(sk1), sk1))), one))))))), one)
0.98/1.14	= { by axiom 5 (converse_idempotence_8) }
0.98/1.14	  join(complement(join(one, complement(meet(complement(complement(composition(converse(sk1), sk1))), join(complement(complement(composition(converse(sk1), sk1))), one))))), one)
0.98/1.14	= { by axiom 1 (maddux1_join_commutativity_1) }
0.98/1.14	  join(complement(join(one, complement(meet(complement(complement(composition(converse(sk1), sk1))), join(one, complement(complement(composition(converse(sk1), sk1)))))))), one)
0.98/1.14	= { by lemma 16 }
0.98/1.14	  join(complement(join(one, complement(meet(join(one, complement(complement(composition(converse(sk1), sk1)))), complement(complement(composition(converse(sk1), sk1))))))), one)
0.98/1.14	= { by lemma 42 }
0.98/1.14	  join(complement(join(one, join(complement(composition(converse(sk1), sk1)), complement(join(one, complement(complement(composition(converse(sk1), sk1)))))))), one)
0.98/1.14	= { by lemma 41 }
0.98/1.14	  join(complement(join(one, join(complement(composition(converse(sk1), sk1)), meet(complement(composition(converse(sk1), sk1)), complement(one))))), one)
0.98/1.14	= { by axiom 6 (maddux2_join_associativity_2) }
0.98/1.14	  join(complement(join(join(one, complement(composition(converse(sk1), sk1))), meet(complement(composition(converse(sk1), sk1)), complement(one)))), one)
0.98/1.14	= { by axiom 1 (maddux1_join_commutativity_1) }
0.98/1.14	  join(complement(join(meet(complement(composition(converse(sk1), sk1)), complement(one)), join(one, complement(composition(converse(sk1), sk1))))), one)
0.98/1.14	= { by lemma 38 }
0.98/1.14	  join(complement(join(meet(complement(composition(converse(sk1), sk1)), complement(one)), meet(join(one, complement(composition(converse(sk1), sk1))), top))), one)
0.98/1.14	= { by lemma 18 }
0.98/1.14	  join(complement(join(meet(complement(composition(converse(sk1), sk1)), complement(one)), complement(join(zero, complement(join(one, complement(composition(converse(sk1), sk1)))))))), one)
0.98/1.14	= { by lemma 41 }
0.98/1.14	  join(meet(join(zero, complement(join(one, complement(composition(converse(sk1), sk1))))), complement(meet(complement(composition(converse(sk1), sk1)), complement(one)))), one)
0.98/1.14	= { by lemma 35 }
0.98/1.14	  join(meet(complement(join(one, complement(composition(converse(sk1), sk1)))), complement(meet(complement(composition(converse(sk1), sk1)), complement(one)))), one)
0.98/1.14	= { by lemma 44 }
0.98/1.14	  join(meet(complement(join(one, complement(composition(converse(sk1), sk1)))), complement(meet(complement(composition(converse(sk1), sk1)), complement(one)))), complement(complement(one)))
0.98/1.14	= { by lemma 45 }
0.98/1.14	  join(meet(complement(join(one, complement(composition(converse(sk1), sk1)))), complement(meet(complement(composition(converse(sk1), sk1)), complement(one)))), complement(join(meet(complement(composition(converse(sk1), sk1)), complement(one)), complement(join(one, complement(composition(converse(sk1), sk1)))))))
0.98/1.14	= { by lemma 45 }
0.98/1.14	  complement(meet(complement(composition(converse(sk1), sk1)), complement(one)))
0.98/1.14	= { by lemma 42 }
0.98/1.14	  join(one, complement(complement(composition(converse(sk1), sk1))))
0.98/1.14	= { by lemma 44 }
0.98/1.14	  join(one, composition(converse(sk1), sk1))
0.98/1.14	= { by axiom 1 (maddux1_join_commutativity_1) }
0.98/1.14	  join(composition(converse(sk1), sk1), one)
0.98/1.14	% SZS output end Proof
0.98/1.14	
0.98/1.14	RESULT: Unsatisfiable (the axioms are contradictory).
0.98/1.15	EOF