0.07/0.12	% 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   : n010.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 13:51:06 EDT 2019
0.12/0.33	% CPUTime    : 
21.30/21.52	% SZS status Unsatisfiable
21.30/21.52	
21.30/21.52	% SZS output start Proof
21.30/21.52	Take the following subset of the input axioms:
21.37/21.54	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
21.37/21.54	  fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
21.37/21.54	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
21.37/21.54	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
21.37/21.54	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
21.37/21.54	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
21.37/21.54	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
21.37/21.54	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
21.37/21.54	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
21.37/21.54	  fof(goals_14, negated_conjecture, complement(sk3)=join(composition(complement(sk1), sk2), complement(sk3))).
21.37/21.54	  fof(goals_15, negated_conjecture, sk1!=join(composition(sk3, converse(sk2)), sk1)).
21.37/21.54	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
21.37/21.54	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
21.37/21.54	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
21.37/21.54	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
21.37/21.54	
21.37/21.54	Now clausify the problem and encode Horn clauses using encoding 3 of
21.37/21.54	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
21.37/21.54	We repeatedly replace C & s=t => u=v by the two clauses:
21.37/21.54	  fresh(y, y, x1...xn) = u
21.37/21.54	  C => fresh(s, t, x1...xn) = v
21.37/21.54	where fresh is a fresh function symbol and x1..xn are the free
21.37/21.54	variables of u and v.
21.37/21.54	A predicate p(X) is encoded as p(X)=true (this is sound, because the
21.37/21.54	input problem has no model of domain size 1).
21.37/21.54	
21.37/21.54	The encoding turns the above axioms into the following unit equations and goals:
21.37/21.54	
21.37/21.54	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
21.37/21.54	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
21.37/21.54	Axiom 3 (composition_identity_6): X = composition(X, one).
21.37/21.54	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
21.37/21.54	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
21.37/21.55	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
21.37/21.55	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
21.37/21.55	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
21.37/21.55	Axiom 9 (def_top_12): join(X, complement(X)) = top.
21.37/21.55	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
21.37/21.55	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
21.37/21.55	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
21.37/21.55	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
21.37/21.55	Axiom 14 (goals_14): complement(sk3) = join(composition(complement(sk1), sk2), complement(sk3)).
21.37/21.55	
21.37/21.55	Lemma 15: meet(X, Y) = meet(Y, X).
21.37/21.55	Proof:
21.37/21.55	  meet(X, Y)
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  complement(join(complement(X), complement(Y)))
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  complement(join(complement(Y), complement(X)))
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  meet(Y, X)
21.37/21.55	
21.37/21.55	Lemma 16: complement(top) = zero.
21.37/21.55	Proof:
21.37/21.55	  complement(top)
21.37/21.55	= { by axiom 9 (def_top_12) }
21.37/21.55	  complement(join(complement(?), complement(complement(?))))
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  meet(?, complement(?))
21.37/21.55	= { by axiom 8 (def_zero_13) }
21.37/21.55	  zero
21.37/21.55	
21.37/21.55	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
21.37/21.55	Proof:
21.37/21.55	  complement(join(zero, complement(X)))
21.37/21.55	= { by lemma 16 }
21.37/21.55	  complement(join(complement(top), complement(X)))
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  meet(top, X)
21.37/21.55	= { by lemma 15 }
21.37/21.55	  meet(X, top)
21.37/21.55	
21.37/21.55	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
21.37/21.55	Proof:
21.37/21.55	  converse(composition(converse(X), Y))
21.37/21.55	= { by axiom 10 (converse_multiplicativity_10) }
21.37/21.55	  composition(converse(Y), converse(converse(X)))
21.37/21.55	= { by axiom 5 (converse_idempotence_8) }
21.37/21.55	  composition(converse(Y), X)
21.37/21.55	
21.37/21.55	Lemma 19: composition(converse(one), X) = X.
21.37/21.55	Proof:
21.37/21.55	  composition(converse(one), X)
21.37/21.55	= { by lemma 18 }
21.37/21.55	  converse(composition(converse(X), one))
21.37/21.55	= { by axiom 3 (composition_identity_6) }
21.37/21.55	  converse(converse(X))
21.37/21.55	= { by axiom 5 (converse_idempotence_8) }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 20: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
21.37/21.55	Proof:
21.37/21.55	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
21.37/21.55	= { by axiom 7 (converse_cancellativity_11) }
21.37/21.55	  complement(Y)
21.37/21.55	
21.37/21.55	Lemma 21: join(complement(X), complement(X)) = complement(X).
21.37/21.55	Proof:
21.37/21.55	  join(complement(X), complement(X))
21.37/21.55	= { by lemma 19 }
21.37/21.55	  join(complement(X), composition(converse(one), complement(X)))
21.37/21.55	= { by lemma 19 }
21.37/21.55	  join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
21.37/21.55	= { by axiom 3 (composition_identity_6) }
21.37/21.55	  join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
21.37/21.55	= { by axiom 12 (composition_associativity_5) }
21.37/21.55	  join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
21.37/21.55	= { by lemma 19 }
21.37/21.55	  join(complement(X), composition(converse(one), complement(composition(one, X))))
21.37/21.55	= { by lemma 20 }
21.37/21.55	  complement(X)
21.37/21.55	
21.37/21.55	Lemma 22: meet(X, X) = complement(complement(X)).
21.37/21.55	Proof:
21.37/21.55	  meet(X, X)
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  complement(join(complement(X), complement(X)))
21.37/21.55	= { by lemma 21 }
21.37/21.55	  complement(complement(X))
21.37/21.55	
21.37/21.55	Lemma 23: join(meet(X, Y), complement(join(complement(X), Y))) = X.
21.37/21.55	Proof:
21.37/21.55	  join(meet(X, Y), complement(join(complement(X), Y)))
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
21.37/21.55	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 24: join(meet(X, Y), meet(X, complement(Y))) = X.
21.37/21.55	Proof:
21.37/21.55	  join(meet(X, Y), meet(X, complement(Y)))
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(meet(X, complement(Y)), meet(X, Y))
21.37/21.55	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.55	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
21.37/21.55	= { by lemma 23 }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 25: join(zero, complement(complement(X))) = X.
21.37/21.55	Proof:
21.37/21.55	  join(zero, complement(complement(X)))
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(complement(complement(X)), zero)
21.37/21.55	= { by lemma 22 }
21.37/21.55	  join(meet(X, X), zero)
21.37/21.55	= { by axiom 8 (def_zero_13) }
21.37/21.55	  join(meet(X, X), meet(X, complement(X)))
21.37/21.55	= { by lemma 24 }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 26: join(X, join(Y, Z)) = join(Z, join(X, Y)).
21.37/21.55	Proof:
21.37/21.55	  join(X, join(Y, Z))
21.37/21.55	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.55	  join(join(X, Y), Z)
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(Z, join(X, Y))
21.37/21.55	
21.37/21.55	Lemma 27: join(X, join(complement(X), Y)) = join(Y, top).
21.37/21.55	Proof:
21.37/21.55	  join(X, join(complement(X), Y))
21.37/21.55	= { by lemma 26 }
21.37/21.55	  join(complement(X), join(Y, X))
21.37/21.55	= { by lemma 26 }
21.37/21.55	  join(Y, join(X, complement(X)))
21.37/21.55	= { by axiom 9 (def_top_12) }
21.37/21.55	  join(Y, top)
21.37/21.55	
21.37/21.55	Lemma 28: join(X, top) = top.
21.37/21.55	Proof:
21.37/21.55	  join(X, top)
21.37/21.55	= { by axiom 9 (def_top_12) }
21.37/21.55	  join(X, join(complement(X), complement(complement(X))))
21.37/21.55	= { by lemma 27 }
21.37/21.55	  join(complement(complement(X)), top)
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(top, complement(complement(X)))
21.37/21.55	= { by axiom 9 (def_top_12) }
21.37/21.55	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
21.37/21.55	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.55	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
21.37/21.55	= { by lemma 21 }
21.37/21.55	  join(complement(X), complement(complement(X)))
21.37/21.55	= { by axiom 9 (def_top_12) }
21.37/21.55	  top
21.37/21.55	
21.37/21.55	Lemma 29: join(zero, meet(X, top)) = X.
21.37/21.55	Proof:
21.37/21.55	  join(zero, meet(X, top))
21.37/21.55	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.55	  join(meet(X, top), zero)
21.37/21.55	= { by lemma 16 }
21.37/21.55	  join(meet(X, top), complement(top))
21.37/21.55	= { by lemma 28 }
21.37/21.55	  join(meet(X, top), complement(join(complement(X), top)))
21.37/21.55	= { by lemma 23 }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 30: join(zero, complement(X)) = complement(X).
21.37/21.55	Proof:
21.37/21.55	  join(zero, complement(X))
21.37/21.55	= { by lemma 25 }
21.37/21.55	  join(zero, complement(join(zero, complement(complement(X)))))
21.37/21.55	= { by lemma 17 }
21.37/21.55	  join(zero, meet(complement(X), top))
21.37/21.55	= { by lemma 29 }
21.37/21.55	  complement(X)
21.37/21.55	
21.37/21.55	Lemma 31: complement(complement(X)) = X.
21.37/21.55	Proof:
21.37/21.55	  complement(complement(X))
21.37/21.55	= { by lemma 30 }
21.37/21.55	  join(zero, complement(complement(X)))
21.37/21.55	= { by lemma 25 }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 32: meet(X, top) = X.
21.37/21.55	Proof:
21.37/21.55	  meet(X, top)
21.37/21.55	= { by lemma 17 }
21.37/21.55	  complement(join(zero, complement(X)))
21.37/21.55	= { by lemma 30 }
21.37/21.55	  join(zero, complement(join(zero, complement(X))))
21.37/21.55	= { by lemma 17 }
21.37/21.55	  join(zero, meet(X, top))
21.37/21.55	= { by lemma 29 }
21.37/21.55	  X
21.37/21.55	
21.37/21.55	Lemma 33: meet(top, X) = X.
21.37/21.55	Proof:
21.37/21.55	  meet(top, X)
21.37/21.55	= { by lemma 15 }
21.37/21.55	  meet(X, top)
21.37/21.55	= { by lemma 32 }
21.37/21.56	  X
21.37/21.56	
21.37/21.56	Lemma 34: complement(join(X, complement(Y))) = meet(Y, complement(X)).
21.37/21.56	Proof:
21.37/21.56	  complement(join(X, complement(Y)))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  complement(join(complement(Y), X))
21.37/21.56	= { by lemma 33 }
21.37/21.56	  complement(join(complement(Y), meet(top, X)))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  complement(join(complement(Y), meet(X, top)))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  complement(join(meet(X, top), complement(Y)))
21.37/21.56	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.56	  complement(join(complement(join(complement(X), complement(top))), complement(Y)))
21.37/21.56	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.56	  meet(join(complement(X), complement(top)), Y)
21.37/21.56	= { by lemma 15 }
21.37/21.56	  meet(Y, join(complement(X), complement(top)))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  meet(Y, join(complement(top), complement(X)))
21.37/21.56	= { by lemma 16 }
21.37/21.56	  meet(Y, join(zero, complement(X)))
21.37/21.56	= { by lemma 30 }
21.37/21.56	  meet(Y, complement(X))
21.37/21.56	
21.37/21.56	Lemma 35: meet(complement(X), complement(Y)) = complement(join(X, Y)).
21.37/21.56	Proof:
21.37/21.56	  meet(complement(X), complement(Y))
21.37/21.56	= { by lemma 30 }
21.37/21.56	  meet(join(zero, complement(X)), complement(Y))
21.37/21.56	= { by lemma 34 }
21.37/21.56	  complement(join(Y, complement(join(zero, complement(X)))))
21.37/21.56	= { by lemma 17 }
21.37/21.56	  complement(join(Y, meet(X, top)))
21.37/21.56	= { by lemma 32 }
21.37/21.56	  complement(join(Y, X))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  complement(join(X, Y))
21.37/21.56	
21.37/21.56	Lemma 36: complement(join(complement(X), Y)) = meet(X, complement(Y)).
21.37/21.56	Proof:
21.37/21.56	  complement(join(complement(X), Y))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  complement(join(Y, complement(X)))
21.37/21.56	= { by lemma 34 }
21.37/21.56	  meet(X, complement(Y))
21.37/21.56	
21.37/21.56	Lemma 37: join(meet(Y, X), meet(X, complement(Y))) = X.
21.37/21.56	Proof:
21.37/21.56	  join(meet(Y, X), meet(X, complement(Y)))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(X, Y), meet(X, complement(Y)))
21.37/21.56	= { by lemma 24 }
21.37/21.56	  X
21.37/21.56	
21.37/21.56	Lemma 38: join(X, zero) = X.
21.37/21.56	Proof:
21.37/21.56	  join(X, zero)
21.37/21.56	= { by lemma 31 }
21.37/21.56	  join(complement(complement(X)), zero)
21.37/21.56	= { by lemma 22 }
21.37/21.56	  join(meet(X, X), zero)
21.37/21.56	= { by axiom 8 (def_zero_13) }
21.37/21.56	  join(meet(X, X), meet(X, complement(X)))
21.37/21.56	= { by lemma 37 }
21.37/21.56	  X
21.37/21.56	
21.37/21.56	Lemma 39: meet(meet(X, Z), Y) = meet(X, meet(Y, Z)).
21.37/21.56	Proof:
21.37/21.56	  meet(meet(X, Z), Y)
21.37/21.56	= { by lemma 15 }
21.37/21.56	  meet(Y, meet(X, Z))
21.37/21.56	= { by lemma 32 }
21.37/21.56	  meet(meet(Y, top), meet(X, Z))
21.37/21.56	= { by lemma 17 }
21.37/21.56	  meet(complement(join(zero, complement(Y))), meet(X, Z))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  meet(meet(X, Z), complement(join(zero, complement(Y))))
21.37/21.56	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.56	  meet(complement(join(complement(X), complement(Z))), complement(join(zero, complement(Y))))
21.37/21.56	= { by lemma 35 }
21.37/21.56	  complement(join(join(complement(X), complement(Z)), join(zero, complement(Y))))
21.37/21.56	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.56	  complement(join(complement(X), join(complement(Z), join(zero, complement(Y)))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  meet(X, complement(join(complement(Z), join(zero, complement(Y)))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  meet(X, meet(Z, complement(join(zero, complement(Y)))))
21.37/21.56	= { by lemma 17 }
21.37/21.56	  meet(X, meet(Z, meet(Y, top)))
21.37/21.56	= { by lemma 32 }
21.37/21.56	  meet(X, meet(Z, Y))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  meet(X, meet(Y, Z))
21.37/21.56	
21.37/21.56	Lemma 40: meet(X, join(X, Y)) = X.
21.37/21.56	Proof:
21.37/21.56	  meet(X, join(X, Y))
21.37/21.56	= { by axiom 5 (converse_idempotence_8) }
21.37/21.56	  meet(converse(converse(X)), join(X, Y))
21.37/21.56	= { by axiom 5 (converse_idempotence_8) }
21.37/21.56	  meet(converse(converse(X)), converse(converse(join(X, Y))))
21.37/21.56	= { by axiom 11 (converse_additivity_9) }
21.37/21.56	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
21.37/21.56	= { by lemma 38 }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), zero)
21.37/21.56	= { by lemma 16 }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(top))
21.37/21.56	= { by lemma 28 }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), top)))
21.37/21.56	= { by axiom 9 (def_top_12) }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), join(converse(converse(X)), complement(converse(converse(X)))))))
21.37/21.56	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(join(converse(converse(Y)), converse(converse(X))), complement(converse(converse(X))))))
21.37/21.56	= { by axiom 11 (converse_additivity_9) }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(join(converse(Y), converse(X))), complement(converse(converse(X))))))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(Y), converse(X))))))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(X), converse(Y))))))
21.37/21.56	= { by lemma 23 }
21.37/21.56	  converse(converse(X))
21.37/21.56	= { by axiom 5 (converse_idempotence_8) }
21.37/21.56	  X
21.37/21.56	
21.37/21.56	Lemma 41: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
21.37/21.56	Proof:
21.37/21.56	  meet(complement(X), join(X, Y))
21.37/21.56	= { by lemma 31 }
21.37/21.56	  meet(complement(X), join(X, complement(complement(Y))))
21.37/21.56	= { by lemma 24 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(X), join(X, complement(complement(Y)))), complement(Y)))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), complement(join(complement(join(X, complement(complement(Y)))), X))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(complement(complement(Y)), join(complement(join(X, complement(complement(Y)))), X))))
21.37/21.56	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(join(complement(complement(Y)), complement(join(X, complement(complement(Y))))), X)))
21.37/21.56	= { by lemma 35 }
21.37/21.56	  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)))
21.37/21.56	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(Y), join(X, complement(complement(Y)))), complement(X)))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(complement(Y), join(X, complement(complement(Y))))))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(join(X, complement(complement(Y))), complement(Y))))
21.37/21.56	= { by lemma 15 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(join(X, complement(complement(Y))), complement(Y)), complement(X)))
21.37/21.56	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(join(complement(join(X, complement(complement(Y)))), complement(complement(Y)))), complement(X)))
21.37/21.56	= { by lemma 35 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(join(complement(join(X, complement(complement(Y)))), complement(complement(Y))), X)))
21.37/21.56	= { by axiom 6 (maddux2_join_associativity_2) }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(complement(join(X, complement(complement(Y)))), join(complement(complement(Y)), X))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(complement(complement(Y)), X))))
21.37/21.56	= { by lemma 36 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
21.37/21.56	= { by lemma 34 }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
21.37/21.56	= { by axiom 8 (def_zero_13) }
21.37/21.56	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
21.37/21.56	= { by lemma 38 }
21.37/21.56	  meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
21.37/21.56	= { by lemma 39 }
21.37/21.56	  meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
21.37/21.56	= { by lemma 31 }
21.37/21.56	  meet(complement(X), meet(Y, join(X, Y)))
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  meet(complement(X), meet(Y, join(Y, X)))
21.37/21.56	= { by lemma 40 }
21.37/21.56	  meet(complement(X), Y)
21.37/21.56	= { by lemma 15 }
21.37/21.56	  meet(Y, complement(X))
21.37/21.56	
21.37/21.56	Lemma 42: join(zero, X) = X.
21.37/21.56	Proof:
21.37/21.56	  join(zero, X)
21.37/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.37/21.56	  join(X, zero)
21.37/21.56	= { by lemma 38 }
21.39/21.56	  X
21.39/21.56	
21.39/21.56	Lemma 43: meet(join(X, Y), join(X, complement(Y))) = X.
21.39/21.56	Proof:
21.39/21.56	  meet(join(X, Y), join(X, complement(Y)))
21.39/21.56	= { by lemma 32 }
21.39/21.56	  meet(join(X, Y), meet(join(X, complement(Y)), top))
21.39/21.56	= { by lemma 17 }
21.39/21.56	  meet(join(X, Y), complement(join(zero, complement(join(X, complement(Y))))))
21.39/21.56	= { by lemma 34 }
21.39/21.56	  meet(join(X, Y), complement(join(zero, meet(Y, complement(X)))))
21.39/21.56	= { by lemma 42 }
21.39/21.56	  meet(join(X, Y), complement(meet(Y, complement(X))))
21.39/21.56	= { by lemma 36 }
21.39/21.56	  complement(join(complement(join(X, Y)), meet(Y, complement(X))))
21.39/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.56	  complement(join(complement(join(Y, X)), meet(Y, complement(X))))
21.39/21.56	= { by lemma 15 }
21.39/21.56	  complement(join(complement(join(Y, X)), meet(complement(X), Y)))
21.39/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.56	  complement(join(meet(complement(X), Y), complement(join(Y, X))))
21.39/21.56	= { by lemma 30 }
21.39/21.56	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, X))))
21.39/21.56	= { by lemma 32 }
21.39/21.56	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top)))))
21.39/21.56	= { by lemma 17 }
21.39/21.56	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X)))))))
21.39/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.56	  complement(join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y))))
21.39/21.56	= { by lemma 23 }
21.39/21.56	  complement(join(zero, complement(X)))
21.39/21.56	= { by lemma 30 }
21.39/21.56	  complement(complement(X))
21.39/21.56	= { by lemma 31 }
21.39/21.56	  X
21.39/21.56	
21.39/21.56	Lemma 44: converse(join(converse(X), Y)) = join(X, converse(Y)).
21.39/21.56	Proof:
21.39/21.56	  converse(join(converse(X), Y))
21.39/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.56	  converse(join(Y, converse(X)))
21.39/21.56	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.56	  converse(join(converse(X), Y))
21.39/21.56	= { by axiom 11 (converse_additivity_9) }
21.39/21.56	  join(converse(converse(X)), converse(Y))
21.39/21.56	= { by axiom 5 (converse_idempotence_8) }
21.39/21.56	  join(X, converse(Y))
21.39/21.56	
21.39/21.56	Lemma 45: join(X, converse(top)) = converse(top).
21.39/21.56	Proof:
21.39/21.56	  join(X, converse(top))
21.39/21.56	= { by lemma 44 }
21.39/21.56	  converse(join(converse(X), top))
21.39/21.56	= { by lemma 28 }
21.39/21.56	  converse(top)
21.39/21.56	
21.39/21.56	Lemma 46: join(X, converse(complement(converse(X)))) = top.
21.39/21.56	Proof:
21.39/21.56	  join(X, converse(complement(converse(X))))
21.39/21.56	= { by lemma 44 }
21.39/21.56	  converse(join(converse(X), complement(converse(X))))
21.39/21.56	= { by axiom 9 (def_top_12) }
21.39/21.56	  converse(top)
21.39/21.56	= { by lemma 45 }
21.39/21.56	  join(?, converse(top))
21.39/21.56	= { by lemma 45 }
21.39/21.56	  join(?, join(complement(?), converse(top)))
21.39/21.56	= { by lemma 27 }
21.39/21.56	  join(converse(top), top)
21.39/21.56	= { by lemma 28 }
21.39/21.57	  top
21.39/21.57	
21.39/21.57	Lemma 47: converse(complement(converse(X))) = complement(X).
21.39/21.57	Proof:
21.39/21.57	  converse(complement(converse(X)))
21.39/21.57	= { by lemma 32 }
21.39/21.57	  converse(complement(converse(meet(X, top))))
21.39/21.57	= { by lemma 17 }
21.39/21.57	  converse(complement(converse(complement(join(zero, complement(X))))))
21.39/21.57	= { by lemma 43 }
21.39/21.57	  meet(join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.57	  meet(join(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by axiom 5 (converse_idempotence_8) }
21.39/21.57	  meet(join(join(zero, complement(X)), converse(complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by lemma 44 }
21.39/21.57	  meet(converse(join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by lemma 33 }
21.39/21.57	  meet(converse(meet(top, join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X)))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by lemma 46 }
21.39/21.57	  meet(converse(meet(join(converse(join(zero, complement(X))), converse(complement(converse(converse(join(zero, complement(X))))))), join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X)))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by lemma 43 }
21.39/21.57	  meet(converse(converse(join(zero, complement(X)))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by axiom 5 (converse_idempotence_8) }
21.39/21.57	  meet(join(zero, complement(X)), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
21.39/21.57	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.57	  meet(join(zero, complement(X)), join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))))
21.39/21.57	= { by lemma 42 }
21.39/21.57	  meet(join(zero, complement(X)), join(zero, join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
21.39/21.57	= { by axiom 6 (maddux2_join_associativity_2) }
21.39/21.57	  meet(join(zero, complement(X)), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))
21.39/21.57	= { by lemma 32 }
21.39/21.57	  meet(join(zero, complement(X)), meet(join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), top))
21.39/21.57	= { by lemma 39 }
21.39/21.57	  meet(meet(join(zero, complement(X)), top), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))
21.39/21.57	= { by lemma 17 }
21.39/21.57	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))
21.39/21.57	= { by lemma 41 }
21.39/21.57	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(join(zero, complement(X))))))
21.39/21.57	= { by lemma 17 }
21.39/21.57	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), meet(join(zero, complement(X)), top))
21.39/21.57	= { by lemma 32 }
21.39/21.57	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X)))
21.39/21.57	= { by lemma 15 }
21.39/21.57	  meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X)))))))
21.39/21.57	= { by axiom 2 (maddux4_definiton_of_meet_4) }
21.39/21.57	  complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))
21.39/21.57	= { by lemma 30 }
21.39/21.57	  join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
21.39/21.57	= { by lemma 16 }
21.39/21.57	  join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
21.39/21.57	= { by lemma 46 }
21.39/21.57	  join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
21.39/21.57	= { by lemma 36 }
21.39/21.57	  join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
21.39/21.57	= { by lemma 23 }
21.39/21.57	  join(zero, complement(X))
21.39/21.57	= { by lemma 30 }
21.39/21.60	  complement(X)
21.39/21.60	
21.39/21.60	Goal 1 (goals_15): sk1 = join(composition(sk3, converse(sk2)), sk1).
21.39/21.60	Proof:
21.39/21.60	  sk1
21.39/21.60	= { by lemma 38 }
21.39/21.60	  join(sk1, zero)
21.39/21.60	= { by lemma 16 }
21.39/21.60	  join(sk1, complement(top))
21.39/21.60	= { by lemma 28 }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), top)))
21.39/21.60	= { by lemma 28 }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(composition(complement(join(complement(complement(composition(complement(sk1), sk2))), sk3)), converse(sk2)), top))))
21.39/21.60	= { by axiom 9 (def_top_12) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(composition(complement(join(complement(complement(composition(complement(sk1), sk2))), sk3)), converse(sk2)), join(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2)), complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2))))))))
21.39/21.60	= { by axiom 6 (maddux2_join_associativity_2) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(join(composition(complement(join(complement(complement(composition(complement(sk1), sk2))), sk3)), converse(sk2)), composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2))), complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2)))))))
21.39/21.60	= { by axiom 13 (composition_distributivity_7) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(composition(join(complement(join(complement(complement(composition(complement(sk1), sk2))), sk3)), meet(complement(composition(complement(sk1), sk2)), sk3)), converse(sk2)), complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2)))))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2))), composition(join(complement(join(complement(complement(composition(complement(sk1), sk2))), sk3)), meet(complement(composition(complement(sk1), sk2)), sk3)), converse(sk2))))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2))), composition(join(meet(complement(composition(complement(sk1), sk2)), sk3), complement(join(complement(complement(composition(complement(sk1), sk2))), sk3))), converse(sk2))))))
21.39/21.60	= { by lemma 23 }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2))), composition(complement(composition(complement(sk1), sk2)), converse(sk2))))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(composition(complement(composition(complement(sk1), sk2)), converse(sk2)), complement(composition(meet(complement(composition(complement(sk1), sk2)), sk3), converse(sk2)))))))
21.39/21.60	= { by lemma 15 }
21.39/21.60	  join(sk1, complement(join(complement(complement(sk1)), join(composition(complement(composition(complement(sk1), sk2)), converse(sk2)), complement(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)))))))
21.39/21.60	= { by axiom 6 (maddux2_join_associativity_2) }
21.39/21.60	  join(sk1, complement(join(join(complement(complement(sk1)), composition(complement(composition(complement(sk1), sk2)), converse(sk2))), complement(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2))))))
21.39/21.60	= { by lemma 34 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(complement(complement(sk1)), composition(complement(composition(complement(sk1), sk2)), converse(sk2))))))
21.39/21.60	= { by lemma 47 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(converse(complement(converse(complement(sk1)))), composition(complement(composition(complement(sk1), sk2)), converse(sk2))))))
21.39/21.60	= { by axiom 5 (converse_idempotence_8) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(converse(complement(converse(complement(sk1)))), composition(complement(composition(complement(sk1), converse(converse(sk2)))), converse(sk2))))))
21.39/21.60	= { by lemma 47 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(converse(complement(converse(complement(sk1)))), composition(converse(complement(converse(composition(complement(sk1), converse(converse(sk2)))))), converse(sk2))))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(composition(converse(complement(converse(composition(complement(sk1), converse(converse(sk2)))))), converse(sk2)), converse(complement(converse(complement(sk1))))))))
21.39/21.60	= { by lemma 18 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(join(converse(composition(converse(converse(sk2)), complement(converse(composition(complement(sk1), converse(converse(sk2))))))), converse(complement(converse(complement(sk1))))))))
21.39/21.60	= { by axiom 11 (converse_additivity_9) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(converse(join(composition(converse(converse(sk2)), complement(converse(composition(complement(sk1), converse(converse(sk2)))))), complement(converse(complement(sk1))))))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(converse(join(complement(converse(complement(sk1))), composition(converse(converse(sk2)), complement(converse(composition(complement(sk1), converse(converse(sk2)))))))))))
21.39/21.60	= { by axiom 5 (converse_idempotence_8) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(converse(join(complement(converse(complement(sk1))), composition(converse(converse(converse(converse(sk2)))), complement(converse(composition(complement(sk1), converse(converse(sk2)))))))))))
21.39/21.60	= { by axiom 10 (converse_multiplicativity_10) }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(converse(join(complement(converse(complement(sk1))), composition(converse(converse(converse(converse(sk2)))), complement(composition(converse(converse(converse(sk2))), converse(complement(sk1))))))))))
21.39/21.60	= { by lemma 20 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(converse(complement(converse(complement(sk1)))))))
21.39/21.60	= { by lemma 47 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(complement(complement(sk1)))))
21.39/21.60	= { by lemma 31 }
21.39/21.60	  join(sk1, meet(composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2)), complement(sk1)))
21.39/21.60	= { by lemma 15 }
21.39/21.60	  join(sk1, meet(complement(sk1), composition(meet(sk3, complement(composition(complement(sk1), sk2))), converse(sk2))))
21.39/21.60	= { by lemma 36 }
21.39/21.60	  join(sk1, meet(complement(sk1), composition(complement(join(complement(sk3), composition(complement(sk1), sk2))), converse(sk2))))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(sk1, meet(complement(sk1), composition(complement(join(composition(complement(sk1), sk2), complement(sk3))), converse(sk2))))
21.39/21.60	= { by axiom 14 (goals_14) }
21.39/21.60	  join(sk1, meet(complement(sk1), composition(complement(complement(sk3)), converse(sk2))))
21.39/21.60	= { by lemma 31 }
21.39/21.60	  join(sk1, meet(complement(sk1), composition(sk3, converse(sk2))))
21.39/21.60	= { by lemma 15 }
21.39/21.60	  join(sk1, meet(composition(sk3, converse(sk2)), complement(sk1)))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(meet(composition(sk3, converse(sk2)), complement(sk1)), sk1)
21.39/21.60	= { by lemma 41 }
21.39/21.60	  join(meet(complement(sk1), join(sk1, composition(sk3, converse(sk2)))), sk1)
21.39/21.60	= { by lemma 40 }
21.39/21.60	  join(meet(complement(sk1), join(sk1, composition(sk3, converse(sk2)))), meet(sk1, join(sk1, composition(sk3, converse(sk2)))))
21.39/21.60	= { by lemma 31 }
21.39/21.60	  join(meet(complement(sk1), join(sk1, composition(sk3, converse(sk2)))), meet(complement(complement(sk1)), join(sk1, composition(sk3, converse(sk2)))))
21.39/21.60	= { by lemma 15 }
21.39/21.60	  join(meet(complement(sk1), join(sk1, composition(sk3, converse(sk2)))), meet(join(sk1, composition(sk3, converse(sk2))), complement(complement(sk1))))
21.39/21.60	= { by lemma 37 }
21.39/21.60	  join(sk1, composition(sk3, converse(sk2)))
21.39/21.60	= { by axiom 1 (maddux1_join_commutativity_1) }
21.39/21.60	  join(composition(sk3, converse(sk2)), sk1)
21.39/21.60	% SZS output end Proof
21.39/21.60	
21.39/21.60	RESULT: Unsatisfiable (the axioms are contradictory).
21.39/21.61	EOF