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   : n017.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 13:04:19 EDT 2019
0.12/0.33	% CPUTime    : 
1.70/1.90	% SZS status Unsatisfiable
1.70/1.90	
1.70/1.90	% SZS output start Proof
1.70/1.90	Take the following subset of the input axioms:
1.70/1.91	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
1.70/1.91	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
1.70/1.91	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
1.70/1.91	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
1.70/1.91	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
1.70/1.91	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
1.70/1.91	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
1.70/1.91	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
1.70/1.91	  fof(goals_14, negated_conjecture, complement(sk1)!=join(composition(complement(composition(sk1, sk2)), converse(sk2)), complement(sk1))).
1.70/1.91	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
1.70/1.91	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
1.70/1.91	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
1.70/1.91	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
1.70/1.91	
1.70/1.91	Now clausify the problem and encode Horn clauses using encoding 3 of
1.70/1.91	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.70/1.91	We repeatedly replace C & s=t => u=v by the two clauses:
1.70/1.91	  fresh(y, y, x1...xn) = u
1.70/1.91	  C => fresh(s, t, x1...xn) = v
1.70/1.91	where fresh is a fresh function symbol and x1..xn are the free
1.70/1.91	variables of u and v.
1.70/1.91	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.70/1.91	input problem has no model of domain size 1).
1.70/1.91	
1.70/1.91	The encoding turns the above axioms into the following unit equations and goals:
1.70/1.91	
1.70/1.91	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
1.70/1.91	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
1.70/1.91	Axiom 3 (composition_identity_6): X = composition(X, one).
1.70/1.91	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
1.70/1.91	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
1.70/1.91	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
1.70/1.91	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
1.70/1.91	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
1.70/1.91	Axiom 9 (def_top_12): join(X, complement(X)) = top.
1.70/1.91	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
1.70/1.91	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
1.78/1.95	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
1.78/1.95	
1.78/1.95	Lemma 13: complement(top) = zero.
1.78/1.95	Proof:
1.78/1.95	  complement(top)
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  complement(join(complement(?), complement(complement(?))))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  meet(?, complement(?))
1.78/1.95	= { by axiom 8 (def_zero_13) }
1.78/1.95	  zero
1.78/1.95	
1.78/1.95	Lemma 14: meet(X, Y) = meet(Y, X).
1.78/1.95	Proof:
1.78/1.95	  meet(X, Y)
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  complement(join(complement(X), complement(Y)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(complement(Y), complement(X)))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  meet(Y, X)
1.78/1.95	
1.78/1.95	Lemma 15: complement(join(zero, complement(X))) = meet(X, top).
1.78/1.95	Proof:
1.78/1.95	  complement(join(zero, complement(X)))
1.78/1.95	= { by lemma 13 }
1.78/1.95	  complement(join(complement(top), complement(X)))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  meet(top, X)
1.78/1.95	= { by lemma 14 }
1.78/1.95	  meet(X, top)
1.78/1.95	
1.78/1.95	Lemma 16: converse(composition(converse(X), Y)) = composition(converse(Y), X).
1.78/1.95	Proof:
1.78/1.95	  converse(composition(converse(X), Y))
1.78/1.95	= { by axiom 10 (converse_multiplicativity_10) }
1.78/1.95	  composition(converse(Y), converse(converse(X)))
1.78/1.95	= { by axiom 5 (converse_idempotence_8) }
1.78/1.95	  composition(converse(Y), X)
1.78/1.95	
1.78/1.95	Lemma 17: composition(converse(one), X) = X.
1.78/1.95	Proof:
1.78/1.95	  composition(converse(one), X)
1.78/1.95	= { by lemma 16 }
1.78/1.95	  converse(composition(converse(X), one))
1.78/1.95	= { by axiom 3 (composition_identity_6) }
1.78/1.95	  converse(converse(X))
1.78/1.95	= { by axiom 5 (converse_idempotence_8) }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 18: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
1.78/1.95	Proof:
1.78/1.95	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
1.78/1.95	= { by axiom 7 (converse_cancellativity_11) }
1.78/1.95	  complement(Y)
1.78/1.95	
1.78/1.95	Lemma 19: join(complement(X), complement(X)) = complement(X).
1.78/1.95	Proof:
1.78/1.95	  join(complement(X), complement(X))
1.78/1.95	= { by lemma 17 }
1.78/1.95	  join(complement(X), composition(converse(one), complement(X)))
1.78/1.95	= { by lemma 17 }
1.78/1.95	  join(complement(X), composition(converse(one), complement(composition(converse(one), X))))
1.78/1.95	= { by axiom 3 (composition_identity_6) }
1.78/1.95	  join(complement(X), composition(converse(one), complement(composition(composition(converse(one), one), X))))
1.78/1.95	= { by axiom 12 (composition_associativity_5) }
1.78/1.95	  join(complement(X), composition(converse(one), complement(composition(converse(one), composition(one, X)))))
1.78/1.95	= { by lemma 17 }
1.78/1.95	  join(complement(X), composition(converse(one), complement(composition(one, X))))
1.78/1.95	= { by lemma 18 }
1.78/1.95	  complement(X)
1.78/1.95	
1.78/1.95	Lemma 20: meet(X, X) = complement(complement(X)).
1.78/1.95	Proof:
1.78/1.95	  meet(X, X)
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  complement(join(complement(X), complement(X)))
1.78/1.95	= { by lemma 19 }
1.78/1.95	  complement(complement(X))
1.78/1.95	
1.78/1.95	Lemma 21: join(meet(X, Y), complement(join(complement(X), Y))) = X.
1.78/1.95	Proof:
1.78/1.95	  join(meet(X, Y), complement(join(complement(X), Y)))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
1.78/1.95	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 22: join(meet(X, Y), meet(X, complement(Y))) = X.
1.78/1.95	Proof:
1.78/1.95	  join(meet(X, Y), meet(X, complement(Y)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(meet(X, complement(Y)), meet(X, Y))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
1.78/1.95	= { by lemma 21 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 23: join(zero, complement(complement(X))) = X.
1.78/1.95	Proof:
1.78/1.95	  join(zero, complement(complement(X)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(complement(complement(X)), zero)
1.78/1.95	= { by lemma 20 }
1.78/1.95	  join(meet(X, X), zero)
1.78/1.95	= { by axiom 8 (def_zero_13) }
1.78/1.95	  join(meet(X, X), meet(X, complement(X)))
1.78/1.95	= { by lemma 22 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 24: join(X, join(Y, Z)) = join(Z, join(X, Y)).
1.78/1.95	Proof:
1.78/1.95	  join(X, join(Y, Z))
1.78/1.95	= { by axiom 6 (maddux2_join_associativity_2) }
1.78/1.95	  join(join(X, Y), Z)
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(Z, join(X, Y))
1.78/1.95	
1.78/1.95	Lemma 25: join(X, join(complement(X), Y)) = join(Y, top).
1.78/1.95	Proof:
1.78/1.95	  join(X, join(complement(X), Y))
1.78/1.95	= { by lemma 24 }
1.78/1.95	  join(complement(X), join(Y, X))
1.78/1.95	= { by lemma 24 }
1.78/1.95	  join(Y, join(X, complement(X)))
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  join(Y, top)
1.78/1.95	
1.78/1.95	Lemma 26: join(X, top) = top.
1.78/1.95	Proof:
1.78/1.95	  join(X, top)
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  join(X, join(complement(X), complement(complement(X))))
1.78/1.95	= { by lemma 25 }
1.78/1.95	  join(complement(complement(X)), top)
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(top, complement(complement(X)))
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
1.78/1.95	= { by axiom 6 (maddux2_join_associativity_2) }
1.78/1.95	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
1.78/1.95	= { by lemma 19 }
1.78/1.95	  join(complement(X), complement(complement(X)))
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  top
1.78/1.95	
1.78/1.95	Lemma 27: join(zero, meet(X, top)) = X.
1.78/1.95	Proof:
1.78/1.95	  join(zero, meet(X, top))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(meet(X, top), zero)
1.78/1.95	= { by lemma 13 }
1.78/1.95	  join(meet(X, top), complement(top))
1.78/1.95	= { by lemma 26 }
1.78/1.95	  join(meet(X, top), complement(join(complement(X), top)))
1.78/1.95	= { by lemma 21 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 28: join(zero, complement(X)) = complement(X).
1.78/1.95	Proof:
1.78/1.95	  join(zero, complement(X))
1.78/1.95	= { by lemma 23 }
1.78/1.95	  join(zero, complement(join(zero, complement(complement(X)))))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  join(zero, meet(complement(X), top))
1.78/1.95	= { by lemma 27 }
1.78/1.95	  complement(X)
1.78/1.95	
1.78/1.95	Lemma 29: meet(X, top) = X.
1.78/1.95	Proof:
1.78/1.95	  meet(X, top)
1.78/1.95	= { by lemma 15 }
1.78/1.95	  complement(join(zero, complement(X)))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  join(zero, complement(join(zero, complement(X))))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  join(zero, meet(X, top))
1.78/1.95	= { by lemma 27 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 30: complement(complement(X)) = X.
1.78/1.95	Proof:
1.78/1.95	  complement(complement(X))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  join(zero, complement(complement(X)))
1.78/1.95	= { by lemma 23 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 31: join(X, zero) = X.
1.78/1.95	Proof:
1.78/1.95	  join(X, zero)
1.78/1.95	= { by lemma 30 }
1.78/1.95	  join(complement(complement(X)), zero)
1.78/1.95	= { by lemma 20 }
1.78/1.95	  join(meet(X, X), zero)
1.78/1.95	= { by axiom 8 (def_zero_13) }
1.78/1.95	  join(meet(X, X), meet(X, complement(X)))
1.78/1.95	= { by lemma 22 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 32: join(zero, X) = X.
1.78/1.95	Proof:
1.78/1.95	  join(zero, X)
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  join(X, zero)
1.78/1.95	= { by lemma 31 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 33: meet(top, X) = X.
1.78/1.95	Proof:
1.78/1.95	  meet(top, X)
1.78/1.95	= { by lemma 14 }
1.78/1.95	  meet(X, top)
1.78/1.95	= { by lemma 29 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 34: complement(join(X, complement(Y))) = meet(Y, complement(X)).
1.78/1.95	Proof:
1.78/1.95	  complement(join(X, complement(Y)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(complement(Y), X))
1.78/1.95	= { by lemma 33 }
1.78/1.95	  complement(join(complement(Y), meet(top, X)))
1.78/1.95	= { by lemma 14 }
1.78/1.95	  complement(join(complement(Y), meet(X, top)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(meet(X, top), complement(Y)))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  complement(join(complement(join(complement(X), complement(top))), complement(Y)))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  meet(join(complement(X), complement(top)), Y)
1.78/1.95	= { by lemma 14 }
1.78/1.95	  meet(Y, join(complement(X), complement(top)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  meet(Y, join(complement(top), complement(X)))
1.78/1.95	= { by lemma 13 }
1.78/1.95	  meet(Y, join(zero, complement(X)))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  meet(Y, complement(X))
1.78/1.95	
1.78/1.95	Lemma 35: complement(join(complement(X), Y)) = meet(X, complement(Y)).
1.78/1.95	Proof:
1.78/1.95	  complement(join(complement(X), Y))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(Y, complement(X)))
1.78/1.95	= { by lemma 34 }
1.78/1.95	  meet(X, complement(Y))
1.78/1.95	
1.78/1.95	Lemma 36: meet(join(X, Y), join(X, complement(Y))) = X.
1.78/1.95	Proof:
1.78/1.95	  meet(join(X, Y), join(X, complement(Y)))
1.78/1.95	= { by lemma 29 }
1.78/1.95	  meet(join(X, Y), meet(join(X, complement(Y)), top))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  meet(join(X, Y), complement(join(zero, complement(join(X, complement(Y))))))
1.78/1.95	= { by lemma 34 }
1.78/1.95	  meet(join(X, Y), complement(join(zero, meet(Y, complement(X)))))
1.78/1.95	= { by lemma 32 }
1.78/1.95	  meet(join(X, Y), complement(meet(Y, complement(X))))
1.78/1.95	= { by lemma 35 }
1.78/1.95	  complement(join(complement(join(X, Y)), meet(Y, complement(X))))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(complement(join(Y, X)), meet(Y, complement(X))))
1.78/1.95	= { by lemma 14 }
1.78/1.95	  complement(join(complement(join(Y, X)), meet(complement(X), Y)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(meet(complement(X), Y), complement(join(Y, X))))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, X))))
1.78/1.95	= { by lemma 29 }
1.78/1.95	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top)))))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X)))))))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(meet(join(zero, complement(X)), Y), complement(join(complement(join(zero, complement(X))), Y))))
1.78/1.95	= { by lemma 21 }
1.78/1.95	  complement(join(zero, complement(X)))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  complement(complement(X))
1.78/1.95	= { by lemma 30 }
1.78/1.95	  X
1.78/1.95	
1.78/1.95	Lemma 37: converse(join(converse(X), Y)) = join(X, converse(Y)).
1.78/1.95	Proof:
1.78/1.95	  converse(join(converse(X), Y))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  converse(join(Y, converse(X)))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  converse(join(converse(X), Y))
1.78/1.95	= { by axiom 11 (converse_additivity_9) }
1.78/1.95	  join(converse(converse(X)), converse(Y))
1.78/1.95	= { by axiom 5 (converse_idempotence_8) }
1.78/1.95	  join(X, converse(Y))
1.78/1.95	
1.78/1.95	Lemma 38: join(X, converse(top)) = converse(top).
1.78/1.95	Proof:
1.78/1.95	  join(X, converse(top))
1.78/1.95	= { by lemma 37 }
1.78/1.95	  converse(join(converse(X), top))
1.78/1.95	= { by lemma 26 }
1.78/1.95	  converse(top)
1.78/1.95	
1.78/1.95	Lemma 39: join(X, converse(complement(converse(X)))) = top.
1.78/1.95	Proof:
1.78/1.95	  join(X, converse(complement(converse(X))))
1.78/1.95	= { by lemma 37 }
1.78/1.95	  converse(join(converse(X), complement(converse(X))))
1.78/1.95	= { by axiom 9 (def_top_12) }
1.78/1.95	  converse(top)
1.78/1.95	= { by lemma 38 }
1.78/1.95	  join(?, converse(top))
1.78/1.95	= { by lemma 38 }
1.78/1.95	  join(?, join(complement(?), converse(top)))
1.78/1.95	= { by lemma 25 }
1.78/1.95	  join(converse(top), top)
1.78/1.95	= { by lemma 26 }
1.78/1.95	  top
1.78/1.95	
1.78/1.95	Lemma 40: meet(complement(X), complement(Y)) = complement(join(X, Y)).
1.78/1.95	Proof:
1.78/1.95	  meet(complement(X), complement(Y))
1.78/1.95	= { by lemma 28 }
1.78/1.95	  meet(join(zero, complement(X)), complement(Y))
1.78/1.95	= { by lemma 34 }
1.78/1.95	  complement(join(Y, complement(join(zero, complement(X)))))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  complement(join(Y, meet(X, top)))
1.78/1.95	= { by lemma 29 }
1.78/1.95	  complement(join(Y, X))
1.78/1.95	= { by axiom 1 (maddux1_join_commutativity_1) }
1.78/1.95	  complement(join(X, Y))
1.78/1.95	
1.78/1.95	Lemma 41: meet(X, meet(Y, complement(Z))) = meet(complement(Z), meet(X, Y)).
1.78/1.95	Proof:
1.78/1.95	  meet(X, meet(Y, complement(Z)))
1.78/1.95	= { by lemma 35 }
1.78/1.95	  meet(X, complement(join(complement(Y), Z)))
1.78/1.95	= { by lemma 35 }
1.78/1.95	  complement(join(complement(X), join(complement(Y), Z)))
1.78/1.95	= { by axiom 6 (maddux2_join_associativity_2) }
1.78/1.95	  complement(join(join(complement(X), complement(Y)), Z))
1.78/1.95	= { by lemma 40 }
1.78/1.95	  meet(complement(join(complement(X), complement(Y))), complement(Z))
1.78/1.95	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.78/1.95	  meet(meet(X, Y), complement(Z))
1.78/1.95	= { by lemma 14 }
1.78/1.95	  meet(complement(Z), meet(X, Y))
1.78/1.95	
1.78/1.95	Lemma 42: meet(meet(X, Z), Y) = meet(X, meet(Y, Z)).
1.78/1.95	Proof:
1.78/1.95	  meet(meet(X, Z), Y)
1.78/1.95	= { by lemma 14 }
1.78/1.95	  meet(Y, meet(X, Z))
1.78/1.95	= { by lemma 29 }
1.78/1.95	  meet(meet(Y, top), meet(X, Z))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  meet(complement(join(zero, complement(Y))), meet(X, Z))
1.78/1.95	= { by lemma 41 }
1.78/1.95	  meet(X, meet(Z, complement(join(zero, complement(Y)))))
1.78/1.95	= { by lemma 15 }
1.78/1.95	  meet(X, meet(Z, meet(Y, top)))
1.78/1.95	= { by lemma 29 }
1.78/1.95	  meet(X, meet(Z, Y))
1.78/1.95	= { by lemma 14 }
1.82/2.00	  meet(X, meet(Y, Z))
1.82/2.00	
1.82/2.00	Lemma 43: converse(complement(converse(X))) = complement(X).
1.82/2.00	Proof:
1.82/2.00	  converse(complement(converse(X)))
1.82/2.00	= { by lemma 29 }
1.82/2.00	  converse(complement(converse(meet(X, top))))
1.82/2.00	= { by lemma 15 }
1.82/2.00	  converse(complement(converse(complement(join(zero, complement(X))))))
1.82/2.00	= { by lemma 36 }
1.82/2.00	  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)))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  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)))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  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)))))
1.82/2.00	= { by lemma 37 }
1.82/2.00	  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)))))
1.82/2.00	= { by lemma 33 }
1.82/2.00	  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)))))
1.82/2.00	= { by lemma 39 }
1.82/2.00	  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)))))
1.82/2.00	= { by lemma 36 }
1.82/2.00	  meet(converse(converse(join(zero, complement(X)))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  meet(join(zero, complement(X)), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X)))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  meet(join(zero, complement(X)), join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 32 }
1.82/2.00	  meet(join(zero, complement(X)), join(zero, join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
1.82/2.00	= { by axiom 6 (maddux2_join_associativity_2) }
1.82/2.00	  meet(join(zero, complement(X)), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 29 }
1.82/2.00	  meet(join(zero, complement(X)), meet(join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))), top))
1.82/2.00	= { by lemma 42 }
1.82/2.00	  meet(meet(join(zero, complement(X)), top), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 15 }
1.82/2.00	  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))))))))
1.82/2.00	= { by lemma 30 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
1.82/2.00	= { by lemma 22 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))))))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), meet(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))), complement(join(zero, complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 35 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), complement(join(complement(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), join(zero, complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 35 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), complement(join(complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))), join(complement(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), join(zero, complement(join(zero, complement(X))))))))
1.82/2.00	= { by axiom 6 (maddux2_join_associativity_2) }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), complement(join(join(complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))))), join(zero, complement(join(zero, complement(X)))))))
1.82/2.00	= { by lemma 40 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))), complement(join(zero, complement(join(zero, complement(X)))))))
1.82/2.00	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), complement(join(zero, complement(join(zero, complement(X)))))))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(join(zero, complement(X))))), meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))))))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(complement(join(zero, complement(join(zero, complement(X))))), meet(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
1.82/2.00	= { by lemma 41 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))), meet(complement(converse(complement(converse(complement(join(zero, complement(X))))))), complement(join(zero, complement(join(zero, complement(X))))))))
1.82/2.00	= { by lemma 34 }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), meet(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))), complement(join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X))))))))))))
1.82/2.00	= { by axiom 8 (def_zero_13) }
1.82/2.00	  join(meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X))))))), zero)
1.82/2.00	= { by lemma 31 }
1.82/2.00	  meet(meet(complement(join(zero, complement(join(zero, complement(X))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))), converse(complement(converse(complement(join(zero, complement(X)))))))
1.82/2.00	= { by lemma 42 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(complement(converse(complement(join(zero, complement(X)))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), join(join(zero, complement(join(zero, complement(X)))), complement(complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
1.82/2.00	= { by lemma 30 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), join(join(zero, complement(join(zero, complement(X)))), converse(complement(converse(complement(join(zero, complement(X)))))))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(join(zero, complement(X)))))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(converse(join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(join(zero, complement(X)))))))))
1.82/2.00	= { by axiom 11 (converse_additivity_9) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))))
1.82/2.00	= { by lemma 31 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), zero))
1.82/2.00	= { by lemma 13 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(top)))
1.82/2.00	= { by lemma 26 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(converse(converse(join(zero, complement(join(zero, complement(X)))))), top))))
1.82/2.00	= { by axiom 9 (def_top_12) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(converse(converse(join(zero, complement(join(zero, complement(X)))))), join(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(converse(converse(converse(complement(converse(complement(join(zero, complement(X))))))))))))))
1.82/2.00	= { by axiom 6 (maddux2_join_associativity_2) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(join(converse(converse(join(zero, complement(join(zero, complement(X)))))), converse(converse(converse(complement(converse(complement(join(zero, complement(X))))))))), complement(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))))))))
1.82/2.00	= { by axiom 11 (converse_additivity_9) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(converse(join(converse(join(zero, complement(join(zero, complement(X))))), converse(converse(complement(converse(complement(join(zero, complement(X))))))))), complement(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))))))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(complement(converse(converse(converse(complement(converse(complement(join(zero, complement(X))))))))), converse(join(converse(join(zero, complement(join(zero, complement(X))))), converse(converse(complement(converse(complement(join(zero, complement(X)))))))))))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), join(meet(converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))), complement(join(complement(converse(converse(converse(complement(converse(complement(join(zero, complement(X))))))))), converse(join(converse(converse(complement(converse(complement(join(zero, complement(X))))))), converse(join(zero, complement(join(zero, complement(X)))))))))))
1.82/2.00	= { by lemma 21 }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), converse(converse(converse(complement(converse(complement(join(zero, complement(X)))))))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  meet(complement(join(zero, complement(join(zero, complement(X))))), converse(complement(converse(complement(join(zero, complement(X)))))))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(join(zero, complement(X))))))
1.82/2.00	= { by lemma 15 }
1.82/2.00	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), meet(join(zero, complement(X)), top))
1.82/2.00	= { by lemma 29 }
1.82/2.00	  meet(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X)))
1.82/2.00	= { by lemma 14 }
1.82/2.00	  meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X)))))))
1.82/2.00	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.82/2.00	  complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))
1.82/2.00	= { by lemma 28 }
1.82/2.00	  join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
1.82/2.00	= { by lemma 13 }
1.82/2.00	  join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
1.82/2.00	= { by lemma 39 }
1.82/2.00	  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))))))))))
1.82/2.00	= { by lemma 35 }
1.82/2.00	  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))))))))))
1.82/2.00	= { by lemma 21 }
1.82/2.00	  join(zero, complement(X))
1.82/2.00	= { by lemma 28 }
1.82/2.00	  complement(X)
1.82/2.00	
1.82/2.00	Goal 1 (goals_14): complement(sk1) = join(composition(complement(composition(sk1, sk2)), converse(sk2)), complement(sk1)).
1.82/2.00	Proof:
1.82/2.00	  complement(sk1)
1.82/2.00	= { by lemma 43 }
1.82/2.00	  converse(complement(converse(sk1)))
1.82/2.00	= { by lemma 18 }
1.82/2.00	  converse(join(complement(converse(sk1)), composition(converse(converse(converse(converse(sk2)))), complement(composition(converse(converse(converse(sk2))), converse(sk1))))))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  converse(join(complement(converse(sk1)), composition(converse(converse(sk2)), complement(composition(converse(converse(converse(sk2))), converse(sk1))))))
1.82/2.00	= { by axiom 10 (converse_multiplicativity_10) }
1.82/2.00	  converse(join(complement(converse(sk1)), composition(converse(converse(sk2)), complement(converse(composition(sk1, converse(converse(sk2))))))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  converse(join(composition(converse(converse(sk2)), complement(converse(composition(sk1, converse(converse(sk2)))))), complement(converse(sk1))))
1.82/2.00	= { by axiom 11 (converse_additivity_9) }
1.82/2.00	  join(converse(composition(converse(converse(sk2)), complement(converse(composition(sk1, converse(converse(sk2))))))), converse(complement(converse(sk1))))
1.82/2.00	= { by lemma 16 }
1.82/2.00	  join(composition(converse(complement(converse(composition(sk1, converse(converse(sk2)))))), converse(sk2)), converse(complement(converse(sk1))))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  join(converse(complement(converse(sk1))), composition(converse(complement(converse(composition(sk1, converse(converse(sk2)))))), converse(sk2)))
1.82/2.00	= { by lemma 43 }
1.82/2.00	  join(complement(sk1), composition(converse(complement(converse(composition(sk1, converse(converse(sk2)))))), converse(sk2)))
1.82/2.00	= { by lemma 43 }
1.82/2.00	  join(complement(sk1), composition(complement(composition(sk1, converse(converse(sk2)))), converse(sk2)))
1.82/2.00	= { by axiom 5 (converse_idempotence_8) }
1.82/2.00	  join(complement(sk1), composition(complement(composition(sk1, sk2)), converse(sk2)))
1.82/2.00	= { by axiom 1 (maddux1_join_commutativity_1) }
1.82/2.00	  join(composition(complement(composition(sk1, sk2)), converse(sk2)), complement(sk1))
1.82/2.00	% SZS output end Proof
1.82/2.00	
1.82/2.00	RESULT: Unsatisfiable (the axioms are contradictory).
1.82/2.01	EOF