0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.34	% Computer   : n005.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 180
0.14/0.34	% DateTime   : Thu Aug 29 12:54:33 EDT 2019
0.14/0.34	% CPUTime    : 
4.60/4.76	% SZS status Unsatisfiable
4.60/4.76	
4.60/4.76	% SZS output start Proof
4.60/4.76	Take the following subset of the input axioms:
4.95/5.10	  fof(composition_associativity_5, axiom, ![A, C, B]: composition(composition(A, B), C)=composition(A, composition(B, C))).
4.95/5.10	  fof(composition_distributivity_7, axiom, ![A, C, B]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
4.95/5.10	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
4.95/5.10	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
4.95/5.10	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
4.95/5.10	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
4.95/5.10	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
4.95/5.10	  fof(dedekind_law_14, axiom, ![A, C, B]: composition(meet(A, composition(C, converse(B))), meet(B, composition(converse(A), C)))=join(meet(composition(A, B), C), composition(meet(A, composition(C, converse(B))), meet(B, composition(converse(A), C))))).
4.95/5.10	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
4.95/5.10	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
4.95/5.10	  fof(goals_17, negated_conjecture, one=join(sk1, one)).
4.95/5.10	  fof(goals_18, negated_conjecture, meet(complement(composition(sk1, top)), one)!=meet(complement(sk1), one)).
4.95/5.10	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
4.95/5.10	  fof(maddux2_join_associativity_2, axiom, ![A, C, B]: join(join(A, B), C)=join(A, join(B, C))).
4.95/5.10	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
4.95/5.10	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
4.95/5.10	
4.95/5.10	Now clausify the problem and encode Horn clauses using encoding 3 of
4.95/5.10	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
4.95/5.10	We repeatedly replace C & s=t => u=v by the two clauses:
4.95/5.10	  fresh(y, y, x1...xn) = u
4.95/5.10	  C => fresh(s, t, x1...xn) = v
4.95/5.10	where fresh is a fresh function symbol and x1..xn are the free
4.95/5.10	variables of u and v.
4.95/5.10	A predicate p(X) is encoded as p(X)=true (this is sound, because the
4.95/5.10	input problem has no model of domain size 1).
4.95/5.10	
4.95/5.10	The encoding turns the above axioms into the following unit equations and goals:
4.95/5.10	
4.95/5.10	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
4.95/5.10	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
4.95/5.10	Axiom 3 (composition_identity_6): X = composition(X, one).
4.95/5.10	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
4.95/5.11	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
4.95/5.11	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
4.95/5.11	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
4.95/5.11	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
4.95/5.11	Axiom 9 (def_top_12): join(X, complement(X)) = top.
4.95/5.11	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
4.95/5.11	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
4.95/5.11	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
4.95/5.11	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
4.95/5.11	Axiom 14 (dedekind_law_14): composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y))) = join(meet(composition(X, Z), Y), composition(meet(X, composition(Y, converse(Z))), meet(Z, composition(converse(X), Y)))).
5.12/5.28	Axiom 15 (goals_17): one = join(sk1, one).
5.12/5.28	
5.12/5.28	Lemma 16: meet(X, Y) = meet(Y, X).
5.12/5.28	Proof:
5.12/5.28	  meet(X, Y)
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  complement(join(complement(X), complement(Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  complement(join(complement(Y), complement(X)))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  meet(Y, X)
5.12/5.28	
5.12/5.28	Lemma 17: complement(top) = zero.
5.12/5.28	Proof:
5.12/5.28	  complement(top)
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  complement(join(complement(?), complement(complement(?))))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  meet(?, complement(?))
5.12/5.28	= { by axiom 8 (def_zero_13) }
5.12/5.28	  zero
5.12/5.28	
5.12/5.28	Lemma 18: complement(join(zero, complement(X))) = meet(X, top).
5.12/5.28	Proof:
5.12/5.28	  complement(join(zero, complement(X)))
5.12/5.28	= { by lemma 17 }
5.12/5.28	  complement(join(complement(top), complement(X)))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  meet(top, X)
5.12/5.28	= { by lemma 16 }
5.12/5.28	  meet(X, top)
5.12/5.28	
5.12/5.28	Lemma 19: composition(converse(one), X) = X.
5.12/5.28	Proof:
5.12/5.28	  composition(converse(one), X)
5.12/5.28	= { by axiom 5 (converse_idempotence_8) }
5.12/5.28	  composition(converse(one), converse(converse(X)))
5.12/5.28	= { by axiom 10 (converse_multiplicativity_10) }
5.12/5.28	  converse(composition(converse(X), one))
5.12/5.28	= { by axiom 3 (composition_identity_6) }
5.12/5.28	  converse(converse(X))
5.12/5.28	= { by axiom 5 (converse_idempotence_8) }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 20: composition(one, X) = X.
5.12/5.28	Proof:
5.12/5.28	  composition(one, X)
5.12/5.28	= { by lemma 19 }
5.12/5.28	  composition(converse(one), composition(one, X))
5.12/5.28	= { by axiom 12 (composition_associativity_5) }
5.12/5.28	  composition(composition(converse(one), one), X)
5.12/5.28	= { by axiom 3 (composition_identity_6) }
5.12/5.28	  composition(converse(one), X)
5.12/5.28	= { by lemma 19 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 21: join(complement(X), complement(X)) = complement(X).
5.12/5.28	Proof:
5.12/5.28	  join(complement(X), complement(X))
5.12/5.28	= { by lemma 19 }
5.12/5.28	  join(complement(X), composition(converse(one), complement(X)))
5.12/5.28	= { by lemma 20 }
5.12/5.28	  join(complement(X), composition(converse(one), complement(composition(one, X))))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(composition(converse(one), complement(composition(one, X))), complement(X))
5.12/5.28	= { by axiom 7 (converse_cancellativity_11) }
5.12/5.28	  complement(X)
5.12/5.28	
5.12/5.28	Lemma 22: join(X, join(Y, Z)) = join(Z, join(X, Y)).
5.12/5.28	Proof:
5.12/5.28	  join(X, join(Y, Z))
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(join(X, Y), Z)
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(Z, join(X, Y))
5.12/5.28	
5.12/5.28	Lemma 23: join(X, join(complement(X), Y)) = join(Y, top).
5.12/5.28	Proof:
5.12/5.28	  join(X, join(complement(X), Y))
5.12/5.28	= { by lemma 22 }
5.12/5.28	  join(complement(X), join(Y, X))
5.12/5.28	= { by lemma 22 }
5.12/5.28	  join(Y, join(X, complement(X)))
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  join(Y, top)
5.12/5.28	
5.12/5.28	Lemma 24: join(top, complement(X)) = top.
5.12/5.28	Proof:
5.12/5.28	  join(top, complement(X))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(complement(X), top)
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(top, complement(X))
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  join(join(X, complement(X)), complement(X))
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(X, join(complement(X), complement(X)))
5.12/5.28	= { by lemma 21 }
5.12/5.28	  join(X, complement(X))
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  top
5.12/5.28	
5.12/5.28	Lemma 25: join(X, top) = top.
5.12/5.28	Proof:
5.12/5.28	  join(X, top)
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  join(X, join(complement(X), complement(complement(X))))
5.12/5.28	= { by lemma 23 }
5.12/5.28	  join(complement(complement(X)), top)
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(top, complement(complement(X)))
5.12/5.28	= { by lemma 24 }
5.12/5.28	  top
5.12/5.28	
5.12/5.28	Lemma 26: join(meet(X, Y), complement(join(complement(X), Y))) = X.
5.12/5.28	Proof:
5.12/5.28	  join(meet(X, Y), complement(join(complement(X), Y)))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
5.12/5.28	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 27: join(zero, meet(X, top)) = X.
5.12/5.28	Proof:
5.12/5.28	  join(zero, meet(X, top))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(meet(X, top), zero)
5.12/5.28	= { by lemma 17 }
5.12/5.28	  join(meet(X, top), complement(top))
5.12/5.28	= { by lemma 25 }
5.12/5.28	  join(meet(X, top), complement(join(complement(X), top)))
5.12/5.28	= { by lemma 26 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 28: join(meet(X, Y), meet(X, complement(Y))) = X.
5.12/5.28	Proof:
5.12/5.28	  join(meet(X, Y), meet(X, complement(Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(meet(X, complement(Y)), meet(X, Y))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
5.12/5.28	= { by lemma 26 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 29: join(X, join(Y, Z)) = join(Y, join(X, Z)).
5.12/5.28	Proof:
5.12/5.28	  join(X, join(Y, Z))
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(join(X, Y), Z)
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(join(Y, X), Z)
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(Y, join(X, Z))
5.12/5.28	
5.12/5.28	Lemma 30: join(zero, meet(X, X)) = X.
5.12/5.28	Proof:
5.12/5.28	  join(zero, meet(X, X))
5.12/5.28	= { by axiom 8 (def_zero_13) }
5.12/5.28	  join(meet(X, complement(X)), meet(X, X))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
5.12/5.28	= { by lemma 26 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 31: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
5.12/5.28	Proof:
5.12/5.28	  join(X, meet(Y, Y))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(meet(Y, Y), X)
5.12/5.28	= { by lemma 30 }
5.12/5.28	  join(meet(Y, Y), join(zero, meet(X, X)))
5.12/5.28	= { by lemma 29 }
5.12/5.28	  join(zero, join(meet(Y, Y), meet(X, X)))
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(join(zero, meet(Y, Y)), meet(X, X))
5.12/5.28	= { by lemma 30 }
5.12/5.28	  join(Y, meet(X, X))
5.12/5.28	
5.12/5.28	Lemma 32: join(X, zero) = X.
5.12/5.28	Proof:
5.12/5.28	  join(X, zero)
5.12/5.28	= { by lemma 17 }
5.12/5.28	  join(X, complement(top))
5.12/5.28	= { by lemma 24 }
5.12/5.28	  join(X, complement(join(top, complement(zero))))
5.12/5.28	= { by lemma 28 }
5.12/5.28	  join(X, complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(zero))))
5.12/5.28	= { by axiom 8 (def_zero_13) }
5.12/5.28	  join(X, complement(join(join(zero, meet(top, complement(complement(top)))), complement(zero))))
5.12/5.28	= { by lemma 21 }
5.12/5.28	  join(X, complement(join(join(zero, meet(top, complement(join(complement(top), complement(top))))), complement(zero))))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  join(X, complement(join(join(zero, meet(top, meet(top, top))), complement(zero))))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  join(X, complement(join(join(zero, meet(meet(top, top), top)), complement(zero))))
5.12/5.28	= { by lemma 27 }
5.12/5.28	  join(X, complement(join(meet(top, top), complement(zero))))
5.12/5.28	= { by lemma 18 }
5.12/5.28	  join(X, complement(join(complement(join(zero, complement(top))), complement(zero))))
5.12/5.28	= { by lemma 17 }
5.12/5.28	  join(X, complement(join(complement(join(complement(top), complement(top))), complement(zero))))
5.12/5.28	= { by lemma 21 }
5.12/5.28	  join(X, complement(join(complement(complement(top)), complement(zero))))
5.12/5.28	= { by lemma 17 }
5.12/5.28	  join(X, complement(join(complement(zero), complement(zero))))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  join(X, meet(zero, zero))
5.12/5.28	= { by lemma 31 }
5.12/5.28	  join(zero, meet(X, X))
5.12/5.28	= { by lemma 30 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 33: join(zero, X) = X.
5.12/5.28	Proof:
5.12/5.28	  join(zero, X)
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(X, zero)
5.12/5.28	= { by lemma 32 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 34: meet(X, X) = X.
5.12/5.28	Proof:
5.12/5.28	  meet(X, X)
5.12/5.28	= { by lemma 33 }
5.12/5.28	  join(zero, meet(X, X))
5.12/5.28	= { by lemma 30 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 35: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
5.12/5.28	Proof:
5.12/5.28	  complement(join(complement(X), meet(Y, Z)))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  complement(join(complement(X), meet(Z, Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  complement(join(meet(Z, Y), complement(X)))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
5.12/5.28	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.28	  meet(join(complement(Z), complement(Y)), X)
5.12/5.28	= { by lemma 16 }
5.12/5.28	  meet(X, join(complement(Z), complement(Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  meet(X, join(complement(Y), complement(Z)))
5.12/5.28	
5.12/5.28	Lemma 36: meet(X, top) = X.
5.12/5.28	Proof:
5.12/5.28	  meet(X, top)
5.12/5.28	= { by lemma 33 }
5.12/5.28	  join(zero, meet(X, top))
5.12/5.28	= { by lemma 27 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 37: meet(top, X) = X.
5.12/5.28	Proof:
5.12/5.28	  meet(top, X)
5.12/5.28	= { by lemma 16 }
5.12/5.28	  meet(X, top)
5.12/5.28	= { by lemma 36 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 38: join(complement(X), complement(Y)) = complement(meet(X, Y)).
5.12/5.28	Proof:
5.12/5.28	  join(complement(X), complement(Y))
5.12/5.28	= { by lemma 37 }
5.12/5.28	  meet(top, join(complement(X), complement(Y)))
5.12/5.28	= { by lemma 35 }
5.12/5.28	  complement(join(complement(top), meet(X, Y)))
5.12/5.28	= { by lemma 17 }
5.12/5.28	  complement(join(zero, meet(X, Y)))
5.12/5.28	= { by lemma 33 }
5.12/5.28	  complement(meet(X, Y))
5.12/5.28	
5.12/5.28	Lemma 39: complement(join(Y, complement(X))) = meet(X, complement(Y)).
5.12/5.28	Proof:
5.12/5.28	  complement(join(Y, complement(X)))
5.12/5.28	= { by lemma 34 }
5.12/5.28	  complement(join(Y, meet(complement(X), complement(X))))
5.12/5.28	= { by lemma 31 }
5.12/5.28	  complement(join(complement(X), meet(Y, Y)))
5.12/5.28	= { by lemma 35 }
5.12/5.28	  meet(X, join(complement(Y), complement(Y)))
5.12/5.28	= { by lemma 38 }
5.12/5.28	  meet(X, complement(meet(Y, Y)))
5.12/5.28	= { by lemma 34 }
5.12/5.28	  meet(X, complement(Y))
5.12/5.28	
5.12/5.28	Lemma 40: complement(join(complement(X), Y)) = meet(X, complement(Y)).
5.12/5.28	Proof:
5.12/5.28	  complement(join(complement(X), Y))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  complement(join(Y, complement(X)))
5.12/5.28	= { by lemma 39 }
5.12/5.28	  meet(X, complement(Y))
5.12/5.28	
5.12/5.28	Lemma 41: join(meet(Y, X), meet(X, complement(Y))) = X.
5.12/5.28	Proof:
5.12/5.28	  join(meet(Y, X), meet(X, complement(Y)))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  join(meet(X, Y), meet(X, complement(Y)))
5.12/5.28	= { by lemma 28 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 42: complement(meet(X, complement(Y))) = join(Y, complement(X)).
5.12/5.28	Proof:
5.12/5.28	  complement(meet(X, complement(Y)))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  complement(meet(complement(Y), X))
5.12/5.28	= { by lemma 33 }
5.12/5.28	  complement(meet(join(zero, complement(Y)), X))
5.12/5.28	= { by lemma 38 }
5.12/5.28	  join(complement(join(zero, complement(Y))), complement(X))
5.12/5.28	= { by lemma 18 }
5.12/5.28	  join(meet(Y, top), complement(X))
5.12/5.28	= { by lemma 36 }
5.12/5.28	  join(Y, complement(X))
5.12/5.28	
5.12/5.28	Lemma 43: converse(join(X, converse(Y))) = join(Y, converse(X)).
5.12/5.28	Proof:
5.12/5.28	  converse(join(X, converse(Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  converse(join(converse(Y), X))
5.12/5.28	= { by axiom 11 (converse_additivity_9) }
5.12/5.28	  join(converse(converse(Y)), converse(X))
5.12/5.28	= { by axiom 5 (converse_idempotence_8) }
5.12/5.28	  join(Y, converse(X))
5.12/5.28	
5.12/5.28	Lemma 44: converse(join(converse(X), Y)) = join(X, converse(Y)).
5.12/5.28	Proof:
5.12/5.28	  converse(join(converse(X), Y))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  converse(join(Y, converse(X)))
5.12/5.28	= { by lemma 43 }
5.12/5.28	  join(X, converse(Y))
5.12/5.28	
5.12/5.28	Lemma 45: join(X, converse(top)) = converse(top).
5.12/5.28	Proof:
5.12/5.28	  join(X, converse(top))
5.12/5.28	= { by lemma 44 }
5.12/5.28	  converse(join(converse(X), top))
5.12/5.28	= { by lemma 25 }
5.12/5.28	  converse(top)
5.12/5.28	
5.12/5.28	Lemma 46: converse(top) = top.
5.12/5.28	Proof:
5.12/5.28	  converse(top)
5.12/5.28	= { by lemma 45 }
5.12/5.28	  join(?, converse(top))
5.12/5.28	= { by lemma 45 }
5.12/5.28	  join(?, join(complement(?), converse(top)))
5.12/5.28	= { by lemma 23 }
5.12/5.28	  join(converse(top), top)
5.12/5.28	= { by lemma 25 }
5.12/5.28	  top
5.12/5.28	
5.12/5.28	Lemma 47: join(X, converse(complement(converse(X)))) = top.
5.12/5.28	Proof:
5.12/5.28	  join(X, converse(complement(converse(X))))
5.12/5.28	= { by lemma 44 }
5.12/5.28	  converse(join(converse(X), complement(converse(X))))
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  converse(top)
5.12/5.28	= { by lemma 46 }
5.12/5.28	  top
5.12/5.28	
5.12/5.28	Lemma 48: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
5.12/5.28	Proof:
5.12/5.28	  join(meet(X, Y), complement(join(Y, complement(X))))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(meet(X, Y), complement(join(complement(X), Y)))
5.12/5.28	= { by lemma 26 }
5.12/5.28	  X
5.12/5.28	
5.12/5.28	Lemma 49: join(meet(Y, complement(X)), complement(join(X, Y))) = complement(X).
5.12/5.28	Proof:
5.12/5.28	  join(meet(Y, complement(X)), complement(join(X, Y)))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  join(meet(complement(X), Y), complement(join(X, Y)))
5.12/5.28	= { by lemma 33 }
5.12/5.28	  join(meet(join(zero, complement(X)), Y), complement(join(X, Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(meet(join(zero, complement(X)), Y), complement(join(Y, X)))
5.12/5.28	= { by lemma 36 }
5.12/5.28	  join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top))))
5.12/5.28	= { by lemma 18 }
5.12/5.28	  join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X))))))
5.12/5.28	= { by lemma 48 }
5.12/5.28	  join(zero, complement(X))
5.12/5.28	= { by lemma 33 }
5.12/5.28	  complement(X)
5.12/5.28	
5.12/5.28	Lemma 50: meet(complement(X), converse(complement(converse(X)))) = complement(X).
5.12/5.28	Proof:
5.12/5.28	  meet(complement(X), converse(complement(converse(X))))
5.12/5.28	= { by lemma 16 }
5.12/5.28	  meet(converse(complement(converse(X))), complement(X))
5.12/5.28	= { by lemma 32 }
5.12/5.28	  join(meet(converse(complement(converse(X))), complement(X)), zero)
5.12/5.28	= { by lemma 17 }
5.12/5.28	  join(meet(converse(complement(converse(X))), complement(X)), complement(top))
5.12/5.28	= { by lemma 47 }
5.12/5.28	  join(meet(converse(complement(converse(X))), complement(X)), complement(join(X, converse(complement(converse(X))))))
5.12/5.28	= { by lemma 49 }
5.12/5.28	  complement(X)
5.12/5.28	
5.12/5.28	Lemma 51: join(complement(converse(X)), converse(join(X, Y))) = top.
5.12/5.28	Proof:
5.12/5.28	  join(complement(converse(X)), converse(join(X, Y)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(complement(converse(X)), converse(join(Y, X)))
5.12/5.28	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.28	  join(converse(join(Y, X)), complement(converse(X)))
5.12/5.28	= { by axiom 11 (converse_additivity_9) }
5.12/5.28	  join(join(converse(Y), converse(X)), complement(converse(X)))
5.12/5.28	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.28	  join(converse(Y), join(converse(X), complement(converse(X))))
5.12/5.28	= { by axiom 9 (def_top_12) }
5.12/5.28	  join(converse(Y), top)
5.12/5.28	= { by lemma 25 }
5.12/5.28	  top
5.12/5.28	
5.12/5.28	Lemma 52: converse(one) = one.
5.12/5.28	Proof:
5.12/5.28	  converse(one)
5.12/5.28	= { by axiom 3 (composition_identity_6) }
5.12/5.28	  composition(converse(one), one)
5.12/5.28	= { by lemma 19 }
5.12/5.28	  one
5.12/5.28	
5.12/5.28	Lemma 53: complement(complement(X)) = X.
5.12/5.28	Proof:
5.12/5.28	  complement(complement(X))
5.12/5.28	= { by lemma 33 }
5.12/5.28	  complement(join(zero, complement(X)))
5.12/5.28	= { by lemma 18 }
5.12/5.28	  meet(X, top)
5.12/5.28	= { by lemma 36 }
5.12/5.29	  X
5.12/5.29	
5.12/5.29	Lemma 54: join(X, complement(join(X, Y))) = join(X, complement(Y)).
5.12/5.29	Proof:
5.12/5.29	  join(X, complement(join(X, Y)))
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(complement(join(X, Y)), X)
5.12/5.29	= { by lemma 53 }
5.12/5.29	  join(complement(join(X, complement(complement(Y)))), X)
5.12/5.29	= { by axiom 5 (converse_idempotence_8) }
5.12/5.29	  join(complement(join(X, complement(converse(converse(complement(Y)))))), X)
5.12/5.29	= { by lemma 26 }
5.12/5.29	  join(complement(join(X, complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(X)))), complement(join(complement(converse(converse(complement(Y)))), converse(join(converse(complement(Y)), converse(X))))))))), X)
5.12/5.29	= { by lemma 51 }
5.12/5.29	  join(complement(join(X, complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(X)))), complement(top))))), X)
5.12/5.29	= { by lemma 17 }
5.12/5.29	  join(complement(join(X, complement(join(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(X)))), zero)))), X)
5.12/5.29	= { by lemma 32 }
5.12/5.29	  join(complement(join(X, complement(meet(converse(converse(complement(Y))), converse(join(converse(complement(Y)), converse(X))))))), X)
5.12/5.29	= { by axiom 5 (converse_idempotence_8) }
5.12/5.29	  join(complement(join(X, complement(meet(complement(Y), converse(join(converse(complement(Y)), converse(X))))))), X)
5.12/5.29	= { by axiom 11 (converse_additivity_9) }
5.12/5.29	  join(complement(join(X, complement(meet(complement(Y), converse(converse(join(complement(Y), X))))))), X)
5.12/5.29	= { by axiom 5 (converse_idempotence_8) }
5.12/5.29	  join(complement(join(X, complement(meet(complement(Y), join(complement(Y), X))))), X)
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(complement(join(X, complement(meet(complement(Y), join(X, complement(Y)))))), X)
5.12/5.29	= { by lemma 16 }
5.12/5.29	  join(complement(join(X, complement(meet(join(X, complement(Y)), complement(Y))))), X)
5.12/5.29	= { by lemma 42 }
5.12/5.29	  join(complement(join(X, join(Y, complement(join(X, complement(Y)))))), X)
5.12/5.29	= { by lemma 39 }
5.12/5.29	  join(complement(join(X, join(Y, meet(Y, complement(X))))), X)
5.12/5.29	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.29	  join(complement(join(join(X, Y), meet(Y, complement(X)))), X)
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(complement(join(meet(Y, complement(X)), join(X, Y))), X)
5.12/5.29	= { by lemma 36 }
5.12/5.29	  join(complement(join(meet(Y, complement(X)), meet(join(X, Y), top))), X)
5.12/5.29	= { by lemma 18 }
5.12/5.29	  join(complement(join(meet(Y, complement(X)), complement(join(zero, complement(join(X, Y)))))), X)
5.12/5.29	= { by lemma 39 }
5.12/5.29	  join(meet(join(zero, complement(join(X, Y))), complement(meet(Y, complement(X)))), X)
5.12/5.29	= { by lemma 33 }
5.12/5.29	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), X)
5.12/5.29	= { by lemma 53 }
5.12/5.29	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), complement(complement(X)))
5.12/5.29	= { by lemma 49 }
5.12/5.29	  join(meet(complement(join(X, Y)), complement(meet(Y, complement(X)))), complement(join(meet(Y, complement(X)), complement(join(X, Y)))))
5.12/5.29	= { by lemma 49 }
5.12/5.29	  complement(meet(Y, complement(X)))
5.12/5.29	= { by lemma 42 }
5.12/5.29	  join(X, complement(Y))
5.12/5.29	
5.12/5.29	Lemma 55: join(one, sk1) = one.
5.12/5.29	Proof:
5.12/5.29	  join(one, sk1)
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(sk1, one)
5.12/5.29	= { by axiom 15 (goals_17) }
5.12/5.29	  one
5.12/5.29	
5.12/5.29	Lemma 56: join(one, join(sk1, X)) = join(X, one).
5.12/5.29	Proof:
5.12/5.29	  join(one, join(sk1, X))
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(one, join(X, sk1))
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(one, join(sk1, X))
5.12/5.29	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.29	  join(join(one, sk1), X)
5.12/5.29	= { by lemma 55 }
5.12/5.29	  join(one, X)
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(X, one)
5.12/5.29	
5.12/5.29	Lemma 57: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
5.12/5.29	Proof:
5.12/5.29	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
5.12/5.29	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.29	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
5.12/5.29	= { by axiom 13 (composition_distributivity_7) }
5.12/5.29	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
5.12/5.29	= { by axiom 6 (maddux2_join_associativity_2) }
5.12/5.29	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
5.12/5.29	= { by axiom 9 (def_top_12) }
5.12/5.29	  join(composition(Z, Y), top)
5.12/5.29	= { by lemma 25 }
5.12/5.30	  top
5.12/5.30	
5.12/5.30	Lemma 58: meet(one, composition(converse(sk1), sk1)) = composition(converse(sk1), sk1).
5.12/5.30	Proof:
5.12/5.30	  meet(one, composition(converse(sk1), sk1))
5.12/5.30	= { by lemma 16 }
5.12/5.30	  meet(composition(converse(sk1), sk1), one)
5.12/5.30	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.30	  complement(join(complement(composition(converse(sk1), sk1)), complement(one)))
5.12/5.30	= { by lemma 54 }
5.12/5.30	  complement(join(complement(composition(converse(sk1), sk1)), complement(join(complement(composition(converse(sk1), sk1)), one))))
5.12/5.30	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(complement(composition(converse(sk1), sk1)), one))
5.12/5.30	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(converse(sk1), sk1))))
5.12/5.30	= { by lemma 48 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(join(one, complement(converse(sk1))))), sk1))))
5.12/5.30	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(join(complement(converse(sk1)), one))), sk1))))
5.12/5.30	= { by lemma 52 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(join(complement(converse(sk1)), converse(one)))), sk1))))
5.12/5.30	= { by lemma 43 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, converse(complement(converse(sk1))))))), sk1))))
5.12/5.30	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(converse(join(converse(complement(converse(sk1))), one)))), sk1))))
5.12/5.30	= { by lemma 56 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, join(sk1, converse(complement(converse(sk1)))))))), sk1))))
5.12/5.30	= { by lemma 47 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(converse(join(one, top)))), sk1))))
5.12/5.30	= { by lemma 25 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(converse(top))), sk1))))
5.12/5.30	= { by lemma 46 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), complement(top)), sk1))))
5.12/5.30	= { by lemma 17 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(join(meet(converse(sk1), one), zero), sk1))))
5.12/5.30	= { by lemma 32 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, complement(composition(meet(converse(sk1), one), sk1))))
5.12/5.30	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(complement(composition(meet(converse(sk1), one), sk1)), one))
5.12/5.30	= { by lemma 56 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, join(sk1, complement(composition(meet(converse(sk1), one), sk1)))))
5.12/5.30	= { by lemma 19 }
5.12/5.30	  meet(composition(converse(sk1), sk1), join(one, join(composition(converse(one), sk1), complement(composition(meet(converse(sk1), one), sk1)))))
5.12/5.30	= { by lemma 16 }
5.12/5.31	  meet(composition(converse(sk1), sk1), join(one, join(composition(converse(one), sk1), complement(composition(meet(one, converse(sk1)), sk1)))))
5.12/5.31	= { by lemma 52 }
5.12/5.31	  meet(composition(converse(sk1), sk1), join(one, join(composition(converse(one), sk1), complement(composition(meet(converse(one), converse(sk1)), sk1)))))
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  meet(composition(converse(sk1), sk1), join(one, join(complement(composition(meet(converse(one), converse(sk1)), sk1)), composition(converse(one), sk1))))
5.12/5.31	= { by lemma 26 }
5.12/5.31	  meet(composition(converse(sk1), sk1), join(one, join(complement(composition(meet(converse(one), converse(sk1)), sk1)), composition(join(meet(converse(one), converse(sk1)), complement(join(complement(converse(one)), converse(sk1)))), sk1))))
5.12/5.31	= { by lemma 57 }
5.12/5.31	  meet(composition(converse(sk1), sk1), join(one, top))
5.12/5.31	= { by lemma 25 }
5.12/5.31	  meet(composition(converse(sk1), sk1), top)
5.12/5.31	= { by lemma 36 }
5.12/5.31	  composition(converse(sk1), sk1)
5.12/5.31	
5.12/5.31	Lemma 59: join(Z, composition(X, composition(Y, Z))) = composition(join(one, composition(X, Y)), Z).
5.12/5.31	Proof:
5.12/5.31	  join(Z, composition(X, composition(Y, Z)))
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  join(composition(X, composition(Y, Z)), Z)
5.12/5.31	= { by axiom 12 (composition_associativity_5) }
5.12/5.31	  join(composition(composition(X, Y), Z), Z)
5.12/5.31	= { by lemma 19 }
5.12/5.31	  join(composition(composition(X, Y), Z), composition(converse(one), Z))
5.12/5.31	= { by axiom 13 (composition_distributivity_7) }
5.12/5.31	  composition(join(composition(X, Y), converse(one)), Z)
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  composition(join(converse(one), composition(X, Y)), Z)
5.12/5.31	= { by lemma 52 }
5.12/5.31	  composition(join(one, composition(X, Y)), Z)
5.12/5.31	
5.12/5.31	Lemma 60: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
5.12/5.31	Proof:
5.12/5.31	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
5.12/5.31	= { by axiom 10 (converse_multiplicativity_10) }
5.12/5.31	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
5.12/5.31	= { by axiom 13 (composition_distributivity_7) }
5.12/5.31	  composition(join(converse(X), Y), converse(Z))
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  composition(join(Y, converse(X)), converse(Z))
5.12/5.31	
5.12/5.31	Lemma 61: join(X, composition(X, Y)) = composition(X, join(Y, one)).
5.12/5.31	Proof:
5.12/5.31	  join(X, composition(X, Y))
5.12/5.31	= { by axiom 5 (converse_idempotence_8) }
5.12/5.31	  converse(converse(join(X, composition(X, Y))))
5.12/5.31	= { by axiom 11 (converse_additivity_9) }
5.12/5.31	  converse(join(converse(X), converse(composition(X, Y))))
5.12/5.31	= { by lemma 19 }
5.12/5.31	  converse(join(composition(converse(one), converse(X)), converse(composition(X, Y))))
5.12/5.31	= { by lemma 60 }
5.12/5.31	  converse(composition(join(converse(one), converse(Y)), converse(X)))
5.12/5.31	= { by axiom 11 (converse_additivity_9) }
5.12/5.31	  converse(composition(converse(join(one, Y)), converse(X)))
5.12/5.31	= { by axiom 10 (converse_multiplicativity_10) }
5.12/5.31	  converse(converse(composition(X, join(one, Y))))
5.12/5.31	= { by axiom 1 (maddux1_join_commutativity_1) }
5.12/5.31	  converse(converse(composition(X, join(Y, one))))
5.12/5.31	= { by axiom 5 (converse_idempotence_8) }
5.29/5.46	  composition(X, join(Y, one))
5.29/5.46	
5.29/5.46	Lemma 62: join(sk1, converse(sk1)) = converse(sk1).
5.29/5.46	Proof:
5.29/5.46	  join(sk1, converse(sk1))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), sk1)
5.29/5.46	= { by axiom 3 (composition_identity_6) }
5.29/5.46	  join(converse(sk1), composition(sk1, one))
5.29/5.46	= { by lemma 32 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, zero)))
5.29/5.46	= { by lemma 17 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(top))))
5.29/5.46	= { by lemma 57 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(join(converse(sk1), one), sk1))))))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(join(one, converse(sk1)), sk1))))))
5.29/5.46	= { by lemma 52 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(join(converse(one), converse(sk1)), sk1))))))
5.29/5.46	= { by axiom 11 (converse_additivity_9) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(converse(join(one, sk1)), sk1))))))
5.29/5.46	= { by lemma 55 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(converse(one), sk1))))))
5.29/5.46	= { by lemma 52 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), composition(one, sk1))))))
5.29/5.46	= { by lemma 20 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(complement(composition(converse(sk1), sk1)), sk1)))))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(join(sk1, complement(composition(converse(sk1), sk1)))))))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(complement(join(sk1, complement(composition(converse(sk1), sk1)))), one)))
5.29/5.46	= { by lemma 56 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, join(sk1, complement(join(sk1, complement(composition(converse(sk1), sk1))))))))
5.29/5.46	= { by lemma 54 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, join(sk1, complement(complement(composition(converse(sk1), sk1)))))))
5.29/5.46	= { by lemma 56 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(complement(complement(composition(converse(sk1), sk1))), one)))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, complement(complement(composition(converse(sk1), sk1))))))
5.29/5.46	= { by lemma 53 }
5.29/5.46	  join(converse(sk1), composition(sk1, join(one, composition(converse(sk1), sk1))))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(sk1, join(composition(converse(sk1), sk1), one)))
5.29/5.46	= { by lemma 61 }
5.29/5.46	  join(converse(sk1), join(sk1, composition(sk1, composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 59 }
5.29/5.46	  join(converse(sk1), composition(join(one, composition(sk1, converse(sk1))), sk1))
5.29/5.46	= { by lemma 34 }
5.29/5.46	  join(converse(sk1), composition(join(one, composition(meet(sk1, sk1), converse(sk1))), sk1))
5.29/5.46	= { by lemma 59 }
5.29/5.46	  join(converse(sk1), join(sk1, composition(meet(sk1, sk1), composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 34 }
5.29/5.46	  join(converse(sk1), join(meet(sk1, sk1), composition(meet(sk1, sk1), composition(converse(sk1), sk1))))
5.29/5.46	= { by axiom 3 (composition_identity_6) }
5.29/5.46	  join(converse(sk1), join(meet(composition(sk1, one), sk1), composition(meet(sk1, sk1), composition(converse(sk1), sk1))))
5.29/5.46	= { by axiom 3 (composition_identity_6) }
5.29/5.46	  join(converse(sk1), join(meet(composition(sk1, one), sk1), composition(meet(sk1, composition(sk1, one)), composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 52 }
5.29/5.46	  join(converse(sk1), join(meet(composition(sk1, one), sk1), composition(meet(sk1, composition(sk1, converse(one))), composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 58 }
5.29/5.46	  join(converse(sk1), join(meet(composition(sk1, one), sk1), composition(meet(sk1, composition(sk1, converse(one))), meet(one, composition(converse(sk1), sk1)))))
5.29/5.46	= { by axiom 14 (dedekind_law_14) }
5.29/5.46	  join(converse(sk1), composition(meet(sk1, composition(sk1, converse(one))), meet(one, composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 52 }
5.29/5.46	  join(converse(sk1), composition(meet(sk1, composition(sk1, one)), meet(one, composition(converse(sk1), sk1))))
5.29/5.46	= { by axiom 3 (composition_identity_6) }
5.29/5.46	  join(converse(sk1), composition(meet(sk1, sk1), meet(one, composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 34 }
5.29/5.46	  join(converse(sk1), composition(sk1, meet(one, composition(converse(sk1), sk1))))
5.29/5.46	= { by lemma 58 }
5.29/5.46	  join(converse(sk1), composition(sk1, composition(converse(sk1), sk1)))
5.29/5.46	= { by axiom 12 (composition_associativity_5) }
5.29/5.46	  join(converse(sk1), composition(composition(sk1, converse(sk1)), sk1))
5.29/5.46	= { by lemma 48 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(converse(sk1), complement(composition(sk1, converse(sk1)))))), sk1))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), converse(sk1)))), sk1))
5.29/5.46	= { by lemma 20 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(one, converse(sk1))))), sk1))
5.29/5.46	= { by lemma 28 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(meet(one, sk1), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 16 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(meet(sk1, one), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 32 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), zero), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 17 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), complement(top)), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 25 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), complement(join(one, top))), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by axiom 9 (def_top_12) }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), complement(join(one, join(sk1, complement(sk1))))), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 56 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), complement(join(complement(sk1), one))), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(join(meet(sk1, one), complement(join(one, complement(sk1)))), meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 48 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(join(complement(composition(sk1, converse(sk1))), composition(join(sk1, meet(one, complement(sk1))), converse(sk1))))), sk1))
5.29/5.46	= { by lemma 57 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), complement(top)), sk1))
5.29/5.46	= { by lemma 17 }
5.29/5.46	  join(converse(sk1), composition(join(meet(composition(sk1, converse(sk1)), converse(sk1)), zero), sk1))
5.29/5.46	= { by lemma 32 }
5.29/5.46	  join(converse(sk1), composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1), converse(sk1))
5.29/5.46	= { by lemma 41 }
5.29/5.46	  join(composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1), join(meet(composition(sk1, converse(sk1)), converse(sk1)), meet(converse(sk1), complement(composition(sk1, converse(sk1))))))
5.29/5.46	= { by lemma 29 }
5.29/5.46	  join(meet(composition(sk1, converse(sk1)), converse(sk1)), join(composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1), meet(converse(sk1), complement(composition(sk1, converse(sk1))))))
5.29/5.46	= { by lemma 22 }
5.29/5.46	  join(composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1), join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), meet(composition(sk1, converse(sk1)), converse(sk1))))
5.29/5.46	= { by lemma 22 }
5.29/5.46	  join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), join(meet(composition(sk1, converse(sk1)), converse(sk1)), composition(meet(composition(sk1, converse(sk1)), converse(sk1)), sk1)))
5.29/5.46	= { by lemma 61 }
5.29/5.46	  join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), composition(meet(composition(sk1, converse(sk1)), converse(sk1)), join(sk1, one)))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), composition(meet(composition(sk1, converse(sk1)), converse(sk1)), join(one, sk1)))
5.29/5.46	= { by lemma 55 }
5.29/5.46	  join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), composition(meet(composition(sk1, converse(sk1)), converse(sk1)), one))
5.29/5.46	= { by axiom 3 (composition_identity_6) }
5.29/5.46	  join(meet(converse(sk1), complement(composition(sk1, converse(sk1)))), meet(composition(sk1, converse(sk1)), converse(sk1)))
5.29/5.46	= { by axiom 1 (maddux1_join_commutativity_1) }
5.29/5.46	  join(meet(composition(sk1, converse(sk1)), converse(sk1)), meet(converse(sk1), complement(composition(sk1, converse(sk1)))))
5.29/5.46	= { by lemma 41 }
5.57/5.81	  converse(sk1)
5.57/5.81	
5.57/5.81	Goal 1 (goals_18): meet(complement(composition(sk1, top)), one) = meet(complement(sk1), one).
5.57/5.81	Proof:
5.57/5.81	  meet(complement(composition(sk1, top)), one)
5.57/5.81	= { by lemma 16 }
5.57/5.81	  meet(one, complement(composition(sk1, top)))
5.57/5.81	= { by lemma 40 }
5.57/5.81	  complement(join(complement(one), composition(sk1, top)))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(converse(converse(sk1)), top)))
5.57/5.81	= { by lemma 62 }
5.57/5.81	  complement(join(complement(one), composition(converse(join(sk1, converse(sk1))), top)))
5.57/5.81	= { by lemma 43 }
5.57/5.81	  complement(join(complement(one), composition(join(sk1, converse(sk1)), top)))
5.57/5.81	= { by lemma 62 }
5.57/5.81	  complement(join(complement(one), composition(converse(sk1), top)))
5.57/5.81	= { by lemma 46 }
5.57/5.81	  complement(join(complement(one), composition(converse(sk1), converse(top))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(join(complement(one), converse(composition(top, sk1))))
5.57/5.81	= { by lemma 44 }
5.57/5.81	  complement(converse(join(converse(complement(one)), composition(top, sk1))))
5.57/5.81	= { by axiom 3 (composition_identity_6) }
5.57/5.81	  complement(converse(join(composition(converse(complement(one)), one), composition(top, sk1))))
5.57/5.81	= { by axiom 9 (def_top_12) }
5.57/5.81	  complement(converse(join(composition(converse(complement(one)), one), composition(join(converse(complement(one)), complement(converse(complement(one)))), sk1))))
5.57/5.81	= { by axiom 13 (composition_distributivity_7) }
5.57/5.81	  complement(converse(join(composition(converse(complement(one)), one), join(composition(converse(complement(one)), sk1), composition(complement(converse(complement(one))), sk1)))))
5.57/5.81	= { by lemma 22 }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), join(composition(converse(complement(one)), one), composition(converse(complement(one)), sk1)))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(converse(join(composition(converse(complement(one)), one), composition(converse(complement(one)), sk1)))))))
5.57/5.81	= { by axiom 1 (maddux1_join_commutativity_1) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(converse(join(composition(converse(complement(one)), sk1), composition(converse(complement(one)), one)))))))
5.57/5.81	= { by axiom 11 (converse_additivity_9) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(join(converse(composition(converse(complement(one)), sk1)), converse(composition(converse(complement(one)), one)))))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(join(composition(converse(sk1), converse(converse(complement(one)))), converse(composition(converse(complement(one)), one)))))))
5.57/5.81	= { by lemma 60 }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(composition(join(converse(sk1), converse(one)), converse(converse(complement(one))))))))
5.57/5.81	= { by axiom 11 (converse_additivity_9) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(composition(converse(join(sk1, one)), converse(converse(complement(one))))))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(converse(composition(converse(complement(one)), join(sk1, one)))))))
5.57/5.81	= { by axiom 1 (maddux1_join_commutativity_1) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(converse(composition(converse(complement(one)), join(one, sk1)))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), composition(converse(complement(one)), join(one, sk1)))))
5.57/5.81	= { by lemma 55 }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), composition(converse(complement(one)), one))))
5.57/5.81	= { by axiom 3 (composition_identity_6) }
5.57/5.81	  complement(converse(join(composition(complement(converse(complement(one))), sk1), converse(complement(one)))))
5.57/5.81	= { by axiom 1 (maddux1_join_commutativity_1) }
5.57/5.81	  complement(converse(join(converse(complement(one)), composition(complement(converse(complement(one))), sk1))))
5.57/5.81	= { by lemma 44 }
5.57/5.81	  complement(join(complement(one), converse(composition(complement(converse(complement(one))), sk1))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(join(complement(one), composition(converse(sk1), converse(complement(converse(complement(one)))))))
5.57/5.81	= { by lemma 62 }
5.57/5.81	  complement(join(complement(one), composition(join(sk1, converse(sk1)), converse(complement(converse(complement(one)))))))
5.57/5.81	= { by lemma 43 }
5.57/5.81	  complement(join(complement(one), composition(converse(join(sk1, converse(sk1))), converse(complement(converse(complement(one)))))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(join(complement(one), converse(composition(complement(converse(complement(one))), join(sk1, converse(sk1))))))
5.57/5.81	= { by lemma 62 }
5.57/5.81	  complement(join(complement(one), converse(composition(complement(converse(complement(one))), converse(sk1)))))
5.57/5.81	= { by axiom 10 (converse_multiplicativity_10) }
5.57/5.81	  complement(join(complement(one), composition(converse(converse(sk1)), converse(complement(converse(complement(one)))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(complement(converse(complement(one)))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(converse(complement(converse(complement(one)))))))))
5.57/5.81	= { by lemma 41 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(complement(one), converse(complement(converse(complement(one))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 16 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(converse(complement(converse(complement(one)))), converse(converse(complement(one)))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 37 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(converse(complement(converse(complement(one)))), meet(top, converse(converse(complement(one))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 36 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(complement(converse(complement(one)))), meet(top, converse(converse(complement(one))))), top), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 18 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, complement(meet(converse(complement(converse(complement(one)))), meet(top, converse(converse(complement(one)))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 2 (maddux4_definiton_of_meet_4) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, complement(meet(converse(complement(converse(complement(one)))), complement(join(complement(top), complement(converse(converse(complement(one)))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 42 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, join(join(complement(top), complement(converse(converse(complement(one))))), complement(converse(complement(converse(complement(one)))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 6 (maddux2_join_associativity_2) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, join(complement(top), join(complement(converse(converse(complement(one)))), complement(converse(complement(converse(complement(one))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 38 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, join(complement(top), complement(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 38 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(complement(join(zero, complement(meet(top, meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 18 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(top, meet(converse(converse(complement(one))), converse(complement(converse(complement(one)))))), top), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 36 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(top, meet(converse(converse(complement(one))), converse(complement(converse(complement(one)))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 16 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), top), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 51 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(complement(converse(meet(converse(complement(converse(converse(complement(one))))), complement(converse(complement(one)))))), converse(join(meet(converse(complement(converse(converse(complement(one))))), complement(converse(complement(one)))), complement(join(complement(converse(complement(converse(converse(complement(one)))))), complement(converse(complement(one))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 26 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(complement(converse(meet(converse(complement(converse(converse(complement(one))))), complement(converse(complement(one)))))), converse(converse(complement(converse(converse(complement(one)))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 1 (maddux1_join_commutativity_1) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(converse(converse(complement(converse(converse(complement(one)))))), complement(converse(meet(converse(complement(converse(converse(complement(one))))), complement(converse(complement(one)))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(complement(converse(converse(complement(one)))), complement(converse(meet(converse(complement(converse(converse(complement(one))))), complement(converse(complement(one)))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 16 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(complement(converse(converse(complement(one)))), complement(converse(meet(complement(converse(complement(one))), converse(complement(converse(converse(complement(one)))))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 50 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), join(complement(converse(converse(complement(one)))), complement(converse(complement(converse(complement(one))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 38 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(meet(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))), complement(meet(converse(converse(complement(one))), converse(complement(converse(complement(one))))))), meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by axiom 8 (def_zero_13) }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(zero, meet(converse(complement(converse(complement(one)))), complement(complement(one)))))))))
5.57/5.81	= { by lemma 16 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(join(zero, meet(complement(complement(one)), converse(complement(converse(complement(one)))))))))))
5.57/5.81	= { by lemma 33 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(meet(complement(complement(one)), converse(complement(converse(complement(one))))))))))
5.57/5.81	= { by lemma 50 }
5.57/5.81	  complement(join(complement(one), composition(sk1, converse(converse(complement(complement(one)))))))
5.57/5.81	= { by axiom 5 (converse_idempotence_8) }
5.57/5.81	  complement(join(complement(one), composition(sk1, complement(complement(one)))))
5.57/5.81	= { by lemma 40 }
5.57/5.81	  meet(one, complement(composition(sk1, complement(complement(one)))))
5.57/5.81	= { by lemma 53 }
5.57/5.81	  meet(one, complement(composition(sk1, one)))
5.57/5.81	= { by axiom 3 (composition_identity_6) }
5.57/5.81	  meet(one, complement(sk1))
5.57/5.81	= { by lemma 16 }
5.57/5.81	  meet(complement(sk1), one)
5.57/5.81	% SZS output end Proof
5.57/5.81	
5.57/5.81	RESULT: Unsatisfiable (the axioms are contradictory).
5.66/5.82	EOF