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
0.12/0.34	% Computer   : n027.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 13:48:09 EDT 2019
0.12/0.34	% CPUTime    : 
1.83/2.04	% SZS status Unsatisfiable
1.83/2.04	
1.83/2.04	% SZS output start Proof
1.83/2.04	Take the following subset of the input axioms:
1.83/2.05	  fof(composition_associativity_5, axiom, ![A, C, B]: composition(composition(A, B), C)=composition(A, composition(B, C))).
1.83/2.05	  fof(composition_distributivity_7, axiom, ![A, C, B]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
1.83/2.05	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
1.83/2.05	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
1.83/2.05	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
1.83/2.05	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
1.83/2.05	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
1.83/2.05	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
1.83/2.05	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
1.83/2.05	  fof(goals_17, negated_conjecture, sk1=composition(sk1, top)).
1.83/2.05	  fof(goals_18, negated_conjecture, sk2=composition(sk2, top)).
1.83/2.05	  fof(goals_19, negated_conjecture, meet(sk1, sk2)!=composition(meet(sk1, sk2), top)).
1.83/2.05	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
1.83/2.05	  fof(maddux2_join_associativity_2, axiom, ![A, C, B]: join(join(A, B), C)=join(A, join(B, C))).
1.83/2.05	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
1.83/2.05	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
1.83/2.05	  fof(modular_law_1_15, axiom, ![A, C, B]: join(meet(composition(A, B), C), meet(composition(A, meet(B, composition(converse(A), C))), C))=meet(composition(A, meet(B, composition(converse(A), C))), C)).
1.83/2.05	
1.83/2.05	Now clausify the problem and encode Horn clauses using encoding 3 of
1.83/2.05	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.83/2.05	We repeatedly replace C & s=t => u=v by the two clauses:
1.83/2.05	  fresh(y, y, x1...xn) = u
1.83/2.05	  C => fresh(s, t, x1...xn) = v
1.83/2.05	where fresh is a fresh function symbol and x1..xn are the free
1.83/2.05	variables of u and v.
1.83/2.05	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.83/2.05	input problem has no model of domain size 1).
1.83/2.05	
1.83/2.05	The encoding turns the above axioms into the following unit equations and goals:
1.83/2.05	
1.83/2.05	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
1.83/2.05	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
1.83/2.05	Axiom 3 (composition_identity_6): X = composition(X, one).
1.83/2.05	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
1.83/2.05	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
1.83/2.05	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
1.83/2.05	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
1.83/2.05	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
1.83/2.05	Axiom 9 (def_top_12): join(X, complement(X)) = top.
1.83/2.05	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
1.83/2.05	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
1.83/2.05	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
1.83/2.05	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
1.83/2.05	Axiom 14 (modular_law_1_15): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
1.83/2.05	Axiom 15 (goals_17): sk1 = composition(sk1, top).
1.92/2.10	Axiom 16 (goals_18): sk2 = composition(sk2, top).
1.92/2.10	
1.92/2.10	Lemma 17: meet(X, Y) = meet(Y, X).
1.92/2.10	Proof:
1.92/2.10	  meet(X, Y)
1.92/2.10	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.10	  complement(join(complement(X), complement(Y)))
1.92/2.10	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.10	  complement(join(complement(Y), complement(X)))
1.92/2.10	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.10	  meet(Y, X)
1.92/2.10	
1.92/2.10	Lemma 18: complement(top) = zero.
1.92/2.10	Proof:
1.92/2.10	  complement(top)
1.92/2.10	= { by axiom 9 (def_top_12) }
1.92/2.10	  complement(join(complement(?), complement(complement(?))))
1.92/2.10	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.10	  meet(?, complement(?))
1.92/2.10	= { by axiom 8 (def_zero_13) }
1.92/2.10	  zero
1.92/2.10	
1.92/2.10	Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
1.92/2.10	Proof:
1.92/2.10	  complement(join(zero, complement(X)))
1.92/2.10	= { by lemma 18 }
1.92/2.10	  complement(join(complement(top), complement(X)))
1.92/2.10	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.10	  meet(top, X)
1.92/2.10	= { by lemma 17 }
1.92/2.10	  meet(X, top)
1.92/2.10	
1.92/2.10	Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
1.92/2.10	Proof:
1.92/2.10	  converse(composition(converse(X), Y))
1.92/2.10	= { by axiom 10 (converse_multiplicativity_10) }
1.92/2.10	  composition(converse(Y), converse(converse(X)))
1.92/2.10	= { by axiom 5 (converse_idempotence_8) }
1.92/2.10	  composition(converse(Y), X)
1.92/2.10	
1.92/2.10	Lemma 21: composition(converse(one), X) = X.
1.92/2.10	Proof:
1.92/2.10	  composition(converse(one), X)
1.92/2.10	= { by lemma 20 }
1.92/2.10	  converse(composition(converse(X), one))
1.92/2.10	= { by axiom 3 (composition_identity_6) }
1.92/2.10	  converse(converse(X))
1.92/2.10	= { by axiom 5 (converse_idempotence_8) }
1.92/2.10	  X
1.92/2.10	
1.92/2.10	Lemma 22: composition(one, X) = X.
1.92/2.10	Proof:
1.92/2.10	  composition(one, X)
1.92/2.10	= { by lemma 21 }
1.92/2.10	  composition(converse(one), composition(one, X))
1.92/2.10	= { by axiom 12 (composition_associativity_5) }
1.92/2.10	  composition(composition(converse(one), one), X)
1.92/2.10	= { by axiom 3 (composition_identity_6) }
1.92/2.10	  composition(converse(one), X)
1.92/2.10	= { by lemma 21 }
1.92/2.10	  X
1.92/2.10	
1.92/2.10	Lemma 23: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
1.92/2.10	Proof:
1.92/2.10	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
1.92/2.10	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.10	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
1.92/2.10	= { by axiom 7 (converse_cancellativity_11) }
1.92/2.10	  complement(Y)
1.92/2.10	
1.92/2.10	Lemma 24: join(complement(X), complement(X)) = complement(X).
1.92/2.10	Proof:
1.92/2.10	  join(complement(X), complement(X))
1.92/2.10	= { by lemma 21 }
1.92/2.10	  join(complement(X), composition(converse(one), complement(X)))
1.92/2.10	= { by lemma 22 }
1.92/2.10	  join(complement(X), composition(converse(one), complement(composition(one, X))))
1.92/2.10	= { by lemma 23 }
1.92/2.10	  complement(X)
1.92/2.10	
1.92/2.10	Lemma 25: join(X, join(Y, Z)) = join(Z, join(X, Y)).
1.92/2.10	Proof:
1.92/2.10	  join(X, join(Y, Z))
1.92/2.10	= { by axiom 6 (maddux2_join_associativity_2) }
1.92/2.10	  join(join(X, Y), Z)
1.92/2.10	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(Z, join(X, Y))
1.92/2.11	
1.92/2.11	Lemma 26: join(X, join(complement(X), Y)) = join(Y, top).
1.92/2.11	Proof:
1.92/2.11	  join(X, join(complement(X), Y))
1.92/2.11	= { by lemma 25 }
1.92/2.11	  join(complement(X), join(Y, X))
1.92/2.11	= { by lemma 25 }
1.92/2.11	  join(Y, join(X, complement(X)))
1.92/2.11	= { by axiom 9 (def_top_12) }
1.92/2.11	  join(Y, top)
1.92/2.11	
1.92/2.11	Lemma 27: join(top, complement(X)) = top.
1.92/2.11	Proof:
1.92/2.11	  join(top, complement(X))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(complement(X), top)
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(top, complement(X))
1.92/2.11	= { by axiom 9 (def_top_12) }
1.92/2.11	  join(join(X, complement(X)), complement(X))
1.92/2.11	= { by axiom 6 (maddux2_join_associativity_2) }
1.92/2.11	  join(X, join(complement(X), complement(X)))
1.92/2.11	= { by lemma 24 }
1.92/2.11	  join(X, complement(X))
1.92/2.11	= { by axiom 9 (def_top_12) }
1.92/2.11	  top
1.92/2.11	
1.92/2.11	Lemma 28: join(X, top) = top.
1.92/2.11	Proof:
1.92/2.11	  join(X, top)
1.92/2.11	= { by axiom 9 (def_top_12) }
1.92/2.11	  join(X, join(complement(X), complement(complement(X))))
1.92/2.11	= { by lemma 26 }
1.92/2.11	  join(complement(complement(X)), top)
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(top, complement(complement(X)))
1.92/2.11	= { by lemma 27 }
1.92/2.11	  top
1.92/2.11	
1.92/2.11	Lemma 29: join(meet(X, Y), complement(join(complement(X), Y))) = X.
1.92/2.11	Proof:
1.92/2.11	  join(meet(X, Y), complement(join(complement(X), Y)))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
1.92/2.11	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 30: join(zero, meet(X, top)) = X.
1.92/2.11	Proof:
1.92/2.11	  join(zero, meet(X, top))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(meet(X, top), zero)
1.92/2.11	= { by lemma 18 }
1.92/2.11	  join(meet(X, top), complement(top))
1.92/2.11	= { by lemma 28 }
1.92/2.11	  join(meet(X, top), complement(join(complement(X), top)))
1.92/2.11	= { by lemma 29 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 31: join(meet(X, Y), meet(X, complement(Y))) = X.
1.92/2.11	Proof:
1.92/2.11	  join(meet(X, Y), meet(X, complement(Y)))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(meet(X, complement(Y)), meet(X, Y))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
1.92/2.11	= { by lemma 29 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 32: complement(zero) = top.
1.92/2.11	Proof:
1.92/2.11	  complement(zero)
1.92/2.11	= { by lemma 18 }
1.92/2.11	  complement(complement(top))
1.92/2.11	= { by lemma 24 }
1.92/2.11	  complement(join(complement(top), complement(top)))
1.92/2.11	= { by lemma 18 }
1.92/2.11	  complement(join(zero, complement(top)))
1.92/2.11	= { by lemma 19 }
1.92/2.11	  meet(top, top)
1.92/2.11	= { by lemma 30 }
1.92/2.11	  join(zero, meet(meet(top, top), top))
1.92/2.11	= { by axiom 8 (def_zero_13) }
1.92/2.11	  join(meet(top, complement(top)), meet(meet(top, top), top))
1.92/2.11	= { by lemma 17 }
1.92/2.11	  join(meet(top, complement(top)), meet(top, meet(top, top)))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  join(meet(top, complement(top)), meet(top, complement(join(complement(top), complement(top)))))
1.92/2.11	= { by lemma 24 }
1.92/2.11	  join(meet(top, complement(top)), meet(top, complement(complement(top))))
1.92/2.11	= { by lemma 31 }
1.92/2.11	  top
1.92/2.11	
1.92/2.11	Lemma 33: meet(X, zero) = zero.
1.92/2.11	Proof:
1.92/2.11	  meet(X, zero)
1.92/2.11	= { by lemma 17 }
1.92/2.11	  meet(zero, X)
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  complement(join(complement(zero), complement(X)))
1.92/2.11	= { by lemma 32 }
1.92/2.11	  complement(join(top, complement(X)))
1.92/2.11	= { by lemma 27 }
1.92/2.11	  complement(top)
1.92/2.11	= { by lemma 18 }
1.92/2.11	  zero
1.92/2.11	
1.92/2.11	Lemma 34: join(zero, meet(X, X)) = X.
1.92/2.11	Proof:
1.92/2.11	  join(zero, meet(X, X))
1.92/2.11	= { by axiom 8 (def_zero_13) }
1.92/2.11	  join(meet(X, complement(X)), meet(X, X))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
1.92/2.11	= { by lemma 29 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 35: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
1.92/2.11	Proof:
1.92/2.11	  join(X, meet(Y, Y))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(meet(Y, Y), X)
1.92/2.11	= { by lemma 34 }
1.92/2.11	  join(meet(Y, Y), join(zero, meet(X, X)))
1.92/2.11	= { by axiom 6 (maddux2_join_associativity_2) }
1.92/2.11	  join(join(meet(Y, Y), zero), meet(X, X))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(join(zero, meet(Y, Y)), meet(X, X))
1.92/2.11	= { by lemma 34 }
1.92/2.11	  join(Y, meet(X, X))
1.92/2.11	
1.92/2.11	Lemma 36: join(X, zero) = X.
1.92/2.11	Proof:
1.92/2.11	  join(X, zero)
1.92/2.11	= { by lemma 33 }
1.92/2.11	  join(X, meet(zero, zero))
1.92/2.11	= { by lemma 35 }
1.92/2.11	  join(zero, meet(X, X))
1.92/2.11	= { by lemma 34 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 37: join(zero, X) = X.
1.92/2.11	Proof:
1.92/2.11	  join(zero, X)
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  join(X, zero)
1.92/2.11	= { by lemma 36 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 38: meet(X, top) = X.
1.92/2.11	Proof:
1.92/2.11	  meet(X, top)
1.92/2.11	= { by lemma 37 }
1.92/2.11	  join(zero, meet(X, top))
1.92/2.11	= { by lemma 30 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 39: complement(complement(X)) = X.
1.92/2.11	Proof:
1.92/2.11	  complement(complement(X))
1.92/2.11	= { by lemma 37 }
1.92/2.11	  complement(join(zero, complement(X)))
1.92/2.11	= { by lemma 19 }
1.92/2.11	  meet(X, top)
1.92/2.11	= { by lemma 38 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 40: meet(top, X) = X.
1.92/2.11	Proof:
1.92/2.11	  meet(top, X)
1.92/2.11	= { by lemma 17 }
1.92/2.11	  meet(X, top)
1.92/2.11	= { by lemma 38 }
1.92/2.11	  X
1.92/2.11	
1.92/2.11	Lemma 41: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
1.92/2.11	Proof:
1.92/2.11	  complement(join(complement(X), meet(Y, Z)))
1.92/2.11	= { by lemma 17 }
1.92/2.11	  complement(join(complement(X), meet(Z, Y)))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  complement(join(meet(Z, Y), complement(X)))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
1.92/2.11	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.92/2.11	  meet(join(complement(Z), complement(Y)), X)
1.92/2.11	= { by lemma 17 }
1.92/2.11	  meet(X, join(complement(Z), complement(Y)))
1.92/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.92/2.11	  meet(X, join(complement(Y), complement(Z)))
1.92/2.11	
1.92/2.11	Lemma 42: join(complement(X), complement(Y)) = complement(meet(X, Y)).
1.92/2.11	Proof:
1.92/2.11	  join(complement(X), complement(Y))
1.92/2.11	= { by lemma 40 }
1.92/2.11	  meet(top, join(complement(X), complement(Y)))
1.92/2.11	= { by lemma 41 }
1.92/2.11	  complement(join(complement(top), meet(X, Y)))
1.92/2.11	= { by lemma 18 }
1.92/2.11	  complement(join(zero, meet(X, Y)))
1.92/2.11	= { by lemma 37 }
1.92/2.11	  complement(meet(X, Y))
1.92/2.11	
1.92/2.11	Lemma 43: join(meet(Y, X), meet(X, complement(Y))) = X.
1.92/2.11	Proof:
1.92/2.11	  join(meet(Y, X), meet(X, complement(Y)))
1.92/2.11	= { by lemma 17 }
1.92/2.11	  join(meet(X, Y), meet(X, complement(Y)))
1.92/2.11	= { by lemma 31 }
1.92/2.11	  X
1.92/2.11	
1.95/2.11	Lemma 44: converse(join(converse(X), Y)) = join(X, converse(Y)).
1.95/2.11	Proof:
1.95/2.11	  converse(join(converse(X), Y))
1.95/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.95/2.11	  converse(join(Y, converse(X)))
1.95/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.95/2.11	  converse(join(converse(X), Y))
1.95/2.11	= { by axiom 11 (converse_additivity_9) }
1.95/2.11	  join(converse(converse(X)), converse(Y))
1.95/2.11	= { by axiom 5 (converse_idempotence_8) }
1.95/2.11	  join(X, converse(Y))
1.95/2.11	
1.95/2.11	Lemma 45: join(X, converse(top)) = converse(top).
1.95/2.11	Proof:
1.95/2.11	  join(X, converse(top))
1.95/2.11	= { by lemma 44 }
1.95/2.11	  converse(join(converse(X), top))
1.95/2.11	= { by lemma 28 }
1.95/2.11	  converse(top)
1.95/2.11	
1.95/2.11	Lemma 46: converse(top) = top.
1.95/2.11	Proof:
1.95/2.11	  converse(top)
1.95/2.11	= { by lemma 45 }
1.95/2.11	  join(?, converse(top))
1.95/2.11	= { by lemma 45 }
1.95/2.11	  join(?, join(complement(?), converse(top)))
1.95/2.11	= { by lemma 26 }
1.95/2.11	  join(converse(top), top)
1.95/2.11	= { by lemma 28 }
1.95/2.11	  top
1.95/2.11	
1.95/2.11	Lemma 47: meet(complement(X), converse(complement(converse(X)))) = complement(X).
1.95/2.11	Proof:
1.95/2.11	  meet(complement(X), converse(complement(converse(X))))
1.95/2.11	= { by lemma 17 }
1.95/2.11	  meet(converse(complement(converse(X))), complement(X))
1.95/2.11	= { by lemma 36 }
1.95/2.11	  join(meet(converse(complement(converse(X))), complement(X)), zero)
1.95/2.11	= { by lemma 17 }
1.95/2.11	  join(meet(complement(X), converse(complement(converse(X)))), zero)
1.95/2.11	= { by lemma 37 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), zero)
1.95/2.11	= { by lemma 18 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(top))
1.95/2.11	= { by lemma 46 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(converse(top)))
1.95/2.11	= { by axiom 9 (def_top_12) }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(converse(join(converse(X), complement(converse(X))))))
1.95/2.11	= { by lemma 44 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(join(X, converse(complement(converse(X))))))
1.95/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(join(converse(complement(converse(X))), X)))
1.95/2.11	= { by lemma 38 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(join(converse(complement(converse(X))), meet(X, top))))
1.95/2.11	= { by lemma 19 }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(join(converse(complement(converse(X))), complement(join(zero, complement(X))))))
1.95/2.11	= { by axiom 1 (maddux1_join_commutativity_1) }
1.95/2.11	  join(meet(join(zero, complement(X)), converse(complement(converse(X)))), complement(join(complement(join(zero, complement(X))), converse(complement(converse(X))))))
1.95/2.11	= { by lemma 29 }
1.95/2.11	  join(zero, complement(X))
1.95/2.11	= { by lemma 37 }
1.97/2.14	  complement(X)
1.97/2.14	
1.97/2.14	Lemma 48: converse(complement(converse(X))) = complement(X).
1.97/2.14	Proof:
1.97/2.14	  converse(complement(converse(X)))
1.97/2.14	= { by lemma 43 }
1.97/2.14	  join(meet(X, converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 5 (converse_idempotence_8) }
1.97/2.14	  join(meet(converse(converse(X)), converse(complement(converse(X)))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 17 }
1.97/2.14	  join(meet(converse(complement(converse(X))), converse(converse(X))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 40 }
1.97/2.14	  join(meet(converse(complement(converse(X))), meet(top, converse(converse(X)))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 38 }
1.97/2.14	  join(meet(meet(converse(complement(converse(X))), meet(top, converse(converse(X)))), top), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 19 }
1.97/2.14	  join(complement(join(zero, complement(meet(converse(complement(converse(X))), meet(top, converse(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.97/2.14	  join(complement(join(zero, complement(meet(converse(complement(converse(X))), complement(join(complement(top), complement(converse(converse(X))))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 17 }
1.97/2.14	  join(complement(join(zero, complement(meet(complement(join(complement(top), complement(converse(converse(X))))), converse(complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 37 }
1.97/2.14	  join(complement(join(zero, complement(meet(join(zero, complement(join(complement(top), complement(converse(converse(X)))))), converse(complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 42 }
1.97/2.14	  join(complement(join(zero, join(complement(join(zero, complement(join(complement(top), complement(converse(converse(X))))))), complement(converse(complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 19 }
1.97/2.14	  join(complement(join(zero, join(meet(join(complement(top), complement(converse(converse(X)))), top), complement(converse(complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 38 }
1.97/2.14	  join(complement(join(zero, join(join(complement(top), complement(converse(converse(X)))), complement(converse(complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 6 (maddux2_join_associativity_2) }
1.97/2.14	  join(complement(join(zero, join(complement(top), join(complement(converse(converse(X))), complement(converse(complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 42 }
1.97/2.14	  join(complement(join(zero, join(complement(top), complement(meet(converse(converse(X)), converse(complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 42 }
1.97/2.14	  join(complement(join(zero, complement(meet(top, meet(converse(converse(X)), converse(complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 19 }
1.97/2.14	  join(meet(meet(top, meet(converse(converse(X)), converse(complement(converse(X))))), top), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 38 }
1.97/2.14	  join(meet(top, meet(converse(converse(X)), converse(complement(converse(X))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 17 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), top), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 28 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))), top)), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 9 (def_top_12) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))), join(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 6 (maddux2_join_associativity_2) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(join(converse(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X))))), converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 11 (converse_additivity_9) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(converse(join(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))), meet(converse(complement(converse(converse(X)))), complement(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(join(complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))), meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(join(meet(converse(complement(converse(converse(X)))), complement(converse(X))), complement(join(complement(converse(complement(converse(converse(X))))), complement(converse(X)))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 29 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))), converse(converse(complement(converse(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(converse(converse(complement(converse(converse(X))))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 5 (converse_idempotence_8) }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(converse(complement(converse(converse(X)))), complement(converse(X))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 17 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(meet(complement(converse(X)), converse(complement(converse(converse(X))))))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 47 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), join(complement(converse(converse(X))), complement(converse(complement(converse(X)))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 42 }
1.97/2.14	  join(meet(meet(converse(converse(X)), converse(complement(converse(X)))), complement(meet(converse(converse(X)), converse(complement(converse(X)))))), meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by axiom 8 (def_zero_13) }
1.97/2.14	  join(zero, meet(converse(complement(converse(X))), complement(X)))
1.97/2.14	= { by lemma 17 }
1.97/2.14	  join(zero, meet(complement(X), converse(complement(converse(X)))))
1.97/2.14	= { by lemma 37 }
1.97/2.14	  meet(complement(X), converse(complement(converse(X))))
1.97/2.14	= { by lemma 47 }
1.97/2.14	  complement(X)
1.97/2.14	
1.97/2.14	Lemma 49: join(top, X) = top.
1.97/2.14	Proof:
1.97/2.14	  join(top, X)
1.97/2.14	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.14	  join(X, top)
1.97/2.14	= { by lemma 28 }
1.97/2.14	  top
1.97/2.14	
1.97/2.14	Lemma 50: join(Y, composition(X, Y)) = composition(join(X, one), Y).
1.97/2.14	Proof:
1.97/2.14	  join(Y, composition(X, Y))
1.97/2.14	= { by lemma 22 }
1.97/2.14	  join(composition(one, Y), composition(X, Y))
1.97/2.14	= { by axiom 13 (composition_distributivity_7) }
1.97/2.14	  composition(join(one, X), Y)
1.97/2.14	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.14	  composition(join(X, one), Y)
1.97/2.14	
1.97/2.14	Lemma 51: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
1.97/2.14	Proof:
1.97/2.14	  converse(composition(X, converse(Y)))
1.97/2.14	= { by axiom 10 (converse_multiplicativity_10) }
1.97/2.14	  composition(converse(converse(Y)), converse(X))
1.97/2.14	= { by axiom 5 (converse_idempotence_8) }
1.97/2.14	  composition(Y, converse(X))
1.97/2.14	
1.97/2.14	Lemma 52: meet(zero, X) = zero.
1.97/2.14	Proof:
1.97/2.14	  meet(zero, X)
1.97/2.14	= { by lemma 17 }
1.97/2.14	  meet(X, zero)
1.97/2.14	= { by lemma 33 }
1.97/2.14	  zero
1.97/2.14	
1.97/2.14	Lemma 53: composition(top, zero) = zero.
1.97/2.14	Proof:
1.97/2.14	  composition(top, zero)
1.97/2.14	= { by lemma 49 }
1.97/2.14	  composition(join(top, one), zero)
1.97/2.14	= { by lemma 46 }
1.97/2.14	  composition(join(converse(top), one), zero)
1.97/2.14	= { by lemma 18 }
1.97/2.14	  composition(join(converse(top), one), complement(top))
1.97/2.14	= { by lemma 50 }
1.97/2.14	  join(complement(top), composition(converse(top), complement(top)))
1.97/2.14	= { by lemma 49 }
1.97/2.14	  join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
1.97/2.14	= { by lemma 50 }
1.97/2.14	  join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
1.97/2.14	= { by lemma 49 }
1.97/2.14	  join(complement(top), composition(converse(top), complement(composition(top, top))))
1.97/2.14	= { by lemma 23 }
1.97/2.14	  complement(top)
1.97/2.14	= { by lemma 18 }
1.97/2.14	  zero
1.97/2.14	
1.97/2.14	Lemma 54: composition(X, zero) = zero.
1.97/2.14	Proof:
1.97/2.14	  composition(X, zero)
1.97/2.14	= { by lemma 37 }
1.97/2.14	  join(zero, composition(X, zero))
1.97/2.14	= { by lemma 53 }
1.97/2.14	  join(composition(top, zero), composition(X, zero))
1.97/2.14	= { by axiom 13 (composition_distributivity_7) }
1.97/2.14	  composition(join(top, X), zero)
1.97/2.14	= { by lemma 49 }
1.97/2.14	  composition(top, zero)
1.97/2.14	= { by lemma 53 }
1.97/2.14	  zero
1.97/2.14	
1.97/2.14	Lemma 55: composition(converse(complement(sk2)), sk2) = zero.
1.97/2.14	Proof:
1.97/2.14	  composition(converse(complement(sk2)), sk2)
1.97/2.14	= { by lemma 20 }
1.97/2.14	  converse(composition(converse(sk2), complement(sk2)))
1.97/2.14	= { by lemma 37 }
1.97/2.14	  converse(join(zero, composition(converse(sk2), complement(sk2))))
1.97/2.14	= { by lemma 18 }
1.97/2.14	  converse(join(complement(top), composition(converse(sk2), complement(sk2))))
1.97/2.14	= { by axiom 16 (goals_18) }
1.97/2.14	  converse(join(complement(top), composition(converse(sk2), complement(composition(sk2, top)))))
1.97/2.14	= { by lemma 23 }
1.97/2.14	  converse(complement(top))
1.97/2.14	= { by lemma 18 }
1.97/2.14	  converse(zero)
1.97/2.14	= { by lemma 37 }
1.97/2.14	  join(zero, converse(zero))
1.97/2.14	= { by lemma 44 }
1.97/2.14	  converse(join(converse(zero), zero))
1.97/2.14	= { by lemma 36 }
1.97/2.14	  converse(converse(zero))
1.97/2.14	= { by axiom 5 (converse_idempotence_8) }
1.97/2.14	  zero
1.97/2.14	
1.97/2.14	Lemma 56: composition(converse(X), complement(composition(X, top))) = zero.
1.97/2.14	Proof:
1.97/2.14	  composition(converse(X), complement(composition(X, top)))
1.97/2.14	= { by lemma 37 }
1.97/2.14	  join(zero, composition(converse(X), complement(composition(X, top))))
1.97/2.14	= { by lemma 18 }
1.97/2.14	  join(complement(top), composition(converse(X), complement(composition(X, top))))
1.97/2.14	= { by lemma 23 }
1.97/2.14	  complement(top)
1.97/2.14	= { by lemma 18 }
1.97/2.14	  zero
1.97/2.14	
1.97/2.14	Lemma 57: meet(X, complement(composition(X, top))) = zero.
1.97/2.14	Proof:
1.97/2.14	  meet(X, complement(composition(X, top)))
1.97/2.14	= { by axiom 3 (composition_identity_6) }
1.97/2.14	  meet(composition(X, one), complement(composition(X, top)))
1.97/2.14	= { by lemma 36 }
1.97/2.14	  join(meet(composition(X, one), complement(composition(X, top))), zero)
1.97/2.14	= { by lemma 52 }
1.97/2.14	  join(meet(composition(X, one), complement(composition(X, top))), meet(zero, complement(composition(X, top))))
1.97/2.14	= { by lemma 54 }
1.97/2.14	  join(meet(composition(X, one), complement(composition(X, top))), meet(composition(X, zero), complement(composition(X, top))))
1.97/2.14	= { by lemma 33 }
1.97/2.14	  join(meet(composition(X, one), complement(composition(X, top))), meet(composition(X, meet(one, zero)), complement(composition(X, top))))
1.97/2.14	= { by lemma 56 }
1.97/2.14	  join(meet(composition(X, one), complement(composition(X, top))), meet(composition(X, meet(one, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top))))
1.97/2.14	= { by axiom 14 (modular_law_1_15) }
1.97/2.14	  meet(composition(X, meet(one, composition(converse(X), complement(composition(X, top))))), complement(composition(X, top)))
1.97/2.14	= { by lemma 56 }
1.97/2.14	  meet(composition(X, meet(one, zero)), complement(composition(X, top)))
1.97/2.14	= { by lemma 33 }
1.97/2.14	  meet(composition(X, zero), complement(composition(X, top)))
1.97/2.14	= { by lemma 54 }
1.97/2.14	  meet(zero, complement(composition(X, top)))
1.97/2.14	= { by lemma 52 }
1.97/2.17	  zero
1.97/2.17	
1.97/2.17	Lemma 58: composition(complement(meet(sk1, sk2)), top) = complement(meet(sk1, sk2)).
1.97/2.17	Proof:
1.97/2.17	  composition(complement(meet(sk1, sk2)), top)
1.97/2.17	= { by lemma 17 }
1.97/2.17	  composition(complement(meet(sk2, sk1)), top)
1.97/2.17	= { by lemma 42 }
1.97/2.17	  composition(join(complement(sk2), complement(sk1)), top)
1.97/2.17	= { by lemma 48 }
1.97/2.17	  composition(join(complement(sk2), converse(complement(converse(sk1)))), top)
1.97/2.17	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.17	  composition(join(converse(complement(converse(sk1))), complement(sk2)), top)
1.97/2.17	= { by lemma 46 }
1.97/2.17	  composition(join(converse(complement(converse(sk1))), complement(sk2)), converse(top))
1.97/2.17	= { by axiom 13 (composition_distributivity_7) }
1.97/2.17	  join(composition(converse(complement(converse(sk1))), converse(top)), composition(complement(sk2), converse(top)))
1.97/2.17	= { by axiom 10 (converse_multiplicativity_10) }
1.97/2.17	  join(converse(composition(top, complement(converse(sk1)))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 49 }
1.97/2.17	  join(converse(composition(join(top, one), complement(converse(sk1)))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 46 }
1.97/2.17	  join(converse(composition(join(converse(top), one), complement(converse(sk1)))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 50 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(converse(sk1))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 21 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(converse(one), converse(sk1)))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 51 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(converse(composition(sk1, converse(converse(one)))))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by axiom 15 (goals_17) }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(converse(composition(composition(sk1, top), converse(converse(one)))))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by axiom 12 (composition_associativity_5) }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(converse(composition(sk1, composition(top, converse(converse(one))))))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by axiom 10 (converse_multiplicativity_10) }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(converse(composition(top, converse(converse(one)))), converse(sk1)))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 51 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(composition(converse(one), converse(top)), converse(sk1)))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by axiom 12 (composition_associativity_5) }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(converse(one), composition(converse(top), converse(sk1))))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 46 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(converse(one), composition(top, converse(sk1))))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 21 }
1.97/2.17	  join(converse(join(complement(converse(sk1)), composition(converse(top), complement(composition(top, converse(sk1)))))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 23 }
1.97/2.17	  join(converse(complement(converse(sk1))), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 48 }
1.97/2.17	  join(complement(sk1), composition(complement(sk2), converse(top)))
1.97/2.17	= { by lemma 46 }
1.97/2.17	  join(complement(sk1), composition(complement(sk2), top))
1.97/2.17	= { by lemma 43 }
1.97/2.17	  join(complement(sk1), join(meet(sk2, composition(complement(sk2), top)), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 17 }
1.97/2.17	  join(complement(sk1), join(meet(composition(complement(sk2), top), sk2), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 36 }
1.97/2.17	  join(complement(sk1), join(join(meet(composition(complement(sk2), top), sk2), zero), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 52 }
1.97/2.17	  join(complement(sk1), join(join(meet(composition(complement(sk2), top), sk2), meet(zero, sk2)), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 54 }
1.97/2.17	  join(complement(sk1), join(join(meet(composition(complement(sk2), top), sk2), meet(composition(complement(sk2), zero), sk2)), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 33 }
1.97/2.17	  join(complement(sk1), join(join(meet(composition(complement(sk2), top), sk2), meet(composition(complement(sk2), meet(top, zero)), sk2)), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 55 }
1.97/2.17	  join(complement(sk1), join(join(meet(composition(complement(sk2), top), sk2), meet(composition(complement(sk2), meet(top, composition(converse(complement(sk2)), sk2))), sk2)), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by axiom 14 (modular_law_1_15) }
1.97/2.17	  join(complement(sk1), join(meet(composition(complement(sk2), meet(top, composition(converse(complement(sk2)), sk2))), sk2), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 55 }
1.97/2.17	  join(complement(sk1), join(meet(composition(complement(sk2), meet(top, zero)), sk2), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 33 }
1.97/2.17	  join(complement(sk1), join(meet(composition(complement(sk2), zero), sk2), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 54 }
1.97/2.17	  join(complement(sk1), join(meet(zero, sk2), meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 52 }
1.97/2.17	  join(complement(sk1), join(zero, meet(composition(complement(sk2), top), complement(sk2))))
1.97/2.17	= { by lemma 37 }
1.97/2.17	  join(complement(sk1), meet(composition(complement(sk2), top), complement(sk2)))
1.97/2.17	= { by lemma 17 }
1.97/2.17	  join(complement(sk1), meet(complement(sk2), composition(complement(sk2), top)))
1.97/2.17	= { by axiom 2 (maddux4_definiton_of_meet_4) }
1.97/2.17	  join(complement(sk1), complement(join(complement(complement(sk2)), complement(composition(complement(sk2), top)))))
1.97/2.17	= { by lemma 37 }
1.97/2.17	  join(complement(sk1), join(zero, complement(join(complement(complement(sk2)), complement(composition(complement(sk2), top))))))
1.97/2.17	= { by lemma 57 }
1.97/2.17	  join(complement(sk1), join(meet(complement(sk2), complement(composition(complement(sk2), top))), complement(join(complement(complement(sk2)), complement(composition(complement(sk2), top))))))
1.97/2.17	= { by lemma 29 }
1.97/2.17	  join(complement(sk1), complement(sk2))
1.97/2.17	= { by lemma 42 }
1.97/2.17	  complement(meet(sk1, sk2))
1.97/2.17	
1.97/2.17	Lemma 59: meet(X, X) = X.
1.97/2.17	Proof:
1.97/2.17	  meet(X, X)
1.97/2.17	= { by lemma 37 }
1.97/2.17	  join(zero, meet(X, X))
1.97/2.17	= { by lemma 34 }
1.97/2.17	  X
1.97/2.17	
1.97/2.17	Lemma 60: complement(join(X, composition(X, top))) = complement(composition(X, top)).
1.97/2.17	Proof:
1.97/2.17	  complement(join(X, composition(X, top)))
1.97/2.17	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.17	  complement(join(composition(X, top), X))
1.97/2.17	= { by axiom 1 (maddux1_join_commutativity_1) }
1.97/2.17	  complement(join(X, composition(X, top)))
1.97/2.17	= { by lemma 38 }
1.97/2.17	  complement(join(X, meet(composition(X, top), top)))
1.97/2.17	= { by lemma 19 }
1.97/2.17	  complement(join(X, complement(join(zero, complement(composition(X, top))))))
1.97/2.17	= { by lemma 59 }
1.97/2.17	  complement(join(X, meet(complement(join(zero, complement(composition(X, top)))), complement(join(zero, complement(composition(X, top)))))))
1.97/2.17	= { by lemma 35 }
1.97/2.17	  complement(join(complement(join(zero, complement(composition(X, top)))), meet(X, X)))
1.97/2.17	= { by lemma 41 }
1.97/2.17	  meet(join(zero, complement(composition(X, top))), join(complement(X), complement(X)))
1.97/2.17	= { by lemma 37 }
1.97/2.17	  meet(complement(composition(X, top)), join(complement(X), complement(X)))
1.97/2.17	= { by lemma 42 }
1.97/2.17	  meet(complement(composition(X, top)), complement(meet(X, X)))
1.97/2.17	= { by lemma 59 }
1.97/2.17	  meet(complement(composition(X, top)), complement(X))
1.97/2.17	= { by lemma 37 }
1.97/2.17	  join(zero, meet(complement(composition(X, top)), complement(X)))
1.97/2.17	= { by lemma 57 }
1.97/2.17	  join(meet(X, complement(composition(X, top))), meet(complement(composition(X, top)), complement(X)))
1.97/2.17	= { by lemma 43 }
1.97/2.17	  complement(composition(X, top))
1.97/2.17	
1.97/2.17	Lemma 61: complement(converse(X)) = converse(complement(X)).
1.97/2.17	Proof:
1.97/2.17	  complement(converse(X))
1.97/2.17	= { by axiom 5 (converse_idempotence_8) }
1.97/2.17	  converse(converse(complement(converse(X))))
1.97/2.17	= { by lemma 48 }
1.97/2.19	  converse(complement(X))
1.97/2.19	
1.97/2.19	Goal 1 (goals_19): meet(sk1, sk2) = composition(meet(sk1, sk2), top).
1.97/2.19	Proof:
1.97/2.19	  meet(sk1, sk2)
1.97/2.19	= { by lemma 39 }
1.97/2.19	  complement(complement(meet(sk1, sk2)))
1.97/2.19	= { by lemma 58 }
1.97/2.19	  complement(composition(complement(meet(sk1, sk2)), top))
1.97/2.19	= { by lemma 60 }
1.97/2.19	  complement(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)))
1.97/2.19	= { by lemma 48 }
1.97/2.19	  converse(complement(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)))))
1.97/2.19	= { by lemma 61 }
1.97/2.19	  complement(converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)))))
1.97/2.19	= { by lemma 23 }
1.97/2.19	  join(complement(converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))), composition(converse(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 61 }
1.97/2.19	  join(converse(complement(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))), composition(converse(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 48 }
1.97/2.19	  join(complement(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 60 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(converse(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 5 (converse_idempotence_8) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 13 (composition_distributivity_7) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(composition(complement(meet(sk1, sk2)), top), composition(composition(complement(meet(sk1, sk2)), top), top))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 39 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(composition(complement(meet(sk1, sk2)), top), top))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 5 (converse_idempotence_8) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), top))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 32 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), complement(zero)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 54 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), complement(composition(converse(top), zero))))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 56 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), complement(composition(converse(top), composition(converse(complement(meet(sk1, sk2))), complement(composition(complement(meet(sk1, sk2)), top)))))))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 12 (composition_associativity_5) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), complement(composition(composition(converse(top), converse(complement(meet(sk1, sk2)))), complement(composition(complement(meet(sk1, sk2)), top))))))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 10 (converse_multiplicativity_10) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(join(complement(complement(composition(complement(meet(sk1, sk2)), top))), composition(converse(converse(composition(complement(meet(sk1, sk2)), top))), complement(composition(converse(composition(complement(meet(sk1, sk2)), top)), complement(composition(complement(meet(sk1, sk2)), top))))))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 23 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(complement(complement(composition(complement(meet(sk1, sk2)), top)))), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by lemma 39 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), complement(composition(converse(complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top))), converse(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))))))))
1.97/2.19	= { by axiom 10 (converse_multiplicativity_10) }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), complement(converse(composition(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))), complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))))))
1.97/2.19	= { by lemma 61 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), converse(complement(composition(converse(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top))), complement(composition(join(complement(meet(sk1, sk2)), composition(complement(meet(sk1, sk2)), top)), top)))))))
1.97/2.19	= { by lemma 56 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), converse(complement(zero))))
1.97/2.19	= { by lemma 32 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), converse(top)))
1.97/2.19	= { by lemma 46 }
1.97/2.19	  join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), top))
1.97/2.19	= { by lemma 38 }
1.97/2.19	  meet(join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), top)), top)
1.97/2.19	= { by lemma 19 }
1.97/2.19	  complement(join(zero, complement(join(complement(composition(complement(meet(sk1, sk2)), top)), composition(complement(composition(complement(meet(sk1, sk2)), top)), top)))))
1.97/2.19	= { by lemma 60 }
1.97/2.19	  complement(join(zero, complement(composition(complement(composition(complement(meet(sk1, sk2)), top)), top))))
1.97/2.19	= { by lemma 19 }
1.97/2.19	  meet(composition(complement(composition(complement(meet(sk1, sk2)), top)), top), top)
1.97/2.19	= { by lemma 38 }
1.97/2.19	  composition(complement(composition(complement(meet(sk1, sk2)), top)), top)
1.97/2.19	= { by lemma 58 }
1.97/2.19	  composition(complement(complement(meet(sk1, sk2))), top)
1.97/2.19	= { by lemma 39 }
1.97/2.19	  composition(meet(sk1, sk2), top)
1.97/2.19	% SZS output end Proof
1.97/2.19	
1.97/2.19	RESULT: Unsatisfiable (the axioms are contradictory).
1.97/2.19	EOF