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
0.12/0.33	% Computer   : n024.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 14:28:50 EDT 2019
0.12/0.33	% CPUTime    : 
43.65/43.87	% SZS status Unsatisfiable
43.65/43.87	
43.71/43.87	% SZS output start Proof
43.71/43.87	Take the following subset of the input axioms:
43.71/43.90	  fof(composition_associativity_5, axiom, ![A, B, C]: composition(composition(A, B), C)=composition(A, composition(B, C))).
43.71/43.90	  fof(composition_distributivity_7, axiom, ![A, B, C]: composition(join(A, B), C)=join(composition(A, C), composition(B, C))).
43.71/43.90	  fof(composition_identity_6, axiom, ![A]: A=composition(A, one)).
43.71/43.90	  fof(converse_additivity_9, axiom, ![A, B]: join(converse(A), converse(B))=converse(join(A, B))).
43.71/43.90	  fof(converse_cancellativity_11, axiom, ![A, B]: join(composition(converse(A), complement(composition(A, B))), complement(B))=complement(B)).
43.71/43.90	  fof(converse_idempotence_8, axiom, ![A]: converse(converse(A))=A).
43.71/43.90	  fof(converse_multiplicativity_10, axiom, ![A, B]: converse(composition(A, B))=composition(converse(B), converse(A))).
43.71/43.90	  fof(def_top_12, axiom, ![A]: join(A, complement(A))=top).
43.71/43.90	  fof(def_zero_13, axiom, ![A]: zero=meet(A, complement(A))).
43.71/43.90	  fof(goals_14, negated_conjecture, sk1=composition(sk1, top)).
43.71/43.90	  fof(goals_15, negated_conjecture, composition(meet(sk1, one), sk2)!=meet(sk1, sk2)).
43.71/43.90	  fof(maddux1_join_commutativity_1, axiom, ![A, B]: join(B, A)=join(A, B)).
43.71/43.90	  fof(maddux2_join_associativity_2, axiom, ![A, B, C]: join(join(A, B), C)=join(A, join(B, C))).
43.71/43.90	  fof(maddux3_a_kind_of_de_Morgan_3, axiom, ![A, B]: join(complement(join(complement(A), complement(B))), complement(join(complement(A), B)))=A).
43.71/43.90	  fof(maddux4_definiton_of_meet_4, axiom, ![A, B]: complement(join(complement(A), complement(B)))=meet(A, B)).
43.71/43.90	
43.71/43.90	Now clausify the problem and encode Horn clauses using encoding 3 of
43.71/43.90	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
43.71/43.90	We repeatedly replace C & s=t => u=v by the two clauses:
43.71/43.90	  fresh(y, y, x1...xn) = u
43.71/43.90	  C => fresh(s, t, x1...xn) = v
43.71/43.90	where fresh is a fresh function symbol and x1..xn are the free
43.71/43.90	variables of u and v.
43.71/43.90	A predicate p(X) is encoded as p(X)=true (this is sound, because the
43.71/43.90	input problem has no model of domain size 1).
43.71/43.90	
43.71/43.90	The encoding turns the above axioms into the following unit equations and goals:
43.71/43.90	
43.71/43.90	Axiom 1 (maddux1_join_commutativity_1): join(X, Y) = join(Y, X).
43.71/43.90	Axiom 2 (maddux4_definiton_of_meet_4): complement(join(complement(X), complement(Y))) = meet(X, Y).
43.71/43.90	Axiom 3 (composition_identity_6): X = composition(X, one).
43.71/43.90	Axiom 4 (maddux3_a_kind_of_de_Morgan_3): join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))) = X.
43.71/43.90	Axiom 5 (converse_idempotence_8): converse(converse(X)) = X.
43.71/43.90	Axiom 6 (maddux2_join_associativity_2): join(join(X, Y), Z) = join(X, join(Y, Z)).
43.71/43.90	Axiom 7 (converse_cancellativity_11): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
43.71/43.90	Axiom 8 (def_zero_13): zero = meet(X, complement(X)).
43.71/43.90	Axiom 9 (def_top_12): join(X, complement(X)) = top.
43.71/43.90	Axiom 10 (converse_multiplicativity_10): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
43.71/43.90	Axiom 11 (converse_additivity_9): join(converse(X), converse(Y)) = converse(join(X, Y)).
43.71/43.90	Axiom 12 (composition_associativity_5): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
43.71/43.90	Axiom 13 (composition_distributivity_7): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
43.71/43.92	Axiom 14 (goals_14): sk1 = composition(sk1, top).
43.71/43.92	
43.71/43.92	Lemma 15: meet(X, Y) = meet(Y, X).
43.71/43.92	Proof:
43.71/43.92	  meet(X, Y)
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  complement(join(complement(X), complement(Y)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  complement(join(complement(Y), complement(X)))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  meet(Y, X)
43.71/43.92	
43.71/43.92	Lemma 16: complement(top) = zero.
43.71/43.92	Proof:
43.71/43.92	  complement(top)
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  complement(join(complement(?), complement(complement(?))))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  meet(?, complement(?))
43.71/43.92	= { by axiom 8 (def_zero_13) }
43.71/43.92	  zero
43.71/43.92	
43.71/43.92	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
43.71/43.92	Proof:
43.71/43.92	  complement(join(zero, complement(X)))
43.71/43.92	= { by lemma 16 }
43.71/43.92	  complement(join(complement(top), complement(X)))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  meet(top, X)
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(X, top)
43.71/43.92	
43.71/43.92	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
43.71/43.92	Proof:
43.71/43.92	  converse(composition(converse(X), Y))
43.71/43.92	= { by axiom 10 (converse_multiplicativity_10) }
43.71/43.92	  composition(converse(Y), converse(converse(X)))
43.71/43.92	= { by axiom 5 (converse_idempotence_8) }
43.71/43.92	  composition(converse(Y), X)
43.71/43.92	
43.71/43.92	Lemma 19: composition(converse(one), X) = X.
43.71/43.92	Proof:
43.71/43.92	  composition(converse(one), X)
43.71/43.92	= { by lemma 18 }
43.71/43.92	  converse(composition(converse(X), one))
43.71/43.92	= { by axiom 3 (composition_identity_6) }
43.71/43.92	  converse(converse(X))
43.71/43.92	= { by axiom 5 (converse_idempotence_8) }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 20: composition(one, X) = X.
43.71/43.92	Proof:
43.71/43.92	  composition(one, X)
43.71/43.92	= { by lemma 19 }
43.71/43.92	  composition(converse(one), composition(one, X))
43.71/43.92	= { by axiom 12 (composition_associativity_5) }
43.71/43.92	  composition(composition(converse(one), one), X)
43.71/43.92	= { by axiom 3 (composition_identity_6) }
43.71/43.92	  composition(converse(one), X)
43.71/43.92	= { by lemma 19 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 21: join(complement(Y), composition(converse(X), complement(composition(X, Y)))) = complement(Y).
43.71/43.92	Proof:
43.71/43.92	  join(complement(Y), composition(converse(X), complement(composition(X, Y))))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(composition(converse(X), complement(composition(X, Y))), complement(Y))
43.71/43.92	= { by axiom 7 (converse_cancellativity_11) }
43.71/43.92	  complement(Y)
43.71/43.92	
43.71/43.92	Lemma 22: join(complement(X), complement(X)) = complement(X).
43.71/43.92	Proof:
43.71/43.92	  join(complement(X), complement(X))
43.71/43.92	= { by lemma 19 }
43.71/43.92	  join(complement(X), composition(converse(one), complement(X)))
43.71/43.92	= { by lemma 20 }
43.71/43.92	  join(complement(X), composition(converse(one), complement(composition(one, X))))
43.71/43.92	= { by lemma 21 }
43.71/43.92	  complement(X)
43.71/43.92	
43.71/43.92	Lemma 23: meet(X, X) = complement(complement(X)).
43.71/43.92	Proof:
43.71/43.92	  meet(X, X)
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  complement(join(complement(X), complement(X)))
43.71/43.92	= { by lemma 22 }
43.71/43.92	  complement(complement(X))
43.71/43.92	
43.71/43.92	Lemma 24: join(meet(X, Y), complement(join(complement(X), Y))) = X.
43.71/43.92	Proof:
43.71/43.92	  join(meet(X, Y), complement(join(complement(X), Y)))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
43.71/43.92	= { by axiom 4 (maddux3_a_kind_of_de_Morgan_3) }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 25: join(meet(X, Y), meet(X, complement(Y))) = X.
43.71/43.92	Proof:
43.71/43.92	  join(meet(X, Y), meet(X, complement(Y)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(meet(X, complement(Y)), meet(X, Y))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
43.71/43.92	= { by lemma 24 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 26: join(zero, complement(complement(X))) = X.
43.71/43.92	Proof:
43.71/43.92	  join(zero, complement(complement(X)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(complement(complement(X)), zero)
43.71/43.92	= { by lemma 23 }
43.71/43.92	  join(meet(X, X), zero)
43.71/43.92	= { by axiom 8 (def_zero_13) }
43.71/43.92	  join(meet(X, X), meet(X, complement(X)))
43.71/43.92	= { by lemma 25 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 27: join(X, join(Y, Z)) = join(Z, join(X, Y)).
43.71/43.92	Proof:
43.71/43.92	  join(X, join(Y, Z))
43.71/43.92	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.92	  join(join(X, Y), Z)
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(Z, join(X, Y))
43.71/43.92	
43.71/43.92	Lemma 28: join(X, join(complement(X), Y)) = join(Y, top).
43.71/43.92	Proof:
43.71/43.92	  join(X, join(complement(X), Y))
43.71/43.92	= { by lemma 27 }
43.71/43.92	  join(complement(X), join(Y, X))
43.71/43.92	= { by lemma 27 }
43.71/43.92	  join(Y, join(X, complement(X)))
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  join(Y, top)
43.71/43.92	
43.71/43.92	Lemma 29: join(X, top) = top.
43.71/43.92	Proof:
43.71/43.92	  join(X, top)
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  join(X, join(complement(X), complement(complement(X))))
43.71/43.92	= { by lemma 28 }
43.71/43.92	  join(complement(complement(X)), top)
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(top, complement(complement(X)))
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
43.71/43.92	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.92	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
43.71/43.92	= { by lemma 22 }
43.71/43.92	  join(complement(X), complement(complement(X)))
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  top
43.71/43.92	
43.71/43.92	Lemma 30: join(zero, meet(X, top)) = X.
43.71/43.92	Proof:
43.71/43.92	  join(zero, meet(X, top))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(meet(X, top), zero)
43.71/43.92	= { by lemma 16 }
43.71/43.92	  join(meet(X, top), complement(top))
43.71/43.92	= { by lemma 29 }
43.71/43.92	  join(meet(X, top), complement(join(complement(X), top)))
43.71/43.92	= { by lemma 24 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 31: join(zero, complement(X)) = complement(X).
43.71/43.92	Proof:
43.71/43.92	  join(zero, complement(X))
43.71/43.92	= { by lemma 26 }
43.71/43.92	  join(zero, complement(join(zero, complement(complement(X)))))
43.71/43.92	= { by lemma 17 }
43.71/43.92	  join(zero, meet(complement(X), top))
43.71/43.92	= { by lemma 30 }
43.71/43.92	  complement(X)
43.71/43.92	
43.71/43.92	Lemma 32: complement(complement(X)) = X.
43.71/43.92	Proof:
43.71/43.92	  complement(complement(X))
43.71/43.92	= { by lemma 31 }
43.71/43.92	  join(zero, complement(complement(X)))
43.71/43.92	= { by lemma 26 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 33: join(meet(Y, X), meet(X, complement(Y))) = X.
43.71/43.92	Proof:
43.71/43.92	  join(meet(Y, X), meet(X, complement(Y)))
43.71/43.92	= { by lemma 15 }
43.71/43.92	  join(meet(X, Y), meet(X, complement(Y)))
43.71/43.92	= { by lemma 25 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 34: join(X, zero) = X.
43.71/43.92	Proof:
43.71/43.92	  join(X, zero)
43.71/43.92	= { by lemma 32 }
43.71/43.92	  join(complement(complement(X)), zero)
43.71/43.92	= { by lemma 23 }
43.71/43.92	  join(meet(X, X), zero)
43.71/43.92	= { by axiom 8 (def_zero_13) }
43.71/43.92	  join(meet(X, X), meet(X, complement(X)))
43.71/43.92	= { by lemma 33 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 35: join(zero, X) = X.
43.71/43.92	Proof:
43.71/43.92	  join(zero, X)
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(X, zero)
43.71/43.92	= { by lemma 34 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 36: meet(X, top) = X.
43.71/43.92	Proof:
43.71/43.92	  meet(X, top)
43.71/43.92	= { by lemma 17 }
43.71/43.92	  complement(join(zero, complement(X)))
43.71/43.92	= { by lemma 31 }
43.71/43.92	  join(zero, complement(join(zero, complement(X))))
43.71/43.92	= { by lemma 17 }
43.71/43.92	  join(zero, meet(X, top))
43.71/43.92	= { by lemma 30 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 37: meet(top, X) = X.
43.71/43.92	Proof:
43.71/43.92	  meet(top, X)
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(X, top)
43.71/43.92	= { by lemma 36 }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 38: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
43.71/43.92	Proof:
43.71/43.92	  meet(X, join(complement(Y), complement(Z)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  meet(X, join(complement(Z), complement(Y)))
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(join(complement(Z), complement(Y)), X)
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  complement(join(meet(Z, Y), complement(X)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  complement(join(complement(X), meet(Z, Y)))
43.71/43.92	= { by lemma 15 }
43.71/43.92	  complement(join(complement(X), meet(Y, Z)))
43.71/43.92	
43.71/43.92	Lemma 39: complement(join(X, complement(Y))) = meet(Y, complement(X)).
43.71/43.92	Proof:
43.71/43.92	  complement(join(X, complement(Y)))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  complement(join(complement(Y), X))
43.71/43.92	= { by lemma 37 }
43.71/43.92	  complement(join(complement(Y), meet(top, X)))
43.71/43.92	= { by lemma 38 }
43.71/43.92	  meet(Y, join(complement(top), complement(X)))
43.71/43.92	= { by lemma 16 }
43.71/43.92	  meet(Y, join(zero, complement(X)))
43.71/43.92	= { by lemma 31 }
43.71/43.92	  meet(Y, complement(X))
43.71/43.92	
43.71/43.92	Lemma 40: complement(join(complement(X), Y)) = meet(X, complement(Y)).
43.71/43.92	Proof:
43.71/43.92	  complement(join(complement(X), Y))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  complement(join(Y, complement(X)))
43.71/43.92	= { by lemma 39 }
43.71/43.92	  meet(X, complement(Y))
43.71/43.92	
43.71/43.92	Lemma 41: meet(complement(X), complement(Y)) = complement(join(X, Y)).
43.71/43.92	Proof:
43.71/43.92	  meet(complement(X), complement(Y))
43.71/43.92	= { by lemma 31 }
43.71/43.92	  meet(join(zero, complement(X)), complement(Y))
43.71/43.92	= { by lemma 39 }
43.71/43.92	  complement(join(Y, complement(join(zero, complement(X)))))
43.71/43.92	= { by lemma 17 }
43.71/43.92	  complement(join(Y, meet(X, top)))
43.71/43.92	= { by lemma 36 }
43.71/43.92	  complement(join(Y, X))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  complement(join(X, Y))
43.71/43.92	
43.71/43.92	Lemma 42: meet(X, meet(Y, complement(Z))) = meet(complement(Z), meet(X, Y)).
43.71/43.92	Proof:
43.71/43.92	  meet(X, meet(Y, complement(Z)))
43.71/43.92	= { by lemma 40 }
43.71/43.92	  meet(X, complement(join(complement(Y), Z)))
43.71/43.92	= { by lemma 40 }
43.71/43.92	  complement(join(complement(X), join(complement(Y), Z)))
43.71/43.92	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.92	  complement(join(join(complement(X), complement(Y)), Z))
43.71/43.92	= { by lemma 41 }
43.71/43.92	  meet(complement(join(complement(X), complement(Y))), complement(Z))
43.71/43.92	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.92	  meet(meet(X, Y), complement(Z))
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(complement(Z), meet(X, Y))
43.71/43.92	
43.71/43.92	Lemma 43: meet(meet(X, Z), Y) = meet(X, meet(Y, Z)).
43.71/43.92	Proof:
43.71/43.92	  meet(meet(X, Z), Y)
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(Y, meet(X, Z))
43.71/43.92	= { by lemma 36 }
43.71/43.92	  meet(meet(Y, top), meet(X, Z))
43.71/43.92	= { by lemma 17 }
43.71/43.92	  meet(complement(join(zero, complement(Y))), meet(X, Z))
43.71/43.92	= { by lemma 42 }
43.71/43.92	  meet(X, meet(Z, complement(join(zero, complement(Y)))))
43.71/43.92	= { by lemma 17 }
43.71/43.92	  meet(X, meet(Z, meet(Y, top)))
43.71/43.92	= { by lemma 36 }
43.71/43.92	  meet(X, meet(Z, Y))
43.71/43.92	= { by lemma 15 }
43.71/43.92	  meet(X, meet(Y, Z))
43.71/43.92	
43.71/43.92	Lemma 44: meet(X, join(X, Y)) = X.
43.71/43.92	Proof:
43.71/43.92	  meet(X, join(X, Y))
43.71/43.92	= { by axiom 5 (converse_idempotence_8) }
43.71/43.92	  meet(converse(converse(X)), join(X, Y))
43.71/43.92	= { by axiom 5 (converse_idempotence_8) }
43.71/43.92	  meet(converse(converse(X)), converse(converse(join(X, Y))))
43.71/43.92	= { by axiom 11 (converse_additivity_9) }
43.71/43.92	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
43.71/43.92	= { by lemma 34 }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), zero)
43.71/43.92	= { by lemma 16 }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(top))
43.71/43.92	= { by lemma 29 }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), top)))
43.71/43.92	= { by axiom 9 (def_top_12) }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(converse(Y)), join(converse(converse(X)), complement(converse(converse(X)))))))
43.71/43.92	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(join(converse(converse(Y)), converse(converse(X))), complement(converse(converse(X))))))
43.71/43.92	= { by axiom 11 (converse_additivity_9) }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(converse(join(converse(Y), converse(X))), complement(converse(converse(X))))))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(Y), converse(X))))))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), complement(join(complement(converse(converse(X))), converse(join(converse(X), converse(Y))))))
43.71/43.92	= { by lemma 24 }
43.71/43.92	  converse(converse(X))
43.71/43.92	= { by axiom 5 (converse_idempotence_8) }
43.71/43.92	  X
43.71/43.92	
43.71/43.92	Lemma 45: meet(X, join(Y, X)) = X.
43.71/43.92	Proof:
43.71/43.92	  meet(X, join(Y, X))
43.71/43.92	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.92	  meet(X, join(X, Y))
43.71/43.92	= { by lemma 44 }
43.71/43.93	  X
43.71/43.93	
43.71/43.93	Lemma 46: meet(complement(X), join(X, Y)) = meet(Y, complement(X)).
43.71/43.93	Proof:
43.71/43.93	  meet(complement(X), join(X, Y))
43.71/43.93	= { by lemma 32 }
43.71/43.93	  meet(complement(X), join(X, complement(complement(Y))))
43.71/43.93	= { by lemma 25 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(X), join(X, complement(complement(Y)))), complement(Y)))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(complement(X), join(X, complement(complement(Y))))))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), meet(join(X, complement(complement(Y))), complement(X))))
43.71/43.93	= { by lemma 40 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(Y), complement(join(complement(join(X, complement(complement(Y)))), X))))
43.71/43.93	= { by lemma 40 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(complement(complement(Y)), join(complement(join(X, complement(complement(Y)))), X))))
43.71/43.93	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), complement(join(join(complement(complement(Y)), complement(join(X, complement(complement(Y))))), X)))
43.71/43.93	= { by lemma 41 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(join(complement(complement(Y)), complement(join(X, complement(complement(Y)))))), complement(X)))
43.71/43.93	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(meet(complement(Y), join(X, complement(complement(Y)))), complement(X)))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(complement(Y), join(X, complement(complement(Y))))))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(complement(X), meet(join(X, complement(complement(Y))), complement(Y))))
43.71/43.93	= { by lemma 42 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), meet(complement(Y), complement(X))))
43.71/43.93	= { by lemma 39 }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), meet(join(X, complement(complement(Y))), complement(join(X, complement(complement(Y))))))
43.71/43.93	= { by axiom 8 (def_zero_13) }
43.71/43.93	  join(meet(meet(complement(X), join(X, complement(complement(Y)))), Y), zero)
43.71/43.93	= { by lemma 34 }
43.71/43.93	  meet(meet(complement(X), join(X, complement(complement(Y)))), Y)
43.71/43.93	= { by lemma 43 }
43.71/43.93	  meet(complement(X), meet(Y, join(X, complement(complement(Y)))))
43.71/43.93	= { by lemma 32 }
43.71/43.93	  meet(complement(X), meet(Y, join(X, Y)))
43.71/43.93	= { by lemma 45 }
43.71/43.93	  meet(complement(X), Y)
43.71/43.93	= { by lemma 15 }
43.71/43.93	  meet(Y, complement(X))
43.71/43.93	
43.71/43.93	Lemma 47: meet(X, join(Y, complement(X))) = meet(X, Y).
43.71/43.93	Proof:
43.71/43.93	  meet(X, join(Y, complement(X)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  meet(X, join(complement(X), Y))
43.71/43.93	= { by lemma 35 }
43.71/43.93	  meet(X, join(zero, join(complement(X), Y)))
43.71/43.93	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.93	  meet(X, join(join(zero, complement(X)), Y))
43.71/43.93	= { by lemma 36 }
43.71/43.93	  meet(X, meet(join(join(zero, complement(X)), Y), top))
43.71/43.93	= { by lemma 43 }
43.71/43.93	  meet(meet(X, top), join(join(zero, complement(X)), Y))
43.71/43.93	= { by lemma 17 }
43.71/43.93	  meet(complement(join(zero, complement(X))), join(join(zero, complement(X)), Y))
43.71/43.93	= { by lemma 46 }
43.71/43.93	  meet(Y, complement(join(zero, complement(X))))
43.71/43.93	= { by lemma 17 }
43.71/43.93	  meet(Y, meet(X, top))
43.71/43.93	= { by lemma 36 }
43.71/43.93	  meet(Y, X)
43.71/43.93	= { by lemma 15 }
43.71/43.93	  meet(X, Y)
43.71/43.93	
43.71/43.93	Lemma 48: meet(X, join(complement(X), Y)) = meet(X, Y).
43.71/43.93	Proof:
43.71/43.93	  meet(X, join(complement(X), Y))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  meet(X, join(Y, complement(X)))
43.71/43.93	= { by lemma 47 }
43.71/43.93	  meet(X, Y)
43.71/43.93	
43.71/43.93	Lemma 49: converse(join(composition(X, Y), composition(X, Z))) = converse(composition(X, join(Y, Z))).
43.71/43.93	Proof:
43.71/43.93	  converse(join(composition(X, Y), composition(X, Z)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  converse(join(composition(X, Z), composition(X, Y)))
43.71/43.93	= { by axiom 11 (converse_additivity_9) }
43.71/43.93	  join(converse(composition(X, Z)), converse(composition(X, Y)))
43.71/43.93	= { by axiom 10 (converse_multiplicativity_10) }
43.71/43.93	  join(composition(converse(Z), converse(X)), converse(composition(X, Y)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(converse(composition(X, Y)), composition(converse(Z), converse(X)))
43.71/43.93	= { by axiom 10 (converse_multiplicativity_10) }
43.71/43.93	  join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X)))
43.71/43.93	= { by axiom 13 (composition_distributivity_7) }
43.71/43.93	  composition(join(converse(Y), converse(Z)), converse(X))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  composition(join(converse(Z), converse(Y)), converse(X))
43.71/43.93	= { by axiom 11 (converse_additivity_9) }
43.71/43.93	  composition(converse(join(Z, Y)), converse(X))
43.71/43.93	= { by axiom 10 (converse_multiplicativity_10) }
43.71/43.93	  converse(composition(X, join(Z, Y)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  converse(composition(X, join(Y, Z)))
43.71/43.93	
43.71/43.93	Lemma 50: converse(one) = one.
43.71/43.93	Proof:
43.71/43.93	  converse(one)
43.71/43.93	= { by axiom 3 (composition_identity_6) }
43.71/43.93	  composition(converse(one), one)
43.71/43.93	= { by lemma 19 }
43.71/43.93	  one
43.71/43.93	
43.71/43.93	Lemma 51: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
43.71/43.93	Proof:
43.71/43.93	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
43.71/43.93	= { by axiom 13 (composition_distributivity_7) }
43.71/43.93	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
43.71/43.93	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.93	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
43.71/43.93	= { by axiom 9 (def_top_12) }
43.71/43.93	  join(composition(Z, Y), top)
43.71/43.93	= { by lemma 29 }
43.71/43.93	  top
43.71/43.93	
43.71/43.93	Lemma 52: join(composition(X, Z), complement(composition(meet(X, Y), Z))) = top.
43.71/43.93	Proof:
43.71/43.93	  join(composition(X, Z), complement(composition(meet(X, Y), Z)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(complement(composition(meet(X, Y), Z)), composition(X, Z))
43.71/43.93	= { by lemma 24 }
43.71/43.93	  join(complement(composition(meet(X, Y), Z)), composition(join(meet(X, Y), complement(join(complement(X), Y))), Z))
43.71/43.93	= { by lemma 51 }
43.71/43.93	  top
43.71/43.93	
43.71/43.93	Lemma 53: join(X, complement(composition(meet(Y, one), X))) = top.
43.71/43.93	Proof:
43.71/43.93	  join(X, complement(composition(meet(Y, one), X)))
43.71/43.93	= { by lemma 19 }
43.71/43.93	  join(composition(converse(one), X), complement(composition(meet(Y, one), X)))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(composition(converse(one), X), complement(composition(meet(one, Y), X)))
43.71/43.93	= { by lemma 50 }
43.71/43.93	  join(composition(converse(one), X), complement(composition(meet(converse(one), Y), X)))
43.71/43.93	= { by lemma 52 }
43.71/43.93	  top
43.71/43.93	
43.71/43.93	Lemma 54: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
43.71/43.93	Proof:
43.71/43.93	  join(meet(X, Y), complement(join(Y, complement(X))))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(meet(X, Y), complement(join(complement(X), Y)))
43.71/43.93	= { by lemma 24 }
43.71/43.93	  X
43.71/43.93	
43.71/43.93	Lemma 55: meet(join(X, Y), join(X, complement(Y))) = X.
43.71/43.93	Proof:
43.71/43.93	  meet(join(X, Y), join(X, complement(Y)))
43.71/43.93	= { by lemma 36 }
43.71/43.93	  meet(join(X, Y), meet(join(X, complement(Y)), top))
43.71/43.93	= { by lemma 17 }
43.71/43.93	  meet(join(X, Y), complement(join(zero, complement(join(X, complement(Y))))))
43.71/43.93	= { by lemma 39 }
43.71/43.93	  meet(join(X, Y), complement(join(zero, meet(Y, complement(X)))))
43.71/43.93	= { by lemma 35 }
43.71/43.93	  meet(join(X, Y), complement(meet(Y, complement(X))))
43.71/43.93	= { by lemma 40 }
43.71/43.93	  complement(join(complement(join(X, Y)), meet(Y, complement(X))))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  complement(join(complement(join(Y, X)), meet(Y, complement(X))))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  complement(join(complement(join(Y, X)), meet(complement(X), Y)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  complement(join(meet(complement(X), Y), complement(join(Y, X))))
43.71/43.93	= { by lemma 31 }
43.71/43.93	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, X))))
43.71/43.93	= { by lemma 36 }
43.71/43.93	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, meet(X, top)))))
43.71/43.93	= { by lemma 17 }
43.71/43.93	  complement(join(meet(join(zero, complement(X)), Y), complement(join(Y, complement(join(zero, complement(X)))))))
43.71/43.93	= { by lemma 54 }
43.71/43.93	  complement(join(zero, complement(X)))
43.71/43.93	= { by lemma 31 }
43.71/43.93	  complement(complement(X))
43.71/43.93	= { by lemma 32 }
43.71/43.93	  X
43.71/43.93	
43.71/43.93	Lemma 56: join(sk1, complement(composition(meet(X, sk1), top))) = top.
43.71/43.93	Proof:
43.71/43.93	  join(sk1, complement(composition(meet(X, sk1), top)))
43.71/43.93	= { by axiom 14 (goals_14) }
43.71/43.93	  join(composition(sk1, top), complement(composition(meet(X, sk1), top)))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(composition(sk1, top), complement(composition(meet(sk1, X), top)))
43.71/43.93	= { by lemma 52 }
43.71/43.93	  top
43.71/43.93	
43.71/43.93	Lemma 57: converse(join(X, converse(Y))) = join(Y, converse(X)).
43.71/43.93	Proof:
43.71/43.93	  converse(join(X, converse(Y)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  converse(join(converse(Y), X))
43.71/43.93	= { by axiom 11 (converse_additivity_9) }
43.71/43.93	  join(converse(converse(Y)), converse(X))
43.71/43.93	= { by axiom 5 (converse_idempotence_8) }
43.71/43.93	  join(Y, converse(X))
43.71/43.93	
43.71/43.93	Lemma 58: converse(join(converse(X), Y)) = join(X, converse(Y)).
43.71/43.93	Proof:
43.71/43.93	  converse(join(converse(X), Y))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  converse(join(Y, converse(X)))
43.71/43.93	= { by lemma 57 }
43.71/43.93	  join(X, converse(Y))
43.71/43.93	
43.71/43.93	Lemma 59: converse(zero) = zero.
43.71/43.93	Proof:
43.71/43.93	  converse(zero)
43.71/43.93	= { by lemma 35 }
43.71/43.93	  join(zero, converse(zero))
43.71/43.93	= { by lemma 58 }
43.71/43.93	  converse(join(converse(zero), zero))
43.71/43.93	= { by lemma 34 }
43.71/43.93	  converse(converse(zero))
43.71/43.93	= { by axiom 5 (converse_idempotence_8) }
43.71/43.93	  zero
43.71/43.93	
43.71/43.93	Lemma 60: composition(join(X, converse(complement(sk1))), sk1) = composition(X, sk1).
43.71/43.93	Proof:
43.71/43.93	  composition(join(X, converse(complement(sk1))), sk1)
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  composition(join(converse(complement(sk1)), X), sk1)
43.71/43.93	= { by axiom 13 (composition_distributivity_7) }
43.71/43.93	  join(composition(converse(complement(sk1)), sk1), composition(X, sk1))
43.71/43.93	= { by lemma 18 }
43.71/43.93	  join(converse(composition(converse(sk1), complement(sk1))), composition(X, sk1))
43.71/43.93	= { by lemma 35 }
43.71/43.93	  join(converse(join(zero, composition(converse(sk1), complement(sk1)))), composition(X, sk1))
43.71/43.93	= { by lemma 16 }
43.71/43.93	  join(converse(join(complement(top), composition(converse(sk1), complement(sk1)))), composition(X, sk1))
43.71/43.93	= { by axiom 14 (goals_14) }
43.71/43.93	  join(converse(join(complement(top), composition(converse(sk1), complement(composition(sk1, top))))), composition(X, sk1))
43.71/43.93	= { by lemma 21 }
43.71/43.93	  join(converse(complement(top)), composition(X, sk1))
43.71/43.93	= { by lemma 16 }
43.71/43.93	  join(converse(zero), composition(X, sk1))
43.71/43.93	= { by lemma 59 }
43.71/43.93	  join(zero, composition(X, sk1))
43.71/43.93	= { by lemma 35 }
43.71/43.93	  composition(X, sk1)
43.71/43.93	
43.71/43.93	Lemma 61: composition(converse(join(complement(sk1), X)), sk1) = composition(converse(X), sk1).
43.71/43.93	Proof:
43.71/43.93	  composition(converse(join(complement(sk1), X)), sk1)
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  composition(converse(join(X, complement(sk1))), sk1)
43.71/43.93	= { by axiom 11 (converse_additivity_9) }
43.71/43.93	  composition(join(converse(X), converse(complement(sk1))), sk1)
43.71/43.93	= { by lemma 60 }
43.71/43.93	  composition(converse(X), sk1)
43.71/43.93	
43.71/43.93	Lemma 62: join(X, meet(Y, complement(X))) = join(X, Y).
43.71/43.93	Proof:
43.71/43.93	  join(X, meet(Y, complement(X)))
43.71/43.93	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.93	  join(meet(Y, complement(X)), X)
43.71/43.93	= { by lemma 46 }
43.71/43.93	  join(meet(complement(X), join(X, Y)), X)
43.71/43.93	= { by lemma 44 }
43.71/43.93	  join(meet(complement(X), join(X, Y)), meet(X, join(X, Y)))
43.71/43.93	= { by lemma 32 }
43.71/43.93	  join(meet(complement(X), join(X, Y)), meet(complement(complement(X)), join(X, Y)))
43.71/43.93	= { by lemma 15 }
43.71/43.93	  join(meet(complement(X), join(X, Y)), meet(join(X, Y), complement(complement(X))))
43.71/43.93	= { by lemma 33 }
43.71/43.93	  join(X, Y)
43.71/43.93	
43.71/43.93	Lemma 63: composition(converse(meet(sk1, X)), sk1) = composition(converse(X), sk1).
43.71/43.93	Proof:
43.71/43.93	  composition(converse(meet(sk1, X)), sk1)
43.71/43.93	= { by lemma 15 }
43.71/43.93	  composition(converse(meet(X, sk1)), sk1)
43.71/43.93	= { by lemma 32 }
43.71/43.93	  composition(converse(meet(X, complement(complement(sk1)))), sk1)
43.71/43.93	= { by lemma 61 }
43.71/43.93	  composition(converse(join(complement(sk1), meet(X, complement(complement(sk1))))), sk1)
43.71/43.93	= { by lemma 62 }
43.71/43.93	  composition(converse(join(complement(sk1), X)), sk1)
43.71/43.93	= { by lemma 61 }
43.71/43.93	  composition(converse(X), sk1)
43.71/43.93	
43.71/43.93	Lemma 64: join(zero, meet(X, X)) = X.
43.71/43.93	Proof:
43.71/43.93	  join(zero, meet(X, X))
43.71/43.93	= { by axiom 8 (def_zero_13) }
43.71/43.93	  join(meet(X, complement(X)), meet(X, X))
43.71/43.93	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.93	  join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
43.71/43.93	= { by lemma 24 }
43.71/43.93	  X
43.71/43.93	
43.71/43.93	Lemma 65: join(complement(X), complement(Y)) = complement(meet(X, Y)).
43.71/43.93	Proof:
43.71/43.93	  join(complement(X), complement(Y))
43.71/43.93	= { by lemma 64 }
43.71/43.93	  join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y))))
43.71/43.93	= { by lemma 38 }
43.71/43.93	  join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
43.71/43.93	= { by lemma 31 }
43.71/43.93	  complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))
43.71/43.93	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.93	  complement(join(meet(X, Y), meet(X, Y)))
43.71/43.93	= { by lemma 64 }
43.71/43.93	  complement(join(join(zero, meet(meet(X, Y), meet(X, Y))), meet(X, Y)))
43.71/43.93	= { by lemma 32 }
43.71/43.93	  complement(join(join(zero, meet(meet(X, Y), meet(X, Y))), complement(complement(meet(X, Y)))))
43.71/43.93	= { by axiom 6 (maddux2_join_associativity_2) }
43.71/43.93	  complement(join(zero, join(meet(meet(X, Y), meet(X, Y)), complement(complement(meet(X, Y))))))
43.71/43.93	= { by lemma 23 }
43.71/43.93	  complement(join(zero, join(complement(complement(meet(X, Y))), complement(complement(meet(X, Y))))))
43.71/43.93	= { by lemma 22 }
43.71/43.93	  complement(join(zero, complement(complement(meet(X, Y)))))
43.71/43.93	= { by lemma 26 }
43.71/43.93	  complement(meet(X, Y))
43.71/43.93	
43.71/43.93	Lemma 66: join(X, converse(top)) = converse(top).
43.71/43.93	Proof:
43.71/43.93	  join(X, converse(top))
43.71/43.93	= { by lemma 58 }
43.71/43.93	  converse(join(converse(X), top))
43.71/43.93	= { by lemma 29 }
43.71/43.93	  converse(top)
43.71/43.93	
43.71/43.93	Lemma 67: converse(top) = top.
43.71/43.93	Proof:
43.71/43.93	  converse(top)
43.71/43.93	= { by lemma 66 }
43.71/43.93	  join(?, converse(top))
43.71/43.93	= { by lemma 66 }
43.71/43.93	  join(?, join(complement(?), converse(top)))
43.71/43.93	= { by lemma 28 }
43.71/43.93	  join(converse(top), top)
43.71/43.93	= { by lemma 29 }
43.71/43.93	  top
43.71/43.93	
43.71/43.93	Lemma 68: join(X, converse(complement(converse(X)))) = top.
43.71/43.93	Proof:
43.71/43.93	  join(X, converse(complement(converse(X))))
43.71/43.93	= { by lemma 58 }
43.71/43.93	  converse(join(converse(X), complement(converse(X))))
43.71/43.93	= { by axiom 9 (def_top_12) }
43.71/43.93	  converse(top)
43.71/43.93	= { by lemma 67 }
43.71/43.94	  top
43.71/43.94	
43.71/43.94	Lemma 69: converse(complement(X)) = complement(converse(X)).
43.71/43.94	Proof:
43.71/43.94	  converse(complement(X))
43.71/43.94	= { by lemma 31 }
43.71/43.94	  converse(join(zero, complement(X)))
43.71/43.94	= { by lemma 24 }
43.71/43.94	  converse(join(meet(join(zero, complement(X)), complement(converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
43.71/43.94	= { by lemma 40 }
43.71/43.94	  converse(join(complement(join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
43.71/43.94	= { by lemma 68 }
43.71/43.94	  converse(join(complement(top), complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
43.71/43.94	= { by lemma 16 }
43.71/43.94	  converse(join(zero, complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X)))))))))))
43.71/43.94	= { by lemma 31 }
43.71/43.94	  converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
43.71/43.94	= { by axiom 2 (maddux4_definiton_of_meet_4) }
43.71/43.94	  converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
43.71/43.94	= { by lemma 47 }
43.71/43.94	  converse(meet(join(zero, complement(X)), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by axiom 5 (converse_idempotence_8) }
43.71/43.94	  converse(meet(converse(converse(join(zero, complement(X)))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by lemma 55 }
43.71/43.94	  converse(meet(converse(meet(join(converse(join(zero, complement(X))), converse(complement(converse(converse(join(zero, complement(X))))))), join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X)))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by lemma 68 }
43.71/43.94	  converse(meet(converse(meet(top, join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X)))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by lemma 37 }
43.71/43.94	  converse(meet(converse(join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by lemma 58 }
43.71/43.94	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by axiom 5 (converse_idempotence_8) }
43.71/43.94	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  converse(meet(join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
43.71/43.94	= { by lemma 55 }
43.71/43.94	  converse(converse(complement(converse(complement(join(zero, complement(X)))))))
43.71/43.94	= { by axiom 5 (converse_idempotence_8) }
43.71/43.94	  complement(converse(complement(join(zero, complement(X)))))
43.71/43.94	= { by lemma 17 }
43.71/43.94	  complement(converse(meet(X, top)))
43.71/43.94	= { by lemma 36 }
43.71/43.94	  complement(converse(X))
43.71/43.94	
43.71/43.94	Lemma 70: join(complement(one), converse(complement(one))) = complement(one).
43.71/43.94	Proof:
43.71/43.94	  join(complement(one), converse(complement(one)))
43.71/43.94	= { by axiom 3 (composition_identity_6) }
43.71/43.94	  join(complement(one), composition(converse(complement(one)), one))
43.71/43.94	= { by lemma 31 }
43.71/43.94	  join(complement(one), composition(converse(join(zero, complement(one))), one))
43.71/43.94	= { by lemma 36 }
43.71/43.94	  join(complement(one), composition(converse(join(zero, complement(one))), meet(one, top)))
43.71/43.94	= { by lemma 17 }
43.71/43.94	  join(complement(one), composition(converse(join(zero, complement(one))), complement(join(zero, complement(one)))))
43.71/43.94	= { by axiom 3 (composition_identity_6) }
43.71/43.94	  join(complement(one), composition(converse(join(zero, complement(one))), complement(composition(join(zero, complement(one)), one))))
43.71/43.94	= { by lemma 21 }
43.71/43.94	  complement(one)
43.71/43.94	
43.71/43.94	Lemma 71: composition(converse(sk1), join(complement(sk1), X)) = composition(converse(sk1), X).
43.71/43.94	Proof:
43.71/43.94	  composition(converse(sk1), join(complement(sk1), X))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  composition(converse(sk1), join(X, complement(sk1)))
43.71/43.94	= { by axiom 5 (converse_idempotence_8) }
43.71/43.94	  composition(converse(sk1), join(X, converse(converse(complement(sk1)))))
43.71/43.94	= { by lemma 57 }
43.71/43.94	  composition(converse(sk1), converse(join(converse(complement(sk1)), converse(X))))
43.71/43.94	= { by axiom 10 (converse_multiplicativity_10) }
43.71/43.94	  converse(composition(join(converse(complement(sk1)), converse(X)), sk1))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  converse(composition(join(converse(X), converse(complement(sk1))), sk1))
43.71/43.94	= { by lemma 60 }
43.71/43.94	  converse(composition(converse(X), sk1))
43.71/43.94	= { by lemma 18 }
43.71/43.94	  composition(converse(sk1), X)
43.71/43.94	
43.71/43.94	Lemma 72: composition(top, meet(X, sk1)) = composition(converse(sk1), X).
43.71/43.94	Proof:
43.71/43.94	  composition(top, meet(X, sk1))
43.71/43.94	= { by lemma 67 }
43.71/43.94	  composition(converse(top), meet(X, sk1))
43.71/43.94	= { by lemma 56 }
43.71/43.94	  composition(converse(join(sk1, complement(composition(meet(X, sk1), top)))), meet(X, sk1))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  composition(converse(join(complement(composition(meet(X, sk1), top)), sk1)), meet(X, sk1))
43.71/43.94	= { by lemma 18 }
43.71/43.94	  converse(composition(converse(meet(X, sk1)), join(complement(composition(meet(X, sk1), top)), sk1)))
43.71/43.94	= { by lemma 49 }
43.71/43.94	  converse(join(composition(converse(meet(X, sk1)), complement(composition(meet(X, sk1), top))), composition(converse(meet(X, sk1)), sk1)))
43.71/43.94	= { by lemma 35 }
43.71/43.94	  converse(join(join(zero, composition(converse(meet(X, sk1)), complement(composition(meet(X, sk1), top)))), composition(converse(meet(X, sk1)), sk1)))
43.71/43.94	= { by lemma 16 }
43.71/43.94	  converse(join(join(complement(top), composition(converse(meet(X, sk1)), complement(composition(meet(X, sk1), top)))), composition(converse(meet(X, sk1)), sk1)))
43.71/43.94	= { by lemma 21 }
43.71/43.94	  converse(join(complement(top), composition(converse(meet(X, sk1)), sk1)))
43.71/43.94	= { by lemma 16 }
43.71/43.94	  converse(join(zero, composition(converse(meet(X, sk1)), sk1)))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  converse(join(composition(converse(meet(X, sk1)), sk1), zero))
43.71/43.94	= { by axiom 11 (converse_additivity_9) }
43.71/43.94	  join(converse(composition(converse(meet(X, sk1)), sk1)), converse(zero))
43.71/43.94	= { by lemma 18 }
43.71/43.94	  join(composition(converse(sk1), meet(X, sk1)), converse(zero))
43.71/43.94	= { by axiom 1 (maddux1_join_commutativity_1) }
43.71/43.94	  join(converse(zero), composition(converse(sk1), meet(X, sk1)))
43.71/43.94	= { by lemma 59 }
43.71/43.94	  join(zero, composition(converse(sk1), meet(X, sk1)))
43.71/43.94	= { by lemma 35 }
43.71/43.94	  composition(converse(sk1), meet(X, sk1))
43.71/43.94	= { by lemma 32 }
43.71/43.94	  composition(converse(sk1), meet(X, complement(complement(sk1))))
43.71/43.94	= { by lemma 71 }
43.71/43.94	  composition(converse(sk1), join(complement(sk1), meet(X, complement(complement(sk1)))))
43.71/43.94	= { by lemma 62 }
43.71/43.94	  composition(converse(sk1), join(complement(sk1), X))
43.71/43.94	= { by lemma 71 }
43.78/43.95	  composition(converse(sk1), X)
43.78/43.95	
43.78/43.95	Lemma 73: composition(meet(sk1, one), sk1) = sk1.
43.78/43.95	Proof:
43.78/43.95	  composition(meet(sk1, one), sk1)
43.78/43.95	= { by axiom 5 (converse_idempotence_8) }
43.78/43.95	  composition(converse(converse(meet(sk1, one))), sk1)
43.78/43.95	= { by lemma 63 }
43.78/43.95	  composition(converse(meet(sk1, converse(meet(sk1, one)))), sk1)
43.78/43.95	= { by lemma 32 }
43.78/43.95	  composition(converse(meet(sk1, complement(complement(converse(meet(sk1, one)))))), sk1)
43.78/43.95	= { by lemma 69 }
43.78/43.95	  composition(converse(meet(sk1, complement(converse(complement(meet(sk1, one)))))), sk1)
43.78/43.95	= { by lemma 65 }
43.78/43.95	  composition(converse(meet(sk1, complement(converse(join(complement(sk1), complement(one)))))), sk1)
43.78/43.95	= { by axiom 1 (maddux1_join_commutativity_1) }
43.78/43.95	  composition(converse(meet(sk1, complement(converse(join(complement(one), complement(sk1)))))), sk1)
43.78/43.95	= { by axiom 11 (converse_additivity_9) }
43.78/43.95	  composition(converse(meet(sk1, complement(join(converse(complement(one)), converse(complement(sk1)))))), sk1)
43.78/43.95	= { by lemma 70 }
43.78/43.95	  composition(converse(meet(sk1, complement(join(converse(join(complement(one), converse(complement(one)))), converse(complement(sk1)))))), sk1)
43.78/43.95	= { by lemma 57 }
43.78/43.95	  composition(converse(meet(sk1, complement(join(join(complement(one), converse(complement(one))), converse(complement(sk1)))))), sk1)
43.78/43.95	= { by lemma 70 }
43.78/43.95	  composition(converse(meet(sk1, complement(join(complement(one), converse(complement(sk1)))))), sk1)
43.78/43.95	= { by lemma 69 }
43.78/43.95	  composition(converse(meet(sk1, complement(join(complement(one), complement(converse(sk1)))))), sk1)
43.78/43.95	= { by lemma 65 }
43.78/43.95	  composition(converse(meet(sk1, complement(complement(meet(one, converse(sk1)))))), sk1)
43.78/43.95	= { by lemma 32 }
43.78/43.95	  composition(converse(meet(sk1, meet(one, converse(sk1)))), sk1)
43.78/43.95	= { by lemma 15 }
43.78/43.95	  composition(converse(meet(sk1, meet(converse(sk1), one))), sk1)
43.78/43.95	= { by lemma 43 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), converse(sk1))), sk1)
43.78/43.95	= { by axiom 3 (composition_identity_6) }
43.78/43.95	  composition(converse(meet(meet(sk1, one), composition(converse(sk1), one))), sk1)
43.78/43.95	= { by lemma 72 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), composition(top, meet(one, sk1)))), sk1)
43.78/43.95	= { by lemma 29 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), composition(join(one, top), meet(one, sk1)))), sk1)
43.78/43.95	= { by axiom 13 (composition_distributivity_7) }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(composition(one, meet(one, sk1)), composition(top, meet(one, sk1))))), sk1)
43.78/43.95	= { by lemma 20 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(meet(one, sk1), composition(top, meet(one, sk1))))), sk1)
43.78/43.95	= { by lemma 72 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(meet(one, sk1), composition(converse(sk1), one)))), sk1)
43.78/43.95	= { by axiom 3 (composition_identity_6) }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(meet(one, sk1), converse(sk1)))), sk1)
43.78/43.95	= { by axiom 1 (maddux1_join_commutativity_1) }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(converse(sk1), meet(one, sk1)))), sk1)
43.78/43.95	= { by lemma 15 }
43.78/43.95	  composition(converse(meet(meet(sk1, one), join(converse(sk1), meet(sk1, one)))), sk1)
43.78/43.95	= { by lemma 45 }
43.78/43.95	  composition(converse(meet(sk1, one)), sk1)
43.78/43.95	= { by lemma 63 }
43.78/43.95	  composition(converse(one), sk1)
43.78/43.95	= { by lemma 50 }
43.78/43.95	  composition(one, sk1)
43.78/43.95	= { by lemma 20 }
43.78/43.95	  sk1
43.78/43.95	
43.78/43.95	Lemma 74: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
43.78/43.95	Proof:
43.78/43.95	  converse(composition(X, converse(Y)))
43.78/43.95	= { by axiom 10 (converse_multiplicativity_10) }
43.78/43.95	  composition(converse(converse(Y)), converse(X))
43.78/43.95	= { by axiom 5 (converse_idempotence_8) }
43.80/43.98	  composition(Y, converse(X))
43.80/43.98	
43.80/43.98	Goal 1 (goals_15): composition(meet(sk1, one), sk2) = meet(sk1, sk2).
43.80/43.98	Proof:
43.80/43.98	  composition(meet(sk1, one), sk2)
43.80/43.98	= { by lemma 54 }
43.80/43.98	  join(meet(composition(meet(sk1, one), sk2), sk2), complement(join(sk2, complement(composition(meet(sk1, one), sk2)))))
43.80/43.98	= { by lemma 53 }
43.80/43.98	  join(meet(composition(meet(sk1, one), sk2), sk2), complement(top))
43.80/43.98	= { by lemma 16 }
43.80/43.98	  join(meet(composition(meet(sk1, one), sk2), sk2), zero)
43.80/43.98	= { by lemma 34 }
43.80/43.98	  meet(composition(meet(sk1, one), sk2), sk2)
43.80/43.98	= { by lemma 15 }
43.80/43.98	  meet(sk2, composition(meet(sk1, one), sk2))
43.80/43.98	= { by lemma 32 }
43.80/43.98	  meet(sk2, composition(meet(sk1, one), complement(complement(sk2))))
43.80/43.98	= { by lemma 48 }
43.80/43.98	  meet(sk2, join(complement(sk2), composition(meet(sk1, one), complement(complement(sk2)))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), complement(sk2)))
43.80/43.98	= { by lemma 55 }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), meet(join(complement(sk2), complement(composition(meet(sk1, one), complement(sk2)))), join(complement(sk2), complement(complement(composition(meet(sk1, one), complement(sk2))))))))
43.80/43.98	= { by lemma 53 }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), meet(top, join(complement(sk2), complement(complement(composition(meet(sk1, one), complement(sk2))))))))
43.80/43.98	= { by lemma 37 }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), join(complement(sk2), complement(complement(composition(meet(sk1, one), complement(sk2)))))))
43.80/43.98	= { by lemma 32 }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), join(complement(sk2), composition(meet(sk1, one), complement(sk2)))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), complement(complement(sk2))), join(composition(meet(sk1, one), complement(sk2)), complement(sk2))))
43.80/43.98	= { by axiom 6 (maddux2_join_associativity_2) }
43.80/43.98	  meet(sk2, join(join(composition(meet(sk1, one), complement(complement(sk2))), composition(meet(sk1, one), complement(sk2))), complement(sk2)))
43.80/43.98	= { by axiom 5 (converse_idempotence_8) }
43.80/43.98	  meet(sk2, join(converse(converse(join(composition(meet(sk1, one), complement(complement(sk2))), composition(meet(sk1, one), complement(sk2))))), complement(sk2)))
43.80/43.98	= { by lemma 49 }
43.80/43.98	  meet(sk2, join(converse(converse(composition(meet(sk1, one), join(complement(complement(sk2)), complement(sk2))))), complement(sk2)))
43.80/43.98	= { by axiom 5 (converse_idempotence_8) }
43.80/43.98	  meet(sk2, join(composition(meet(sk1, one), join(complement(complement(sk2)), complement(sk2))), complement(sk2)))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, join(complement(sk2), composition(meet(sk1, one), join(complement(complement(sk2)), complement(sk2)))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, join(complement(sk2), composition(meet(sk1, one), join(complement(sk2), complement(complement(sk2))))))
43.80/43.98	= { by axiom 9 (def_top_12) }
43.80/43.98	  meet(sk2, join(complement(sk2), composition(meet(sk1, one), top)))
43.80/43.98	= { by lemma 48 }
43.80/43.98	  meet(sk2, composition(meet(sk1, one), top))
43.80/43.98	= { by lemma 15 }
43.80/43.98	  meet(sk2, composition(meet(one, sk1), top))
43.80/43.98	= { by lemma 54 }
43.80/43.98	  meet(sk2, join(meet(composition(meet(one, sk1), top), sk1), complement(join(sk1, complement(composition(meet(one, sk1), top))))))
43.80/43.98	= { by lemma 56 }
43.80/43.98	  meet(sk2, join(meet(composition(meet(one, sk1), top), sk1), complement(top)))
43.80/43.98	= { by lemma 16 }
43.80/43.98	  meet(sk2, join(meet(composition(meet(one, sk1), top), sk1), zero))
43.80/43.98	= { by lemma 34 }
43.80/43.98	  meet(sk2, meet(composition(meet(one, sk1), top), sk1))
43.80/43.98	= { by lemma 15 }
43.80/43.98	  meet(sk2, meet(sk1, composition(meet(one, sk1), top)))
43.80/43.98	= { by lemma 73 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), composition(meet(one, sk1), top)))
43.80/43.98	= { by lemma 15 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), composition(meet(sk1, one), top)))
43.80/43.98	= { by lemma 47 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(composition(meet(sk1, one), top), complement(composition(meet(sk1, one), sk1)))))
43.80/43.98	= { by axiom 5 (converse_idempotence_8) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(composition(meet(sk1, one), top), complement(composition(meet(sk1, one), converse(converse(sk1)))))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(complement(composition(meet(sk1, one), converse(converse(sk1)))), composition(meet(sk1, one), top))))
43.80/43.98	= { by lemma 74 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(complement(converse(composition(converse(sk1), converse(meet(sk1, one))))), composition(meet(sk1, one), top))))
43.80/43.98	= { by lemma 69 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(converse(complement(composition(converse(sk1), converse(meet(sk1, one))))), composition(meet(sk1, one), top))))
43.80/43.98	= { by lemma 67 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(converse(complement(composition(converse(sk1), converse(meet(sk1, one))))), composition(meet(sk1, one), converse(top)))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(composition(meet(sk1, one), converse(top)), converse(complement(composition(converse(sk1), converse(meet(sk1, one))))))))
43.80/43.98	= { by lemma 74 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), join(converse(composition(top, converse(meet(sk1, one)))), converse(complement(composition(converse(sk1), converse(meet(sk1, one))))))))
43.80/43.98	= { by axiom 11 (converse_additivity_9) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), converse(join(composition(top, converse(meet(sk1, one))), complement(composition(converse(sk1), converse(meet(sk1, one))))))))
43.80/43.98	= { by axiom 1 (maddux1_join_commutativity_1) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), converse(join(complement(composition(converse(sk1), converse(meet(sk1, one)))), composition(top, converse(meet(sk1, one)))))))
43.80/43.98	= { by axiom 9 (def_top_12) }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), converse(join(complement(composition(converse(sk1), converse(meet(sk1, one)))), composition(join(converse(sk1), complement(converse(sk1))), converse(meet(sk1, one)))))))
43.80/43.98	= { by lemma 51 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), converse(top)))
43.80/43.98	= { by lemma 67 }
43.80/43.98	  meet(sk2, meet(composition(meet(sk1, one), sk1), top))
43.80/43.98	= { by lemma 36 }
43.80/43.98	  meet(sk2, composition(meet(sk1, one), sk1))
43.80/43.98	= { by lemma 73 }
43.80/43.98	  meet(sk2, sk1)
43.80/43.98	= { by lemma 15 }
43.80/43.98	  meet(sk1, sk2)
43.80/43.98	% SZS output end Proof
43.80/43.98	
43.80/43.98	RESULT: Unsatisfiable (the axioms are contradictory).
43.80/43.99	EOF