0.02/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.02/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.23	% Computer   : n065.star.cs.uiowa.edu
0.03/0.23	% Model      : x86_64 x86_64
0.03/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.23	% Memory     : 32218.625MB
0.03/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.23	% CPULimit   : 300
0.03/0.23	% DateTime   : Sat Jul 14 05:17:09 CDT 2018
0.03/0.23	% CPUTime    : 
17.11/17.29	% SZS status Theorem
17.11/17.29	
17.11/17.34	% SZS output start Proof
17.11/17.34	Take the following subset of the input axioms:
17.19/17.37	  fof(composition_associativity, axiom,
17.19/17.37	      ![X0, X1, X2]:
17.19/17.37	        composition(X0, composition(X1, X2))=composition(composition(X0,
17.19/17.37	                                                                     X1),
17.19/17.37	                                                         X2)).
17.19/17.37	  fof(composition_distributivity, axiom,
17.19/17.37	      ![X0, X1, X2]:
17.19/17.37	        composition(join(X0, X1), X2)=join(composition(X0, X2),
17.19/17.37	                                           composition(X1, X2))).
17.19/17.37	  fof(composition_identity, axiom, ![X0]: composition(X0, one)=X0).
17.19/17.37	  fof(converse_additivity, axiom,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        join(converse(X0), converse(X1))=converse(join(X0, X1))).
17.19/17.37	  fof(converse_cancellativity, axiom,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        join(composition(converse(X0), complement(composition(X0, X1))),
17.19/17.37	             complement(X1))=complement(X1)).
17.19/17.37	  fof(converse_idempotence, axiom, ![X0]: converse(converse(X0))=X0).
17.19/17.37	  fof(converse_multiplicativity, axiom,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        converse(composition(X0, X1))=composition(converse(X1),
17.19/17.37	                                                  converse(X0))).
17.19/17.37	  fof(def_top, axiom, ![X0]: top=join(X0, complement(X0))).
17.19/17.37	  fof(def_zero, axiom, ![X0]: zero=meet(X0, complement(X0))).
17.19/17.37	  fof(goals, conjecture,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        ((X0=composition(X0, top) & composition(X1, top)=X1)
17.19/17.37	           => composition(meet(X0, X1), top)=meet(X0, X1))).
17.19/17.37	  fof(maddux1_join_commutativity, axiom,
17.19/17.37	      ![X0, X1]: join(X0, X1)=join(X1, X0)).
17.19/17.37	  fof(maddux2_join_associativity, axiom,
17.19/17.37	      ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
17.19/17.37	  fof(maddux3_a_kind_of_de_Morgan, axiom,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        X0=join(complement(join(complement(X0), complement(X1))),
17.19/17.37	                complement(join(complement(X0), X1)))).
17.19/17.37	  fof(maddux4_definiton_of_meet, axiom,
17.19/17.37	      ![X0, X1]:
17.19/17.37	        meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
17.19/17.37	
17.19/17.37	Now clausify the problem and encode Horn clauses using encoding 3 of
17.19/17.37	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
17.19/17.37	We repeatedly replace C & s=t => u=v by the two clauses:
17.19/17.37	  $$fresh(y, y, x1...xn) = u
17.19/17.37	  C => $$fresh(s, t, x1...xn) = v
17.19/17.37	where $$fresh is a fresh function symbol and x1..xn are the free
17.19/17.37	variables of u and v.
17.19/17.37	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
17.19/17.37	input problem has no model of domain size 1).
17.19/17.37	
17.19/17.37	The encoding turns the above axioms into the following unit equations and goals:
17.19/17.37	
17.19/17.37	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
17.19/17.37	Axiom 2 (def_zero): zero = meet(X, complement(X)).
17.19/17.37	Axiom 3 (def_top): top = join(X, complement(X)).
17.19/17.37	Axiom 4 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
17.19/17.37	Axiom 5 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
17.19/17.37	Axiom 6 (converse_idempotence): converse(converse(X)) = X.
17.19/17.37	Axiom 7 (converse_additivity): join(converse(X), converse(Y)) = converse(join(X, Y)).
17.19/17.37	Axiom 8 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
17.19/17.37	Axiom 9 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
17.19/17.37	Axiom 10 (composition_identity): composition(X, one) = X.
17.19/17.37	Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
17.19/17.37	Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
17.19/17.37	Axiom 13 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
17.19/17.37	Axiom 14 (goals): composition(sK2_goals_X1, top) = sK2_goals_X1.
17.19/17.37	Axiom 15 (goals_1): sK1_goals_X0 = composition(sK1_goals_X0, top).
17.19/17.37	
17.19/17.37	Lemma 16: complement(top) = zero.
17.19/17.37	Proof:
17.19/17.37	  complement(top)
17.19/17.37	= { by axiom 3 (def_top) }
17.19/17.37	  complement(join(complement(?), complement(complement(?))))
17.19/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.19/17.37	  meet(?, complement(?))
17.19/17.37	= { by axiom 2 (def_zero) }
17.19/17.37	  zero
17.19/17.37	
17.19/17.37	Lemma 17: meet(X, Y) = meet(Y, X).
17.19/17.37	Proof:
17.19/17.37	  meet(X, Y)
17.19/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.19/17.37	  complement(join(complement(X), complement(Y)))
17.19/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.19/17.37	  complement(join(complement(Y), complement(X)))
17.19/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.19/17.37	  meet(Y, X)
17.19/17.37	
17.19/17.37	Lemma 18: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
17.19/17.37	Proof:
17.19/17.37	  converse(composition(X, converse(Y)))
17.19/17.37	= { by axiom 4 (converse_multiplicativity) }
17.19/17.37	  composition(converse(converse(Y)), converse(X))
17.19/17.37	= { by axiom 6 (converse_idempotence) }
17.19/17.37	  composition(Y, converse(X))
17.19/17.37	
17.19/17.37	Lemma 19: converse(composition(converse(X), Y)) = composition(converse(Y), X).
17.19/17.37	Proof:
17.19/17.37	  converse(composition(converse(X), Y))
17.19/17.37	= { by axiom 4 (converse_multiplicativity) }
17.19/17.37	  composition(converse(Y), converse(converse(X)))
17.19/17.37	= { by axiom 6 (converse_idempotence) }
17.19/17.37	  composition(converse(Y), X)
17.19/17.37	
17.19/17.37	Lemma 20: composition(converse(one), X) = X.
17.19/17.37	Proof:
17.19/17.37	  composition(converse(one), X)
17.19/17.37	= { by lemma 19 }
17.19/17.37	  converse(composition(converse(X), one))
17.19/17.37	= { by axiom 10 (composition_identity) }
17.19/17.37	  converse(converse(X))
17.19/17.37	= { by axiom 6 (converse_idempotence) }
17.19/17.37	  X
17.19/17.37	
17.19/17.37	Lemma 21: composition(one, X) = X.
17.19/17.37	Proof:
17.19/17.37	  composition(one, X)
17.19/17.37	= { by lemma 20 }
17.19/17.37	  composition(converse(one), composition(one, X))
17.19/17.37	= { by axiom 5 (composition_associativity) }
17.19/17.37	  composition(composition(converse(one), one), X)
17.19/17.37	= { by axiom 10 (composition_identity) }
17.20/17.37	  composition(converse(one), X)
17.20/17.37	= { by lemma 20 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 22: join(X, join(Y, Z)) = join(Z, join(X, Y)).
17.20/17.37	Proof:
17.20/17.37	  join(X, join(Y, Z))
17.20/17.37	= { by axiom 13 (maddux2_join_associativity) }
17.20/17.37	  join(join(X, Y), Z)
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(Z, join(X, Y))
17.20/17.37	
17.20/17.37	Lemma 23: complement(join(zero, complement(X))) = meet(X, top).
17.20/17.37	Proof:
17.20/17.37	  complement(join(zero, complement(X)))
17.20/17.37	= { by lemma 16 }
17.20/17.37	  complement(join(complement(top), complement(X)))
17.20/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.37	  meet(top, X)
17.20/17.37	= { by lemma 17 }
17.20/17.37	  meet(X, top)
17.20/17.37	
17.20/17.37	Lemma 24: converse(join(converse(X), Y)) = join(X, converse(Y)).
17.20/17.37	Proof:
17.20/17.37	  converse(join(converse(X), Y))
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  converse(join(Y, converse(X)))
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  converse(join(converse(X), Y))
17.20/17.37	= { by axiom 7 (converse_additivity) }
17.20/17.37	  join(converse(converse(X)), converse(Y))
17.20/17.37	= { by axiom 6 (converse_idempotence) }
17.20/17.37	  join(X, converse(Y))
17.20/17.37	
17.20/17.37	Lemma 25: join(complement(X), complement(X)) = complement(X).
17.20/17.37	Proof:
17.20/17.37	  join(complement(X), complement(X))
17.20/17.37	= { by lemma 20 }
17.20/17.37	  join(composition(converse(one), complement(X)), complement(X))
17.20/17.37	= { by lemma 21 }
17.20/17.37	  join(composition(converse(one), complement(composition(one, X))), complement(X))
17.20/17.37	= { by axiom 8 (converse_cancellativity) }
17.20/17.37	  complement(X)
17.20/17.37	
17.20/17.37	Lemma 26: meet(X, X) = complement(complement(X)).
17.20/17.37	Proof:
17.20/17.37	  meet(X, X)
17.20/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.37	  complement(join(complement(X), complement(X)))
17.20/17.37	= { by lemma 25 }
17.20/17.37	  complement(complement(X))
17.20/17.37	
17.20/17.37	Lemma 27: join(X, join(complement(X), Y)) = join(Y, top).
17.20/17.37	Proof:
17.20/17.37	  join(X, join(complement(X), Y))
17.20/17.37	= { by lemma 22 }
17.20/17.37	  join(complement(X), join(Y, X))
17.20/17.37	= { by lemma 22 }
17.20/17.37	  join(Y, join(X, complement(X)))
17.20/17.37	= { by axiom 3 (def_top) }
17.20/17.37	  join(Y, top)
17.20/17.37	
17.20/17.37	Lemma 28: join(X, top) = top.
17.20/17.37	Proof:
17.20/17.37	  join(X, top)
17.20/17.37	= { by axiom 3 (def_top) }
17.20/17.37	  join(X, join(complement(X), complement(complement(X))))
17.20/17.37	= { by lemma 27 }
17.20/17.37	  join(complement(complement(X)), top)
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(top, complement(complement(X)))
17.20/17.37	= { by axiom 3 (def_top) }
17.20/17.37	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
17.20/17.37	= { by axiom 13 (maddux2_join_associativity) }
17.20/17.37	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
17.20/17.37	= { by lemma 25 }
17.20/17.37	  join(complement(X), complement(complement(X)))
17.20/17.37	= { by axiom 3 (def_top) }
17.20/17.37	  top
17.20/17.37	
17.20/17.37	Lemma 29: join(top, X) = top.
17.20/17.37	Proof:
17.20/17.37	  join(top, X)
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(X, top)
17.20/17.37	= { by lemma 28 }
17.20/17.37	  top
17.20/17.37	
17.20/17.37	Lemma 30: join(X, converse(top)) = converse(top).
17.20/17.37	Proof:
17.20/17.37	  join(X, converse(top))
17.20/17.37	= { by lemma 24 }
17.20/17.37	  converse(join(converse(X), top))
17.20/17.37	= { by lemma 28 }
17.20/17.37	  converse(top)
17.20/17.37	
17.20/17.37	Lemma 31: converse(top) = top.
17.20/17.37	Proof:
17.20/17.37	  converse(top)
17.20/17.37	= { by lemma 30 }
17.20/17.37	  join(?, converse(top))
17.20/17.37	= { by lemma 30 }
17.20/17.37	  join(?, join(complement(?), converse(top)))
17.20/17.37	= { by lemma 27 }
17.20/17.37	  join(converse(top), top)
17.20/17.37	= { by lemma 28 }
17.20/17.37	  top
17.20/17.37	
17.20/17.37	Lemma 32: join(zero, meet(X, top)) = X.
17.20/17.37	Proof:
17.20/17.37	  join(zero, meet(X, top))
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(meet(X, top), zero)
17.20/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.37	  join(complement(join(complement(X), complement(top))), zero)
17.20/17.37	= { by lemma 16 }
17.20/17.37	  join(complement(join(complement(X), complement(top))), complement(top))
17.20/17.37	= { by lemma 28 }
17.20/17.37	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
17.20/17.37	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 33: join(meet(X, Y), meet(X, complement(Y))) = X.
17.20/17.37	Proof:
17.20/17.37	  join(meet(X, Y), meet(X, complement(Y)))
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(meet(X, complement(Y)), meet(X, Y))
17.20/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.37	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
17.20/17.37	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.37	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
17.20/17.37	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 34: join(zero, complement(complement(X))) = X.
17.20/17.37	Proof:
17.20/17.37	  join(zero, complement(complement(X)))
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(complement(complement(X)), zero)
17.20/17.37	= { by lemma 26 }
17.20/17.37	  join(meet(X, X), zero)
17.20/17.37	= { by axiom 2 (def_zero) }
17.20/17.37	  join(meet(X, X), meet(X, complement(X)))
17.20/17.37	= { by lemma 33 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 35: join(X, converse(complement(converse(X)))) = top.
17.20/17.37	Proof:
17.20/17.37	  join(X, converse(complement(converse(X))))
17.20/17.37	= { by lemma 24 }
17.20/17.37	  converse(join(converse(X), complement(converse(X))))
17.20/17.37	= { by axiom 3 (def_top) }
17.20/17.37	  converse(top)
17.20/17.37	= { by lemma 31 }
17.20/17.37	  top
17.20/17.37	
17.20/17.37	Lemma 36: join(zero, meet(top, X)) = X.
17.20/17.37	Proof:
17.20/17.37	  join(zero, meet(top, X))
17.20/17.37	= { by lemma 17 }
17.20/17.37	  join(zero, meet(X, top))
17.20/17.37	= { by lemma 32 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 37: complement(zero) = top.
17.20/17.37	Proof:
17.20/17.37	  complement(zero)
17.20/17.37	= { by lemma 16 }
17.20/17.37	  complement(complement(top))
17.20/17.37	= { by lemma 36 }
17.20/17.37	  join(zero, meet(top, complement(complement(top))))
17.20/17.37	= { by axiom 2 (def_zero) }
17.20/17.37	  join(meet(top, complement(top)), meet(top, complement(complement(top))))
17.20/17.37	= { by lemma 33 }
17.20/17.37	  top
17.20/17.37	
17.20/17.37	Lemma 38: join(zero, complement(X)) = complement(X).
17.20/17.37	Proof:
17.20/17.37	  join(zero, complement(X))
17.20/17.37	= { by lemma 34 }
17.20/17.37	  join(zero, complement(join(zero, complement(complement(X)))))
17.20/17.37	= { by lemma 23 }
17.20/17.37	  join(zero, meet(complement(X), top))
17.20/17.37	= { by lemma 17 }
17.20/17.37	  join(zero, meet(top, complement(X)))
17.20/17.37	= { by lemma 36 }
17.20/17.37	  complement(X)
17.20/17.37	
17.20/17.37	Lemma 39: meet(X, top) = X.
17.20/17.37	Proof:
17.20/17.37	  meet(X, top)
17.20/17.37	= { by lemma 23 }
17.20/17.37	  complement(join(zero, complement(X)))
17.20/17.37	= { by lemma 38 }
17.20/17.37	  join(zero, complement(join(zero, complement(X))))
17.20/17.37	= { by lemma 23 }
17.20/17.37	  join(zero, meet(X, top))
17.20/17.37	= { by lemma 32 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 40: complement(complement(X)) = X.
17.20/17.37	Proof:
17.20/17.37	  complement(complement(X))
17.20/17.37	= { by lemma 38 }
17.20/17.37	  join(zero, complement(complement(X)))
17.20/17.37	= { by lemma 34 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 41: join(meet(Y, X), meet(X, complement(Y))) = X.
17.20/17.37	Proof:
17.20/17.37	  join(meet(Y, X), meet(X, complement(Y)))
17.20/17.37	= { by lemma 17 }
17.20/17.37	  join(meet(X, Y), meet(X, complement(Y)))
17.20/17.37	= { by lemma 33 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 42: join(X, zero) = X.
17.20/17.37	Proof:
17.20/17.37	  join(X, zero)
17.20/17.37	= { by lemma 40 }
17.20/17.37	  join(complement(complement(X)), zero)
17.20/17.37	= { by lemma 26 }
17.20/17.37	  join(meet(X, X), zero)
17.20/17.37	= { by axiom 2 (def_zero) }
17.20/17.37	  join(meet(X, X), meet(X, complement(X)))
17.20/17.37	= { by lemma 41 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 43: join(zero, X) = X.
17.20/17.37	Proof:
17.20/17.37	  join(zero, X)
17.20/17.37	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.37	  join(X, zero)
17.20/17.37	= { by lemma 42 }
17.20/17.37	  X
17.20/17.37	
17.20/17.37	Lemma 44: converse(zero) = zero.
17.20/17.37	Proof:
17.20/17.37	  converse(zero)
17.20/17.37	= { by lemma 43 }
17.20/17.37	  join(zero, converse(zero))
17.20/17.37	= { by lemma 24 }
17.20/17.37	  converse(join(converse(zero), zero))
17.20/17.37	= { by lemma 42 }
17.20/17.37	  converse(converse(zero))
17.20/17.38	= { by axiom 6 (converse_idempotence) }
17.20/17.38	  zero
17.20/17.38	
17.20/17.38	Lemma 45: meet(top, X) = X.
17.20/17.38	Proof:
17.20/17.38	  meet(top, X)
17.20/17.38	= { by lemma 17 }
17.20/17.38	  meet(X, top)
17.20/17.38	= { by lemma 39 }
17.20/17.38	  X
17.20/17.38	
17.20/17.38	Lemma 46: join(X, composition(top, X)) = composition(top, X).
17.20/17.38	Proof:
17.20/17.38	  join(X, composition(top, X))
17.20/17.38	= { by lemma 21 }
17.20/17.38	  join(composition(one, X), composition(top, X))
17.20/17.38	= { by axiom 12 (composition_distributivity) }
17.20/17.38	  composition(join(one, top), X)
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  composition(join(top, one), X)
17.20/17.38	= { by lemma 29 }
17.20/17.38	  composition(top, X)
17.20/17.38	
17.20/17.38	Lemma 47: composition(top, zero) = zero.
17.20/17.38	Proof:
17.20/17.38	  composition(top, zero)
17.20/17.38	= { by lemma 42 }
17.20/17.38	  join(composition(top, zero), zero)
17.20/17.38	= { by lemma 31 }
17.20/17.38	  join(composition(converse(top), zero), zero)
17.20/17.38	= { by lemma 16 }
17.20/17.38	  join(composition(converse(top), complement(top)), zero)
17.20/17.38	= { by lemma 29 }
17.20/17.38	  join(composition(converse(top), complement(join(top, composition(top, top)))), zero)
17.20/17.38	= { by lemma 46 }
17.20/17.38	  join(composition(converse(top), complement(composition(top, top))), zero)
17.20/17.38	= { by lemma 16 }
17.20/17.38	  join(composition(converse(top), complement(composition(top, top))), complement(top))
17.20/17.38	= { by axiom 8 (converse_cancellativity) }
17.20/17.38	  complement(top)
17.20/17.38	= { by lemma 16 }
17.20/17.38	  zero
17.20/17.38	
17.20/17.38	Lemma 48: meet(X, join(X, Y)) = X.
17.20/17.38	Proof:
17.20/17.38	  meet(X, join(X, Y))
17.20/17.38	= { by axiom 6 (converse_idempotence) }
17.20/17.38	  meet(converse(converse(X)), join(X, Y))
17.20/17.38	= { by axiom 6 (converse_idempotence) }
17.20/17.38	  meet(converse(converse(X)), converse(converse(join(X, Y))))
17.20/17.38	= { by axiom 7 (converse_additivity) }
17.20/17.38	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
17.20/17.38	= { by lemma 42 }
17.20/17.38	  join(meet(converse(converse(X)), converse(join(converse(X), converse(Y)))), zero)
17.20/17.38	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), zero)
17.20/17.38	= { by lemma 16 }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(top))
17.20/17.38	= { by lemma 28 }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(converse(converse(Y)), top)))
17.20/17.38	= { by axiom 3 (def_top) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(converse(converse(Y)), join(converse(converse(X)), complement(converse(converse(X)))))))
17.20/17.38	= { by axiom 13 (maddux2_join_associativity) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(join(converse(converse(Y)), converse(converse(X))), complement(converse(converse(X))))))
17.20/17.38	= { by axiom 7 (converse_additivity) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(converse(join(converse(Y), converse(X))), complement(converse(converse(X))))))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(complement(converse(converse(X))), converse(join(converse(Y), converse(X))))))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  join(complement(join(complement(converse(converse(X))), complement(converse(join(converse(X), converse(Y)))))), complement(join(complement(converse(converse(X))), converse(join(converse(X), converse(Y))))))
17.20/17.38	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
17.20/17.38	  converse(converse(X))
17.20/17.38	= { by axiom 6 (converse_idempotence) }
17.20/17.38	  X
17.20/17.38	
17.20/17.38	Lemma 49: meet(X, join(Y, X)) = X.
17.20/17.38	Proof:
17.20/17.38	  meet(X, join(Y, X))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  meet(X, join(X, Y))
17.20/17.38	= { by lemma 48 }
17.20/17.38	  X
17.20/17.38	
17.20/17.38	Lemma 50: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
17.20/17.38	Proof:
17.20/17.38	  meet(X, join(complement(Y), complement(Z)))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  meet(X, join(complement(Z), complement(Y)))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  meet(join(complement(Z), complement(Y)), X)
17.20/17.38	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.38	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
17.20/17.38	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.38	  complement(join(meet(Z, Y), complement(X)))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  complement(join(complement(X), meet(Z, Y)))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  complement(join(complement(X), meet(Y, Z)))
17.20/17.38	
17.20/17.38	Lemma 51: complement(join(X, complement(Y))) = meet(Y, complement(X)).
17.20/17.38	Proof:
17.20/17.38	  complement(join(X, complement(Y)))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  complement(join(complement(Y), X))
17.20/17.38	= { by lemma 45 }
17.20/17.38	  complement(join(complement(Y), meet(top, X)))
17.20/17.38	= { by lemma 50 }
17.20/17.38	  meet(Y, join(complement(top), complement(X)))
17.20/17.38	= { by lemma 16 }
17.20/17.38	  meet(Y, join(zero, complement(X)))
17.20/17.38	= { by lemma 38 }
17.20/17.38	  meet(Y, complement(X))
17.20/17.38	
17.20/17.38	Lemma 52: complement(join(complement(X), Y)) = meet(X, complement(Y)).
17.20/17.38	Proof:
17.20/17.38	  complement(join(complement(X), Y))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  complement(join(Y, complement(X)))
17.20/17.38	= { by lemma 51 }
17.20/17.38	  meet(X, complement(Y))
17.20/17.38	
17.20/17.38	Lemma 53: complement(meet(Y, complement(X))) = join(X, complement(Y)).
17.20/17.38	Proof:
17.20/17.38	  complement(meet(Y, complement(X)))
17.20/17.38	= { by lemma 43 }
17.20/17.38	  complement(join(zero, meet(Y, complement(X))))
17.20/17.38	= { by lemma 51 }
17.20/17.38	  complement(join(zero, complement(join(X, complement(Y)))))
17.20/17.38	= { by lemma 23 }
17.20/17.38	  meet(join(X, complement(Y)), top)
17.20/17.38	= { by lemma 39 }
17.20/17.38	  join(X, complement(Y))
17.20/17.38	
17.20/17.38	Lemma 54: complement(meet(complement(X), Y)) = join(X, complement(Y)).
17.20/17.38	Proof:
17.20/17.38	  complement(meet(complement(X), Y))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  complement(meet(Y, complement(X)))
17.20/17.38	= { by lemma 53 }
17.20/17.38	  join(X, complement(Y))
17.20/17.38	
17.20/17.38	Lemma 55: join(complement(X), complement(Y)) = complement(meet(X, Y)).
17.20/17.38	Proof:
17.20/17.38	  join(complement(X), complement(Y))
17.20/17.38	= { by lemma 43 }
17.20/17.38	  join(zero, join(complement(X), complement(Y)))
17.20/17.38	= { by axiom 13 (maddux2_join_associativity) }
17.20/17.38	  join(join(zero, complement(X)), complement(Y))
17.20/17.38	= { by lemma 53 }
17.20/17.38	  complement(meet(Y, complement(join(zero, complement(X)))))
17.20/17.38	= { by lemma 23 }
17.20/17.38	  complement(meet(Y, meet(X, top)))
17.20/17.38	= { by lemma 39 }
17.20/17.38	  complement(meet(Y, X))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  complement(meet(X, Y))
17.20/17.38	
17.20/17.38	Lemma 56: composition(converse(complement(sK2_goals_X1)), composition(sK2_goals_X1, X)) = zero.
17.20/17.38	Proof:
17.20/17.38	  composition(converse(complement(sK2_goals_X1)), composition(sK2_goals_X1, X))
17.20/17.38	= { by lemma 19 }
17.20/17.38	  converse(composition(converse(composition(sK2_goals_X1, X)), complement(sK2_goals_X1)))
17.20/17.38	= { by axiom 4 (converse_multiplicativity) }
17.20/17.38	  converse(composition(composition(converse(X), converse(sK2_goals_X1)), complement(sK2_goals_X1)))
17.20/17.38	= { by axiom 5 (composition_associativity) }
17.20/17.38	  converse(composition(converse(X), composition(converse(sK2_goals_X1), complement(sK2_goals_X1))))
17.20/17.38	= { by lemma 42 }
17.20/17.38	  converse(composition(converse(X), join(composition(converse(sK2_goals_X1), complement(sK2_goals_X1)), zero)))
17.20/17.38	= { by axiom 14 (goals) }
17.20/17.38	  converse(composition(converse(X), join(composition(converse(sK2_goals_X1), complement(composition(sK2_goals_X1, top))), zero)))
17.20/17.38	= { by lemma 16 }
17.20/17.38	  converse(composition(converse(X), join(composition(converse(sK2_goals_X1), complement(composition(sK2_goals_X1, top))), complement(top))))
17.20/17.38	= { by axiom 8 (converse_cancellativity) }
17.20/17.38	  converse(composition(converse(X), complement(top)))
17.20/17.38	= { by lemma 16 }
17.20/17.38	  converse(composition(converse(X), zero))
17.20/17.38	= { by lemma 43 }
17.20/17.38	  converse(join(zero, composition(converse(X), zero)))
17.20/17.38	= { by lemma 47 }
17.20/17.38	  converse(join(composition(top, zero), composition(converse(X), zero)))
17.20/17.38	= { by axiom 12 (composition_distributivity) }
17.20/17.38	  converse(composition(join(top, converse(X)), zero))
17.20/17.38	= { by lemma 29 }
17.20/17.38	  converse(composition(top, zero))
17.20/17.38	= { by lemma 47 }
17.20/17.38	  converse(zero)
17.20/17.38	= { by lemma 44 }
17.20/17.38	  zero
17.20/17.38	
17.20/17.38	Lemma 57: converse(composition(top, X)) = composition(converse(X), top).
17.20/17.38	Proof:
17.20/17.38	  converse(composition(top, X))
17.20/17.38	= { by axiom 4 (converse_multiplicativity) }
17.20/17.38	  composition(converse(X), converse(top))
17.20/17.38	= { by lemma 31 }
17.20/17.38	  composition(converse(X), top)
17.20/17.38	
17.20/17.38	Lemma 58: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
17.20/17.38	Proof:
17.20/17.38	  meet(join(X, complement(Y)), complement(meet(X, Y)))
17.20/17.38	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.38	  meet(join(complement(Y), X), complement(meet(X, Y)))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  meet(join(complement(Y), X), complement(meet(Y, X)))
17.20/17.38	= { by lemma 17 }
17.20/17.38	  meet(complement(meet(Y, X)), join(complement(Y), X))
17.20/17.38	= { by lemma 55 }
17.20/17.38	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
17.20/17.38	= { by axiom 11 (maddux4_definiton_of_meet) }
17.20/17.38	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
17.20/17.38	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
17.20/17.39	  complement(Y)
17.20/17.39	
17.20/17.39	Lemma 59: meet(X, join(Y, complement(X))) = meet(X, Y).
17.20/17.39	Proof:
17.20/17.39	  meet(X, join(Y, complement(X)))
17.20/17.39	= { by lemma 40 }
17.20/17.39	  meet(complement(complement(X)), join(Y, complement(X)))
17.20/17.39	= { by lemma 48 }
17.20/17.39	  meet(complement(meet(complement(X), join(complement(X), join(zero, complement(Y))))), join(Y, complement(X)))
17.20/17.39	= { by lemma 54 }
17.20/17.39	  meet(join(X, complement(join(complement(X), join(zero, complement(Y))))), join(Y, complement(X)))
17.20/17.39	= { by lemma 52 }
17.20/17.39	  meet(join(X, meet(X, complement(join(zero, complement(Y))))), join(Y, complement(X)))
17.20/17.39	= { by lemma 23 }
17.20/17.39	  meet(join(X, meet(X, meet(Y, top))), join(Y, complement(X)))
17.20/17.39	= { by lemma 39 }
17.20/17.39	  meet(join(X, meet(X, Y)), join(Y, complement(X)))
17.20/17.39	= { by lemma 40 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), join(Y, complement(X)))
17.20/17.39	= { by lemma 53 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(Y))))
17.20/17.39	= { by lemma 52 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(complement(join(complement(X), Y))))
17.20/17.39	= { by lemma 58 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(complement(join(complement(X), complement(Y))), join(complement(X), Y))))))
17.20/17.39	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(meet(complement(join(complement(X), complement(Y))), join(complement(X), Y))))))
17.20/17.39	= { by lemma 54 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))))
17.20/17.39	= { by axiom 13 (maddux2_join_associativity) }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))))
17.20/17.39	= { by lemma 55 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, join(complement(X), complement(meet(Y, join(complement(X), Y)))))))
17.20/17.39	= { by lemma 50 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(complement(join(complement(X), meet(X, meet(Y, join(complement(X), Y)))))))
17.20/17.39	= { by lemma 52 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(meet(X, meet(Y, join(complement(X), Y)))))))
17.20/17.39	= { by lemma 49 }
17.20/17.39	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(meet(X, Y)))))
17.20/17.39	= { by lemma 58 }
17.20/17.39	  complement(complement(meet(X, Y)))
17.20/17.39	= { by lemma 40 }
17.20/17.39	  meet(X, Y)
17.20/17.39	
17.20/17.39	Lemma 60: meet(join(X, Y), join(X, complement(Y))) = X.
17.20/17.39	Proof:
17.20/17.39	  meet(join(X, Y), join(X, complement(Y)))
17.20/17.39	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.39	  meet(join(Y, X), join(X, complement(Y)))
17.20/17.39	= { by lemma 39 }
17.20/17.39	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
17.20/17.39	= { by lemma 23 }
17.20/17.39	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
17.20/17.39	= { by lemma 53 }
17.20/17.39	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, complement(X))))
17.20/17.39	= { by lemma 38 }
17.20/17.39	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, join(zero, complement(X)))))
17.20/17.39	= { by lemma 58 }
17.20/17.39	  complement(join(zero, complement(X)))
17.20/17.39	= { by lemma 23 }
17.20/17.39	  meet(X, top)
17.20/17.39	= { by lemma 39 }
17.20/17.39	  X
17.20/17.39	
17.20/17.39	Lemma 61: converse(complement(X)) = complement(converse(X)).
17.20/17.39	Proof:
17.20/17.39	  converse(complement(X))
17.20/17.39	= { by lemma 38 }
17.20/17.39	  converse(join(zero, complement(X)))
17.20/17.39	= { by lemma 39 }
17.20/17.39	  converse(meet(join(zero, complement(X)), top))
17.20/17.39	= { by axiom 6 (converse_idempotence) }
17.20/17.39	  converse(meet(converse(converse(join(zero, complement(X)))), top))
17.20/17.39	= { by lemma 60 }
17.20/17.39	  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)))))))))), top))
17.20/17.39	= { by lemma 35 }
17.20/17.39	  converse(meet(converse(meet(top, join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X)))))))))), top))
17.20/17.39	= { by lemma 45 }
17.20/17.39	  converse(meet(converse(join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X))))))))), top))
17.20/17.39	= { by lemma 24 }
17.20/17.39	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(converse(converse(join(zero, complement(X))))))))), top))
17.20/17.39	= { by axiom 6 (converse_idempotence) }
17.20/17.39	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), top))
17.20/17.39	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.39	  converse(meet(join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X))), top))
17.20/17.39	= { by lemma 35 }
17.20/17.39	  converse(meet(join(converse(complement(converse(complement(join(zero, complement(X)))))), join(zero, complement(X))), join(complement(join(zero, complement(X))), converse(complement(converse(complement(join(zero, complement(X)))))))))
17.20/17.39	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.39	  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))))))
17.20/17.39	= { by lemma 60 }
17.20/17.39	  converse(converse(complement(converse(complement(join(zero, complement(X)))))))
17.20/17.39	= { by axiom 6 (converse_idempotence) }
17.20/17.39	  complement(converse(complement(join(zero, complement(X)))))
17.20/17.39	= { by lemma 23 }
17.20/17.39	  complement(converse(meet(X, top)))
17.20/17.39	= { by lemma 39 }
17.20/17.39	  complement(converse(X))
17.20/17.39	
17.20/17.39	Lemma 62: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
17.20/17.39	Proof:
17.20/17.39	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
17.20/17.39	= { by axiom 9 (maddux1_join_commutativity) }
17.20/17.39	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
17.20/17.39	= { by axiom 4 (converse_multiplicativity) }
17.20/17.39	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
17.20/17.39	= { by axiom 12 (composition_distributivity) }
17.20/17.39	  composition(join(converse(X), Y), converse(Z))
17.20/17.39	= { by axiom 9 (maddux1_join_commutativity) }
17.23/17.43	  composition(join(Y, converse(X)), converse(Z))
17.23/17.43	
17.23/17.43	Lemma 63: composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top) = complement(meet(sK2_goals_X1, sK1_goals_X0)).
17.23/17.43	Proof:
17.23/17.43	  composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)
17.23/17.43	= { by lemma 17 }
17.23/17.43	  composition(complement(meet(sK1_goals_X0, sK2_goals_X1)), top)
17.23/17.43	= { by lemma 55 }
17.23/17.43	  composition(join(complement(sK1_goals_X0), complement(sK2_goals_X1)), top)
17.23/17.43	= { by axiom 6 (converse_idempotence) }
17.23/17.43	  composition(join(complement(sK1_goals_X0), complement(converse(converse(sK2_goals_X1)))), top)
17.23/17.43	= { by lemma 61 }
17.23/17.43	  composition(join(complement(sK1_goals_X0), converse(complement(converse(sK2_goals_X1)))), top)
17.23/17.43	= { by lemma 31 }
17.23/17.43	  composition(join(complement(sK1_goals_X0), converse(complement(converse(sK2_goals_X1)))), converse(top))
17.23/17.43	= { by lemma 62 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), converse(top)), converse(composition(top, complement(converse(sK2_goals_X1)))))
17.23/17.43	= { by lemma 31 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(composition(top, complement(converse(sK2_goals_X1)))))
17.23/17.43	= { by lemma 46 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(top, complement(converse(sK2_goals_X1))))))
17.23/17.43	= { by lemma 31 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(converse(sK2_goals_X1))))))
17.23/17.43	= { by lemma 20 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(converse(one), converse(sK2_goals_X1)))))))
17.23/17.43	= { by lemma 18 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(converse(composition(sK2_goals_X1, converse(converse(one)))))))))
17.23/17.43	= { by axiom 14 (goals) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(converse(composition(composition(sK2_goals_X1, top), converse(converse(one)))))))))
17.23/17.43	= { by axiom 5 (composition_associativity) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(converse(composition(sK2_goals_X1, composition(top, converse(converse(one))))))))))
17.23/17.43	= { by axiom 4 (converse_multiplicativity) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(converse(composition(top, converse(converse(one)))), converse(sK2_goals_X1)))))))
17.23/17.43	= { by lemma 18 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(composition(converse(one), converse(top)), converse(sK2_goals_X1)))))))
17.23/17.43	= { by axiom 5 (composition_associativity) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(converse(one), composition(converse(top), converse(sK2_goals_X1))))))))
17.23/17.43	= { by lemma 31 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(converse(one), composition(top, converse(sK2_goals_X1))))))))
17.23/17.43	= { by lemma 20 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(complement(converse(sK2_goals_X1)), composition(converse(top), complement(composition(top, converse(sK2_goals_X1)))))))
17.23/17.43	= { by axiom 9 (maddux1_join_commutativity) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(join(composition(converse(top), complement(composition(top, converse(sK2_goals_X1)))), complement(converse(sK2_goals_X1)))))
17.23/17.43	= { by axiom 8 (converse_cancellativity) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), converse(complement(converse(sK2_goals_X1))))
17.23/17.43	= { by lemma 61 }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), complement(converse(converse(sK2_goals_X1))))
17.23/17.43	= { by axiom 6 (converse_idempotence) }
17.23/17.43	  join(composition(complement(sK1_goals_X0), top), complement(sK2_goals_X1))
17.23/17.43	= { by axiom 9 (maddux1_join_commutativity) }
17.23/17.43	  join(complement(sK2_goals_X1), composition(complement(sK1_goals_X0), top))
17.23/17.43	= { by axiom 6 (converse_idempotence) }
17.23/17.43	  join(complement(sK2_goals_X1), composition(converse(converse(complement(sK1_goals_X0))), top))
17.23/17.43	= { by lemma 57 }
17.23/17.43	  join(complement(sK2_goals_X1), converse(composition(top, converse(complement(sK1_goals_X0)))))
17.23/17.43	= { by lemma 46 }
17.23/17.43	  join(complement(sK2_goals_X1), converse(join(converse(complement(sK1_goals_X0)), composition(top, converse(complement(sK1_goals_X0))))))
17.23/17.43	= { by lemma 24 }
17.23/17.43	  join(complement(sK2_goals_X1), join(complement(sK1_goals_X0), converse(composition(top, converse(complement(sK1_goals_X0))))))
17.23/17.43	= { by lemma 57 }
17.23/17.43	  join(complement(sK2_goals_X1), join(complement(sK1_goals_X0), composition(converse(converse(complement(sK1_goals_X0))), top)))
17.23/17.43	= { by axiom 6 (converse_idempotence) }
17.23/17.43	  join(complement(sK2_goals_X1), join(complement(sK1_goals_X0), composition(complement(sK1_goals_X0), top)))
17.23/17.43	= { by axiom 9 (maddux1_join_commutativity) }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(complement(sK1_goals_X0), top), complement(sK1_goals_X0)))
17.23/17.43	= { by axiom 6 (converse_idempotence) }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), top), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 37 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(zero)), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 44 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(zero))), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 16 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(complement(top)))), complement(sK1_goals_X0)))
17.23/17.43	= { by axiom 8 (converse_cancellativity) }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(join(composition(converse(sK1_goals_X0), complement(composition(sK1_goals_X0, top))), complement(top))))), complement(sK1_goals_X0)))
17.23/17.43	= { by axiom 15 (goals_1) }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(join(composition(converse(sK1_goals_X0), complement(sK1_goals_X0)), complement(top))))), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 16 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(join(composition(converse(sK1_goals_X0), complement(sK1_goals_X0)), zero)))), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 42 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(converse(composition(converse(sK1_goals_X0), complement(sK1_goals_X0))))), complement(sK1_goals_X0)))
17.23/17.43	= { by lemma 19 }
17.23/17.43	  join(complement(sK2_goals_X1), join(composition(converse(converse(complement(sK1_goals_X0))), complement(composition(converse(complement(sK1_goals_X0)), sK1_goals_X0))), complement(sK1_goals_X0)))
17.23/17.43	= { by axiom 8 (converse_cancellativity) }
17.23/17.43	  join(complement(sK2_goals_X1), complement(sK1_goals_X0))
17.23/17.43	= { by lemma 55 }
17.26/17.46	  complement(meet(sK2_goals_X1, sK1_goals_X0))
17.26/17.46	
17.26/17.46	Goal 1 (goals_2): composition(meet(sK1_goals_X0, sK2_goals_X1), top) = meet(sK1_goals_X0, sK2_goals_X1).
17.26/17.46	Proof:
17.26/17.46	  composition(meet(sK1_goals_X0, sK2_goals_X1), top)
17.26/17.46	= { by lemma 17 }
17.26/17.46	  composition(meet(sK2_goals_X1, sK1_goals_X0), top)
17.26/17.46	= { by lemma 40 }
17.26/17.46	  composition(complement(complement(meet(sK2_goals_X1, sK1_goals_X0))), top)
17.26/17.46	= { by lemma 63 }
17.26/17.46	  composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)
17.26/17.46	= { by lemma 29 }
17.26/17.46	  composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), join(top, one))
17.26/17.46	= { by axiom 6 (converse_idempotence) }
17.26/17.46	  converse(converse(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), join(top, one))))
17.26/17.46	= { by axiom 9 (maddux1_join_commutativity) }
17.26/17.46	  converse(converse(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), join(one, top))))
17.26/17.46	= { by axiom 4 (converse_multiplicativity) }
17.26/17.46	  converse(composition(converse(join(one, top)), converse(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)))))
17.26/17.46	= { by axiom 7 (converse_additivity) }
17.26/17.46	  converse(composition(join(converse(one), converse(top)), converse(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)))))
17.26/17.46	= { by lemma 62 }
17.26/17.46	  converse(join(composition(converse(one), converse(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)))), converse(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 20 }
17.26/17.46	  converse(join(converse(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))), converse(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by axiom 7 (converse_additivity) }
17.26/17.46	  converse(converse(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by axiom 6 (converse_idempotence) }
17.26/17.46	  join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))
17.26/17.46	= { by lemma 40 }
17.26/17.46	  join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 41 }
17.26/17.46	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 40 }
17.26/17.46	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 49 }
17.26/17.46	  join(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 40 }
17.26/17.46	  join(complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by axiom 8 (converse_cancellativity) }
17.26/17.46	  join(complement(join(composition(converse(converse(complement(meet(sK2_goals_X1, sK1_goals_X0)))), complement(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 51 }
17.26/17.46	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(converse(converse(complement(meet(sK2_goals_X1, sK1_goals_X0)))), complement(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by axiom 6 (converse_idempotence) }
17.26/17.46	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by axiom 5 (composition_associativity) }
17.26/17.46	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.46	= { by lemma 42 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(join(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))), zero), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 16 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(join(composition(converse(complement(meet(sK2_goals_X1, sK1_goals_X0))), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))), complement(top)), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by axiom 8 (converse_cancellativity) }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(complement(top), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 16 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(zero, top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 56 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(composition(converse(complement(sK2_goals_X1)), composition(sK2_goals_X1, ?)), top))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by axiom 5 (composition_associativity) }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(converse(complement(sK2_goals_X1)), composition(composition(sK2_goals_X1, ?), top)))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by axiom 5 (composition_associativity) }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(composition(converse(complement(sK2_goals_X1)), composition(sK2_goals_X1, composition(?, top))))))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 56 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), complement(zero)))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 37 }
17.26/17.47	  join(meet(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 17 }
17.26/17.47	  join(meet(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), meet(join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)))), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 17 }
17.26/17.47	  join(meet(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), meet(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), join(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))))
17.26/17.47	= { by lemma 59 }
17.26/17.47	  join(meet(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), meet(complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))))
17.26/17.47	= { by lemma 17 }
17.26/17.47	  join(meet(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top)), meet(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), complement(composition(complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top)), top))))
17.26/17.47	= { by lemma 33 }
17.26/17.47	  complement(composition(complement(meet(sK2_goals_X1, sK1_goals_X0)), top))
17.26/17.47	= { by lemma 63 }
17.26/17.47	  complement(complement(meet(sK2_goals_X1, sK1_goals_X0)))
17.26/17.47	= { by lemma 40 }
17.26/17.47	  meet(sK2_goals_X1, sK1_goals_X0)
17.26/17.47	= { by lemma 17 }
17.26/17.47	  meet(sK1_goals_X0, sK2_goals_X1)
17.26/17.47	% SZS output end Proof
17.26/17.47	
17.26/17.47	RESULT: Theorem (the conjecture is true).
17.26/17.48	EOF