0.00/0.04	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.00/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.26	% Computer   : n067.star.cs.uiowa.edu
0.03/0.26	% Model      : x86_64 x86_64
0.03/0.26	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.26	% Memory     : 32218.625MB
0.03/0.26	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.26	% CPULimit   : 300
0.03/0.26	% DateTime   : Sat Jul 14 04:16:44 CDT 2018
0.03/0.26	% CPUTime    : 
8.85/9.11	% SZS status Theorem
8.85/9.11	
8.85/9.12	% SZS output start Proof
8.85/9.12	Take the following subset of the input axioms:
9.40/9.63	  fof(composition_associativity, axiom,
9.40/9.63	      ![X0, X1, X2]:
9.40/9.63	        composition(X0, composition(X1, X2))=composition(composition(X0,
9.40/9.63	                                                                     X1),
9.40/9.63	                                                         X2)).
9.40/9.63	  fof(composition_distributivity, axiom,
9.40/9.63	      ![X0, X1, X2]:
9.40/9.63	        composition(join(X0, X1), X2)=join(composition(X0, X2),
9.40/9.63	                                           composition(X1, X2))).
9.40/9.63	  fof(composition_identity, axiom, ![X0]: composition(X0, one)=X0).
9.40/9.63	  fof(converse_additivity, axiom,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        join(converse(X0), converse(X1))=converse(join(X0, X1))).
9.40/9.63	  fof(converse_cancellativity, axiom,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        join(composition(converse(X0), complement(composition(X0, X1))),
9.40/9.63	             complement(X1))=complement(X1)).
9.40/9.63	  fof(converse_idempotence, axiom, ![X0]: converse(converse(X0))=X0).
9.40/9.63	  fof(converse_multiplicativity, axiom,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        converse(composition(X0, X1))=composition(converse(X1),
9.40/9.63	                                                  converse(X0))).
9.40/9.63	  fof(def_top, axiom, ![X0]: top=join(X0, complement(X0))).
9.40/9.63	  fof(def_zero, axiom, ![X0]: zero=meet(X0, complement(X0))).
9.40/9.63	  fof(goals, conjecture,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        (composition(X0, X1)=meet(X0, X1)
9.40/9.63	           <= (one=join(X0, one) & join(X1, one)=one))).
9.40/9.63	  fof(maddux1_join_commutativity, axiom,
9.40/9.63	      ![X0, X1]: join(X0, X1)=join(X1, X0)).
9.40/9.63	  fof(maddux2_join_associativity, axiom,
9.40/9.63	      ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
9.40/9.63	  fof(maddux3_a_kind_of_de_Morgan, axiom,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        X0=join(complement(join(complement(X0), complement(X1))),
9.40/9.63	                complement(join(complement(X0), X1)))).
9.40/9.63	  fof(maddux4_definiton_of_meet, axiom,
9.40/9.63	      ![X0, X1]:
9.40/9.63	        meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
9.40/9.63	
9.40/9.63	Now clausify the problem and encode Horn clauses using encoding 3 of
9.40/9.63	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
9.40/9.63	We repeatedly replace C & s=t => u=v by the two clauses:
9.40/9.63	  $$fresh(y, y, x1...xn) = u
9.40/9.63	  C => $$fresh(s, t, x1...xn) = v
9.40/9.63	where $$fresh is a fresh function symbol and x1..xn are the free
9.40/9.63	variables of u and v.
9.40/9.63	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
9.40/9.63	input problem has no model of domain size 1).
9.40/9.63	
9.40/9.63	The encoding turns the above axioms into the following unit equations and goals:
9.40/9.63	
9.40/9.63	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
9.40/9.63	Axiom 2 (def_zero): zero = meet(X, complement(X)).
9.40/9.63	Axiom 3 (def_top): top = join(X, complement(X)).
9.40/9.63	Axiom 4 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
9.40/9.63	Axiom 5 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
9.40/9.63	Axiom 6 (converse_idempotence): converse(converse(X)) = X.
9.40/9.63	Axiom 7 (converse_additivity): join(converse(X), converse(Y)) = converse(join(X, Y)).
9.40/9.63	Axiom 8 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
9.40/9.63	Axiom 9 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
9.40/9.63	Axiom 10 (composition_identity): composition(X, one) = X.
9.40/9.63	Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
9.40/9.63	Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
9.40/9.63	Axiom 13 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
9.40/9.63	Axiom 17 (goals): join(sK2_goals_X1, one) = one.
9.40/9.63	Axiom 18 (goals_1): one = join(sK1_goals_X0, one).
9.40/9.63	
9.40/9.63	Lemma 19: join(one, sK2_goals_X1) = one.
9.40/9.63	Proof:
9.40/9.63	  join(one, sK2_goals_X1)
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(sK2_goals_X1, one)
9.40/9.63	= { by axiom 17 (goals) }
9.40/9.63	  one
9.40/9.63	
9.40/9.63	Lemma 20: complement(top) = zero.
9.40/9.63	Proof:
9.40/9.63	  complement(top)
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  complement(join(complement(?), complement(complement(?))))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  meet(?, complement(?))
9.40/9.63	= { by axiom 2 (def_zero) }
9.40/9.63	  zero
9.40/9.63	
9.40/9.63	Lemma 21: meet(X, Y) = meet(Y, X).
9.40/9.63	Proof:
9.40/9.63	  meet(X, Y)
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  complement(join(complement(X), complement(Y)))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  complement(join(complement(Y), complement(X)))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  meet(Y, X)
9.40/9.63	
9.40/9.63	Lemma 22: composition(converse(one), X) = X.
9.40/9.63	Proof:
9.40/9.63	  composition(converse(one), X)
9.40/9.63	= { by axiom 6 (converse_idempotence) }
9.40/9.63	  composition(converse(one), converse(converse(X)))
9.40/9.63	= { by axiom 4 (converse_multiplicativity) }
9.40/9.63	  converse(composition(converse(X), one))
9.40/9.63	= { by axiom 10 (composition_identity) }
9.40/9.63	  converse(converse(X))
9.40/9.63	= { by axiom 6 (converse_idempotence) }
9.40/9.63	  X
9.40/9.63	
9.40/9.63	Lemma 23: converse(one) = one.
9.40/9.63	Proof:
9.40/9.63	  converse(one)
9.40/9.63	= { by axiom 10 (composition_identity) }
9.40/9.63	  composition(converse(one), one)
9.40/9.63	= { by lemma 22 }
9.40/9.63	  one
9.40/9.63	
9.40/9.63	Lemma 24: composition(one, X) = X.
9.40/9.63	Proof:
9.40/9.63	  composition(one, X)
9.40/9.63	= { by lemma 22 }
9.40/9.63	  composition(converse(one), composition(one, X))
9.40/9.63	= { by axiom 5 (composition_associativity) }
9.40/9.63	  composition(composition(converse(one), one), X)
9.40/9.63	= { by axiom 10 (composition_identity) }
9.40/9.63	  composition(converse(one), X)
9.40/9.63	= { by lemma 22 }
9.40/9.63	  X
9.40/9.63	
9.40/9.63	Lemma 25: join(X, join(Y, Z)) = join(Z, join(X, Y)).
9.40/9.63	Proof:
9.40/9.63	  join(X, join(Y, Z))
9.40/9.63	= { by axiom 13 (maddux2_join_associativity) }
9.40/9.63	  join(join(X, Y), Z)
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(Z, join(X, Y))
9.40/9.63	
9.40/9.63	Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
9.40/9.63	Proof:
9.40/9.63	  join(X, join(Y, complement(X)))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(X, join(complement(X), Y))
9.40/9.63	= { by axiom 13 (maddux2_join_associativity) }
9.40/9.63	  join(join(X, complement(X)), Y)
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  join(top, Y)
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(Y, top)
9.40/9.63	
9.40/9.63	Lemma 27: complement(join(zero, complement(X))) = meet(X, top).
9.40/9.63	Proof:
9.40/9.63	  complement(join(zero, complement(X)))
9.40/9.63	= { by lemma 20 }
9.40/9.63	  complement(join(complement(top), complement(X)))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  meet(top, X)
9.40/9.63	= { by lemma 21 }
9.40/9.63	  meet(X, top)
9.40/9.63	
9.40/9.63	Lemma 28: converse(join(converse(X), Y)) = join(X, converse(Y)).
9.40/9.63	Proof:
9.40/9.63	  converse(join(converse(X), Y))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  converse(join(Y, converse(X)))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  converse(join(converse(X), Y))
9.40/9.63	= { by axiom 7 (converse_additivity) }
9.40/9.63	  join(converse(converse(X)), converse(Y))
9.40/9.63	= { by axiom 6 (converse_idempotence) }
9.40/9.63	  join(X, converse(Y))
9.40/9.63	
9.40/9.63	Lemma 29: join(complement(X), complement(X)) = complement(X).
9.40/9.63	Proof:
9.40/9.63	  join(complement(X), complement(X))
9.40/9.63	= { by lemma 22 }
9.40/9.63	  join(composition(converse(one), complement(X)), complement(X))
9.40/9.63	= { by lemma 24 }
9.40/9.63	  join(composition(converse(one), complement(composition(one, X))), complement(X))
9.40/9.63	= { by axiom 8 (converse_cancellativity) }
9.40/9.63	  complement(X)
9.40/9.63	
9.40/9.63	Lemma 31: complement(complement(X)) = meet(X, X).
9.40/9.63	Proof:
9.40/9.63	  complement(complement(X))
9.40/9.63	= { by lemma 29 }
9.40/9.63	  complement(join(complement(X), complement(X)))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  meet(X, X)
9.40/9.63	
9.40/9.63	Lemma 31: meet(X, X) = complement(complement(X)).
9.40/9.63	Proof:
9.40/9.63	  meet(X, X)
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  complement(join(complement(X), complement(X)))
9.40/9.63	= { by lemma 29 }
9.40/9.63	  complement(complement(X))
9.40/9.63	
9.40/9.63	Lemma 32: join(top, complement(X)) = top.
9.40/9.63	Proof:
9.40/9.63	  join(top, complement(X))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(complement(X), top)
9.40/9.63	= { by lemma 26 }
9.40/9.63	  join(X, join(complement(X), complement(X)))
9.40/9.63	= { by lemma 29 }
9.40/9.63	  join(X, complement(X))
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  top
9.40/9.63	
9.40/9.63	Lemma 33: join(X, join(complement(X), Y)) = join(Y, top).
9.40/9.63	Proof:
9.40/9.63	  join(X, join(complement(X), Y))
9.40/9.63	= { by lemma 25 }
9.40/9.63	  join(complement(X), join(Y, X))
9.40/9.63	= { by lemma 25 }
9.40/9.63	  join(Y, join(X, complement(X)))
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  join(Y, top)
9.40/9.63	
9.40/9.63	Lemma 34: join(X, top) = top.
9.40/9.63	Proof:
9.40/9.63	  join(X, top)
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  join(X, join(complement(X), complement(complement(X))))
9.40/9.63	= { by lemma 33 }
9.40/9.63	  join(complement(complement(X)), top)
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(top, complement(complement(X)))
9.40/9.63	= { by lemma 32 }
9.40/9.63	  top
9.40/9.63	
9.40/9.63	Lemma 35: join(X, converse(top)) = converse(top).
9.40/9.63	Proof:
9.40/9.63	  join(X, converse(top))
9.40/9.63	= { by lemma 28 }
9.40/9.63	  converse(join(converse(X), top))
9.40/9.63	= { by lemma 34 }
9.40/9.63	  converse(top)
9.40/9.63	
9.40/9.63	Lemma 36: converse(top) = top.
9.40/9.63	Proof:
9.40/9.63	  converse(top)
9.40/9.63	= { by lemma 35 }
9.40/9.63	  join(?, converse(top))
9.40/9.63	= { by lemma 35 }
9.40/9.63	  join(?, join(complement(?), converse(top)))
9.40/9.63	= { by lemma 33 }
9.40/9.63	  join(converse(top), top)
9.40/9.63	= { by lemma 34 }
9.40/9.63	  top
9.40/9.63	
9.40/9.63	Lemma 37: join(zero, meet(X, top)) = X.
9.40/9.63	Proof:
9.40/9.63	  join(zero, meet(X, top))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(meet(X, top), zero)
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  join(complement(join(complement(X), complement(top))), zero)
9.40/9.63	= { by lemma 20 }
9.40/9.63	  join(complement(join(complement(X), complement(top))), complement(top))
9.40/9.63	= { by lemma 34 }
9.40/9.63	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
9.40/9.63	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.40/9.63	  X
9.40/9.63	
9.40/9.63	Lemma 38: join(meet(X, Y), meet(X, complement(Y))) = X.
9.40/9.63	Proof:
9.40/9.63	  join(meet(X, Y), meet(X, complement(Y)))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(meet(X, complement(Y)), meet(X, Y))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
9.40/9.63	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.63	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
9.40/9.63	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.40/9.63	  X
9.40/9.63	
9.40/9.63	Lemma 39: join(zero, meet(X, X)) = X.
9.40/9.63	Proof:
9.40/9.63	  join(zero, meet(X, X))
9.40/9.63	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.63	  join(meet(X, X), zero)
9.40/9.63	= { by axiom 2 (def_zero) }
9.40/9.63	  join(meet(X, X), meet(X, complement(X)))
9.40/9.63	= { by lemma 38 }
9.40/9.63	  X
9.40/9.63	
9.40/9.63	Lemma 40: join(X, converse(complement(converse(X)))) = top.
9.40/9.63	Proof:
9.40/9.63	  join(X, converse(complement(converse(X))))
9.40/9.63	= { by lemma 28 }
9.40/9.63	  converse(join(converse(X), complement(converse(X))))
9.40/9.63	= { by axiom 3 (def_top) }
9.40/9.63	  converse(top)
9.40/9.63	= { by lemma 36 }
9.40/9.64	  top
9.40/9.64	
9.40/9.64	Lemma 41: join(X, zero) = X.
9.40/9.64	Proof:
9.40/9.64	  join(X, zero)
9.40/9.64	= { by lemma 39 }
9.40/9.64	  join(join(zero, meet(X, X)), zero)
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  join(join(meet(X, X), zero), zero)
9.40/9.64	= { by lemma 20 }
9.40/9.64	  join(join(meet(X, X), zero), complement(top))
9.40/9.64	= { by lemma 32 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(top, complement(zero))))
9.40/9.64	= { by lemma 38 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(zero))))
9.40/9.64	= { by axiom 2 (def_zero) }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(join(zero, meet(top, complement(complement(top)))), complement(zero))))
9.40/9.64	= { by lemma 31 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(join(zero, meet(top, meet(top, top))), complement(zero))))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(join(zero, meet(meet(top, top), top)), complement(zero))))
9.40/9.64	= { by lemma 37 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(meet(top, top), complement(zero))))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(complement(join(zero, complement(top))), complement(zero))))
9.40/9.64	= { by lemma 20 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(complement(join(complement(top), complement(top))), complement(zero))))
9.40/9.64	= { by lemma 29 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(complement(complement(top)), complement(zero))))
9.40/9.64	= { by lemma 20 }
9.40/9.64	  join(join(meet(X, X), zero), complement(join(complement(zero), complement(zero))))
9.40/9.64	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.64	  join(join(meet(X, X), zero), meet(zero, zero))
9.40/9.64	= { by axiom 13 (maddux2_join_associativity) }
9.40/9.64	  join(meet(X, X), join(zero, meet(zero, zero)))
9.40/9.64	= { by lemma 39 }
9.40/9.64	  join(meet(X, X), zero)
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  join(zero, meet(X, X))
9.40/9.64	= { by lemma 39 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 42: join(zero, X) = X.
9.40/9.64	Proof:
9.40/9.64	  join(zero, X)
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  join(X, zero)
9.40/9.64	= { by lemma 41 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 43: meet(X, top) = X.
9.40/9.64	Proof:
9.40/9.64	  meet(X, top)
9.40/9.64	= { by lemma 42 }
9.40/9.64	  join(zero, meet(X, top))
9.40/9.64	= { by lemma 37 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 44: complement(complement(X)) = X.
9.40/9.64	Proof:
9.40/9.64	  complement(complement(X))
9.40/9.64	= { by lemma 42 }
9.40/9.64	  join(zero, complement(complement(X)))
9.40/9.64	= { by lemma 31 }
9.40/9.64	  join(zero, meet(X, X))
9.40/9.64	= { by lemma 39 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 45: meet(top, X) = X.
9.40/9.64	Proof:
9.40/9.64	  meet(top, X)
9.40/9.64	= { by lemma 21 }
9.40/9.64	  meet(X, top)
9.40/9.64	= { by lemma 43 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 46: meet(join(X, complement(Y)), join(complement(X), complement(Y))) = complement(Y).
9.40/9.64	Proof:
9.40/9.64	  meet(join(X, complement(Y)), join(complement(X), complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(join(complement(Y), X), join(complement(X), complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(join(complement(Y), X), join(complement(Y), complement(X)))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
9.40/9.64	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.64	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
9.40/9.64	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.40/9.64	  complement(Y)
9.40/9.64	
9.40/9.64	Lemma 47: meet(join(X, Y), join(X, complement(Y))) = X.
9.40/9.64	Proof:
9.40/9.64	  meet(join(X, Y), join(X, complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(join(Y, X), join(X, complement(Y)))
9.40/9.64	= { by lemma 43 }
9.40/9.64	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), X))
9.40/9.64	= { by lemma 43 }
9.40/9.64	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), meet(X, top)))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), complement(join(zero, complement(X)))))
9.40/9.64	= { by lemma 46 }
9.40/9.64	  complement(join(zero, complement(X)))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  meet(X, top)
9.40/9.64	= { by lemma 43 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 48: join(X, complement(converse(complement(converse(X))))) = X.
9.40/9.64	Proof:
9.40/9.64	  join(X, complement(converse(complement(converse(X)))))
9.40/9.64	= { by lemma 45 }
9.40/9.64	  meet(top, join(X, complement(converse(complement(converse(X))))))
9.40/9.64	= { by lemma 40 }
9.40/9.64	  meet(join(X, converse(complement(converse(X)))), join(X, complement(converse(complement(converse(X))))))
9.40/9.64	= { by lemma 47 }
9.40/9.64	  X
9.40/9.64	
9.40/9.64	Lemma 49: join(complement(X), complement(Y)) = complement(meet(X, Y)).
9.40/9.64	Proof:
9.40/9.64	  join(complement(X), complement(Y))
9.40/9.64	= { by lemma 45 }
9.40/9.64	  meet(top, join(complement(X), complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(top, join(complement(Y), complement(X)))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  meet(join(complement(Y), complement(X)), top)
9.40/9.64	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.64	  complement(join(complement(join(complement(Y), complement(X))), complement(top)))
9.40/9.64	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.64	  complement(join(meet(Y, X), complement(top)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  complement(join(complement(top), meet(Y, X)))
9.40/9.64	= { by lemma 20 }
9.40/9.64	  complement(join(zero, meet(Y, X)))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  complement(join(zero, meet(X, Y)))
9.40/9.64	= { by lemma 42 }
9.40/9.64	  complement(meet(X, Y))
9.40/9.64	
9.40/9.64	Lemma 50: complement(meet(X, complement(Y))) = join(Y, complement(X)).
9.40/9.64	Proof:
9.40/9.64	  complement(meet(X, complement(Y)))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  complement(meet(complement(Y), X))
9.40/9.64	= { by lemma 42 }
9.40/9.64	  complement(meet(join(zero, complement(Y)), X))
9.40/9.64	= { by lemma 49 }
9.40/9.64	  join(complement(join(zero, complement(Y))), complement(X))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  join(meet(Y, top), complement(X))
9.40/9.64	= { by lemma 43 }
9.40/9.64	  join(Y, complement(X))
9.40/9.64	
9.40/9.64	Lemma 51: meet(X, join(Y, complement(X))) = meet(X, Y).
9.40/9.64	Proof:
9.40/9.64	  meet(X, join(Y, complement(X)))
9.40/9.64	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), join(Y, complement(X)))
9.40/9.64	= { by lemma 50 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(Y))))
9.40/9.64	= { by lemma 44 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(complement(complement(Y))))))
9.40/9.64	= { by lemma 47 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(meet(join(complement(complement(Y)), join(complement(X), complement(complement(complement(Y))))), join(complement(complement(Y)), complement(join(complement(X), complement(complement(complement(Y)))))))))))
9.40/9.64	= { by lemma 26 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(meet(join(complement(X), top), join(complement(complement(Y)), complement(join(complement(X), complement(complement(complement(Y)))))))))))
9.40/9.64	= { by lemma 34 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(meet(top, join(complement(complement(Y)), complement(join(complement(X), complement(complement(complement(Y)))))))))))
9.40/9.64	= { by lemma 45 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, complement(join(complement(complement(Y)), complement(join(complement(X), complement(complement(complement(Y))))))))))
9.40/9.64	= { by axiom 11 (maddux4_definiton_of_meet) }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, meet(complement(Y), join(complement(X), complement(complement(complement(Y))))))))
9.40/9.64	= { by lemma 44 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(X, meet(complement(Y), join(complement(X), complement(Y))))))
9.40/9.64	= { by lemma 49 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), join(complement(X), complement(meet(complement(Y), join(complement(X), complement(Y))))))
9.40/9.64	= { by lemma 49 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), join(complement(X), join(complement(complement(Y)), complement(join(complement(X), complement(Y))))))
9.40/9.64	= { by axiom 13 (maddux2_join_associativity) }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), join(join(complement(X), complement(complement(Y))), complement(join(complement(X), complement(Y)))))
9.40/9.64	= { by lemma 50 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(join(complement(X), complement(Y)), complement(join(complement(X), complement(complement(Y)))))))
9.40/9.64	= { by lemma 21 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), complement(meet(complement(join(complement(X), complement(complement(Y)))), join(complement(X), complement(Y)))))
9.40/9.64	= { by lemma 49 }
9.40/9.64	  meet(join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y)))), join(complement(complement(join(complement(X), complement(complement(Y))))), complement(join(complement(X), complement(Y)))))
9.40/9.64	= { by lemma 46 }
9.40/9.64	  complement(join(complement(X), complement(Y)))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  complement(join(complement(Y), complement(X)))
9.40/9.64	= { by lemma 42 }
9.40/9.64	  complement(join(zero, join(complement(Y), complement(X))))
9.40/9.64	= { by lemma 50 }
9.40/9.64	  complement(join(zero, complement(meet(X, complement(complement(Y))))))
9.40/9.64	= { by lemma 27 }
9.40/9.64	  meet(meet(X, complement(complement(Y))), top)
9.40/9.64	= { by lemma 43 }
9.40/9.64	  meet(X, complement(complement(Y)))
9.40/9.64	= { by lemma 44 }
9.40/9.64	  meet(X, Y)
9.40/9.64	
9.40/9.64	Lemma 52: meet(X, join(complement(X), Y)) = meet(X, Y).
9.40/9.64	Proof:
9.40/9.64	  meet(X, join(complement(X), Y))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  meet(X, join(Y, complement(X)))
9.40/9.64	= { by lemma 51 }
9.40/9.64	  meet(X, Y)
9.40/9.64	
9.40/9.64	Lemma 53: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
9.40/9.64	Proof:
9.40/9.64	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
9.40/9.64	= { by axiom 9 (maddux1_join_commutativity) }
9.40/9.64	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
9.40/9.65	= { by axiom 12 (composition_distributivity) }
9.40/9.65	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
9.40/9.65	= { by axiom 13 (maddux2_join_associativity) }
9.40/9.65	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
9.40/9.65	= { by axiom 3 (def_top) }
9.40/9.65	  join(composition(Z, Y), top)
9.40/9.65	= { by lemma 34 }
9.94/10.16	  top
9.94/10.16	
9.94/10.16	Goal 1 (goals_2): composition(sK1_goals_X0, sK2_goals_X1) = meet(sK1_goals_X0, sK2_goals_X1).
9.94/10.16	Proof:
9.94/10.16	  composition(sK1_goals_X0, sK2_goals_X1)
9.94/10.16	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.94/10.16	  join(complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), complement(sK2_goals_X1))), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), sK2_goals_X1)))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), sK2_goals_X1)))
9.94/10.16	= { by lemma 24 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(one, sK2_goals_X1))))
9.94/10.16	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(join(complement(one), complement(sK1_goals_X0)))), sK2_goals_X1))))
9.94/10.16	= { by lemma 45 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(top, join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by lemma 34 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(join(one, top), join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by axiom 3 (def_top) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(join(one, join(sK1_goals_X0, complement(sK1_goals_X0))), join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by axiom 13 (maddux2_join_associativity) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(join(join(one, sK1_goals_X0), complement(sK1_goals_X0)), join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(join(join(sK1_goals_X0, one), complement(sK1_goals_X0)), join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by axiom 18 (goals_1) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(meet(join(one, complement(sK1_goals_X0)), join(complement(one), complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by lemma 46 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(complement(join(complement(one), complement(complement(sK1_goals_X0)))), complement(complement(sK1_goals_X0))), sK2_goals_X1))))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(complement(meet(join(complement(one), complement(complement(sK1_goals_X0))), complement(sK1_goals_X0))), sK2_goals_X1))))
9.94/10.16	= { by lemma 50 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(sK1_goals_X0, complement(join(complement(one), complement(complement(sK1_goals_X0))))), sK2_goals_X1))))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(join(complement(composition(sK1_goals_X0, sK2_goals_X1)), composition(join(sK1_goals_X0, meet(one, complement(sK1_goals_X0))), sK2_goals_X1))))
9.94/10.16	= { by lemma 53 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), complement(top))
9.94/10.16	= { by lemma 20 }
9.94/10.16	  join(meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1), zero)
9.94/10.16	= { by lemma 41 }
9.94/10.16	  meet(composition(sK1_goals_X0, sK2_goals_X1), sK2_goals_X1)
9.94/10.16	= { by lemma 21 }
9.94/10.16	  meet(sK2_goals_X1, composition(sK1_goals_X0, sK2_goals_X1))
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), top))
9.94/10.16	= { by lemma 36 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), converse(top)))
9.94/10.16	= { by lemma 34 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), converse(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), top))))
9.94/10.16	= { by lemma 26 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), converse(join(converse(sK1_goals_X0), join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), complement(converse(sK1_goals_X0)))))))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), converse(join(converse(sK1_goals_X0), complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), converse(sK1_goals_X0)))))))
9.94/10.16	= { by lemma 21 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), converse(join(converse(sK1_goals_X0), complement(meet(converse(sK1_goals_X0), composition(converse(sK2_goals_X1), converse(sK1_goals_X0))))))))
9.94/10.16	= { by lemma 28 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(converse(sK1_goals_X0), composition(converse(sK2_goals_X1), converse(sK1_goals_X0))))))))
9.94/10.16	= { by lemma 21 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), converse(sK1_goals_X0)))))))
9.94/10.16	= { by lemma 24 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(one, converse(sK1_goals_X0))))))))
9.94/10.16	= { by lemma 23 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(converse(one), converse(sK1_goals_X0))))))))
9.94/10.16	= { by lemma 19 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(converse(join(one, sK2_goals_X1)), converse(sK1_goals_X0))))))))
9.94/10.16	= { by axiom 7 (converse_additivity) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(join(converse(one), converse(sK2_goals_X1)), converse(sK1_goals_X0))))))))
9.94/10.16	= { by lemma 23 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(join(one, converse(sK2_goals_X1)), converse(sK1_goals_X0))))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(meet(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)), composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0))))))))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(complement(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), complement(composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0))))))))))
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(complement(meet(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), complement(composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0)))), top)))))))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(join(complement(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), complement(composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0))))), complement(top)))))))
9.94/10.16	= { by lemma 53 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(join(complement(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), complement(composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0))))), complement(join(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0))), composition(join(converse(sK2_goals_X1), one), converse(sK1_goals_X0))))))))))
9.94/10.16	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(composition(converse(sK2_goals_X1), converse(sK1_goals_X0)))))))
9.94/10.16	= { by axiom 4 (converse_multiplicativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(converse(composition(sK1_goals_X0, sK2_goals_X1)))))))
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(converse(meet(composition(sK1_goals_X0, sK2_goals_X1), top)))))))
9.94/10.16	= { by lemma 27 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))))))
9.94/10.16	= { by lemma 47 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(join(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))), converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by axiom 6 (converse_idempotence) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(join(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))), converse(complement(converse(complement(converse(converse(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by lemma 28 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(converse(join(converse(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))), complement(converse(complement(converse(converse(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by lemma 48 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(converse(converse(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by axiom 6 (converse_idempotence) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))), join(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))), complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))
9.94/10.16	= { by lemma 51 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, meet(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))), converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, complement(join(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))), complement(converse(complement(converse(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1)))))))))))))
9.94/10.16	= { by lemma 48 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, complement(complement(join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))))
9.94/10.16	= { by lemma 44 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, join(zero, complement(composition(sK1_goals_X0, sK2_goals_X1))))))
9.94/10.16	= { by lemma 42 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), join(sK1_goals_X0, complement(composition(sK1_goals_X0, sK2_goals_X1)))))
9.94/10.16	= { by lemma 51 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(sK1_goals_X0, sK2_goals_X1), sK1_goals_X0))
9.94/10.16	= { by lemma 21 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, composition(sK1_goals_X0, sK2_goals_X1)))
9.94/10.16	= { by lemma 44 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, composition(complement(complement(sK1_goals_X0)), sK2_goals_X1)))
9.94/10.16	= { by lemma 52 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(complement(sK1_goals_X0), composition(complement(complement(sK1_goals_X0)), sK2_goals_X1))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), complement(sK1_goals_X0))))
9.94/10.16	= { by axiom 10 (composition_identity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), composition(complement(sK1_goals_X0), one))))
9.94/10.16	= { by lemma 19 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), composition(complement(sK1_goals_X0), join(one, sK2_goals_X1)))))
9.94/10.16	= { by axiom 6 (converse_idempotence) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(converse(composition(complement(sK1_goals_X0), join(one, sK2_goals_X1)))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(converse(composition(complement(sK1_goals_X0), join(sK2_goals_X1, one)))))))
9.94/10.16	= { by axiom 4 (converse_multiplicativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(composition(converse(join(sK2_goals_X1, one)), converse(complement(sK1_goals_X0)))))))
9.94/10.16	= { by axiom 7 (converse_additivity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(composition(join(converse(sK2_goals_X1), converse(one)), converse(complement(sK1_goals_X0)))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(composition(join(converse(one), converse(sK2_goals_X1)), converse(complement(sK1_goals_X0)))))))
9.94/10.16	= { by axiom 12 (composition_distributivity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(join(composition(converse(one), converse(complement(sK1_goals_X0))), composition(converse(sK2_goals_X1), converse(complement(sK1_goals_X0))))))))
9.94/10.16	= { by axiom 4 (converse_multiplicativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(join(converse(composition(complement(sK1_goals_X0), one)), composition(converse(sK2_goals_X1), converse(complement(sK1_goals_X0))))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(join(composition(converse(sK2_goals_X1), converse(complement(sK1_goals_X0))), converse(composition(complement(sK1_goals_X0), one)))))))
9.94/10.16	= { by axiom 4 (converse_multiplicativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(join(converse(composition(complement(sK1_goals_X0), sK2_goals_X1)), converse(composition(complement(sK1_goals_X0), one)))))))
9.94/10.16	= { by axiom 7 (converse_additivity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(converse(join(composition(complement(sK1_goals_X0), sK2_goals_X1), composition(complement(sK1_goals_X0), one)))))))
9.94/10.16	= { by axiom 9 (maddux1_join_commutativity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), converse(converse(join(composition(complement(sK1_goals_X0), one), composition(complement(sK1_goals_X0), sK2_goals_X1)))))))
9.94/10.16	= { by axiom 6 (converse_idempotence) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(complement(sK1_goals_X0)), sK2_goals_X1), join(composition(complement(sK1_goals_X0), one), composition(complement(sK1_goals_X0), sK2_goals_X1)))))
9.94/10.16	= { by lemma 25 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(composition(complement(sK1_goals_X0), one), join(composition(complement(sK1_goals_X0), sK2_goals_X1), composition(complement(complement(sK1_goals_X0)), sK2_goals_X1)))))
9.94/10.16	= { by axiom 10 (composition_identity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(complement(sK1_goals_X0), join(composition(complement(sK1_goals_X0), sK2_goals_X1), composition(complement(complement(sK1_goals_X0)), sK2_goals_X1)))))
9.94/10.16	= { by axiom 12 (composition_distributivity) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(complement(sK1_goals_X0), composition(join(complement(sK1_goals_X0), complement(complement(sK1_goals_X0))), sK2_goals_X1))))
9.94/10.16	= { by axiom 3 (def_top) }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, join(complement(sK1_goals_X0), composition(top, sK2_goals_X1))))
9.94/10.16	= { by lemma 52 }
9.94/10.16	  meet(sK2_goals_X1, meet(sK1_goals_X0, composition(top, sK2_goals_X1)))
9.94/10.16	= { by lemma 21 }
9.94/10.16	  meet(sK2_goals_X1, meet(composition(top, sK2_goals_X1), sK1_goals_X0))
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(meet(sK2_goals_X1, meet(composition(top, sK2_goals_X1), sK1_goals_X0)), top)
9.94/10.16	= { by lemma 27 }
9.94/10.16	  complement(join(zero, complement(meet(sK2_goals_X1, meet(composition(top, sK2_goals_X1), sK1_goals_X0)))))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  complement(join(zero, join(complement(sK2_goals_X1), complement(meet(composition(top, sK2_goals_X1), sK1_goals_X0)))))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  complement(join(zero, join(complement(sK2_goals_X1), join(complement(composition(top, sK2_goals_X1)), complement(sK1_goals_X0)))))
9.94/10.16	= { by axiom 13 (maddux2_join_associativity) }
9.94/10.16	  complement(join(zero, join(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1))), complement(sK1_goals_X0))))
9.94/10.16	= { by lemma 50 }
9.94/10.16	  complement(join(zero, complement(meet(sK1_goals_X0, complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1))))))))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  complement(join(zero, complement(meet(sK1_goals_X0, meet(sK2_goals_X1, composition(top, sK2_goals_X1))))))
9.94/10.16	= { by lemma 27 }
9.94/10.16	  meet(meet(sK1_goals_X0, meet(sK2_goals_X1, composition(top, sK2_goals_X1))), top)
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(sK1_goals_X0, meet(sK2_goals_X1, composition(top, sK2_goals_X1)))
9.94/10.16	= { by axiom 11 (maddux4_definiton_of_meet) }
9.94/10.16	  meet(sK1_goals_X0, complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))))
9.94/10.16	= { by lemma 43 }
9.94/10.16	  meet(sK1_goals_X0, complement(meet(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1))), top)))
9.94/10.16	= { by lemma 49 }
9.94/10.16	  meet(sK1_goals_X0, join(complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))), complement(top)))
9.94/10.16	= { by lemma 53 }
9.94/10.16	  meet(sK1_goals_X0, join(complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))), complement(join(complement(composition(one, sK2_goals_X1)), composition(join(one, converse(complement(one))), sK2_goals_X1)))))
9.94/10.16	= { by lemma 24 }
9.94/10.16	  meet(sK1_goals_X0, join(complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))), complement(join(complement(sK2_goals_X1), composition(join(one, converse(complement(one))), sK2_goals_X1)))))
9.94/10.16	= { by lemma 23 }
9.94/10.16	  meet(sK1_goals_X0, join(complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))), complement(join(complement(sK2_goals_X1), composition(join(one, converse(complement(converse(one)))), sK2_goals_X1)))))
9.94/10.16	= { by lemma 40 }
9.94/10.16	  meet(sK1_goals_X0, join(complement(join(complement(sK2_goals_X1), complement(composition(top, sK2_goals_X1)))), complement(join(complement(sK2_goals_X1), composition(top, sK2_goals_X1)))))
9.94/10.16	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
9.94/10.16	  meet(sK1_goals_X0, sK2_goals_X1)
9.94/10.16	% SZS output end Proof
9.94/10.16	
9.94/10.16	RESULT: Theorem (the conjecture is true).
9.94/10.18	EOF