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
0.03/0.24	% Computer   : n172.star.cs.uiowa.edu
0.03/0.24	% Model      : x86_64 x86_64
0.03/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.24	% Memory     : 32218.625MB
0.03/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.24	% CPULimit   : 300
0.03/0.24	% DateTime   : Sat Jul 14 06:07:55 CDT 2018
0.03/0.24	% CPUTime    : 
57.57/57.77	% SZS status Theorem
57.57/57.77	
57.64/57.89	% SZS output start Proof
57.64/57.89	Take the following subset of the input axioms:
57.72/57.91	  fof(composition_associativity, axiom,
57.72/57.91	      ![X0, X1, X2]:
57.72/57.91	        composition(X0, composition(X1, X2))=composition(composition(X0,
57.72/57.91	                                                                     X1),
57.72/57.91	                                                         X2)).
57.72/57.91	  fof(composition_distributivity, axiom,
57.72/57.91	      ![X0, X1, X2]:
57.72/57.91	        composition(join(X0, X1), X2)=join(composition(X0, X2),
57.72/57.91	                                           composition(X1, X2))).
57.72/57.91	  fof(composition_identity, axiom, ![X0]: composition(X0, one)=X0).
57.72/57.91	  fof(converse_additivity, axiom,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        join(converse(X0), converse(X1))=converse(join(X0, X1))).
57.72/57.91	  fof(converse_cancellativity, axiom,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        join(composition(converse(X0), complement(composition(X0, X1))),
57.72/57.91	             complement(X1))=complement(X1)).
57.72/57.91	  fof(converse_idempotence, axiom, ![X0]: converse(converse(X0))=X0).
57.72/57.91	  fof(converse_multiplicativity, axiom,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        converse(composition(X0, X1))=composition(converse(X1),
57.72/57.91	                                                  converse(X0))).
57.72/57.91	  fof(def_top, axiom, ![X0]: top=join(X0, complement(X0))).
57.72/57.91	  fof(def_zero, axiom, ![X0]: zero=meet(X0, complement(X0))).
57.72/57.91	  fof(goals, conjecture,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        (composition(X0, top)=X0
57.72/57.91	           => join(meet(X0, X1),
57.72/57.91	                   composition(meet(X0, one), X1))=composition(meet(X0, one), X1))).
57.72/57.91	  fof(maddux1_join_commutativity, axiom,
57.72/57.91	      ![X0, X1]: join(X0, X1)=join(X1, X0)).
57.72/57.91	  fof(maddux2_join_associativity, axiom,
57.72/57.91	      ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
57.72/57.91	  fof(maddux3_a_kind_of_de_Morgan, axiom,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        X0=join(complement(join(complement(X0), complement(X1))),
57.72/57.91	                complement(join(complement(X0), X1)))).
57.72/57.91	  fof(maddux4_definiton_of_meet, axiom,
57.72/57.91	      ![X0, X1]:
57.72/57.91	        meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
57.72/57.91	
57.72/57.91	Now clausify the problem and encode Horn clauses using encoding 3 of
57.72/57.91	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
57.72/57.91	We repeatedly replace C & s=t => u=v by the two clauses:
57.72/57.91	  $$fresh(y, y, x1...xn) = u
57.72/57.91	  C => $$fresh(s, t, x1...xn) = v
57.72/57.91	where $$fresh is a fresh function symbol and x1..xn are the free
57.72/57.91	variables of u and v.
57.72/57.91	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
57.72/57.91	input problem has no model of domain size 1).
57.72/57.91	
57.72/57.91	The encoding turns the above axioms into the following unit equations and goals:
57.72/57.91	
57.72/57.91	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
57.72/57.91	Axiom 2 (def_zero): zero = meet(X, complement(X)).
57.72/57.91	Axiom 3 (def_top): top = join(X, complement(X)).
57.72/57.91	Axiom 4 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
57.72/57.91	Axiom 5 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
57.72/57.91	Axiom 6 (converse_idempotence): converse(converse(X)) = X.
57.72/57.91	Axiom 7 (converse_additivity): join(converse(X), converse(Y)) = converse(join(X, Y)).
57.72/57.91	Axiom 8 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
57.72/57.91	Axiom 9 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
57.72/57.91	Axiom 10 (composition_identity): composition(X, one) = X.
57.72/57.91	Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
57.72/57.91	Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
57.72/57.91	Axiom 13 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
57.72/57.91	Axiom 14 (goals): composition(sK2_goals_X0, top) = sK2_goals_X0.
57.72/57.91	
57.72/57.91	Lemma 15: complement(top) = zero.
57.72/57.91	Proof:
57.72/57.91	  complement(top)
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  complement(join(complement(?), complement(complement(?))))
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  meet(?, complement(?))
57.72/57.91	= { by axiom 2 (def_zero) }
57.72/57.91	  zero
57.72/57.91	
57.72/57.91	Lemma 16: meet(X, Y) = meet(Y, X).
57.72/57.91	Proof:
57.72/57.91	  meet(X, Y)
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  complement(join(complement(X), complement(Y)))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  complement(join(complement(Y), complement(X)))
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  meet(Y, X)
57.72/57.91	
57.72/57.91	Lemma 17: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
57.72/57.91	Proof:
57.72/57.91	  converse(composition(X, converse(Y)))
57.72/57.91	= { by axiom 4 (converse_multiplicativity) }
57.72/57.91	  composition(converse(converse(Y)), converse(X))
57.72/57.91	= { by axiom 6 (converse_idempotence) }
57.72/57.91	  composition(Y, converse(X))
57.72/57.91	
57.72/57.91	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
57.72/57.91	Proof:
57.72/57.91	  converse(composition(converse(X), Y))
57.72/57.91	= { by axiom 4 (converse_multiplicativity) }
57.72/57.91	  composition(converse(Y), converse(converse(X)))
57.72/57.91	= { by axiom 6 (converse_idempotence) }
57.72/57.91	  composition(converse(Y), X)
57.72/57.91	
57.72/57.91	Lemma 19: composition(converse(one), X) = X.
57.72/57.91	Proof:
57.72/57.91	  composition(converse(one), X)
57.72/57.91	= { by lemma 18 }
57.72/57.91	  converse(composition(converse(X), one))
57.72/57.91	= { by axiom 10 (composition_identity) }
57.72/57.91	  converse(converse(X))
57.72/57.91	= { by axiom 6 (converse_idempotence) }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 20: converse(one) = one.
57.72/57.91	Proof:
57.72/57.91	  converse(one)
57.72/57.91	= { by axiom 10 (composition_identity) }
57.72/57.91	  composition(converse(one), one)
57.72/57.91	= { by lemma 19 }
57.72/57.91	  one
57.72/57.91	
57.72/57.91	Lemma 21: composition(one, X) = X.
57.72/57.91	Proof:
57.72/57.91	  composition(one, X)
57.72/57.91	= { by lemma 19 }
57.72/57.91	  composition(converse(one), composition(one, X))
57.72/57.91	= { by axiom 5 (composition_associativity) }
57.72/57.91	  composition(composition(converse(one), one), X)
57.72/57.91	= { by axiom 10 (composition_identity) }
57.72/57.91	  composition(converse(one), X)
57.72/57.91	= { by lemma 19 }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 22: converse(join(X, converse(Y))) = join(Y, converse(X)).
57.72/57.91	Proof:
57.72/57.91	  converse(join(X, converse(Y)))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  converse(join(converse(Y), X))
57.72/57.91	= { by axiom 7 (converse_additivity) }
57.72/57.91	  join(converse(converse(Y)), converse(X))
57.72/57.91	= { by axiom 6 (converse_idempotence) }
57.72/57.91	  join(Y, converse(X))
57.72/57.91	
57.72/57.91	Lemma 24: join(X, join(Y, Z)) = join(Z, join(X, Y)).
57.72/57.91	Proof:
57.72/57.91	  join(X, join(Y, Z))
57.72/57.91	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.91	  join(join(X, Y), Z)
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(Z, join(X, Y))
57.72/57.91	
57.72/57.91	Lemma 24: join(Z, join(X, Y)) = join(X, join(Y, Z)).
57.72/57.91	Proof:
57.72/57.91	  join(Z, join(X, Y))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(join(X, Y), Z)
57.72/57.91	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.91	  join(X, join(Y, Z))
57.72/57.91	
57.72/57.91	Lemma 25: join(X, join(Y, Z)) = join(Y, join(X, Z)).
57.72/57.91	Proof:
57.72/57.91	  join(X, join(Y, Z))
57.72/57.91	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.91	  join(join(X, Y), Z)
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(join(Y, X), Z)
57.72/57.91	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.91	  join(Y, join(X, Z))
57.72/57.91	
57.72/57.91	Lemma 26: complement(join(zero, complement(X))) = meet(X, top).
57.72/57.91	Proof:
57.72/57.91	  complement(join(zero, complement(X)))
57.72/57.91	= { by lemma 15 }
57.72/57.91	  complement(join(complement(top), complement(X)))
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  meet(top, X)
57.72/57.91	= { by lemma 16 }
57.72/57.91	  meet(X, top)
57.72/57.91	
57.72/57.91	Lemma 27: converse(join(converse(X), Y)) = join(X, converse(Y)).
57.72/57.91	Proof:
57.72/57.91	  converse(join(converse(X), Y))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  converse(join(Y, converse(X)))
57.72/57.91	= { by lemma 22 }
57.72/57.91	  join(X, converse(Y))
57.72/57.91	
57.72/57.91	Lemma 28: join(complement(X), complement(X)) = complement(X).
57.72/57.91	Proof:
57.72/57.91	  join(complement(X), complement(X))
57.72/57.91	= { by lemma 19 }
57.72/57.91	  join(composition(converse(one), complement(X)), complement(X))
57.72/57.91	= { by lemma 21 }
57.72/57.91	  join(composition(converse(one), complement(composition(one, X))), complement(X))
57.72/57.91	= { by axiom 8 (converse_cancellativity) }
57.72/57.91	  complement(X)
57.72/57.91	
57.72/57.91	Lemma 29: meet(X, X) = complement(complement(X)).
57.72/57.91	Proof:
57.72/57.91	  meet(X, X)
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  complement(join(complement(X), complement(X)))
57.72/57.91	= { by lemma 28 }
57.72/57.91	  complement(complement(X))
57.72/57.91	
57.72/57.91	Lemma 30: join(X, join(complement(X), Y)) = join(Y, top).
57.72/57.91	Proof:
57.72/57.91	  join(X, join(complement(X), Y))
57.72/57.91	= { by lemma 24 }
57.72/57.91	  join(complement(X), join(Y, X))
57.72/57.91	= { by lemma 24 }
57.72/57.91	  join(Y, join(X, complement(X)))
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  join(Y, top)
57.72/57.91	
57.72/57.91	Lemma 31: join(X, top) = top.
57.72/57.91	Proof:
57.72/57.91	  join(X, top)
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  join(X, join(complement(X), complement(complement(X))))
57.72/57.91	= { by lemma 30 }
57.72/57.91	  join(complement(complement(X)), top)
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(top, complement(complement(X)))
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  join(join(complement(X), complement(complement(X))), complement(complement(X)))
57.72/57.91	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.91	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
57.72/57.91	= { by lemma 28 }
57.72/57.91	  join(complement(X), complement(complement(X)))
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  top
57.72/57.91	
57.72/57.91	Lemma 32: join(X, converse(top)) = converse(top).
57.72/57.91	Proof:
57.72/57.91	  join(X, converse(top))
57.72/57.91	= { by lemma 27 }
57.72/57.91	  converse(join(converse(X), top))
57.72/57.91	= { by lemma 31 }
57.72/57.91	  converse(top)
57.72/57.91	
57.72/57.91	Lemma 33: converse(top) = top.
57.72/57.91	Proof:
57.72/57.91	  converse(top)
57.72/57.91	= { by lemma 32 }
57.72/57.91	  join(?, converse(top))
57.72/57.91	= { by lemma 32 }
57.72/57.91	  join(?, join(complement(?), converse(top)))
57.72/57.91	= { by lemma 30 }
57.72/57.91	  join(converse(top), top)
57.72/57.91	= { by lemma 31 }
57.72/57.91	  top
57.72/57.91	
57.72/57.91	Lemma 34: join(zero, meet(X, top)) = X.
57.72/57.91	Proof:
57.72/57.91	  join(zero, meet(X, top))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(meet(X, top), zero)
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  join(complement(join(complement(X), complement(top))), zero)
57.72/57.91	= { by lemma 15 }
57.72/57.91	  join(complement(join(complement(X), complement(top))), complement(top))
57.72/57.91	= { by lemma 31 }
57.72/57.91	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
57.72/57.91	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 35: join(meet(X, Y), meet(X, complement(Y))) = X.
57.72/57.91	Proof:
57.72/57.91	  join(meet(X, Y), meet(X, complement(Y)))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(meet(X, complement(Y)), meet(X, Y))
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
57.72/57.91	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.91	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
57.72/57.91	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 36: join(zero, complement(complement(X))) = X.
57.72/57.91	Proof:
57.72/57.91	  join(zero, complement(complement(X)))
57.72/57.91	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.91	  join(complement(complement(X)), zero)
57.72/57.91	= { by lemma 29 }
57.72/57.91	  join(meet(X, X), zero)
57.72/57.91	= { by axiom 2 (def_zero) }
57.72/57.91	  join(meet(X, X), meet(X, complement(X)))
57.72/57.91	= { by lemma 35 }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 37: join(X, converse(complement(converse(X)))) = top.
57.72/57.91	Proof:
57.72/57.91	  join(X, converse(complement(converse(X))))
57.72/57.91	= { by lemma 27 }
57.72/57.91	  converse(join(converse(X), complement(converse(X))))
57.72/57.91	= { by axiom 3 (def_top) }
57.72/57.91	  converse(top)
57.72/57.91	= { by lemma 33 }
57.72/57.91	  top
57.72/57.91	
57.72/57.91	Lemma 38: join(zero, complement(X)) = complement(X).
57.72/57.91	Proof:
57.72/57.91	  join(zero, complement(X))
57.72/57.91	= { by lemma 36 }
57.72/57.91	  join(zero, complement(join(zero, complement(complement(X)))))
57.72/57.91	= { by lemma 26 }
57.72/57.91	  join(zero, meet(complement(X), top))
57.72/57.91	= { by lemma 34 }
57.72/57.91	  complement(X)
57.72/57.91	
57.72/57.91	Lemma 39: meet(X, top) = X.
57.72/57.91	Proof:
57.72/57.91	  meet(X, top)
57.72/57.91	= { by lemma 26 }
57.72/57.91	  complement(join(zero, complement(X)))
57.72/57.91	= { by lemma 38 }
57.72/57.91	  join(zero, complement(join(zero, complement(X))))
57.72/57.91	= { by lemma 26 }
57.72/57.91	  join(zero, meet(X, top))
57.72/57.91	= { by lemma 34 }
57.72/57.91	  X
57.72/57.91	
57.72/57.91	Lemma 40: complement(complement(X)) = X.
57.72/57.91	Proof:
57.72/57.91	  complement(complement(X))
57.72/57.91	= { by lemma 38 }
57.72/57.91	  join(zero, complement(complement(X)))
57.72/57.91	= { by lemma 36 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 41: join(X, X) = X.
57.72/57.92	Proof:
57.72/57.92	  join(X, X)
57.72/57.92	= { by lemma 40 }
57.72/57.92	  join(complement(complement(X)), X)
57.72/57.92	= { by lemma 40 }
57.72/57.92	  join(complement(complement(X)), complement(complement(X)))
57.72/57.92	= { by lemma 28 }
57.72/57.92	  complement(complement(X))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 42: join(meet(Y, X), meet(X, complement(Y))) = X.
57.72/57.92	Proof:
57.72/57.92	  join(meet(Y, X), meet(X, complement(Y)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  join(meet(X, Y), meet(X, complement(Y)))
57.72/57.92	= { by lemma 35 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 43: join(X, zero) = X.
57.72/57.92	Proof:
57.72/57.92	  join(X, zero)
57.72/57.92	= { by lemma 40 }
57.72/57.92	  join(complement(complement(X)), zero)
57.72/57.92	= { by lemma 29 }
57.72/57.92	  join(meet(X, X), zero)
57.72/57.92	= { by axiom 2 (def_zero) }
57.72/57.92	  join(meet(X, X), meet(X, complement(X)))
57.72/57.92	= { by lemma 42 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 44: join(zero, X) = X.
57.72/57.92	Proof:
57.72/57.92	  join(zero, X)
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(X, zero)
57.72/57.92	= { by lemma 43 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 45: converse(zero) = zero.
57.72/57.92	Proof:
57.72/57.92	  converse(zero)
57.72/57.92	= { by lemma 44 }
57.72/57.92	  join(zero, converse(zero))
57.72/57.92	= { by lemma 27 }
57.72/57.92	  converse(join(converse(zero), zero))
57.72/57.92	= { by lemma 43 }
57.72/57.92	  converse(converse(zero))
57.72/57.92	= { by axiom 6 (converse_idempotence) }
57.72/57.92	  zero
57.72/57.92	
57.72/57.92	Lemma 46: meet(top, X) = X.
57.72/57.92	Proof:
57.72/57.92	  meet(top, X)
57.72/57.92	= { by lemma 16 }
57.72/57.92	  meet(X, top)
57.72/57.92	= { by lemma 39 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 48: join(Y, composition(X, Y)) = composition(join(X, one), Y).
57.72/57.92	Proof:
57.72/57.92	  join(Y, composition(X, Y))
57.72/57.92	= { by lemma 21 }
57.72/57.92	  join(composition(one, Y), composition(X, Y))
57.72/57.92	= { by axiom 12 (composition_distributivity) }
57.72/57.92	  composition(join(one, X), Y)
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  composition(join(X, one), Y)
57.72/57.92	
57.72/57.92	Lemma 48: composition(join(X, one), Y) = join(Y, composition(X, Y)).
57.72/57.92	Proof:
57.72/57.92	  composition(join(X, one), Y)
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  composition(join(one, X), Y)
57.72/57.92	= { by axiom 12 (composition_distributivity) }
57.72/57.92	  join(composition(one, Y), composition(X, Y))
57.72/57.92	= { by lemma 21 }
57.72/57.92	  join(Y, composition(X, Y))
57.72/57.92	
57.72/57.92	Lemma 49: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
57.72/57.92	Proof:
57.72/57.92	  join(meet(X, Y), complement(join(Y, complement(X))))
57.72/57.92	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.92	  join(complement(join(complement(X), complement(Y))), complement(join(Y, complement(X))))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
57.72/57.92	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 50: meet(converse(X), converse(join(X, Y))) = converse(X).
57.72/57.92	Proof:
57.72/57.92	  meet(converse(X), converse(join(X, Y)))
57.72/57.92	= { by lemma 43 }
57.72/57.92	  join(meet(converse(X), converse(join(X, Y))), zero)
57.72/57.92	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), zero)
57.72/57.92	= { by lemma 15 }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(top))
57.72/57.92	= { by lemma 31 }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(converse(Y), top)))
57.72/57.92	= { by axiom 3 (def_top) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(converse(Y), join(converse(X), complement(converse(X))))))
57.72/57.92	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(join(converse(Y), converse(X)), complement(converse(X)))))
57.72/57.92	= { by axiom 7 (converse_additivity) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(converse(join(Y, X)), complement(converse(X)))))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(complement(converse(X)), converse(join(Y, X)))))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(complement(converse(X)), converse(join(X, Y)))))
57.72/57.92	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.92	  converse(X)
57.72/57.92	
57.72/57.92	Lemma 51: meet(X, join(X, Y)) = X.
57.72/57.92	Proof:
57.72/57.92	  meet(X, join(X, Y))
57.72/57.92	= { by axiom 6 (converse_idempotence) }
57.72/57.92	  meet(converse(converse(X)), join(X, Y))
57.72/57.92	= { by axiom 6 (converse_idempotence) }
57.72/57.92	  meet(converse(converse(X)), converse(converse(join(X, Y))))
57.72/57.92	= { by axiom 7 (converse_additivity) }
57.72/57.92	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
57.72/57.92	= { by lemma 50 }
57.72/57.92	  converse(converse(X))
57.72/57.92	= { by axiom 6 (converse_idempotence) }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 52: meet(X, join(Y, X)) = X.
57.72/57.92	Proof:
57.72/57.92	  meet(X, join(Y, X))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  meet(X, join(X, Y))
57.72/57.92	= { by lemma 51 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 53: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
57.72/57.92	Proof:
57.72/57.92	  meet(X, join(complement(Y), complement(Z)))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  meet(X, join(complement(Z), complement(Y)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  meet(join(complement(Z), complement(Y)), X)
57.72/57.92	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.92	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
57.72/57.92	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.92	  complement(join(meet(Z, Y), complement(X)))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  complement(join(complement(X), meet(Z, Y)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  complement(join(complement(X), meet(Y, Z)))
57.72/57.92	
57.72/57.92	Lemma 54: complement(join(X, complement(Y))) = meet(Y, complement(X)).
57.72/57.92	Proof:
57.72/57.92	  complement(join(X, complement(Y)))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  complement(join(complement(Y), X))
57.72/57.92	= { by lemma 46 }
57.72/57.92	  complement(join(complement(Y), meet(top, X)))
57.72/57.92	= { by lemma 53 }
57.72/57.92	  meet(Y, join(complement(top), complement(X)))
57.72/57.92	= { by lemma 15 }
57.72/57.92	  meet(Y, join(zero, complement(X)))
57.72/57.92	= { by lemma 38 }
57.72/57.92	  meet(Y, complement(X))
57.72/57.92	
57.72/57.92	Lemma 55: complement(join(complement(X), Y)) = meet(X, complement(Y)).
57.72/57.92	Proof:
57.72/57.92	  complement(join(complement(X), Y))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  complement(join(Y, complement(X)))
57.72/57.92	= { by lemma 54 }
57.72/57.92	  meet(X, complement(Y))
57.72/57.92	
57.72/57.92	Lemma 56: meet(complement(X), complement(Y)) = complement(join(X, Y)).
57.72/57.92	Proof:
57.72/57.92	  meet(complement(X), complement(Y))
57.72/57.92	= { by lemma 38 }
57.72/57.92	  meet(join(zero, complement(X)), complement(Y))
57.72/57.92	= { by lemma 54 }
57.72/57.92	  complement(join(Y, complement(join(zero, complement(X)))))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  complement(join(Y, meet(X, top)))
57.72/57.92	= { by lemma 39 }
57.72/57.92	  complement(join(Y, X))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  complement(join(X, Y))
57.72/57.92	
57.72/57.92	Lemma 57: complement(meet(Y, complement(X))) = join(X, complement(Y)).
57.72/57.92	Proof:
57.72/57.92	  complement(meet(Y, complement(X)))
57.72/57.92	= { by lemma 44 }
57.72/57.92	  complement(join(zero, meet(Y, complement(X))))
57.72/57.92	= { by lemma 54 }
57.72/57.92	  complement(join(zero, complement(join(X, complement(Y)))))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  meet(join(X, complement(Y)), top)
57.72/57.92	= { by lemma 39 }
57.72/57.92	  join(X, complement(Y))
57.72/57.92	
57.72/57.92	Lemma 58: complement(meet(complement(X), Y)) = join(X, complement(Y)).
57.72/57.92	Proof:
57.72/57.92	  complement(meet(complement(X), Y))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  complement(meet(Y, complement(X)))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  join(X, complement(Y))
57.72/57.92	
57.72/57.92	Lemma 59: join(complement(X), complement(Y)) = complement(meet(X, Y)).
57.72/57.92	Proof:
57.72/57.92	  join(complement(X), complement(Y))
57.72/57.92	= { by lemma 44 }
57.72/57.92	  join(zero, join(complement(X), complement(Y)))
57.72/57.92	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.92	  join(join(zero, complement(X)), complement(Y))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  complement(meet(Y, complement(join(zero, complement(X)))))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  complement(meet(Y, meet(X, top)))
57.72/57.92	= { by lemma 39 }
57.72/57.92	  complement(meet(Y, X))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  complement(meet(X, Y))
57.72/57.92	
57.72/57.92	Lemma 60: join(X, meet(X, Y)) = X.
57.72/57.92	Proof:
57.72/57.92	  join(X, meet(X, Y))
57.72/57.92	= { by lemma 39 }
57.72/57.92	  join(X, meet(X, meet(Y, top)))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  join(X, meet(X, complement(join(zero, complement(Y)))))
57.72/57.92	= { by lemma 55 }
57.72/57.92	  join(X, complement(join(complement(X), join(zero, complement(Y)))))
57.72/57.92	= { by lemma 58 }
57.72/57.92	  complement(meet(complement(X), join(complement(X), join(zero, complement(Y)))))
57.72/57.92	= { by lemma 51 }
57.72/57.92	  complement(complement(X))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 61: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
57.72/57.92	Proof:
57.72/57.92	  meet(join(X, complement(Y)), complement(meet(X, Y)))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  meet(join(complement(Y), X), complement(meet(X, Y)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  meet(join(complement(Y), X), complement(meet(Y, X)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  meet(complement(meet(Y, X)), join(complement(Y), X))
57.72/57.92	= { by lemma 59 }
57.72/57.92	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
57.72/57.92	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.92	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
57.72/57.92	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.92	  complement(Y)
57.72/57.92	
57.72/57.92	Lemma 62: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
57.72/57.92	Proof:
57.72/57.92	  meet(X, complement(meet(X, Y)))
57.72/57.92	= { by lemma 52 }
57.72/57.92	  meet(X, complement(meet(X, meet(Y, join(complement(X), Y)))))
57.72/57.92	= { by lemma 55 }
57.72/57.92	  complement(join(complement(X), meet(X, meet(Y, join(complement(X), Y)))))
57.72/57.92	= { by lemma 53 }
57.72/57.92	  meet(X, join(complement(X), complement(meet(Y, join(complement(X), Y)))))
57.72/57.92	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.92	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), complement(meet(Y, join(complement(X), Y)))))
57.72/57.92	= { by lemma 59 }
57.72/57.92	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
57.72/57.92	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.92	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
57.72/57.92	= { by lemma 58 }
57.72/57.92	  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))))
57.72/57.92	= { by lemma 61 }
57.72/57.92	  complement(join(complement(X), Y))
57.72/57.92	= { by lemma 55 }
57.72/57.92	  meet(X, complement(Y))
57.72/57.92	
57.72/57.92	Lemma 63: meet(X, join(Y, complement(X))) = meet(X, Y).
57.72/57.92	Proof:
57.72/57.92	  meet(X, join(Y, complement(X)))
57.72/57.92	= { by lemma 60 }
57.72/57.92	  meet(join(X, meet(X, Y)), join(Y, complement(X)))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  meet(join(X, complement(complement(meet(X, Y)))), join(Y, complement(X)))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(Y))))
57.72/57.92	= { by lemma 62 }
57.72/57.92	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(meet(X, Y)))))
57.72/57.92	= { by lemma 61 }
57.72/57.92	  complement(complement(meet(X, Y)))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  meet(X, Y)
57.72/57.92	
57.72/57.92	Lemma 64: meet(X, join(complement(X), Y)) = meet(X, Y).
57.72/57.92	Proof:
57.72/57.92	  meet(X, join(complement(X), Y))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  meet(X, join(Y, complement(X)))
57.72/57.92	= { by lemma 63 }
57.72/57.92	  meet(X, Y)
57.72/57.92	
57.72/57.92	Lemma 65: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
57.72/57.92	Proof:
57.72/57.92	  join(complement(X), meet(X, Y))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(meet(X, Y), complement(X))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  complement(meet(X, complement(meet(X, Y))))
57.72/57.92	= { by lemma 62 }
57.72/57.92	  complement(meet(X, complement(Y)))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  join(Y, complement(X))
57.72/57.92	
57.72/57.92	Lemma 66: join(meet(Y, X), meet(complement(Y), X)) = X.
57.72/57.92	Proof:
57.72/57.92	  join(meet(Y, X), meet(complement(Y), X))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  join(meet(Y, X), meet(X, complement(Y)))
57.72/57.92	= { by lemma 42 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 67: join(X, meet(Y, complement(X))) = join(X, Y).
57.72/57.92	Proof:
57.72/57.92	  join(X, meet(Y, complement(X)))
57.72/57.92	= { by lemma 52 }
57.72/57.92	  join(meet(X, join(Y, X)), meet(Y, complement(X)))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  join(meet(X, join(Y, complement(complement(X)))), meet(Y, complement(X)))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  join(meet(X, join(Y, complement(complement(X)))), meet(complement(X), Y))
57.72/57.92	= { by lemma 63 }
57.72/57.92	  join(meet(X, join(Y, complement(complement(X)))), meet(complement(X), join(Y, complement(complement(X)))))
57.72/57.92	= { by lemma 66 }
57.72/57.92	  join(Y, complement(complement(X)))
57.72/57.92	= { by lemma 40 }
57.72/57.92	  join(Y, X)
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  join(X, Y)
57.72/57.92	
57.72/57.92	Lemma 68: meet(join(X, Y), join(X, complement(Y))) = X.
57.72/57.92	Proof:
57.72/57.92	  meet(join(X, Y), join(X, complement(Y)))
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  meet(join(Y, X), join(X, complement(Y)))
57.72/57.92	= { by lemma 39 }
57.72/57.92	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
57.72/57.92	= { by lemma 57 }
57.72/57.92	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, complement(X))))
57.72/57.92	= { by lemma 38 }
57.72/57.92	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, join(zero, complement(X)))))
57.72/57.92	= { by lemma 61 }
57.72/57.92	  complement(join(zero, complement(X)))
57.72/57.92	= { by lemma 26 }
57.72/57.92	  meet(X, top)
57.72/57.92	= { by lemma 39 }
57.72/57.92	  X
57.72/57.92	
57.72/57.92	Lemma 69: composition(join(X, converse(complement(sK2_goals_X0))), sK2_goals_X0) = composition(X, sK2_goals_X0).
57.72/57.92	Proof:
57.72/57.92	  composition(join(X, converse(complement(sK2_goals_X0))), sK2_goals_X0)
57.72/57.92	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.92	  composition(join(converse(complement(sK2_goals_X0)), X), sK2_goals_X0)
57.72/57.92	= { by axiom 12 (composition_distributivity) }
57.72/57.92	  join(composition(converse(complement(sK2_goals_X0)), sK2_goals_X0), composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 18 }
57.72/57.92	  join(converse(composition(converse(sK2_goals_X0), complement(sK2_goals_X0))), composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 43 }
57.72/57.92	  join(converse(join(composition(converse(sK2_goals_X0), complement(sK2_goals_X0)), zero)), composition(X, sK2_goals_X0))
57.72/57.92	= { by axiom 14 (goals) }
57.72/57.92	  join(converse(join(composition(converse(sK2_goals_X0), complement(composition(sK2_goals_X0, top))), zero)), composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 15 }
57.72/57.92	  join(converse(join(composition(converse(sK2_goals_X0), complement(composition(sK2_goals_X0, top))), complement(top))), composition(X, sK2_goals_X0))
57.72/57.92	= { by axiom 8 (converse_cancellativity) }
57.72/57.92	  join(converse(complement(top)), composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 15 }
57.72/57.92	  join(converse(zero), composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 45 }
57.72/57.92	  join(zero, composition(X, sK2_goals_X0))
57.72/57.92	= { by lemma 44 }
57.72/57.92	  composition(X, sK2_goals_X0)
57.72/57.92	
57.72/57.92	Lemma 70: join(X, meet(complement(X), Y)) = join(X, Y).
57.72/57.92	Proof:
57.72/57.92	  join(X, meet(complement(X), Y))
57.72/57.92	= { by lemma 16 }
57.72/57.92	  join(X, meet(Y, complement(X)))
57.72/57.92	= { by lemma 67 }
57.72/57.93	  join(X, Y)
57.72/57.93	
57.72/57.93	Lemma 71: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
57.72/57.93	Proof:
57.72/57.93	  meet(X, meet(Y, Z))
57.72/57.93	= { by lemma 39 }
57.72/57.93	  meet(meet(X, top), meet(Y, Z))
57.72/57.93	= { by lemma 26 }
57.72/57.93	  meet(complement(join(zero, complement(X))), meet(Y, Z))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  meet(meet(Y, Z), complement(join(zero, complement(X))))
57.72/57.93	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.93	  meet(complement(join(complement(Y), complement(Z))), complement(join(zero, complement(X))))
57.72/57.93	= { by lemma 56 }
57.72/57.93	  complement(join(join(complement(Y), complement(Z)), join(zero, complement(X))))
57.72/57.93	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.93	  complement(join(complement(Y), join(complement(Z), join(zero, complement(X)))))
57.72/57.93	= { by lemma 55 }
57.72/57.93	  meet(Y, complement(join(complement(Z), join(zero, complement(X)))))
57.72/57.93	= { by lemma 55 }
57.72/57.93	  meet(Y, meet(Z, complement(join(zero, complement(X)))))
57.72/57.93	= { by lemma 26 }
57.72/57.93	  meet(Y, meet(Z, meet(X, top)))
57.72/57.93	= { by lemma 39 }
57.72/57.93	  meet(Y, meet(Z, X))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  meet(Y, meet(X, Z))
57.72/57.93	
57.72/57.93	Lemma 72: complement(converse(complement(X))) = converse(X).
57.72/57.93	Proof:
57.72/57.93	  complement(converse(complement(X)))
57.72/57.93	= { by axiom 6 (converse_idempotence) }
57.72/57.93	  converse(converse(complement(converse(complement(X)))))
57.72/57.93	= { by lemma 68 }
57.72/57.93	  converse(meet(join(converse(complement(converse(complement(X)))), X), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  converse(meet(join(X, converse(complement(converse(complement(X))))), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by axiom 6 (converse_idempotence) }
57.72/57.93	  converse(meet(join(X, converse(complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by lemma 27 }
57.72/57.93	  converse(meet(converse(join(converse(X), complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by lemma 46 }
57.72/57.93	  converse(meet(converse(meet(top, join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by lemma 37 }
57.72/57.93	  converse(meet(converse(meet(join(converse(X), converse(complement(converse(converse(X))))), join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by lemma 68 }
57.72/57.93	  converse(meet(converse(converse(X)), join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by axiom 6 (converse_idempotence) }
57.72/57.93	  converse(meet(X, join(converse(complement(converse(complement(X)))), complement(X))))
57.72/57.93	= { by lemma 63 }
57.72/57.93	  converse(meet(X, converse(complement(converse(complement(X))))))
57.72/57.93	= { by lemma 64 }
57.72/57.93	  converse(meet(X, join(complement(X), converse(complement(converse(complement(X)))))))
57.72/57.93	= { by lemma 37 }
57.72/57.93	  converse(meet(X, top))
57.72/57.93	= { by lemma 39 }
57.72/57.93	  converse(X)
57.72/57.93	
57.72/57.93	Lemma 73: converse(complement(X)) = complement(converse(X)).
57.72/57.93	Proof:
57.72/57.93	  converse(complement(X))
57.72/57.93	= { by lemma 38 }
57.72/57.93	  converse(join(zero, complement(X)))
57.72/57.93	= { by lemma 72 }
57.72/57.93	  complement(converse(complement(join(zero, complement(X)))))
57.72/57.93	= { by lemma 26 }
57.72/57.93	  complement(converse(meet(X, top)))
57.72/57.93	= { by lemma 39 }
57.72/57.93	  complement(converse(X))
57.72/57.93	
57.72/57.93	Lemma 74: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
57.72/57.93	Proof:
57.72/57.93	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
57.72/57.93	= { by axiom 12 (composition_distributivity) }
57.72/57.93	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
57.72/57.93	= { by axiom 13 (maddux2_join_associativity) }
57.72/57.93	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
57.72/57.93	= { by axiom 3 (def_top) }
57.72/57.93	  join(composition(Z, Y), top)
57.72/57.93	= { by lemma 31 }
57.72/57.93	  top
57.72/57.93	
57.72/57.93	Lemma 75: composition(converse(join(X, complement(sK2_goals_X0))), sK2_goals_X0) = composition(converse(X), sK2_goals_X0).
57.72/57.93	Proof:
57.72/57.93	  composition(converse(join(X, complement(sK2_goals_X0))), sK2_goals_X0)
57.72/57.93	= { by axiom 7 (converse_additivity) }
57.72/57.93	  composition(join(converse(X), converse(complement(sK2_goals_X0))), sK2_goals_X0)
57.72/57.93	= { by lemma 69 }
57.72/57.93	  composition(converse(X), sK2_goals_X0)
57.72/57.93	
57.72/57.93	Lemma 76: composition(converse(sK2_goals_X0), join(X, complement(sK2_goals_X0))) = composition(converse(sK2_goals_X0), X).
57.72/57.93	Proof:
57.72/57.93	  composition(converse(sK2_goals_X0), join(X, complement(sK2_goals_X0)))
57.72/57.93	= { by axiom 6 (converse_idempotence) }
57.72/57.93	  composition(converse(sK2_goals_X0), join(X, converse(converse(complement(sK2_goals_X0)))))
57.72/57.93	= { by lemma 22 }
57.72/57.93	  composition(converse(sK2_goals_X0), converse(join(converse(complement(sK2_goals_X0)), converse(X))))
57.72/57.93	= { by axiom 4 (converse_multiplicativity) }
57.72/57.93	  converse(composition(join(converse(complement(sK2_goals_X0)), converse(X)), sK2_goals_X0))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  converse(composition(join(converse(X), converse(complement(sK2_goals_X0))), sK2_goals_X0))
57.72/57.93	= { by lemma 69 }
57.72/57.93	  converse(composition(converse(X), sK2_goals_X0))
57.72/57.93	= { by lemma 18 }
57.72/57.93	  composition(converse(sK2_goals_X0), X)
57.72/57.93	
57.72/57.93	Lemma 77: converse(join(composition(X, Y), composition(X, Z))) = converse(composition(X, join(Y, Z))).
57.72/57.93	Proof:
57.72/57.93	  converse(join(composition(X, Y), composition(X, Z)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  converse(join(composition(X, Z), composition(X, Y)))
57.72/57.93	= { by axiom 7 (converse_additivity) }
57.72/57.93	  join(converse(composition(X, Z)), converse(composition(X, Y)))
57.72/57.93	= { by axiom 4 (converse_multiplicativity) }
57.72/57.93	  join(composition(converse(Z), converse(X)), converse(composition(X, Y)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  join(converse(composition(X, Y)), composition(converse(Z), converse(X)))
57.72/57.93	= { by axiom 4 (converse_multiplicativity) }
57.72/57.93	  join(composition(converse(Y), converse(X)), composition(converse(Z), converse(X)))
57.72/57.93	= { by axiom 12 (composition_distributivity) }
57.72/57.93	  composition(join(converse(Y), converse(Z)), converse(X))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  composition(join(converse(Z), converse(Y)), converse(X))
57.72/57.93	= { by axiom 7 (converse_additivity) }
57.72/57.93	  composition(converse(join(Z, Y)), converse(X))
57.72/57.93	= { by axiom 4 (converse_multiplicativity) }
57.72/57.93	  converse(composition(X, join(Z, Y)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  converse(composition(X, join(Y, Z)))
57.72/57.93	
57.72/57.93	Lemma 78: join(composition(X, Z), complement(composition(meet(X, Y), Z))) = top.
57.72/57.93	Proof:
57.72/57.93	  join(composition(X, Z), complement(composition(meet(X, Y), Z)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  join(complement(composition(meet(X, Y), Z)), composition(X, Z))
57.72/57.93	= { by axiom 11 (maddux4_definiton_of_meet) }
57.72/57.93	  join(complement(composition(complement(join(complement(X), complement(Y))), Z)), composition(X, Z))
57.72/57.93	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.72/57.93	  join(complement(composition(complement(join(complement(X), complement(Y))), Z)), composition(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), Z))
57.72/57.93	= { by lemma 74 }
57.72/57.93	  top
57.72/57.93	
57.72/57.93	Lemma 79: join(X, complement(composition(meet(Y, one), X))) = top.
57.72/57.93	Proof:
57.72/57.93	  join(X, complement(composition(meet(Y, one), X)))
57.72/57.93	= { by lemma 19 }
57.72/57.93	  join(composition(converse(one), X), complement(composition(meet(Y, one), X)))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  join(composition(converse(one), X), complement(composition(meet(one, Y), X)))
57.72/57.93	= { by lemma 20 }
57.72/57.93	  join(composition(converse(one), X), complement(composition(meet(converse(one), Y), X)))
57.72/57.93	= { by lemma 78 }
57.72/57.93	  top
57.72/57.93	
57.72/57.93	Lemma 80: join(sK2_goals_X0, complement(composition(meet(X, sK2_goals_X0), top))) = top.
57.72/57.93	Proof:
57.72/57.93	  join(sK2_goals_X0, complement(composition(meet(X, sK2_goals_X0), top)))
57.72/57.93	= { by axiom 14 (goals) }
57.72/57.93	  join(composition(sK2_goals_X0, top), complement(composition(meet(X, sK2_goals_X0), top)))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  join(composition(sK2_goals_X0, top), complement(composition(meet(sK2_goals_X0, X), top)))
57.72/57.93	= { by lemma 78 }
57.72/57.93	  top
57.72/57.93	
57.72/57.93	Lemma 81: composition(top, meet(X, sK2_goals_X0)) = composition(converse(sK2_goals_X0), X).
57.72/57.93	Proof:
57.72/57.93	  composition(top, meet(X, sK2_goals_X0))
57.72/57.93	= { by lemma 33 }
57.72/57.93	  composition(converse(top), meet(X, sK2_goals_X0))
57.72/57.93	= { by lemma 80 }
57.72/57.93	  composition(converse(join(sK2_goals_X0, complement(composition(meet(X, sK2_goals_X0), top)))), meet(X, sK2_goals_X0))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  composition(converse(join(complement(composition(meet(X, sK2_goals_X0), top)), sK2_goals_X0)), meet(X, sK2_goals_X0))
57.72/57.93	= { by lemma 18 }
57.72/57.93	  converse(composition(converse(meet(X, sK2_goals_X0)), join(complement(composition(meet(X, sK2_goals_X0), top)), sK2_goals_X0)))
57.72/57.93	= { by lemma 77 }
57.72/57.93	  converse(join(composition(converse(meet(X, sK2_goals_X0)), complement(composition(meet(X, sK2_goals_X0), top))), composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)))
57.72/57.93	= { by lemma 43 }
57.72/57.93	  converse(join(join(composition(converse(meet(X, sK2_goals_X0)), complement(composition(meet(X, sK2_goals_X0), top))), zero), composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)))
57.72/57.93	= { by lemma 15 }
57.72/57.93	  converse(join(join(composition(converse(meet(X, sK2_goals_X0)), complement(composition(meet(X, sK2_goals_X0), top))), complement(top)), composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)))
57.72/57.93	= { by axiom 8 (converse_cancellativity) }
57.72/57.93	  converse(join(complement(top), composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)))
57.72/57.93	= { by lemma 15 }
57.72/57.93	  converse(join(zero, composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  converse(join(composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0), zero))
57.72/57.93	= { by axiom 7 (converse_additivity) }
57.72/57.93	  join(converse(composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)), converse(zero))
57.72/57.93	= { by lemma 18 }
57.72/57.93	  join(composition(converse(sK2_goals_X0), meet(X, sK2_goals_X0)), converse(zero))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  join(converse(zero), composition(converse(sK2_goals_X0), meet(X, sK2_goals_X0)))
57.72/57.93	= { by lemma 45 }
57.72/57.93	  join(zero, composition(converse(sK2_goals_X0), meet(X, sK2_goals_X0)))
57.72/57.93	= { by lemma 44 }
57.72/57.93	  composition(converse(sK2_goals_X0), meet(X, sK2_goals_X0))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  composition(converse(sK2_goals_X0), meet(sK2_goals_X0, X))
57.72/57.93	= { by lemma 76 }
57.72/57.93	  composition(converse(sK2_goals_X0), join(meet(sK2_goals_X0, X), complement(sK2_goals_X0)))
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  composition(converse(sK2_goals_X0), join(complement(sK2_goals_X0), meet(sK2_goals_X0, X)))
57.72/57.93	= { by lemma 65 }
57.72/57.93	  composition(converse(sK2_goals_X0), join(X, complement(sK2_goals_X0)))
57.72/57.93	= { by lemma 76 }
57.72/57.93	  composition(converse(sK2_goals_X0), X)
57.72/57.93	
57.72/57.93	Lemma 82: composition(converse(meet(sK2_goals_X0, X)), sK2_goals_X0) = composition(converse(X), sK2_goals_X0).
57.72/57.93	Proof:
57.72/57.93	  composition(converse(meet(sK2_goals_X0, X)), sK2_goals_X0)
57.72/57.93	= { by lemma 16 }
57.72/57.93	  composition(converse(meet(X, sK2_goals_X0)), sK2_goals_X0)
57.72/57.93	= { by lemma 16 }
57.72/57.93	  composition(converse(meet(sK2_goals_X0, X)), sK2_goals_X0)
57.72/57.93	= { by lemma 75 }
57.72/57.93	  composition(converse(join(meet(sK2_goals_X0, X), complement(sK2_goals_X0))), sK2_goals_X0)
57.72/57.93	= { by axiom 9 (maddux1_join_commutativity) }
57.72/57.93	  composition(converse(join(complement(sK2_goals_X0), meet(sK2_goals_X0, X))), sK2_goals_X0)
57.72/57.93	= { by lemma 65 }
57.72/57.93	  composition(converse(join(X, complement(sK2_goals_X0))), sK2_goals_X0)
57.72/57.93	= { by lemma 75 }
57.72/57.93	  composition(converse(X), sK2_goals_X0)
57.72/57.93	
57.72/57.93	Lemma 83: meet(X, join(Y, join(complement(X), Z))) = meet(X, join(Y, Z)).
57.72/57.93	Proof:
57.72/57.93	  meet(X, join(Y, join(complement(X), Z)))
57.72/57.93	= { by lemma 25 }
57.72/57.93	  meet(X, join(complement(X), join(Y, Z)))
57.72/57.93	= { by lemma 64 }
57.72/57.93	  meet(X, join(Y, Z))
57.72/57.93	
57.72/57.93	Lemma 84: meet(X, join(Y, meet(X, Z))) = meet(X, join(Y, Z)).
57.72/57.93	Proof:
57.72/57.93	  meet(X, join(Y, meet(X, Z)))
57.72/57.93	= { by lemma 16 }
57.72/57.93	  meet(X, join(Y, meet(Z, X)))
57.72/57.93	= { by lemma 40 }
57.72/57.93	  meet(X, join(Y, meet(Z, complement(complement(X)))))
57.72/57.93	= { by lemma 83 }
57.72/57.93	  meet(X, join(Y, join(complement(X), meet(Z, complement(complement(X))))))
57.72/57.93	= { by lemma 67 }
57.72/57.93	  meet(X, join(Y, join(complement(X), Z)))
57.72/57.93	= { by lemma 83 }
57.79/57.99	  meet(X, join(Y, Z))
57.79/57.99	
57.79/57.99	Lemma 85: composition(meet(sK2_goals_X0, one), sK2_goals_X0) = sK2_goals_X0.
57.79/57.99	Proof:
57.79/57.99	  composition(meet(sK2_goals_X0, one), sK2_goals_X0)
57.79/57.99	= { by axiom 6 (converse_idempotence) }
57.79/57.99	  composition(converse(converse(meet(sK2_goals_X0, one))), sK2_goals_X0)
57.79/57.99	= { by lemma 82 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, converse(meet(sK2_goals_X0, one)))), sK2_goals_X0)
57.79/57.99	= { by lemma 20 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, converse(meet(sK2_goals_X0, converse(one))))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, converse(meet(converse(one), sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by lemma 72 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, complement(converse(complement(meet(converse(one), sK2_goals_X0)))))), sK2_goals_X0)
57.79/57.99	= { by lemma 59 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, complement(converse(join(complement(converse(one)), complement(sK2_goals_X0)))))), sK2_goals_X0)
57.79/57.99	= { by lemma 73 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, converse(complement(join(complement(converse(one)), complement(sK2_goals_X0)))))), sK2_goals_X0)
57.79/57.99	= { by lemma 50 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(converse(complement(join(complement(converse(one)), complement(sK2_goals_X0)))), converse(join(complement(join(complement(converse(one)), complement(sK2_goals_X0))), complement(join(complement(converse(one)), sK2_goals_X0))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 73 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(complement(converse(join(complement(converse(one)), complement(sK2_goals_X0)))), converse(join(complement(join(complement(converse(one)), complement(sK2_goals_X0))), complement(join(complement(converse(one)), sK2_goals_X0))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 59 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(complement(converse(complement(meet(converse(one), sK2_goals_X0)))), converse(join(complement(join(complement(converse(one)), complement(sK2_goals_X0))), complement(join(complement(converse(one)), sK2_goals_X0))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 72 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(converse(meet(converse(one), sK2_goals_X0)), converse(join(complement(join(complement(converse(one)), complement(sK2_goals_X0))), complement(join(complement(converse(one)), sK2_goals_X0))))))), sK2_goals_X0)
57.79/57.99	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(converse(meet(converse(one), sK2_goals_X0)), converse(converse(one))))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(converse(converse(one)), converse(meet(converse(one), sK2_goals_X0))))), sK2_goals_X0)
57.79/57.99	= { by axiom 6 (converse_idempotence) }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(meet(converse(one), sK2_goals_X0))))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(meet(sK2_goals_X0, converse(one)))))), sK2_goals_X0)
57.79/57.99	= { by lemma 40 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(meet(sK2_goals_X0, complement(complement(converse(one)))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 73 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(meet(sK2_goals_X0, complement(converse(complement(one)))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 64 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, join(complement(one), converse(meet(sK2_goals_X0, complement(converse(complement(one))))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 27 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(join(converse(complement(one)), meet(sK2_goals_X0, complement(converse(complement(one))))))))), sK2_goals_X0)
57.79/57.99	= { by lemma 67 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(join(converse(complement(one)), sK2_goals_X0))))), sK2_goals_X0)
57.79/57.99	= { by lemma 27 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, join(complement(one), converse(sK2_goals_X0))))), sK2_goals_X0)
57.79/57.99	= { by lemma 64 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(one, converse(sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, meet(converse(sK2_goals_X0), one))), sK2_goals_X0)
57.79/57.99	= { by lemma 71 }
57.79/57.99	  composition(converse(meet(converse(sK2_goals_X0), meet(sK2_goals_X0, one))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), converse(sK2_goals_X0))), sK2_goals_X0)
57.79/57.99	= { by axiom 10 (composition_identity) }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), composition(converse(sK2_goals_X0), one))), sK2_goals_X0)
57.79/57.99	= { by lemma 81 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), composition(top, meet(one, sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by lemma 31 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), composition(join(one, top), meet(one, sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by axiom 9 (maddux1_join_commutativity) }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), composition(join(top, one), meet(one, sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by lemma 48 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), join(meet(one, sK2_goals_X0), composition(top, meet(one, sK2_goals_X0))))), sK2_goals_X0)
57.79/57.99	= { by lemma 81 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), join(meet(one, sK2_goals_X0), composition(converse(sK2_goals_X0), one)))), sK2_goals_X0)
57.79/57.99	= { by axiom 10 (composition_identity) }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), join(meet(one, sK2_goals_X0), converse(sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by axiom 9 (maddux1_join_commutativity) }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), join(converse(sK2_goals_X0), meet(one, sK2_goals_X0)))), sK2_goals_X0)
57.79/57.99	= { by lemma 16 }
57.79/57.99	  composition(converse(meet(meet(sK2_goals_X0, one), join(converse(sK2_goals_X0), meet(sK2_goals_X0, one)))), sK2_goals_X0)
57.79/57.99	= { by lemma 52 }
57.79/57.99	  composition(converse(meet(sK2_goals_X0, one)), sK2_goals_X0)
57.79/57.99	= { by lemma 82 }
57.79/57.99	  composition(converse(one), sK2_goals_X0)
57.79/57.99	= { by lemma 20 }
57.79/57.99	  composition(one, sK2_goals_X0)
57.79/57.99	= { by lemma 21 }
57.79/58.05	  sK2_goals_X0
57.79/58.05	
57.79/58.05	Lemma 86: composition(meet(sK2_goals_X0, one), X) = meet(X, sK2_goals_X0).
57.79/58.05	Proof:
57.79/58.05	  composition(meet(sK2_goals_X0, one), X)
57.79/58.05	= { by lemma 49 }
57.79/58.05	  join(meet(composition(meet(sK2_goals_X0, one), X), X), complement(join(X, complement(composition(meet(sK2_goals_X0, one), X)))))
57.79/58.05	= { by lemma 79 }
57.79/58.05	  join(meet(composition(meet(sK2_goals_X0, one), X), X), complement(top))
57.79/58.05	= { by lemma 15 }
57.79/58.05	  join(meet(composition(meet(sK2_goals_X0, one), X), X), zero)
57.79/58.05	= { by lemma 43 }
57.79/58.05	  meet(composition(meet(sK2_goals_X0, one), X), X)
57.79/58.05	= { by lemma 16 }
57.79/58.05	  meet(X, composition(meet(sK2_goals_X0, one), X))
57.79/58.05	= { by lemma 40 }
57.79/58.05	  meet(X, composition(meet(sK2_goals_X0, one), complement(complement(X))))
57.79/58.05	= { by lemma 64 }
57.79/58.05	  meet(X, join(complement(X), composition(meet(sK2_goals_X0, one), complement(complement(X)))))
57.79/58.05	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), complement(X)))
57.79/58.05	= { by lemma 43 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(complement(X), zero)))
57.79/58.05	= { by lemma 51 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), join(complement(X), complement(complement(composition(meet(sK2_goals_X0, one), complement(X)))))), zero)))
57.79/58.05	= { by lemma 58 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))), zero)))
57.79/58.05	= { by lemma 15 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))), complement(top))))
57.79/58.05	= { by lemma 79 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))), complement(join(complement(X), complement(composition(meet(sK2_goals_X0, one), complement(X))))))))
57.79/58.05	= { by lemma 56 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))), meet(complement(complement(X)), complement(complement(composition(meet(sK2_goals_X0, one), complement(X))))))))
57.79/58.05	= { by lemma 62 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(meet(complement(X), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))), meet(complement(complement(X)), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))))))
57.79/58.05	= { by lemma 66 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), complement(meet(complement(complement(X)), complement(composition(meet(sK2_goals_X0, one), complement(X)))))))
57.79/58.05	= { by lemma 58 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(complement(X), complement(complement(composition(meet(sK2_goals_X0, one), complement(X)))))))
57.79/58.05	= { by lemma 40 }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(complement(X), composition(meet(sK2_goals_X0, one), complement(X)))))
57.79/58.05	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), complement(complement(X))), join(composition(meet(sK2_goals_X0, one), complement(X)), complement(X))))
57.79/58.05	= { by axiom 13 (maddux2_join_associativity) }
57.79/58.05	  meet(X, join(join(composition(meet(sK2_goals_X0, one), complement(complement(X))), composition(meet(sK2_goals_X0, one), complement(X))), complement(X)))
57.79/58.05	= { by axiom 6 (converse_idempotence) }
57.79/58.05	  meet(X, join(converse(converse(join(composition(meet(sK2_goals_X0, one), complement(complement(X))), composition(meet(sK2_goals_X0, one), complement(X))))), complement(X)))
57.79/58.05	= { by lemma 77 }
57.79/58.05	  meet(X, join(converse(converse(composition(meet(sK2_goals_X0, one), join(complement(complement(X)), complement(X))))), complement(X)))
57.79/58.05	= { by axiom 6 (converse_idempotence) }
57.79/58.05	  meet(X, join(composition(meet(sK2_goals_X0, one), join(complement(complement(X)), complement(X))), complement(X)))
57.79/58.05	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.05	  meet(X, join(complement(X), composition(meet(sK2_goals_X0, one), join(complement(complement(X)), complement(X)))))
57.79/58.05	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.05	  meet(X, join(complement(X), composition(meet(sK2_goals_X0, one), join(complement(X), complement(complement(X))))))
57.79/58.05	= { by axiom 3 (def_top) }
57.79/58.05	  meet(X, join(complement(X), composition(meet(sK2_goals_X0, one), top)))
57.79/58.05	= { by lemma 64 }
57.79/58.05	  meet(X, composition(meet(sK2_goals_X0, one), top))
57.79/58.05	= { by lemma 16 }
57.79/58.05	  meet(X, composition(meet(one, sK2_goals_X0), top))
57.79/58.05	= { by lemma 49 }
57.79/58.05	  meet(X, join(meet(composition(meet(one, sK2_goals_X0), top), sK2_goals_X0), complement(join(sK2_goals_X0, complement(composition(meet(one, sK2_goals_X0), top))))))
57.79/58.05	= { by lemma 80 }
57.79/58.05	  meet(X, join(meet(composition(meet(one, sK2_goals_X0), top), sK2_goals_X0), complement(top)))
57.79/58.05	= { by lemma 15 }
57.79/58.05	  meet(X, join(meet(composition(meet(one, sK2_goals_X0), top), sK2_goals_X0), zero))
57.79/58.05	= { by lemma 43 }
57.79/58.05	  meet(X, meet(composition(meet(one, sK2_goals_X0), top), sK2_goals_X0))
57.79/58.05	= { by lemma 16 }
57.79/58.05	  meet(X, meet(sK2_goals_X0, composition(meet(one, sK2_goals_X0), top)))
57.79/58.05	= { by lemma 85 }
57.79/58.05	  meet(X, meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(one, sK2_goals_X0), top)))
57.79/58.05	= { by lemma 16 }
57.79/58.05	  meet(X, meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)))
57.79/58.05	= { by lemma 43 }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), zero))
57.79/58.05	= { by lemma 15 }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(top)))
57.79/58.05	= { by lemma 33 }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(converse(top))))
57.79/58.05	= { by lemma 74 }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(converse(join(complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one)))), composition(join(converse(sK2_goals_X0), complement(converse(sK2_goals_X0))), converse(meet(sK2_goals_X0, one))))))))
57.79/58.05	= { by axiom 3 (def_top) }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(converse(join(complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one)))), composition(top, converse(meet(sK2_goals_X0, one))))))))
57.79/58.05	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(converse(join(composition(top, converse(meet(sK2_goals_X0, one))), complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one)))))))))
57.79/58.05	= { by axiom 7 (converse_additivity) }
57.79/58.05	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(converse(composition(top, converse(meet(sK2_goals_X0, one)))), converse(complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one)))))))))
57.79/58.05	= { by lemma 17 }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(composition(meet(sK2_goals_X0, one), converse(top)), converse(complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one)))))))))
57.79/58.06	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(converse(complement(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one))))), composition(meet(sK2_goals_X0, one), converse(top))))))
57.79/58.06	= { by lemma 73 }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(complement(converse(composition(converse(sK2_goals_X0), converse(meet(sK2_goals_X0, one))))), composition(meet(sK2_goals_X0, one), converse(top))))))
57.79/58.06	= { by lemma 17 }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(complement(composition(meet(sK2_goals_X0, one), converse(converse(sK2_goals_X0)))), composition(meet(sK2_goals_X0, one), converse(top))))))
57.79/58.06	= { by lemma 33 }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(complement(composition(meet(sK2_goals_X0, one), converse(converse(sK2_goals_X0)))), composition(meet(sK2_goals_X0, one), top)))))
57.79/58.06	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(composition(meet(sK2_goals_X0, one), top), complement(composition(meet(sK2_goals_X0, one), converse(converse(sK2_goals_X0))))))))
57.79/58.06	= { by axiom 6 (converse_idempotence) }
57.79/58.06	  meet(X, join(meet(composition(meet(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, one), top)), complement(join(composition(meet(sK2_goals_X0, one), top), complement(composition(meet(sK2_goals_X0, one), sK2_goals_X0))))))
57.79/58.06	= { by lemma 49 }
57.79/58.06	  meet(X, composition(meet(sK2_goals_X0, one), sK2_goals_X0))
57.79/58.06	= { by lemma 85 }
57.79/58.07	  meet(X, sK2_goals_X0)
57.79/58.07	
57.79/58.07	Goal 1 (goals_1): join(meet(sK2_goals_X0, sK1_goals_X1), composition(meet(sK2_goals_X0, one), sK1_goals_X1)) = composition(meet(sK2_goals_X0, one), sK1_goals_X1).
57.79/58.07	Proof:
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(meet(sK2_goals_X0, one), sK1_goals_X1))
57.79/58.07	= { by lemma 86 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK1_goals_X1, sK2_goals_X0))
57.79/58.07	= { by lemma 16 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, sK1_goals_X1))
57.79/58.07	= { by lemma 70 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(complement(meet(sK2_goals_X0, sK1_goals_X1)), meet(sK2_goals_X0, sK1_goals_X1)))
57.79/58.07	= { by lemma 71 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, meet(complement(meet(sK2_goals_X0, sK1_goals_X1)), sK1_goals_X1)))
57.79/58.07	= { by lemma 63 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, meet(complement(meet(sK2_goals_X0, sK1_goals_X1)), join(sK1_goals_X1, complement(complement(meet(sK2_goals_X0, sK1_goals_X1)))))))
57.79/58.07	= { by lemma 71 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(complement(meet(sK2_goals_X0, sK1_goals_X1)), meet(sK2_goals_X0, join(sK1_goals_X1, complement(complement(meet(sK2_goals_X0, sK1_goals_X1)))))))
57.79/58.07	= { by lemma 70 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(sK1_goals_X1, complement(complement(meet(sK2_goals_X0, sK1_goals_X1))))))
57.79/58.07	= { by lemma 40 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(sK1_goals_X1, meet(sK2_goals_X0, sK1_goals_X1))))
57.79/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))
57.79/58.07	= { by lemma 16 }
57.79/58.07	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0))
57.79/58.07	= { by lemma 39 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), top)
57.79/58.07	= { by lemma 31 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0)), top))
57.79/58.07	= { by axiom 3 (def_top) }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0)), join(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1))), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))))))
57.79/58.07	= { by axiom 13 (maddux2_join_associativity) }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1))))))))
57.79/58.07	= { by lemma 24 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(join(meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1))))), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0)))))
57.79/58.07	= { by axiom 13 (maddux2_join_associativity) }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), join(complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0))))))
57.79/58.07	= { by lemma 25 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))), join(meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(sK2_goals_X0))))))
57.79/58.07	= { by lemma 42 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))), join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1))))
57.79/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))))))
57.79/58.07	= { by lemma 16 }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0))))))
57.79/58.07	= { by axiom 13 (maddux2_join_associativity) }
57.79/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(join(meet(sK2_goals_X0, sK1_goals_X1), join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), complement(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))))
57.79/58.07	= { by lemma 63 }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)))
57.88/58.07	= { by lemma 16 }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1)), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by lemma 21 }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), join(composition(one, meet(sK2_goals_X0, sK1_goals_X1)), sK1_goals_X1)), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), join(sK1_goals_X1, composition(one, meet(sK2_goals_X0, sK1_goals_X1)))), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by lemma 21 }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), join(sK1_goals_X1, meet(sK2_goals_X0, sK1_goals_X1))), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by lemma 24 }
57.88/58.07	  meet(join(sK1_goals_X1, join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK2_goals_X0, sK1_goals_X1))), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by lemma 41 }
57.88/58.07	  meet(join(sK1_goals_X1, meet(sK2_goals_X0, sK1_goals_X1)), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), join(meet(sK2_goals_X0, sK1_goals_X1), meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), sK2_goals_X0)))
57.88/58.07	= { by lemma 84 }
57.88/58.07	  meet(join(meet(sK2_goals_X0, sK1_goals_X1), sK1_goals_X1), join(meet(sK2_goals_X0, sK1_goals_X1), sK2_goals_X0))
57.88/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.88/58.07	  meet(join(sK1_goals_X1, meet(sK2_goals_X0, sK1_goals_X1)), join(meet(sK2_goals_X0, sK1_goals_X1), sK2_goals_X0))
57.88/58.07	= { by lemma 16 }
57.88/58.07	  meet(join(sK1_goals_X1, meet(sK1_goals_X1, sK2_goals_X0)), join(meet(sK2_goals_X0, sK1_goals_X1), sK2_goals_X0))
57.88/58.07	= { by lemma 60 }
57.88/58.07	  meet(sK1_goals_X1, join(meet(sK2_goals_X0, sK1_goals_X1), sK2_goals_X0))
57.88/58.07	= { by lemma 16 }
57.88/58.07	  meet(sK1_goals_X1, join(meet(sK1_goals_X1, sK2_goals_X0), sK2_goals_X0))
57.88/58.07	= { by axiom 9 (maddux1_join_commutativity) }
57.88/58.07	  meet(sK1_goals_X1, join(sK2_goals_X0, meet(sK1_goals_X1, sK2_goals_X0)))
57.88/58.07	= { by lemma 84 }
57.88/58.07	  meet(sK1_goals_X1, join(sK2_goals_X0, sK2_goals_X0))
57.88/58.07	= { by lemma 41 }
57.88/58.07	  meet(sK1_goals_X1, sK2_goals_X0)
57.88/58.07	= { by lemma 86 }
57.88/58.07	  composition(meet(sK2_goals_X0, one), sK1_goals_X1)
57.88/58.07	% SZS output end Proof
57.88/58.07	
57.88/58.07	RESULT: Theorem (the conjecture is true).
57.91/58.09	EOF