0.00/0.03	% 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
0.02/0.23	% Computer   : n172.star.cs.uiowa.edu
0.02/0.23	% Model      : x86_64 x86_64
0.02/0.23	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.02/0.23	% Memory     : 32218.625MB
0.02/0.23	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.02/0.23	% CPULimit   : 300
0.02/0.23	% DateTime   : Sat Jul 14 06:07:24 CDT 2018
0.02/0.24	% CPUTime    : 
114.71/114.91	% SZS status Theorem
114.71/114.91	
114.89/115.14	% SZS output start Proof
114.89/115.14	Take the following subset of the input axioms:
115.01/115.24	  fof(composition_associativity, axiom,
115.01/115.24	      ![X0, X1, X2]:
115.01/115.24	        composition(X0, composition(X1, X2))=composition(composition(X0,
115.01/115.24	                                                                     X1),
115.01/115.24	                                                         X2)).
115.01/115.24	  fof(composition_distributivity, axiom,
115.01/115.24	      ![X0, X1, X2]:
115.01/115.24	        composition(join(X0, X1), X2)=join(composition(X0, X2),
115.01/115.24	                                           composition(X1, X2))).
115.01/115.24	  fof(composition_identity, axiom, ![X0]: composition(X0, one)=X0).
115.01/115.24	  fof(converse_additivity, axiom,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        join(converse(X0), converse(X1))=converse(join(X0, X1))).
115.01/115.24	  fof(converse_cancellativity, axiom,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        join(composition(converse(X0), complement(composition(X0, X1))),
115.01/115.24	             complement(X1))=complement(X1)).
115.01/115.24	  fof(converse_idempotence, axiom, ![X0]: converse(converse(X0))=X0).
115.01/115.24	  fof(converse_multiplicativity, axiom,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        converse(composition(X0, X1))=composition(converse(X1),
115.01/115.24	                                                  converse(X0))).
115.01/115.24	  fof(def_top, axiom, ![X0]: top=join(X0, complement(X0))).
115.01/115.24	  fof(def_zero, axiom, ![X0]: zero=meet(X0, complement(X0))).
115.01/115.24	  fof(goals, conjecture,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        (meet(X0, X1)=join(composition(meet(X0, one), X1), meet(X0, X1))
115.01/115.24	           <= composition(X0, top)=X0)).
115.01/115.24	  fof(maddux1_join_commutativity, axiom,
115.01/115.24	      ![X0, X1]: join(X0, X1)=join(X1, X0)).
115.01/115.24	  fof(maddux2_join_associativity, axiom,
115.01/115.24	      ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
115.01/115.24	  fof(maddux3_a_kind_of_de_Morgan, axiom,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        X0=join(complement(join(complement(X0), complement(X1))),
115.01/115.24	                complement(join(complement(X0), X1)))).
115.01/115.24	  fof(maddux4_definiton_of_meet, axiom,
115.01/115.24	      ![X0, X1]:
115.01/115.24	        meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
115.01/115.24	  fof(modular_law_1, axiom,
115.01/115.24	      ![X0, X1, X2]:
115.01/115.24	        join(meet(composition(X0, X1), X2),
115.01/115.24	             meet(composition(X0, meet(X1, composition(converse(X0), X2))),
115.01/115.24	                  X2))=meet(composition(X0, meet(X1, composition(converse(X0), X2))),
115.01/115.24	                            X2)).
115.01/115.24	
115.01/115.24	Now clausify the problem and encode Horn clauses using encoding 3 of
115.01/115.24	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
115.01/115.24	We repeatedly replace C & s=t => u=v by the two clauses:
115.01/115.24	  $$fresh(y, y, x1...xn) = u
115.01/115.24	  C => $$fresh(s, t, x1...xn) = v
115.01/115.24	where $$fresh is a fresh function symbol and x1..xn are the free
115.01/115.24	variables of u and v.
115.01/115.24	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
115.01/115.24	input problem has no model of domain size 1).
115.01/115.24	
115.01/115.24	The encoding turns the above axioms into the following unit equations and goals:
115.01/115.24	
115.01/115.24	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
115.01/115.24	Axiom 2 (def_zero): zero = meet(X, complement(X)).
115.01/115.24	Axiom 3 (def_top): top = join(X, complement(X)).
115.01/115.24	Axiom 4 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
115.01/115.24	Axiom 5 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
115.01/115.24	Axiom 6 (converse_idempotence): converse(converse(X)) = X.
115.01/115.24	Axiom 7 (converse_additivity): join(converse(X), converse(Y)) = converse(join(X, Y)).
115.01/115.24	Axiom 8 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
115.01/115.24	Axiom 9 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
115.01/115.24	Axiom 10 (composition_identity): composition(X, one) = X.
115.01/115.24	Axiom 11 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
115.01/115.24	Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
115.01/115.24	Axiom 13 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
115.01/115.24	Axiom 14 (modular_law_1): join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z)) = meet(composition(X, meet(Y, composition(converse(X), Z))), Z).
115.01/115.24	Axiom 17 (goals): composition(sK2_goals_X0, top) = sK2_goals_X0.
115.01/115.24	
115.01/115.24	Lemma 18: complement(top) = zero.
115.01/115.24	Proof:
115.01/115.24	  complement(top)
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  complement(join(complement(?), complement(complement(?))))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  meet(?, complement(?))
115.01/115.24	= { by axiom 2 (def_zero) }
115.01/115.24	  zero
115.01/115.24	
115.01/115.24	Lemma 19: meet(X, Y) = meet(Y, X).
115.01/115.24	Proof:
115.01/115.24	  meet(X, Y)
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  complement(join(complement(X), complement(Y)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  complement(join(complement(Y), complement(X)))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  meet(Y, X)
115.01/115.24	
115.01/115.24	Lemma 20: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
115.01/115.24	Proof:
115.01/115.24	  converse(composition(X, converse(Y)))
115.01/115.24	= { by axiom 4 (converse_multiplicativity) }
115.01/115.24	  composition(converse(converse(Y)), converse(X))
115.01/115.24	= { by axiom 6 (converse_idempotence) }
115.01/115.24	  composition(Y, converse(X))
115.01/115.24	
115.01/115.24	Lemma 21: converse(composition(converse(X), Y)) = composition(converse(Y), X).
115.01/115.24	Proof:
115.01/115.24	  converse(composition(converse(X), Y))
115.01/115.24	= { by axiom 4 (converse_multiplicativity) }
115.01/115.24	  composition(converse(Y), converse(converse(X)))
115.01/115.24	= { by axiom 6 (converse_idempotence) }
115.01/115.24	  composition(converse(Y), X)
115.01/115.24	
115.01/115.24	Lemma 22: composition(converse(one), X) = X.
115.01/115.24	Proof:
115.01/115.24	  composition(converse(one), X)
115.01/115.24	= { by lemma 21 }
115.01/115.24	  converse(composition(converse(X), one))
115.01/115.24	= { by axiom 10 (composition_identity) }
115.01/115.24	  converse(converse(X))
115.01/115.24	= { by axiom 6 (converse_idempotence) }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 23: converse(one) = one.
115.01/115.24	Proof:
115.01/115.24	  converse(one)
115.01/115.24	= { by axiom 10 (composition_identity) }
115.01/115.24	  composition(converse(one), one)
115.01/115.24	= { by lemma 22 }
115.01/115.24	  one
115.01/115.24	
115.01/115.24	Lemma 24: composition(one, X) = X.
115.01/115.24	Proof:
115.01/115.24	  composition(one, X)
115.01/115.24	= { by lemma 22 }
115.01/115.24	  composition(converse(one), composition(one, X))
115.01/115.24	= { by axiom 5 (composition_associativity) }
115.01/115.24	  composition(composition(converse(one), one), X)
115.01/115.24	= { by axiom 10 (composition_identity) }
115.01/115.24	  composition(converse(one), X)
115.01/115.24	= { by lemma 22 }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 25: converse(join(X, converse(Y))) = join(Y, converse(X)).
115.01/115.24	Proof:
115.01/115.24	  converse(join(X, converse(Y)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  converse(join(converse(Y), X))
115.01/115.24	= { by axiom 7 (converse_additivity) }
115.01/115.24	  join(converse(converse(Y)), converse(X))
115.01/115.24	= { by axiom 6 (converse_idempotence) }
115.01/115.24	  join(Y, converse(X))
115.01/115.24	
115.01/115.24	Lemma 26: join(X, join(Y, Z)) = join(Z, join(X, Y)).
115.01/115.24	Proof:
115.01/115.24	  join(X, join(Y, Z))
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(join(X, Y), Z)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(Z, join(X, Y))
115.01/115.24	
115.01/115.24	Lemma 27: join(X, join(Y, Z)) = join(Y, join(X, Z)).
115.01/115.24	Proof:
115.01/115.24	  join(X, join(Y, Z))
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(join(X, Y), Z)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(join(Y, X), Z)
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(Y, join(X, Z))
115.01/115.24	
115.01/115.24	Lemma 28: join(X, join(Y, complement(X))) = join(Y, top).
115.01/115.24	Proof:
115.01/115.24	  join(X, join(Y, complement(X)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(X, join(complement(X), Y))
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(join(X, complement(X)), Y)
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  join(top, Y)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(Y, top)
115.01/115.24	
115.01/115.24	Lemma 29: complement(join(zero, complement(X))) = meet(X, top).
115.01/115.24	Proof:
115.01/115.24	  complement(join(zero, complement(X)))
115.01/115.24	= { by lemma 18 }
115.01/115.24	  complement(join(complement(top), complement(X)))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  meet(top, X)
115.01/115.24	= { by lemma 19 }
115.01/115.24	  meet(X, top)
115.01/115.24	
115.01/115.24	Lemma 30: converse(join(converse(X), Y)) = join(X, converse(Y)).
115.01/115.24	Proof:
115.01/115.24	  converse(join(converse(X), Y))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  converse(join(Y, converse(X)))
115.01/115.24	= { by lemma 25 }
115.01/115.24	  join(X, converse(Y))
115.01/115.24	
115.01/115.24	Lemma 31: join(complement(X), complement(X)) = complement(X).
115.01/115.24	Proof:
115.01/115.24	  join(complement(X), complement(X))
115.01/115.24	= { by lemma 22 }
115.01/115.24	  join(composition(converse(one), complement(X)), complement(X))
115.01/115.24	= { by lemma 24 }
115.01/115.24	  join(composition(converse(one), complement(composition(one, X))), complement(X))
115.01/115.24	= { by axiom 8 (converse_cancellativity) }
115.01/115.24	  complement(X)
115.01/115.24	
115.01/115.24	Lemma 32: join(zero, zero) = zero.
115.01/115.24	Proof:
115.01/115.24	  join(zero, zero)
115.01/115.24	= { by lemma 18 }
115.01/115.24	  join(complement(top), zero)
115.01/115.24	= { by lemma 18 }
115.01/115.24	  join(complement(top), complement(top))
115.01/115.24	= { by lemma 31 }
115.01/115.24	  complement(top)
115.01/115.24	= { by lemma 18 }
115.01/115.24	  zero
115.01/115.24	
115.01/115.24	Lemma 34: complement(complement(X)) = meet(X, X).
115.01/115.24	Proof:
115.01/115.24	  complement(complement(X))
115.01/115.24	= { by lemma 31 }
115.01/115.24	  complement(join(complement(X), complement(X)))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  meet(X, X)
115.01/115.24	
115.01/115.24	Lemma 34: meet(X, X) = complement(complement(X)).
115.01/115.24	Proof:
115.01/115.24	  meet(X, X)
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  complement(join(complement(X), complement(X)))
115.01/115.24	= { by lemma 31 }
115.01/115.24	  complement(complement(X))
115.01/115.24	
115.01/115.24	Lemma 35: join(top, complement(X)) = top.
115.01/115.24	Proof:
115.01/115.24	  join(top, complement(X))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(complement(X), top)
115.01/115.24	= { by lemma 28 }
115.01/115.24	  join(X, join(complement(X), complement(X)))
115.01/115.24	= { by lemma 31 }
115.01/115.24	  join(X, complement(X))
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 36: join(X, join(complement(X), Y)) = join(Y, top).
115.01/115.24	Proof:
115.01/115.24	  join(X, join(complement(X), Y))
115.01/115.24	= { by lemma 26 }
115.01/115.24	  join(complement(X), join(Y, X))
115.01/115.24	= { by lemma 26 }
115.01/115.24	  join(Y, join(X, complement(X)))
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  join(Y, top)
115.01/115.24	
115.01/115.24	Lemma 37: join(X, top) = top.
115.01/115.24	Proof:
115.01/115.24	  join(X, top)
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  join(X, join(complement(X), complement(complement(X))))
115.01/115.24	= { by lemma 36 }
115.01/115.24	  join(complement(complement(X)), top)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(top, complement(complement(X)))
115.01/115.24	= { by lemma 35 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 38: join(top, X) = top.
115.01/115.24	Proof:
115.01/115.24	  join(top, X)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(X, top)
115.01/115.24	= { by lemma 37 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 39: join(X, converse(top)) = converse(top).
115.01/115.24	Proof:
115.01/115.24	  join(X, converse(top))
115.01/115.24	= { by lemma 30 }
115.01/115.24	  converse(join(converse(X), top))
115.01/115.24	= { by lemma 37 }
115.01/115.24	  converse(top)
115.01/115.24	
115.01/115.24	Lemma 40: converse(top) = top.
115.01/115.24	Proof:
115.01/115.24	  converse(top)
115.01/115.24	= { by lemma 39 }
115.01/115.24	  join(?, converse(top))
115.01/115.24	= { by lemma 39 }
115.01/115.24	  join(?, join(complement(?), converse(top)))
115.01/115.24	= { by lemma 36 }
115.01/115.24	  join(converse(top), top)
115.01/115.24	= { by lemma 37 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 41: join(zero, meet(X, top)) = X.
115.01/115.24	Proof:
115.01/115.24	  join(zero, meet(X, top))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(meet(X, top), zero)
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  join(complement(join(complement(X), complement(top))), zero)
115.01/115.24	= { by lemma 18 }
115.01/115.24	  join(complement(join(complement(X), complement(top))), complement(top))
115.01/115.24	= { by lemma 37 }
115.01/115.24	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
115.01/115.24	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 42: join(meet(X, Y), meet(X, complement(Y))) = X.
115.01/115.24	Proof:
115.01/115.24	  join(meet(X, Y), meet(X, complement(Y)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(meet(X, complement(Y)), meet(X, Y))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
115.01/115.24	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 43: join(zero, meet(X, X)) = X.
115.01/115.24	Proof:
115.01/115.24	  join(zero, meet(X, X))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(meet(X, X), zero)
115.01/115.24	= { by axiom 2 (def_zero) }
115.01/115.24	  join(meet(X, X), meet(X, complement(X)))
115.01/115.24	= { by lemma 42 }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 44: join(X, converse(complement(converse(X)))) = top.
115.01/115.24	Proof:
115.01/115.24	  join(X, converse(complement(converse(X))))
115.01/115.24	= { by lemma 30 }
115.01/115.24	  converse(join(converse(X), complement(converse(X))))
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  converse(top)
115.01/115.24	= { by lemma 40 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 45: join(meet(Y, X), meet(X, complement(Y))) = X.
115.01/115.24	Proof:
115.01/115.24	  join(meet(Y, X), meet(X, complement(Y)))
115.01/115.24	= { by lemma 19 }
115.01/115.24	  join(meet(X, Y), meet(X, complement(Y)))
115.01/115.24	= { by lemma 42 }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 46: join(Y, composition(X, Y)) = composition(join(X, one), Y).
115.01/115.24	Proof:
115.01/115.24	  join(Y, composition(X, Y))
115.01/115.24	= { by lemma 24 }
115.01/115.24	  join(composition(one, Y), composition(X, Y))
115.01/115.24	= { by axiom 12 (composition_distributivity) }
115.01/115.24	  composition(join(one, X), Y)
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  composition(join(X, one), Y)
115.01/115.24	
115.01/115.24	Lemma 47: join(complement(converse(X)), converse(join(X, Y))) = top.
115.01/115.24	Proof:
115.01/115.24	  join(complement(converse(X)), converse(join(X, Y)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(complement(converse(X)), converse(join(Y, X)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(converse(join(Y, X)), complement(converse(X)))
115.01/115.24	= { by axiom 7 (converse_additivity) }
115.01/115.24	  join(join(converse(Y), converse(X)), complement(converse(X)))
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(converse(Y), join(converse(X), complement(converse(X))))
115.01/115.24	= { by axiom 3 (def_top) }
115.01/115.24	  join(converse(Y), top)
115.01/115.24	= { by lemma 37 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 48: complement(converse(zero)) = top.
115.01/115.24	Proof:
115.01/115.24	  complement(converse(zero))
115.01/115.24	= { by axiom 8 (converse_cancellativity) }
115.01/115.24	  join(composition(converse(?), complement(composition(?, converse(zero)))), complement(converse(zero)))
115.01/115.24	= { by axiom 6 (converse_idempotence) }
115.01/115.24	  join(converse(converse(composition(converse(?), complement(composition(?, converse(zero)))))), complement(converse(zero)))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(complement(converse(zero)), converse(converse(composition(converse(?), complement(composition(?, converse(zero)))))))
115.01/115.24	= { by lemma 43 }
115.01/115.24	  join(complement(converse(zero)), converse(join(zero, meet(converse(composition(converse(?), complement(composition(?, converse(zero))))), converse(composition(converse(?), complement(composition(?, converse(zero)))))))))
115.01/115.24	= { by lemma 47 }
115.01/115.24	  top
115.01/115.24	
115.01/115.24	Lemma 49: converse(zero) = zero.
115.01/115.24	Proof:
115.01/115.24	  converse(zero)
115.01/115.24	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.24	  join(complement(join(complement(converse(zero)), complement(?))), complement(join(complement(converse(zero)), ?)))
115.01/115.24	= { by lemma 48 }
115.01/115.24	  join(complement(join(top, complement(?))), complement(join(complement(converse(zero)), ?)))
115.01/115.24	= { by lemma 35 }
115.01/115.24	  join(complement(top), complement(join(complement(converse(zero)), ?)))
115.01/115.24	= { by lemma 18 }
115.01/115.24	  join(zero, complement(join(complement(converse(zero)), ?)))
115.01/115.24	= { by lemma 48 }
115.01/115.24	  join(zero, complement(join(top, ?)))
115.01/115.24	= { by lemma 38 }
115.01/115.24	  join(zero, complement(top))
115.01/115.24	= { by lemma 18 }
115.01/115.24	  join(zero, zero)
115.01/115.24	= { by lemma 32 }
115.01/115.24	  zero
115.01/115.24	
115.01/115.24	Lemma 50: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
115.01/115.24	Proof:
115.01/115.24	  join(meet(X, Y), complement(join(Y, complement(X))))
115.01/115.24	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.24	  join(complement(join(complement(X), complement(Y))), complement(join(Y, complement(X))))
115.01/115.24	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.24	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
115.01/115.24	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.24	  X
115.01/115.24	
115.01/115.24	Lemma 51: join(zero, join(meet(X, X), Y)) = join(X, Y).
115.01/115.24	Proof:
115.01/115.24	  join(zero, join(meet(X, X), Y))
115.01/115.24	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.24	  join(join(zero, meet(X, X)), Y)
115.01/115.24	= { by lemma 43 }
115.01/115.24	  join(X, Y)
115.01/115.24	
115.01/115.24	Lemma 52: join(zero, join(X, meet(Y, Y))) = join(X, Y).
115.01/115.24	Proof:
115.01/115.24	  join(zero, join(X, meet(Y, Y)))
115.01/115.24	= { by lemma 27 }
115.01/115.24	  join(X, join(zero, meet(Y, Y)))
115.01/115.24	= { by lemma 43 }
115.01/115.25	  join(X, Y)
115.01/115.25	
115.01/115.25	Lemma 53: join(X, zero) = X.
115.01/115.25	Proof:
115.01/115.25	  join(X, zero)
115.01/115.25	= { by lemma 18 }
115.01/115.25	  join(X, complement(top))
115.01/115.25	= { by lemma 35 }
115.01/115.25	  join(X, complement(join(top, complement(zero))))
115.01/115.25	= { by lemma 42 }
115.01/115.25	  join(X, complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(zero))))
115.01/115.25	= { by axiom 2 (def_zero) }
115.01/115.25	  join(X, complement(join(join(zero, meet(top, complement(complement(top)))), complement(zero))))
115.01/115.25	= { by lemma 34 }
115.01/115.25	  join(X, complement(join(join(zero, meet(top, meet(top, top))), complement(zero))))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  join(X, complement(join(join(zero, meet(meet(top, top), top)), complement(zero))))
115.01/115.25	= { by lemma 41 }
115.01/115.25	  join(X, complement(join(meet(top, top), complement(zero))))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  join(X, complement(join(complement(join(zero, complement(top))), complement(zero))))
115.01/115.25	= { by lemma 18 }
115.01/115.25	  join(X, complement(join(complement(join(zero, zero)), complement(zero))))
115.01/115.25	= { by lemma 32 }
115.01/115.25	  join(X, complement(join(complement(zero), complement(zero))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(X, meet(zero, zero))
115.01/115.25	= { by lemma 51 }
115.01/115.25	  join(zero, join(meet(X, X), meet(zero, zero)))
115.01/115.25	= { by lemma 52 }
115.01/115.25	  join(meet(X, X), zero)
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  join(zero, meet(X, X))
115.01/115.25	= { by lemma 43 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 54: join(zero, X) = X.
115.01/115.25	Proof:
115.01/115.25	  join(zero, X)
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  join(X, zero)
115.01/115.25	= { by lemma 53 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 55: meet(X, top) = X.
115.01/115.25	Proof:
115.01/115.25	  meet(X, top)
115.01/115.25	= { by lemma 54 }
115.01/115.25	  join(zero, meet(X, top))
115.01/115.25	= { by lemma 41 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 56: complement(complement(X)) = X.
115.01/115.25	Proof:
115.01/115.25	  complement(complement(X))
115.01/115.25	= { by lemma 54 }
115.01/115.25	  join(zero, complement(complement(X)))
115.01/115.25	= { by lemma 34 }
115.01/115.25	  join(zero, meet(X, X))
115.01/115.25	= { by lemma 43 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 57: join(X, X) = X.
115.01/115.25	Proof:
115.01/115.25	  join(X, X)
115.01/115.25	= { by lemma 52 }
115.01/115.25	  join(zero, join(X, meet(X, X)))
115.01/115.25	= { by lemma 51 }
115.01/115.25	  join(zero, join(zero, join(meet(X, X), meet(X, X))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(zero, join(zero, join(complement(join(complement(X), complement(X))), meet(X, X))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(zero, join(zero, join(complement(join(complement(X), complement(X))), complement(join(complement(X), complement(X))))))
115.01/115.25	= { by lemma 31 }
115.01/115.25	  join(zero, join(zero, complement(join(complement(X), complement(X)))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(zero, join(zero, meet(X, X)))
115.01/115.25	= { by lemma 43 }
115.01/115.25	  join(zero, X)
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  join(X, zero)
115.01/115.25	= { by lemma 53 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 58: meet(top, X) = X.
115.01/115.25	Proof:
115.01/115.25	  meet(top, X)
115.01/115.25	= { by lemma 19 }
115.01/115.25	  meet(X, top)
115.01/115.25	= { by lemma 55 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 59: meet(join(X, complement(Y)), join(complement(X), complement(Y))) = complement(Y).
115.01/115.25	Proof:
115.01/115.25	  meet(join(X, complement(Y)), join(complement(X), complement(Y)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(join(complement(Y), X), join(complement(X), complement(Y)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(join(complement(Y), X), join(complement(Y), complement(X)))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
115.01/115.25	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.25	  complement(Y)
115.01/115.25	
115.01/115.25	Lemma 60: composition(X, composition(top, converse(sK2_goals_X0))) = composition(X, converse(sK2_goals_X0)).
115.01/115.25	Proof:
115.01/115.25	  composition(X, composition(top, converse(sK2_goals_X0)))
115.01/115.25	= { by lemma 40 }
115.01/115.25	  composition(X, composition(converse(top), converse(sK2_goals_X0)))
115.01/115.25	= { by axiom 5 (composition_associativity) }
115.01/115.25	  composition(composition(X, converse(top)), converse(sK2_goals_X0))
115.01/115.25	= { by lemma 20 }
115.01/115.25	  composition(converse(composition(top, converse(X))), converse(sK2_goals_X0))
115.01/115.25	= { by axiom 4 (converse_multiplicativity) }
115.01/115.25	  converse(composition(sK2_goals_X0, composition(top, converse(X))))
115.01/115.25	= { by axiom 5 (composition_associativity) }
115.01/115.25	  converse(composition(composition(sK2_goals_X0, top), converse(X)))
115.01/115.25	= { by axiom 17 (goals) }
115.01/115.25	  converse(composition(sK2_goals_X0, converse(X)))
115.01/115.25	= { by lemma 20 }
115.01/115.25	  composition(X, converse(sK2_goals_X0))
115.01/115.25	
115.01/115.25	Lemma 61: composition(top, converse(sK2_goals_X0)) = converse(sK2_goals_X0).
115.01/115.25	Proof:
115.01/115.25	  composition(top, converse(sK2_goals_X0))
115.01/115.25	= { by lemma 22 }
115.01/115.25	  composition(converse(one), composition(top, converse(sK2_goals_X0)))
115.01/115.25	= { by lemma 60 }
115.01/115.25	  composition(converse(one), converse(sK2_goals_X0))
115.01/115.25	= { by lemma 22 }
115.01/115.25	  converse(sK2_goals_X0)
115.01/115.25	
115.01/115.25	Lemma 62: meet(join(X, Y), join(X, complement(Y))) = X.
115.01/115.25	Proof:
115.01/115.25	  meet(join(X, Y), join(X, complement(Y)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(join(Y, X), join(X, complement(Y)))
115.01/115.25	= { by lemma 55 }
115.01/115.25	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), X))
115.01/115.25	= { by lemma 55 }
115.01/115.25	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), meet(X, top)))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  meet(join(Y, complement(join(zero, complement(X)))), join(complement(Y), complement(join(zero, complement(X)))))
115.01/115.25	= { by lemma 59 }
115.01/115.25	  complement(join(zero, complement(X)))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  meet(X, top)
115.01/115.25	= { by lemma 55 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 63: join(X, complement(join(Y, complement(X)))) = X.
115.01/115.25	Proof:
115.01/115.25	  join(X, complement(join(Y, complement(X))))
115.01/115.25	= { by lemma 58 }
115.01/115.25	  meet(top, join(X, complement(join(Y, complement(X)))))
115.01/115.25	= { by lemma 37 }
115.01/115.25	  meet(join(Y, top), join(X, complement(join(Y, complement(X)))))
115.01/115.25	= { by lemma 28 }
115.01/115.25	  meet(join(X, join(Y, complement(X))), join(X, complement(join(Y, complement(X)))))
115.01/115.25	= { by lemma 62 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 64: meet(X, join(Y, X)) = X.
115.01/115.25	Proof:
115.01/115.25	  meet(X, join(Y, X))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(X, join(X, Y))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(X, join(Y, X))
115.01/115.25	= { by lemma 56 }
115.01/115.25	  meet(X, join(Y, complement(complement(X))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  complement(join(complement(X), complement(join(Y, complement(complement(X))))))
115.01/115.25	= { by lemma 63 }
115.01/115.25	  complement(complement(X))
115.01/115.25	= { by lemma 56 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 65: join(X, meet(Y, X)) = X.
115.01/115.25	Proof:
115.01/115.25	  join(X, meet(Y, X))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  join(X, meet(X, Y))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  join(X, meet(Y, X))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(X, complement(join(complement(Y), complement(X))))
115.01/115.25	= { by lemma 63 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 66: join(X, complement(converse(complement(converse(X))))) = X.
115.01/115.25	Proof:
115.01/115.25	  join(X, complement(converse(complement(converse(X)))))
115.01/115.25	= { by lemma 58 }
115.01/115.25	  meet(top, join(X, complement(converse(complement(converse(X))))))
115.01/115.25	= { by lemma 44 }
115.01/115.25	  meet(join(X, converse(complement(converse(X)))), join(X, complement(converse(complement(converse(X))))))
115.01/115.25	= { by lemma 62 }
115.01/115.25	  X
115.01/115.25	
115.01/115.25	Lemma 67: join(complement(X), complement(Y)) = complement(meet(X, Y)).
115.01/115.25	Proof:
115.01/115.25	  join(complement(X), complement(Y))
115.01/115.25	= { by lemma 58 }
115.01/115.25	  meet(top, join(complement(X), complement(Y)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  meet(top, join(complement(Y), complement(X)))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  meet(join(complement(Y), complement(X)), top)
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  complement(join(complement(join(complement(Y), complement(X))), complement(top)))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  complement(join(meet(Y, X), complement(top)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  complement(join(complement(top), meet(Y, X)))
115.01/115.25	= { by lemma 18 }
115.01/115.25	  complement(join(zero, meet(Y, X)))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  complement(join(zero, meet(X, Y)))
115.01/115.25	= { by lemma 54 }
115.01/115.25	  complement(meet(X, Y))
115.01/115.25	
115.01/115.25	Lemma 68: complement(meet(X, complement(Y))) = join(Y, complement(X)).
115.01/115.25	Proof:
115.01/115.25	  complement(meet(X, complement(Y)))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  complement(meet(complement(Y), X))
115.01/115.25	= { by lemma 54 }
115.01/115.25	  complement(meet(join(zero, complement(Y)), X))
115.01/115.25	= { by lemma 67 }
115.01/115.25	  join(complement(join(zero, complement(Y))), complement(X))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  join(meet(Y, top), complement(X))
115.01/115.25	= { by lemma 55 }
115.01/115.25	  join(Y, complement(X))
115.01/115.25	
115.01/115.25	Lemma 69: complement(meet(complement(X), Y)) = join(X, complement(Y)).
115.01/115.25	Proof:
115.01/115.25	  complement(meet(complement(X), Y))
115.01/115.25	= { by lemma 19 }
115.01/115.25	  complement(meet(Y, complement(X)))
115.01/115.25	= { by lemma 68 }
115.01/115.25	  join(X, complement(Y))
115.01/115.25	
115.01/115.25	Lemma 70: complement(join(Y, complement(X))) = meet(X, complement(Y)).
115.01/115.25	Proof:
115.01/115.25	  complement(join(Y, complement(X)))
115.01/115.25	= { by lemma 54 }
115.01/115.25	  complement(join(zero, join(Y, complement(X))))
115.01/115.25	= { by lemma 68 }
115.01/115.25	  complement(join(zero, complement(meet(X, complement(Y)))))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  meet(meet(X, complement(Y)), top)
115.01/115.25	= { by lemma 55 }
115.01/115.25	  meet(X, complement(Y))
115.01/115.25	
115.01/115.25	Lemma 71: join(converse(X), complement(converse(meet(X, Y)))) = top.
115.01/115.25	Proof:
115.01/115.25	  join(converse(X), complement(converse(meet(X, Y))))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  join(complement(converse(meet(X, Y))), converse(X))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  join(complement(converse(complement(join(complement(X), complement(Y))))), converse(X))
115.01/115.25	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.25	  join(complement(converse(complement(join(complement(X), complement(Y))))), converse(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))))
115.01/115.25	= { by lemma 47 }
115.01/115.25	  top
115.01/115.25	
115.01/115.25	Lemma 72: meet(X, meet(Y, Z)) = meet(Z, meet(X, Y)).
115.01/115.25	Proof:
115.01/115.25	  meet(X, meet(Y, Z))
115.01/115.25	= { by lemma 55 }
115.01/115.25	  meet(meet(X, meet(Y, Z)), top)
115.01/115.25	= { by lemma 29 }
115.01/115.25	  complement(join(zero, complement(meet(X, meet(Y, Z)))))
115.01/115.25	= { by lemma 67 }
115.01/115.25	  complement(join(zero, join(complement(X), complement(meet(Y, Z)))))
115.01/115.25	= { by lemma 67 }
115.01/115.25	  complement(join(zero, join(complement(X), join(complement(Y), complement(Z)))))
115.01/115.25	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.25	  complement(join(zero, join(join(complement(X), complement(Y)), complement(Z))))
115.01/115.25	= { by lemma 68 }
115.01/115.25	  complement(join(zero, complement(meet(Z, complement(join(complement(X), complement(Y)))))))
115.01/115.25	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.25	  complement(join(zero, complement(meet(Z, meet(X, Y)))))
115.01/115.25	= { by lemma 29 }
115.01/115.25	  meet(meet(Z, meet(X, Y)), top)
115.01/115.25	= { by lemma 55 }
115.01/115.25	  meet(Z, meet(X, Y))
115.01/115.25	
115.01/115.25	Lemma 73: join(complement(one), converse(complement(one))) = converse(complement(one)).
115.01/115.25	Proof:
115.01/115.25	  join(complement(one), converse(complement(one)))
115.01/115.25	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.25	  join(converse(complement(one)), complement(one))
115.01/115.25	= { by lemma 62 }
115.01/115.25	  join(converse(complement(one)), complement(meet(join(one, converse(complement(one))), join(one, complement(converse(complement(one)))))))
115.01/115.25	= { by lemma 23 }
115.01/115.25	  join(converse(complement(one)), complement(meet(join(one, converse(complement(converse(one)))), join(one, complement(converse(complement(one)))))))
115.01/115.25	= { by lemma 44 }
115.01/115.25	  join(converse(complement(one)), complement(meet(top, join(one, complement(converse(complement(one)))))))
115.01/115.25	= { by lemma 58 }
115.01/115.25	  join(converse(complement(one)), complement(join(one, complement(converse(complement(one))))))
115.01/115.25	= { by lemma 63 }
115.01/115.25	  converse(complement(one))
115.01/115.25	
115.01/115.25	Lemma 74: meet(meet(Z, X), Y) = meet(X, meet(Y, Z)).
115.01/115.25	Proof:
115.01/115.25	  meet(meet(Z, X), Y)
115.01/115.25	= { by lemma 19 }
115.01/115.25	  meet(Y, meet(Z, X))
115.01/115.25	= { by lemma 72 }
115.01/115.25	  meet(X, meet(Y, Z))
115.01/115.25	
115.01/115.25	Lemma 75: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
115.01/115.25	Proof:
115.01/115.25	  meet(X, meet(Y, Z))
115.01/115.25	= { by lemma 72 }
115.01/115.25	  meet(Y, meet(Z, X))
115.01/115.25	= { by lemma 19 }
115.01/115.26	  meet(Y, meet(X, Z))
115.01/115.26	
115.01/115.26	Lemma 76: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
115.01/115.26	Proof:
115.01/115.26	  meet(X, complement(meet(X, Y)))
115.01/115.26	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, Y)))
115.01/115.26	= { by lemma 64 }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, join(complement(X), Y)))))
115.01/115.26	= { by lemma 67 }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), complement(meet(Y, join(complement(X), Y)))))
115.01/115.26	= { by lemma 67 }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
115.01/115.26	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
115.01/115.26	= { by lemma 68 }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(join(complement(X), Y), complement(join(complement(X), complement(Y))))))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  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))))
115.01/115.26	= { by lemma 67 }
115.01/115.26	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(complement(join(complement(X), complement(Y)))), complement(join(complement(X), Y))))
115.01/115.26	= { by lemma 59 }
115.01/115.26	  complement(join(complement(X), Y))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  complement(join(Y, complement(X)))
115.01/115.26	= { by lemma 70 }
115.01/115.26	  meet(X, complement(Y))
115.01/115.26	
115.01/115.26	Lemma 77: meet(X, join(Y, complement(X))) = meet(X, Y).
115.01/115.26	Proof:
115.01/115.26	  meet(X, join(Y, complement(X)))
115.01/115.26	= { by lemma 68 }
115.01/115.26	  meet(X, complement(meet(X, complement(Y))))
115.01/115.26	= { by lemma 76 }
115.01/115.26	  meet(X, complement(complement(Y)))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  meet(X, Y)
115.01/115.26	
115.01/115.26	Lemma 78: join(X, meet(Y, complement(X))) = join(X, Y).
115.01/115.26	Proof:
115.01/115.26	  join(X, meet(Y, complement(X)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  join(X, meet(complement(X), Y))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  join(X, complement(complement(meet(complement(X), Y))))
115.01/115.26	= { by lemma 69 }
115.01/115.26	  complement(meet(complement(X), complement(meet(complement(X), Y))))
115.01/115.26	= { by lemma 76 }
115.01/115.26	  complement(meet(complement(X), complement(Y)))
115.01/115.26	= { by lemma 69 }
115.01/115.26	  join(X, complement(complement(Y)))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  join(X, Y)
115.01/115.26	
115.01/115.26	Lemma 79: meet(X, join(complement(X), Y)) = meet(X, Y).
115.01/115.26	Proof:
115.01/115.26	  meet(X, join(complement(X), Y))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  meet(X, join(Y, complement(X)))
115.01/115.26	= { by lemma 77 }
115.01/115.26	  meet(X, Y)
115.01/115.26	
115.01/115.26	Lemma 80: join(complement(X), join(Y, complement(Z))) = join(Y, complement(meet(X, Z))).
115.01/115.26	Proof:
115.01/115.26	  join(complement(X), join(Y, complement(Z)))
115.01/115.26	= { by lemma 26 }
115.01/115.26	  join(Y, join(complement(Z), complement(X)))
115.01/115.26	= { by lemma 67 }
115.01/115.26	  join(Y, complement(meet(Z, X)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  join(Y, complement(meet(X, Z)))
115.01/115.26	
115.01/115.26	Lemma 81: meet(join(Y, X), join(X, complement(Y))) = X.
115.01/115.26	Proof:
115.01/115.26	  meet(join(Y, X), join(X, complement(Y)))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  meet(join(X, Y), join(X, complement(Y)))
115.01/115.26	= { by lemma 62 }
115.01/115.26	  X
115.01/115.26	
115.01/115.26	Lemma 82: complement(converse(X)) = converse(complement(X)).
115.01/115.26	Proof:
115.01/115.26	  complement(converse(X))
115.01/115.26	= { by lemma 55 }
115.01/115.26	  complement(converse(meet(X, top)))
115.01/115.26	= { by lemma 29 }
115.01/115.26	  complement(converse(complement(join(zero, complement(X)))))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  converse(converse(complement(converse(complement(join(zero, complement(X)))))))
115.01/115.26	= { by lemma 81 }
115.01/115.26	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  converse(meet(join(join(zero, complement(X)), converse(complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
115.01/115.26	= { by lemma 30 }
115.01/115.26	  converse(meet(converse(join(converse(join(zero, complement(X))), complement(converse(complement(converse(converse(join(zero, complement(X))))))))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
115.01/115.26	= { by lemma 66 }
115.01/115.26	  converse(meet(converse(converse(join(zero, complement(X)))), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  converse(meet(join(zero, complement(X)), join(converse(complement(converse(complement(join(zero, complement(X)))))), complement(join(zero, complement(X))))))
115.01/115.26	= { by lemma 77 }
115.01/115.26	  converse(meet(join(zero, complement(X)), converse(complement(converse(complement(join(zero, complement(X))))))))
115.01/115.26	= { by axiom 11 (maddux4_definiton_of_meet) }
115.01/115.26	  converse(complement(join(complement(join(zero, complement(X))), complement(converse(complement(converse(complement(join(zero, complement(X))))))))))
115.01/115.26	= { by lemma 66 }
115.01/115.26	  converse(complement(complement(join(zero, complement(X)))))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  converse(join(zero, complement(X)))
115.01/115.26	= { by lemma 54 }
115.01/115.26	  converse(complement(X))
115.01/115.26	
115.01/115.26	Lemma 83: join(X, meet(complement(X), Y)) = join(X, Y).
115.01/115.26	Proof:
115.01/115.26	  join(X, meet(complement(X), Y))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  join(X, meet(Y, complement(X)))
115.01/115.26	= { by lemma 78 }
115.01/115.26	  join(X, Y)
115.01/115.26	
115.01/115.26	Lemma 84: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
115.01/115.26	Proof:
115.01/115.26	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
115.01/115.26	= { by axiom 12 (composition_distributivity) }
115.01/115.26	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
115.01/115.26	= { by axiom 13 (maddux2_join_associativity) }
115.01/115.26	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
115.01/115.26	= { by axiom 3 (def_top) }
115.01/115.26	  join(composition(Z, Y), top)
115.01/115.26	= { by lemma 37 }
115.01/115.26	  top
115.01/115.26	
115.01/115.26	Lemma 85: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
115.01/115.26	Proof:
115.01/115.26	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
115.01/115.26	= { by axiom 4 (converse_multiplicativity) }
115.01/115.26	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
115.01/115.26	= { by axiom 12 (composition_distributivity) }
115.01/115.26	  composition(join(converse(X), Y), converse(Z))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  composition(join(Y, converse(X)), converse(Z))
115.01/115.26	
115.01/115.26	Lemma 86: join(composition(X, Y), composition(X, Z)) = composition(X, join(Y, Z)).
115.01/115.26	Proof:
115.01/115.26	  join(composition(X, Y), composition(X, Z))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  converse(converse(join(composition(X, Y), composition(X, Z))))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  converse(converse(join(composition(X, Z), composition(X, Y))))
115.01/115.26	= { by axiom 7 (converse_additivity) }
115.01/115.26	  converse(join(converse(composition(X, Z)), converse(composition(X, Y))))
115.01/115.26	= { by axiom 4 (converse_multiplicativity) }
115.01/115.26	  converse(join(composition(converse(Z), converse(X)), converse(composition(X, Y))))
115.01/115.26	= { by lemma 85 }
115.01/115.26	  converse(composition(join(converse(Z), converse(Y)), converse(X)))
115.01/115.26	= { by axiom 7 (converse_additivity) }
115.01/115.26	  converse(composition(converse(join(Z, Y)), converse(X)))
115.01/115.26	= { by axiom 4 (converse_multiplicativity) }
115.01/115.26	  converse(converse(composition(X, join(Z, Y))))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  converse(converse(composition(X, join(Y, Z))))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  composition(X, join(Y, Z))
115.01/115.26	
115.01/115.26	Lemma 87: complement(meet(X, meet(Y, complement(Z)))) = join(Z, complement(meet(X, Y))).
115.01/115.26	Proof:
115.01/115.26	  complement(meet(X, meet(Y, complement(Z))))
115.01/115.26	= { by lemma 67 }
115.01/115.26	  join(complement(X), complement(meet(Y, complement(Z))))
115.01/115.26	= { by lemma 68 }
115.01/115.26	  join(complement(X), join(Z, complement(Y)))
115.01/115.26	= { by lemma 80 }
115.01/115.26	  join(Z, complement(meet(X, Y)))
115.01/115.26	
115.01/115.26	Lemma 88: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
115.01/115.26	Proof:
115.01/115.26	  join(composition(X, Y), join(Z, composition(W, Y)))
115.01/115.26	= { by lemma 26 }
115.01/115.26	  join(Z, join(composition(W, Y), composition(X, Y)))
115.01/115.26	= { by axiom 12 (composition_distributivity) }
115.01/115.26	  join(Z, composition(join(W, X), Y))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  join(Z, composition(join(X, W), Y))
115.01/115.26	
115.01/115.26	Lemma 89: join(meet(X, Y), complement(meet(Y, Z))) = join(X, complement(meet(Y, Z))).
115.01/115.26	Proof:
115.01/115.26	  join(meet(X, Y), complement(meet(Y, Z)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  join(meet(X, Y), complement(meet(Z, Y)))
115.01/115.26	= { by lemma 87 }
115.01/115.26	  complement(meet(Z, meet(Y, complement(meet(X, Y)))))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  complement(meet(Z, meet(Y, complement(meet(Y, X)))))
115.01/115.26	= { by lemma 76 }
115.01/115.26	  complement(meet(Z, meet(Y, complement(X))))
115.01/115.26	= { by lemma 87 }
115.01/115.26	  join(X, complement(meet(Z, Y)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  join(X, complement(meet(Y, Z)))
115.01/115.26	
115.01/115.26	Lemma 90: converse(meet(one, complement(X))) = meet(one, converse(complement(X))).
115.01/115.26	Proof:
115.01/115.26	  converse(meet(one, complement(X)))
115.01/115.26	= { by lemma 70 }
115.01/115.26	  converse(complement(join(X, complement(one))))
115.01/115.26	= { by lemma 82 }
115.01/115.26	  complement(converse(join(X, complement(one))))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  complement(converse(join(complement(one), X)))
115.01/115.26	= { by axiom 7 (converse_additivity) }
115.01/115.26	  complement(join(converse(complement(one)), converse(X)))
115.01/115.26	= { by lemma 73 }
115.01/115.26	  complement(join(join(complement(one), converse(complement(one))), converse(X)))
115.01/115.26	= { by lemma 25 }
115.01/115.26	  complement(join(converse(join(complement(one), converse(complement(one)))), converse(X)))
115.01/115.26	= { by lemma 73 }
115.01/115.26	  complement(join(converse(converse(complement(one))), converse(X)))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  complement(join(complement(one), converse(X)))
115.01/115.26	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.26	  complement(join(converse(X), complement(one)))
115.01/115.26	= { by lemma 70 }
115.01/115.26	  meet(one, complement(converse(X)))
115.01/115.26	= { by lemma 82 }
115.01/115.26	  meet(one, converse(complement(X)))
115.01/115.26	
115.01/115.26	Lemma 91: converse(meet(X, one)) = meet(one, converse(X)).
115.01/115.26	Proof:
115.01/115.26	  converse(meet(X, one))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  converse(meet(one, X))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  converse(meet(one, complement(complement(X))))
115.01/115.26	= { by lemma 90 }
115.01/115.26	  meet(one, converse(complement(complement(X))))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  meet(one, converse(X))
115.01/115.26	
115.01/115.26	Lemma 92: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
115.01/115.26	Proof:
115.01/115.26	  converse(meet(X, converse(Y)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  converse(meet(converse(Y), X))
115.01/115.26	= { by lemma 50 }
115.01/115.26	  join(meet(converse(meet(converse(Y), X)), converse(converse(Y))), complement(join(converse(converse(Y)), complement(converse(meet(converse(Y), X))))))
115.01/115.26	= { by lemma 71 }
115.01/115.26	  join(meet(converse(meet(converse(Y), X)), converse(converse(Y))), complement(top))
115.01/115.26	= { by lemma 18 }
115.01/115.26	  join(meet(converse(meet(converse(Y), X)), converse(converse(Y))), zero)
115.01/115.26	= { by lemma 53 }
115.01/115.26	  meet(converse(meet(converse(Y), X)), converse(converse(Y)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  meet(converse(converse(Y)), converse(meet(converse(Y), X)))
115.01/115.26	= { by axiom 6 (converse_idempotence) }
115.01/115.26	  meet(Y, converse(meet(converse(Y), X)))
115.01/115.26	= { by lemma 19 }
115.01/115.26	  meet(Y, converse(meet(X, converse(Y))))
115.01/115.26	= { by lemma 56 }
115.01/115.26	  meet(Y, converse(meet(X, converse(complement(complement(Y))))))
115.01/115.26	= { by lemma 79 }
115.01/115.26	  meet(Y, join(complement(Y), converse(meet(X, converse(complement(complement(Y)))))))
115.01/115.26	= { by lemma 82 }
115.01/115.26	  meet(Y, join(complement(Y), converse(meet(X, complement(converse(complement(Y)))))))
115.01/115.26	= { by lemma 30 }
115.01/115.26	  meet(Y, converse(join(converse(complement(Y)), meet(X, complement(converse(complement(Y)))))))
115.01/115.26	= { by lemma 78 }
115.01/115.26	  meet(Y, converse(join(converse(complement(Y)), X)))
115.01/115.26	= { by lemma 30 }
115.01/115.26	  meet(Y, join(complement(Y), converse(X)))
115.01/115.26	= { by lemma 79 }
115.01/115.27	  meet(Y, converse(X))
115.01/115.27	
115.01/115.27	Lemma 93: composition(X, converse(join(complement(composition(top, X)), Y))) = composition(X, converse(Y)).
115.01/115.27	Proof:
115.01/115.27	  composition(X, converse(join(complement(composition(top, X)), Y)))
115.01/115.27	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.27	  composition(X, converse(join(Y, complement(composition(top, X)))))
115.01/115.27	= { by lemma 20 }
115.01/115.27	  converse(composition(join(Y, complement(composition(top, X))), converse(X)))
115.01/115.27	= { by axiom 6 (converse_idempotence) }
115.01/115.27	  converse(composition(join(Y, converse(converse(complement(composition(top, X))))), converse(X)))
115.01/115.27	= { by lemma 85 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(composition(X, converse(complement(composition(top, X)))))))
115.01/115.27	= { by lemma 54 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(zero, composition(X, converse(complement(composition(top, X))))))))
115.01/115.27	= { by lemma 49 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(converse(zero), composition(X, converse(complement(composition(top, X))))))))
115.01/115.27	= { by lemma 18 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(converse(complement(top)), composition(X, converse(complement(composition(top, X))))))))
115.01/115.27	= { by axiom 9 (maddux1_join_commutativity) }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(composition(X, converse(complement(composition(top, X)))), converse(complement(top))))))
115.01/115.27	= { by axiom 6 (converse_idempotence) }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(composition(converse(converse(X)), converse(complement(composition(top, X)))), converse(complement(top))))))
115.01/115.27	= { by lemma 82 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(composition(converse(converse(X)), complement(converse(composition(top, X)))), converse(complement(top))))))
115.01/115.27	= { by axiom 4 (converse_multiplicativity) }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(composition(converse(converse(X)), complement(composition(converse(X), converse(top)))), converse(complement(top))))))
115.01/115.27	= { by lemma 82 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(join(composition(converse(converse(X)), complement(composition(converse(X), converse(top)))), complement(converse(top))))))
115.01/115.27	= { by axiom 8 (converse_cancellativity) }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(complement(converse(top)))))
115.01/115.27	= { by lemma 82 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(converse(complement(top)))))
115.01/115.27	= { by lemma 18 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(converse(zero))))
115.01/115.27	= { by lemma 49 }
115.01/115.27	  converse(join(composition(Y, converse(X)), converse(zero)))
115.01/115.27	= { by lemma 49 }
115.01/115.27	  converse(join(composition(Y, converse(X)), zero))
115.01/115.27	= { by lemma 53 }
115.01/115.27	  converse(composition(Y, converse(X)))
115.01/115.27	= { by lemma 20 }
115.19/115.45	  composition(X, converse(Y))
115.19/115.45	
115.19/115.45	Lemma 94: meet(X, meet(one, converse(X))) = meet(X, one).
115.19/115.45	Proof:
115.19/115.45	  meet(X, meet(one, converse(X)))
115.19/115.45	= { by lemma 75 }
115.19/115.45	  meet(one, meet(X, converse(X)))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(one, meet(converse(X), X))
115.19/115.45	= { by lemma 74 }
115.19/115.45	  meet(meet(X, one), converse(X))
115.19/115.45	= { by lemma 77 }
115.19/115.45	  meet(meet(X, one), join(converse(X), complement(meet(X, one))))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(meet(X, one), join(converse(X), complement(meet(one, X))))
115.19/115.45	= { by lemma 89 }
115.19/115.45	  meet(meet(X, one), join(meet(converse(X), one), complement(meet(one, X))))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(meet(X, one), join(meet(one, converse(X)), complement(meet(one, X))))
115.19/115.45	= { by lemma 22 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(meet(one, X))))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(meet(X, one))))
115.19/115.45	= { by axiom 10 (composition_identity) }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(X, one), one))))
115.19/115.45	= { by lemma 23 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(X, one), converse(one)))))
115.19/115.45	= { by lemma 93 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(X, one), converse(join(complement(composition(top, meet(X, one))), one))))))
115.19/115.45	= { by lemma 78 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(X, one), converse(join(complement(composition(top, meet(X, one))), meet(one, complement(complement(composition(top, meet(X, one)))))))))))
115.19/115.45	= { by lemma 93 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(X, one), converse(meet(one, complement(complement(composition(top, meet(X, one))))))))))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(one, X), converse(meet(one, complement(complement(composition(top, meet(X, one))))))))))
115.19/115.45	= { by lemma 23 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, complement(complement(composition(top, meet(X, one))))))))))
115.19/115.45	= { by lemma 56 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, composition(top, meet(X, one))))))))
115.19/115.45	= { by lemma 79 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(top, meet(X, one)))))))))
115.19/115.45	= { by lemma 71 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(join(converse(one), complement(converse(meet(one, X)))), meet(X, one)))))))))
115.19/115.45	= { by lemma 23 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(join(one, complement(converse(meet(one, X)))), meet(X, one)))))))))
115.19/115.45	= { by lemma 82 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(one, X)))), meet(X, one)))))))))
115.19/115.45	= { by lemma 19 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(X, one)))), meet(X, one)))))))))
115.19/115.45	= { by lemma 56 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(X, one)))), complement(complement(meet(X, one)))))))))))
115.19/115.45	= { by lemma 88 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(complement(one), composition(converse(complement(meet(X, one))), complement(complement(meet(X, one))))))))))))
115.19/115.45	= { by axiom 9 (maddux1_join_commutativity) }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(composition(converse(complement(meet(X, one))), complement(complement(meet(X, one)))), complement(one)))))))))
115.19/115.45	= { by axiom 10 (composition_identity) }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(composition(converse(complement(meet(X, one))), complement(composition(complement(meet(X, one)), one))), complement(one)))))))))
115.19/115.45	= { by axiom 8 (converse_cancellativity) }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), complement(one))))))))
115.19/115.45	= { by axiom 9 (maddux1_join_commutativity) }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), composition(one, complement(complement(meet(X, one)))))))))))
115.19/115.45	= { by lemma 24 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), complement(complement(meet(X, one))))))))))
115.19/115.45	= { by lemma 67 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, complement(meet(one, complement(meet(X, one))))))))))
115.19/115.45	= { by lemma 68 }
115.19/115.45	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(meet(X, one), complement(one))))))))
115.19/115.45	= { by axiom 9 (maddux1_join_commutativity) }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), meet(X, one))))))))
115.19/115.46	= { by lemma 19 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(complement(one), meet(one, X))))))))
115.19/115.46	= { by axiom 9 (maddux1_join_commutativity) }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(meet(one, X), complement(one))))))))
115.19/115.46	= { by lemma 68 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, complement(meet(one, complement(meet(one, X))))))))))
115.19/115.46	= { by lemma 76 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, complement(meet(one, complement(X)))))))))
115.19/115.46	= { by lemma 68 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, join(X, complement(one))))))))
115.19/115.46	= { by lemma 77 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(one, X))))))
115.19/115.46	= { by lemma 19 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), converse(meet(X, one))))))
115.19/115.46	= { by lemma 91 }
115.19/115.46	  meet(meet(X, one), join(composition(converse(one), meet(one, converse(X))), complement(composition(meet(converse(one), X), meet(one, converse(X))))))
115.19/115.46	= { by axiom 9 (maddux1_join_commutativity) }
115.19/115.46	  meet(meet(X, one), join(complement(composition(meet(converse(one), X), meet(one, converse(X)))), composition(converse(one), meet(one, converse(X)))))
115.19/115.46	= { by axiom 11 (maddux4_definiton_of_meet) }
115.19/115.46	  meet(meet(X, one), join(complement(composition(complement(join(complement(converse(one)), complement(X))), meet(one, converse(X)))), composition(converse(one), meet(one, converse(X)))))
115.19/115.46	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
115.19/115.46	  meet(meet(X, one), join(complement(composition(complement(join(complement(converse(one)), complement(X))), meet(one, converse(X)))), composition(join(complement(join(complement(converse(one)), complement(X))), complement(join(complement(converse(one)), X))), meet(one, converse(X)))))
115.19/115.46	= { by lemma 84 }
115.19/115.46	  meet(meet(X, one), top)
115.19/115.46	= { by lemma 74 }
115.19/115.46	  meet(one, meet(top, X))
115.19/115.46	= { by lemma 58 }
115.19/115.46	  meet(one, X)
115.19/115.46	= { by lemma 19 }
115.19/115.46	  meet(X, one)
115.19/115.46	
115.19/115.46	Lemma 95: meet(one, converse(X)) = meet(X, one).
115.19/115.46	Proof:
115.19/115.46	  meet(one, converse(X))
115.19/115.46	= { by lemma 91 }
115.19/115.46	  converse(meet(X, one))
115.19/115.46	= { by lemma 94 }
115.19/115.46	  converse(meet(X, meet(one, converse(X))))
115.19/115.46	= { by lemma 75 }
115.19/115.46	  converse(meet(one, meet(X, converse(X))))
115.19/115.46	= { by lemma 19 }
115.19/115.46	  converse(meet(one, meet(converse(X), X)))
115.19/115.46	= { by lemma 74 }
115.19/115.46	  converse(meet(meet(X, one), converse(X)))
115.19/115.46	= { by lemma 92 }
115.19/115.46	  meet(X, converse(meet(X, one)))
115.19/115.46	= { by lemma 91 }
115.19/115.46	  meet(X, meet(one, converse(X)))
115.19/115.46	= { by lemma 94 }
115.28/115.49	  meet(X, one)
115.28/115.49	
115.28/115.49	Lemma 96: join(meet(X, Y), meet(Y, Z)) = meet(Y, join(X, Z)).
115.28/115.49	Proof:
115.28/115.49	  join(meet(X, Y), meet(Y, Z))
115.28/115.49	= { by lemma 56 }
115.28/115.49	  join(meet(X, Y), meet(Y, complement(complement(Z))))
115.28/115.49	= { by lemma 19 }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), Y))
115.28/115.49	= { by lemma 83 }
115.28/115.49	  join(meet(X, Y), meet(complement(meet(X, Y)), meet(complement(complement(Z)), Y)))
115.28/115.49	= { by lemma 75 }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), meet(complement(meet(X, Y)), Y)))
115.28/115.49	= { by lemma 77 }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), meet(complement(meet(X, Y)), join(Y, complement(complement(meet(X, Y)))))))
115.28/115.49	= { by lemma 75 }
115.28/115.49	  join(meet(X, Y), meet(complement(meet(X, Y)), meet(complement(complement(Z)), join(Y, complement(complement(meet(X, Y)))))))
115.28/115.49	= { by lemma 83 }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), join(Y, complement(complement(meet(X, Y))))))
115.28/115.49	= { by lemma 56 }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), join(Y, meet(X, Y))))
115.28/115.49	= { by axiom 9 (maddux1_join_commutativity) }
115.28/115.49	  join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y)))
115.28/115.49	= { by lemma 81 }
115.28/115.49	  meet(join(meet(complement(Z), join(meet(X, Y), Y)), join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y)))), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 27 }
115.28/115.49	  meet(join(meet(X, Y), join(meet(complement(Z), join(meet(X, Y), Y)), meet(complement(complement(Z)), join(meet(X, Y), Y)))), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 19 }
115.28/115.49	  meet(join(meet(X, Y), join(meet(complement(Z), join(meet(X, Y), Y)), meet(join(meet(X, Y), Y), complement(complement(Z))))), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 45 }
115.28/115.49	  meet(join(meet(X, Y), join(meet(X, Y), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 24 }
115.28/115.49	  meet(join(meet(X, Y), join(meet(X, Y), composition(one, Y))), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 57 }
115.28/115.49	  meet(join(meet(X, Y), join(meet(X, Y), composition(join(one, one), Y))), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.49	  meet(join(join(meet(X, Y), meet(X, Y)), composition(join(one, one), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 22 }
115.28/115.49	  meet(join(join(meet(X, Y), composition(converse(one), meet(X, Y))), composition(join(one, one), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 46 }
115.28/115.49	  meet(join(composition(join(converse(one), one), meet(X, Y)), composition(join(one, one), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 23 }
115.28/115.49	  meet(join(composition(join(one, one), meet(X, Y)), composition(join(one, one), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 86 }
115.28/115.49	  meet(composition(join(one, one), join(meet(X, Y), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 57 }
115.28/115.49	  meet(composition(one, join(meet(X, Y), Y)), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by lemma 24 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(join(meet(X, Y), meet(complement(complement(Z)), join(meet(X, Y), Y))), complement(meet(complement(Z), join(meet(X, Y), Y)))))
115.28/115.49	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), join(meet(complement(complement(Z)), join(meet(X, Y), Y)), complement(meet(complement(Z), join(meet(X, Y), Y))))))
115.28/115.49	= { by lemma 19 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), join(meet(complement(complement(Z)), join(meet(X, Y), Y)), complement(meet(join(meet(X, Y), Y), complement(Z))))))
115.28/115.49	= { by lemma 89 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), join(complement(complement(Z)), complement(meet(join(meet(X, Y), Y), complement(Z))))))
115.28/115.49	= { by lemma 67 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), complement(meet(complement(Z), meet(join(meet(X, Y), Y), complement(Z))))))
115.28/115.49	= { by lemma 72 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), complement(meet(join(meet(X, Y), Y), meet(complement(Z), complement(Z))))))
115.28/115.49	= { by lemma 34 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), complement(meet(join(meet(X, Y), Y), complement(complement(complement(Z)))))))
115.28/115.49	= { by lemma 56 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), complement(meet(join(meet(X, Y), Y), complement(Z)))))
115.28/115.49	= { by lemma 80 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(complement(join(meet(X, Y), Y)), join(meet(X, Y), complement(complement(Z)))))
115.28/115.49	= { by lemma 79 }
115.28/115.49	  meet(join(meet(X, Y), Y), join(meet(X, Y), complement(complement(Z))))
115.28/115.49	= { by axiom 9 (maddux1_join_commutativity) }
115.28/115.49	  meet(join(Y, meet(X, Y)), join(meet(X, Y), complement(complement(Z))))
115.28/115.49	= { by lemma 65 }
115.28/115.49	  meet(Y, join(meet(X, Y), complement(complement(Z))))
115.28/115.49	= { by lemma 19 }
115.28/115.49	  meet(Y, join(meet(Y, X), complement(complement(Z))))
115.28/115.49	= { by lemma 56 }
115.28/115.49	  meet(Y, join(meet(Y, X), Z))
115.28/115.49	= { by axiom 9 (maddux1_join_commutativity) }
115.28/115.49	  meet(Y, join(Z, meet(Y, X)))
115.28/115.49	= { by lemma 19 }
115.28/115.49	  meet(Y, join(Z, meet(X, Y)))
115.28/115.49	= { by lemma 56 }
115.28/115.49	  meet(Y, join(Z, meet(X, complement(complement(Y)))))
115.28/115.49	= { by lemma 79 }
115.28/115.49	  meet(Y, join(complement(Y), join(Z, meet(X, complement(complement(Y))))))
115.28/115.49	= { by axiom 9 (maddux1_join_commutativity) }
115.28/115.49	  meet(Y, join(complement(Y), join(meet(X, complement(complement(Y))), Z)))
115.28/115.49	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.49	  meet(Y, join(join(complement(Y), meet(X, complement(complement(Y)))), Z))
115.28/115.49	= { by lemma 78 }
115.28/115.49	  meet(Y, join(join(complement(Y), X), Z))
115.28/115.49	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.49	  meet(Y, join(complement(Y), join(X, Z)))
115.28/115.49	= { by lemma 79 }
115.28/115.50	  meet(Y, join(X, Z))
115.28/115.50	
115.28/115.50	Lemma 97: meet(Z, composition(X, meet(Y, composition(converse(X), Z)))) = meet(Z, composition(X, Y)).
115.28/115.50	Proof:
115.28/115.50	  meet(Z, composition(X, meet(Y, composition(converse(X), Z))))
115.28/115.50	= { by lemma 19 }
115.28/115.50	  meet(composition(X, meet(Y, composition(converse(X), Z))), Z)
115.28/115.50	= { by axiom 14 (modular_law_1) }
115.28/115.50	  join(meet(composition(X, Y), Z), meet(composition(X, meet(Y, composition(converse(X), Z))), Z))
115.28/115.50	= { by lemma 19 }
115.28/115.50	  join(meet(composition(X, Y), Z), meet(Z, composition(X, meet(Y, composition(converse(X), Z)))))
115.28/115.50	= { by lemma 96 }
115.28/115.50	  meet(Z, join(composition(X, Y), composition(X, meet(Y, composition(converse(X), Z)))))
115.28/115.50	= { by lemma 86 }
115.28/115.50	  meet(Z, composition(X, join(Y, meet(Y, composition(converse(X), Z)))))
115.28/115.50	= { by lemma 45 }
115.28/115.50	  meet(Z, composition(X, join(Y, meet(Y, composition(join(meet(?, converse(X)), meet(converse(X), complement(?))), Z)))))
115.28/115.50	= { by axiom 12 (composition_distributivity) }
115.28/115.50	  meet(Z, composition(X, join(Y, meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))))))
115.28/115.50	= { by axiom 9 (maddux1_join_commutativity) }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), Y)))
115.28/115.50	= { by lemma 62 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(Y, join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by lemma 27 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(Y, join(composition(meet(converse(X), complement(?)), Z), join(composition(meet(?, converse(X)), Z), complement(Y)))), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(Y, join(join(composition(meet(converse(X), complement(?)), Z), composition(meet(?, converse(X)), Z)), complement(Y))), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by lemma 28 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(join(composition(meet(converse(X), complement(?)), Z), composition(meet(?, converse(X)), Z)), top), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(composition(meet(converse(X), complement(?)), Z), join(composition(meet(?, converse(X)), Z), top)), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by lemma 37 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(join(composition(meet(converse(X), complement(?)), Z), top), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by lemma 37 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(top, join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y)))))))))
115.28/115.50	= { by lemma 58 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), join(Y, complement(join(composition(meet(?, converse(X)), Z), join(composition(meet(converse(X), complement(?)), Z), complement(Y))))))))
115.28/115.50	= { by axiom 13 (maddux2_join_associativity) }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), join(Y, complement(join(join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z)), complement(Y)))))))
115.28/115.50	= { by lemma 70 }
115.28/115.50	  meet(Z, composition(X, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), join(Y, meet(Y, complement(join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))))))))
115.28/115.50	= { by lemma 27 }
115.28/115.50	  meet(Z, composition(X, join(Y, join(meet(Y, join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))), meet(Y, complement(join(composition(meet(?, converse(X)), Z), composition(meet(converse(X), complement(?)), Z))))))))
115.28/115.50	= { by lemma 42 }
115.28/115.50	  meet(Z, composition(X, join(Y, Y)))
115.28/115.50	= { by lemma 57 }
115.31/115.59	  meet(Z, composition(X, Y))
115.31/115.59	
115.31/115.59	Goal 1 (goals_1): meet(sK2_goals_X0, sK1_goals_X1) = join(composition(meet(sK2_goals_X0, one), sK1_goals_X1), meet(sK2_goals_X0, sK1_goals_X1)).
115.31/115.59	Proof:
115.31/115.59	  meet(sK2_goals_X0, sK1_goals_X1)
115.31/115.59	= { by lemma 19 }
115.31/115.59	  meet(sK1_goals_X1, sK2_goals_X0)
115.31/115.59	= { by lemma 57 }
115.31/115.59	  meet(sK1_goals_X1, join(sK2_goals_X0, sK2_goals_X0))
115.31/115.59	= { by lemma 96 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK1_goals_X1, sK2_goals_X0))
115.31/115.59	= { by axiom 6 (converse_idempotence) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), meet(sK1_goals_X1, converse(converse(sK2_goals_X0))))
115.31/115.59	= { by lemma 92 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), converse(sK1_goals_X1))))
115.31/115.59	= { by axiom 10 (composition_identity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), one))))
115.31/115.59	= { by lemma 97 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(one, composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 19 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), one)))))
115.31/115.59	= { by lemma 50 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(converse(sK2_goals_X0), complement(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)))))), one)))))
115.31/115.59	= { by lemma 61 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(composition(top, converse(sK2_goals_X0)), complement(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)))))), one)))))
115.31/115.59	= { by lemma 60 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(composition(top, composition(top, converse(sK2_goals_X0))), complement(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)))))), one)))))
115.31/115.59	= { by lemma 60 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(composition(top, composition(top, converse(sK2_goals_X0))), complement(composition(converse(converse(sK1_goals_X1)), composition(top, converse(sK2_goals_X0))))))), one)))))
115.31/115.59	= { by axiom 9 (maddux1_join_commutativity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(complement(composition(converse(converse(sK1_goals_X1)), composition(top, converse(sK2_goals_X0)))), composition(top, composition(top, converse(sK2_goals_X0)))))), one)))))
115.31/115.59	= { by axiom 3 (def_top) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(join(complement(composition(converse(converse(sK1_goals_X1)), composition(top, converse(sK2_goals_X0)))), composition(join(converse(converse(sK1_goals_X1)), complement(converse(converse(sK1_goals_X1)))), composition(top, converse(sK2_goals_X0)))))), one)))))
115.31/115.59	= { by lemma 84 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), complement(top)), one)))))
115.31/115.59	= { by lemma 18 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(join(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), zero), one)))))
115.31/115.59	= { by lemma 53 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), converse(sK2_goals_X0)), one)))))
115.31/115.59	= { by lemma 19 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(meet(converse(sK2_goals_X0), composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0))), one)))))
115.31/115.59	= { by lemma 74 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)), meet(one, converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 19 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(meet(one, converse(sK2_goals_X0)), composition(converse(converse(sK1_goals_X1)), converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 97 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))))))
115.31/115.59	= { by lemma 19 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), converse(sK2_goals_X0))))
115.31/115.59	= { by lemma 61 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(top, converse(sK2_goals_X0)))))
115.31/115.59	= { by lemma 38 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(join(top, converse(sK1_goals_X1)), converse(sK2_goals_X0)))))
115.31/115.59	= { by axiom 12 (composition_distributivity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(top, converse(sK2_goals_X0)), composition(converse(sK1_goals_X1), converse(sK2_goals_X0))))))
115.31/115.59	= { by lemma 65 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(top, converse(sK2_goals_X0)), composition(converse(sK1_goals_X1), join(converse(sK2_goals_X0), meet(one, converse(sK2_goals_X0))))))))
115.31/115.59	= { by axiom 9 (maddux1_join_commutativity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(top, converse(sK2_goals_X0)), composition(converse(sK1_goals_X1), join(meet(one, converse(sK2_goals_X0)), converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 86 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(top, converse(sK2_goals_X0)), join(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(converse(sK1_goals_X1), converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 88 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(join(top, converse(sK1_goals_X1)), converse(sK2_goals_X0))))))
115.31/115.59	= { by axiom 9 (maddux1_join_commutativity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(join(converse(sK1_goals_X1), top), converse(sK2_goals_X0))))))
115.31/115.59	= { by lemma 37 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), composition(top, converse(sK2_goals_X0))))))
115.31/115.59	= { by lemma 61 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), converse(sK2_goals_X0)))))
115.31/115.59	= { by axiom 9 (maddux1_join_commutativity) }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(meet(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0))), join(converse(sK2_goals_X0), composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0)))))))
115.31/115.59	= { by lemma 64 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(composition(converse(sK1_goals_X1), meet(one, converse(sK2_goals_X0)))))
115.31/115.59	= { by lemma 95 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), converse(composition(converse(sK1_goals_X1), meet(sK2_goals_X0, one))))
115.31/115.59	= { by lemma 21 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(converse(meet(sK2_goals_X0, one)), sK1_goals_X1))
115.31/115.59	= { by lemma 19 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(converse(meet(one, sK2_goals_X0)), sK1_goals_X1))
115.31/115.59	= { by lemma 56 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(converse(meet(one, complement(complement(sK2_goals_X0)))), sK1_goals_X1))
115.31/115.59	= { by lemma 90 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(meet(one, converse(complement(complement(sK2_goals_X0)))), sK1_goals_X1))
115.31/115.59	= { by lemma 56 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(meet(one, converse(sK2_goals_X0)), sK1_goals_X1))
115.31/115.59	= { by lemma 95 }
115.31/115.59	  join(meet(sK2_goals_X0, sK1_goals_X1), composition(meet(sK2_goals_X0, one), sK1_goals_X1))
115.31/115.59	= { by axiom 9 (maddux1_join_commutativity) }
115.31/115.59	  join(composition(meet(sK2_goals_X0, one), sK1_goals_X1), meet(sK2_goals_X0, sK1_goals_X1))
115.31/115.59	% SZS output end Proof
115.31/115.59	
115.31/115.59	RESULT: Theorem (the conjecture is true).
115.41/115.63	EOF