0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n009.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 15:27:12 EDT 2019
0.12/0.33	% CPUTime    : 
59.18/59.37	% SZS status Theorem
59.18/59.37	
59.18/59.37	% SZS output start Proof
59.18/59.37	Take the following subset of the input axioms:
59.18/59.38	  fof(composition_associativity, axiom, ![X0, X1, X2]: composition(composition(X0, X1), X2)=composition(X0, composition(X1, X2))).
59.18/59.38	  fof(composition_distributivity, axiom, ![X0, X1, X2]: composition(join(X0, X1), X2)=join(composition(X0, X2), composition(X1, X2))).
59.18/59.38	  fof(composition_identity, axiom, ![X0]: X0=composition(X0, one)).
59.18/59.38	  fof(converse_additivity, axiom, ![X0, X1]: converse(join(X0, X1))=join(converse(X0), converse(X1))).
59.18/59.38	  fof(converse_cancellativity, axiom, ![X0, X1]: complement(X1)=join(composition(converse(X0), complement(composition(X0, X1))), complement(X1))).
59.18/59.38	  fof(converse_idempotence, axiom, ![X0]: X0=converse(converse(X0))).
59.18/59.38	  fof(converse_multiplicativity, axiom, ![X0, X1]: converse(composition(X0, X1))=composition(converse(X1), converse(X0))).
59.18/59.38	  fof(def_top, axiom, ![X0]: join(X0, complement(X0))=top).
59.18/59.38	  fof(def_zero, axiom, ![X0]: meet(X0, complement(X0))=zero).
59.18/59.38	  fof(goals, conjecture, ![X0]: (join(composition(converse(X0), X0), one)=one => ![X1]: zero=meet(composition(X0, X1), composition(X0, complement(X1))))).
59.18/59.38	  fof(maddux1_join_commutativity, axiom, ![X0, X1]: join(X1, X0)=join(X0, X1)).
59.18/59.38	  fof(maddux2_join_associativity, axiom, ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
59.18/59.38	  fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0, X1]: X0=join(complement(join(complement(X0), complement(X1))), complement(join(complement(X0), X1)))).
59.18/59.38	  fof(maddux4_definiton_of_meet, axiom, ![X0, X1]: meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
59.18/59.38	
59.18/59.38	Now clausify the problem and encode Horn clauses using encoding 3 of
59.18/59.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
59.18/59.38	We repeatedly replace C & s=t => u=v by the two clauses:
59.18/59.38	  fresh(y, y, x1...xn) = u
59.18/59.38	  C => fresh(s, t, x1...xn) = v
59.18/59.38	where fresh is a fresh function symbol and x1..xn are the free
59.18/59.38	variables of u and v.
59.18/59.38	A predicate p(X) is encoded as p(X)=true (this is sound, because the
59.18/59.38	input problem has no model of domain size 1).
59.18/59.38	
59.18/59.38	The encoding turns the above axioms into the following unit equations and goals:
59.18/59.38	
59.18/59.38	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
59.18/59.38	Axiom 2 (composition_identity): X = composition(X, one).
59.18/59.38	Axiom 3 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
59.18/59.38	Axiom 4 (def_zero): meet(X, complement(X)) = zero.
59.18/59.38	Axiom 5 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
59.18/59.38	Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
59.18/59.38	Axiom 7 (def_top): join(X, complement(X)) = top.
59.18/59.38	Axiom 8 (composition_associativity): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
59.18/59.38	Axiom 9 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
59.18/59.38	Axiom 10 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
59.18/59.38	Axiom 11 (converse_idempotence): X = converse(converse(X)).
59.18/59.38	Axiom 12 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
59.18/59.38	Axiom 13 (converse_cancellativity): complement(X) = join(composition(converse(Y), complement(composition(Y, X))), complement(X)).
59.76/59.95	Axiom 14 (goals): join(composition(converse(sK2_goals_X0), sK2_goals_X0), one) = one.
59.76/59.95	
59.76/59.95	Lemma 15: meet(X, Y) = meet(Y, X).
59.76/59.95	Proof:
59.76/59.95	  meet(X, Y)
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  complement(join(complement(X), complement(Y)))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  complement(join(complement(Y), complement(X)))
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  meet(Y, X)
59.76/59.95	
59.76/59.95	Lemma 16: complement(top) = zero.
59.76/59.95	Proof:
59.76/59.95	  complement(top)
59.76/59.95	= { by axiom 7 (def_top) }
59.76/59.95	  complement(join(complement(?), complement(complement(?))))
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  meet(?, complement(?))
59.76/59.95	= { by axiom 4 (def_zero) }
59.76/59.95	  zero
59.76/59.95	
59.76/59.95	Lemma 17: complement(join(zero, complement(X))) = meet(X, top).
59.76/59.95	Proof:
59.76/59.95	  complement(join(zero, complement(X)))
59.76/59.95	= { by lemma 16 }
59.76/59.95	  complement(join(complement(top), complement(X)))
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  meet(top, X)
59.76/59.95	= { by lemma 15 }
59.76/59.95	  meet(X, top)
59.76/59.95	
59.76/59.95	Lemma 18: converse(composition(converse(X), Y)) = composition(converse(Y), X).
59.76/59.95	Proof:
59.76/59.95	  converse(composition(converse(X), Y))
59.76/59.95	= { by axiom 3 (converse_multiplicativity) }
59.76/59.95	  composition(converse(Y), converse(converse(X)))
59.76/59.95	= { by axiom 11 (converse_idempotence) }
59.76/59.95	  composition(converse(Y), X)
59.76/59.95	
59.76/59.95	Lemma 19: composition(converse(one), X) = X.
59.76/59.95	Proof:
59.76/59.95	  composition(converse(one), X)
59.76/59.95	= { by lemma 18 }
59.76/59.95	  converse(composition(converse(X), one))
59.76/59.95	= { by axiom 2 (composition_identity) }
59.76/59.95	  converse(converse(X))
59.76/59.95	= { by axiom 11 (converse_idempotence) }
59.76/59.95	  X
59.76/59.95	
59.76/59.95	Lemma 20: composition(one, X) = X.
59.76/59.95	Proof:
59.76/59.95	  composition(one, X)
59.76/59.95	= { by lemma 19 }
59.76/59.95	  composition(converse(one), composition(one, X))
59.76/59.95	= { by axiom 8 (composition_associativity) }
59.76/59.95	  composition(composition(converse(one), one), X)
59.76/59.95	= { by axiom 2 (composition_identity) }
59.76/59.95	  composition(converse(one), X)
59.76/59.95	= { by lemma 19 }
59.76/59.95	  X
59.76/59.95	
59.76/59.95	Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
59.76/59.95	Proof:
59.76/59.95	  join(complement(X), composition(converse(Y), complement(composition(Y, X))))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(composition(converse(Y), complement(composition(Y, X))), complement(X))
59.76/59.95	= { by axiom 13 (converse_cancellativity) }
59.76/59.95	  complement(X)
59.76/59.95	
59.76/59.95	Lemma 22: join(complement(X), complement(X)) = complement(X).
59.76/59.95	Proof:
59.76/59.95	  join(complement(X), complement(X))
59.76/59.95	= { by lemma 19 }
59.76/59.95	  join(complement(X), composition(converse(one), complement(X)))
59.76/59.95	= { by lemma 20 }
59.76/59.95	  join(complement(X), composition(converse(one), complement(composition(one, X))))
59.76/59.95	= { by lemma 21 }
59.76/59.95	  complement(X)
59.76/59.95	
59.76/59.95	Lemma 23: meet(X, X) = complement(complement(X)).
59.76/59.95	Proof:
59.76/59.95	  meet(X, X)
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  complement(join(complement(X), complement(X)))
59.76/59.95	= { by lemma 22 }
59.76/59.95	  complement(complement(X))
59.76/59.95	
59.76/59.95	Lemma 24: join(meet(X, Y), meet(X, complement(Y))) = X.
59.76/59.95	Proof:
59.76/59.95	  join(meet(X, Y), meet(X, complement(Y)))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(meet(X, complement(Y)), meet(X, Y))
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
59.76/59.95	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.95	  X
59.76/59.95	
59.76/59.95	Lemma 25: join(zero, complement(complement(X))) = X.
59.76/59.95	Proof:
59.76/59.95	  join(zero, complement(complement(X)))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(complement(complement(X)), zero)
59.76/59.95	= { by lemma 23 }
59.76/59.95	  join(meet(X, X), zero)
59.76/59.95	= { by axiom 4 (def_zero) }
59.76/59.95	  join(meet(X, X), meet(X, complement(X)))
59.76/59.95	= { by lemma 24 }
59.76/59.95	  X
59.76/59.95	
59.76/59.95	Lemma 26: join(X, join(Y, Z)) = join(Z, join(X, Y)).
59.76/59.95	Proof:
59.76/59.95	  join(X, join(Y, Z))
59.76/59.95	= { by axiom 12 (maddux2_join_associativity) }
59.76/59.95	  join(join(X, Y), Z)
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(Z, join(X, Y))
59.76/59.95	
59.76/59.95	Lemma 27: join(X, join(complement(X), Y)) = join(Y, top).
59.76/59.95	Proof:
59.76/59.95	  join(X, join(complement(X), Y))
59.76/59.95	= { by lemma 26 }
59.76/59.95	  join(complement(X), join(Y, X))
59.76/59.95	= { by lemma 26 }
59.76/59.95	  join(Y, join(X, complement(X)))
59.76/59.95	= { by axiom 7 (def_top) }
59.76/59.95	  join(Y, top)
59.76/59.95	
59.76/59.95	Lemma 28: join(X, join(Y, complement(X))) = join(Y, top).
59.76/59.95	Proof:
59.76/59.95	  join(X, join(Y, complement(X)))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(X, join(complement(X), Y))
59.76/59.95	= { by axiom 12 (maddux2_join_associativity) }
59.76/59.95	  join(join(X, complement(X)), Y)
59.76/59.95	= { by axiom 7 (def_top) }
59.76/59.95	  join(top, Y)
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(Y, top)
59.76/59.95	
59.76/59.95	Lemma 29: join(X, top) = top.
59.76/59.95	Proof:
59.76/59.95	  join(X, top)
59.76/59.95	= { by axiom 7 (def_top) }
59.76/59.95	  join(X, join(complement(X), complement(complement(X))))
59.76/59.95	= { by lemma 27 }
59.76/59.95	  join(complement(complement(X)), top)
59.76/59.95	= { by lemma 28 }
59.76/59.95	  join(complement(X), join(complement(complement(X)), complement(complement(X))))
59.76/59.95	= { by lemma 22 }
59.76/59.95	  join(complement(X), complement(complement(X)))
59.76/59.95	= { by axiom 7 (def_top) }
59.76/59.95	  top
59.76/59.95	
59.76/59.95	Lemma 30: join(zero, meet(X, top)) = X.
59.76/59.95	Proof:
59.76/59.95	  join(zero, meet(X, top))
59.76/59.95	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.95	  join(meet(X, top), zero)
59.76/59.95	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.95	  join(complement(join(complement(X), complement(top))), zero)
59.76/59.96	= { by lemma 16 }
59.76/59.96	  join(complement(join(complement(X), complement(top))), complement(top))
59.76/59.96	= { by lemma 29 }
59.76/59.96	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 31: join(zero, complement(X)) = complement(X).
59.76/59.96	Proof:
59.76/59.96	  join(zero, complement(X))
59.76/59.96	= { by lemma 25 }
59.76/59.96	  join(zero, complement(join(zero, complement(complement(X)))))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  join(zero, meet(complement(X), top))
59.76/59.96	= { by lemma 30 }
59.76/59.96	  complement(X)
59.76/59.96	
59.76/59.96	Lemma 32: complement(complement(X)) = X.
59.76/59.96	Proof:
59.76/59.96	  complement(complement(X))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  join(zero, complement(complement(X)))
59.76/59.96	= { by lemma 25 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 33: join(meet(Y, X), meet(X, complement(Y))) = X.
59.76/59.96	Proof:
59.76/59.96	  join(meet(Y, X), meet(X, complement(Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  join(meet(X, Y), meet(X, complement(Y)))
59.76/59.96	= { by lemma 24 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 34: join(X, zero) = X.
59.76/59.96	Proof:
59.76/59.96	  join(X, zero)
59.76/59.96	= { by lemma 32 }
59.76/59.96	  join(complement(complement(X)), zero)
59.76/59.96	= { by lemma 23 }
59.76/59.96	  join(meet(X, X), zero)
59.76/59.96	= { by axiom 4 (def_zero) }
59.76/59.96	  join(meet(X, X), meet(X, complement(X)))
59.76/59.96	= { by lemma 33 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 35: converse(join(X, converse(Y))) = join(Y, converse(X)).
59.76/59.96	Proof:
59.76/59.96	  converse(join(X, converse(Y)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  converse(join(converse(Y), X))
59.76/59.96	= { by axiom 6 (converse_additivity) }
59.76/59.96	  join(converse(converse(Y)), converse(X))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  join(Y, converse(X))
59.76/59.96	
59.76/59.96	Lemma 36: converse(join(converse(X), Y)) = join(X, converse(Y)).
59.76/59.96	Proof:
59.76/59.96	  converse(join(converse(X), Y))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  converse(join(Y, converse(X)))
59.76/59.96	= { by lemma 35 }
59.76/59.96	  join(X, converse(Y))
59.76/59.96	
59.76/59.96	Lemma 37: join(X, converse(top)) = converse(top).
59.76/59.96	Proof:
59.76/59.96	  join(X, converse(top))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  converse(join(converse(X), top))
59.76/59.96	= { by lemma 29 }
59.76/59.96	  converse(top)
59.76/59.96	
59.76/59.96	Lemma 38: converse(top) = top.
59.76/59.96	Proof:
59.76/59.96	  converse(top)
59.76/59.96	= { by lemma 37 }
59.76/59.96	  join(?, converse(top))
59.76/59.96	= { by lemma 37 }
59.76/59.96	  join(?, join(complement(?), converse(top)))
59.76/59.96	= { by lemma 27 }
59.76/59.96	  join(converse(top), top)
59.76/59.96	= { by lemma 29 }
59.76/59.96	  top
59.76/59.96	
59.76/59.96	Lemma 39: converse(composition(X, top)) = composition(top, converse(X)).
59.76/59.96	Proof:
59.76/59.96	  converse(composition(X, top))
59.76/59.96	= { by axiom 3 (converse_multiplicativity) }
59.76/59.96	  composition(converse(top), converse(X))
59.76/59.96	= { by lemma 38 }
59.76/59.96	  composition(top, converse(X))
59.76/59.96	
59.76/59.96	Lemma 40: meet(X, top) = X.
59.76/59.96	Proof:
59.76/59.96	  meet(X, top)
59.76/59.96	= { by lemma 17 }
59.76/59.96	  complement(join(zero, complement(X)))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  join(zero, complement(join(zero, complement(X))))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  join(zero, meet(X, top))
59.76/59.96	= { by lemma 30 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 41: join(zero, X) = X.
59.76/59.96	Proof:
59.76/59.96	  join(zero, X)
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  join(X, zero)
59.76/59.96	= { by lemma 34 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 42: meet(top, X) = X.
59.76/59.96	Proof:
59.76/59.96	  meet(top, X)
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(X, top)
59.76/59.96	= { by lemma 40 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 43: meet(X, join(complement(Y), complement(Z))) = complement(join(complement(X), meet(Y, Z))).
59.76/59.96	Proof:
59.76/59.96	  meet(X, join(complement(Y), complement(Z)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  meet(X, join(complement(Z), complement(Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(join(complement(Z), complement(Y)), X)
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  complement(join(meet(Z, Y), complement(X)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  complement(join(complement(X), meet(Z, Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  complement(join(complement(X), meet(Y, Z)))
59.76/59.96	
59.76/59.96	Lemma 44: complement(join(X, complement(Y))) = meet(Y, complement(X)).
59.76/59.96	Proof:
59.76/59.96	  complement(join(X, complement(Y)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  complement(join(complement(Y), X))
59.76/59.96	= { by lemma 42 }
59.76/59.96	  complement(join(complement(Y), meet(top, X)))
59.76/59.96	= { by lemma 43 }
59.76/59.96	  meet(Y, join(complement(top), complement(X)))
59.76/59.96	= { by lemma 16 }
59.76/59.96	  meet(Y, join(zero, complement(X)))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  meet(Y, complement(X))
59.76/59.96	
59.76/59.96	Lemma 45: complement(meet(Y, complement(X))) = join(X, complement(Y)).
59.76/59.96	Proof:
59.76/59.96	  complement(meet(Y, complement(X)))
59.76/59.96	= { by lemma 41 }
59.76/59.96	  complement(join(zero, meet(Y, complement(X))))
59.76/59.96	= { by lemma 44 }
59.76/59.96	  complement(join(zero, complement(join(X, complement(Y)))))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  meet(join(X, complement(Y)), top)
59.76/59.96	= { by lemma 40 }
59.76/59.96	  join(X, complement(Y))
59.76/59.96	
59.76/59.96	Lemma 46: join(zero, meet(X, X)) = X.
59.76/59.96	Proof:
59.76/59.96	  join(zero, meet(X, X))
59.76/59.96	= { by lemma 16 }
59.76/59.96	  join(complement(top), meet(X, X))
59.76/59.96	= { by axiom 7 (def_top) }
59.76/59.96	  join(complement(join(complement(X), complement(complement(X)))), meet(X, X))
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  join(complement(join(complement(X), complement(complement(X)))), complement(join(complement(X), complement(X))))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 47: join(complement(X), complement(Y)) = complement(meet(X, Y)).
59.76/59.96	Proof:
59.76/59.96	  join(complement(X), complement(Y))
59.76/59.96	= { by lemma 46 }
59.76/59.96	  join(zero, meet(join(complement(X), complement(Y)), join(complement(X), complement(Y))))
59.76/59.96	= { by lemma 43 }
59.76/59.96	  join(zero, complement(join(complement(join(complement(X), complement(Y))), meet(X, Y))))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  complement(join(complement(join(complement(X), complement(Y))), meet(X, Y)))
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  complement(join(meet(X, Y), meet(X, Y)))
59.76/59.96	= { by lemma 46 }
59.76/59.96	  complement(join(join(zero, meet(meet(X, Y), meet(X, Y))), meet(X, Y)))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  complement(join(join(zero, meet(meet(X, Y), meet(X, Y))), complement(complement(meet(X, Y)))))
59.76/59.96	= { by axiom 12 (maddux2_join_associativity) }
59.76/59.96	  complement(join(zero, join(meet(meet(X, Y), meet(X, Y)), complement(complement(meet(X, Y))))))
59.76/59.96	= { by lemma 23 }
59.76/59.96	  complement(join(zero, join(complement(complement(meet(X, Y))), complement(complement(meet(X, Y))))))
59.76/59.96	= { by lemma 22 }
59.76/59.96	  complement(join(zero, complement(complement(meet(X, Y)))))
59.76/59.96	= { by lemma 25 }
59.76/59.96	  complement(meet(X, Y))
59.76/59.96	
59.76/59.96	Lemma 48: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
59.76/59.96	Proof:
59.76/59.96	  meet(join(X, complement(Y)), complement(meet(X, Y)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  meet(join(complement(Y), X), complement(meet(X, Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(join(complement(Y), X), complement(meet(Y, X)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(complement(meet(Y, X)), join(complement(Y), X))
59.76/59.96	= { by lemma 47 }
59.76/59.96	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  complement(Y)
59.76/59.96	
59.76/59.96	Lemma 49: meet(join(X, Y), join(X, complement(Y))) = X.
59.76/59.96	Proof:
59.76/59.96	  meet(join(X, Y), join(X, complement(Y)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  meet(join(Y, X), join(X, complement(Y)))
59.76/59.96	= { by lemma 40 }
59.76/59.96	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
59.76/59.96	= { by lemma 45 }
59.76/59.96	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, complement(X))))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, join(zero, complement(X)))))
59.76/59.96	= { by lemma 48 }
59.76/59.96	  complement(join(zero, complement(X)))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  meet(X, top)
59.76/59.96	= { by lemma 40 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 50: join(X, converse(complement(converse(X)))) = top.
59.76/59.96	Proof:
59.76/59.96	  join(X, converse(complement(converse(X))))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  converse(join(converse(X), complement(converse(X))))
59.76/59.96	= { by axiom 7 (def_top) }
59.76/59.96	  converse(top)
59.76/59.96	= { by lemma 38 }
59.76/59.96	  top
59.76/59.96	
59.76/59.96	Lemma 51: complement(join(complement(X), Y)) = meet(X, complement(Y)).
59.76/59.96	Proof:
59.76/59.96	  complement(join(complement(X), Y))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  complement(join(Y, complement(X)))
59.76/59.96	= { by lemma 44 }
59.76/59.96	  meet(X, complement(Y))
59.76/59.96	
59.76/59.96	Lemma 52: complement(meet(complement(X), Y)) = join(X, complement(Y)).
59.76/59.96	Proof:
59.76/59.96	  complement(meet(complement(X), Y))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  complement(meet(Y, complement(X)))
59.76/59.96	= { by lemma 45 }
59.76/59.96	  join(X, complement(Y))
59.76/59.96	
59.76/59.96	Lemma 53: join(complement(converse(X)), converse(join(X, Y))) = top.
59.76/59.96	Proof:
59.76/59.96	  join(complement(converse(X)), converse(join(X, Y)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  join(complement(converse(X)), converse(join(Y, X)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  join(converse(join(Y, X)), complement(converse(X)))
59.76/59.96	= { by axiom 6 (converse_additivity) }
59.76/59.96	  join(join(converse(Y), converse(X)), complement(converse(X)))
59.76/59.96	= { by axiom 12 (maddux2_join_associativity) }
59.76/59.96	  join(converse(Y), join(converse(X), complement(converse(X))))
59.76/59.96	= { by axiom 7 (def_top) }
59.76/59.96	  join(converse(Y), top)
59.76/59.96	= { by lemma 29 }
59.76/59.96	  top
59.76/59.96	
59.76/59.96	Lemma 54: meet(converse(X), converse(join(X, Y))) = converse(X).
59.76/59.96	Proof:
59.76/59.96	  meet(converse(X), converse(join(X, Y)))
59.76/59.96	= { by lemma 34 }
59.76/59.96	  join(meet(converse(X), converse(join(X, Y))), zero)
59.76/59.96	= { by axiom 5 (maddux4_definiton_of_meet) }
59.76/59.96	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), zero)
59.76/59.96	= { by lemma 16 }
59.76/59.96	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(top))
59.76/59.96	= { by lemma 53 }
59.76/59.96	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(complement(converse(X)), converse(join(X, Y)))))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  converse(X)
59.76/59.96	
59.76/59.96	Lemma 55: meet(X, join(X, Y)) = X.
59.76/59.96	Proof:
59.76/59.96	  meet(X, join(X, Y))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  meet(converse(converse(X)), join(X, Y))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  meet(converse(converse(X)), converse(converse(join(X, Y))))
59.76/59.96	= { by axiom 6 (converse_additivity) }
59.76/59.96	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
59.76/59.96	= { by lemma 54 }
59.76/59.96	  converse(converse(X))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 56: join(X, meet(X, complement(Y))) = X.
59.76/59.96	Proof:
59.76/59.96	  join(X, meet(X, complement(Y)))
59.76/59.96	= { by lemma 51 }
59.76/59.96	  join(X, complement(join(complement(X), Y)))
59.76/59.96	= { by lemma 52 }
59.76/59.96	  complement(meet(complement(X), join(complement(X), Y)))
59.76/59.96	= { by lemma 55 }
59.76/59.96	  complement(complement(X))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 57: meet(X, join(Y, X)) = X.
59.76/59.96	Proof:
59.76/59.96	  meet(X, join(Y, X))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  meet(X, join(X, Y))
59.76/59.96	= { by lemma 55 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 58: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
59.76/59.96	Proof:
59.76/59.96	  meet(X, complement(meet(X, Y)))
59.76/59.96	= { by lemma 57 }
59.76/59.96	  meet(X, complement(meet(X, meet(Y, join(complement(X), Y)))))
59.76/59.96	= { by lemma 51 }
59.76/59.96	  complement(join(complement(X), meet(X, meet(Y, join(complement(X), Y)))))
59.76/59.96	= { by lemma 43 }
59.76/59.96	  meet(X, join(complement(X), complement(meet(Y, join(complement(X), Y)))))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), complement(meet(Y, join(complement(X), Y)))))
59.76/59.96	= { by lemma 47 }
59.76/59.96	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
59.76/59.96	= { by axiom 12 (maddux2_join_associativity) }
59.76/59.96	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
59.76/59.96	= { by lemma 52 }
59.76/59.96	  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))))
59.76/59.96	= { by lemma 48 }
59.76/59.96	  complement(join(complement(X), Y))
59.76/59.96	= { by lemma 51 }
59.76/59.96	  meet(X, complement(Y))
59.76/59.96	
59.76/59.96	Lemma 59: meet(X, join(Y, complement(X))) = meet(X, Y).
59.76/59.96	Proof:
59.76/59.96	  meet(X, join(Y, complement(X)))
59.76/59.96	= { by lemma 56 }
59.76/59.96	  meet(join(X, meet(X, complement(join(zero, complement(Y))))), join(Y, complement(X)))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  meet(join(X, meet(X, meet(Y, top))), join(Y, complement(X)))
59.76/59.96	= { by lemma 40 }
59.76/59.96	  meet(join(X, meet(X, Y)), join(Y, complement(X)))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  meet(join(X, complement(complement(meet(X, Y)))), join(Y, complement(X)))
59.76/59.96	= { by lemma 45 }
59.76/59.96	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(Y))))
59.76/59.96	= { by lemma 58 }
59.76/59.96	  meet(join(X, complement(complement(meet(X, Y)))), complement(meet(X, complement(meet(X, Y)))))
59.76/59.96	= { by lemma 48 }
59.76/59.96	  complement(complement(meet(X, Y)))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  meet(X, Y)
59.76/59.96	
59.76/59.96	Lemma 60: meet(X, join(complement(X), Y)) = meet(X, Y).
59.76/59.96	Proof:
59.76/59.96	  meet(X, join(complement(X), Y))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  meet(X, join(Y, complement(X)))
59.76/59.96	= { by lemma 59 }
59.76/59.96	  meet(X, Y)
59.76/59.96	
59.76/59.96	Lemma 61: complement(converse(complement(X))) = converse(X).
59.76/59.96	Proof:
59.76/59.96	  complement(converse(complement(X)))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  converse(converse(complement(converse(complement(X)))))
59.76/59.96	= { by lemma 49 }
59.76/59.96	  converse(meet(join(converse(complement(converse(complement(X)))), X), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  converse(meet(join(X, converse(complement(converse(complement(X))))), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  converse(meet(join(X, converse(complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  converse(meet(converse(join(converse(X), complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by lemma 42 }
59.76/59.96	  converse(meet(converse(meet(top, join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by lemma 50 }
59.76/59.96	  converse(meet(converse(meet(join(converse(X), converse(complement(converse(converse(X))))), join(converse(X), complement(converse(complement(converse(converse(X)))))))), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by lemma 49 }
59.76/59.96	  converse(meet(converse(converse(X)), join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  converse(meet(X, join(converse(complement(converse(complement(X)))), complement(X))))
59.76/59.96	= { by lemma 59 }
59.76/59.96	  converse(meet(X, converse(complement(converse(complement(X))))))
59.76/59.96	= { by lemma 60 }
59.76/59.96	  converse(meet(X, join(complement(X), converse(complement(converse(complement(X)))))))
59.76/59.96	= { by lemma 50 }
59.76/59.96	  converse(meet(X, top))
59.76/59.96	= { by lemma 40 }
59.76/59.96	  converse(X)
59.76/59.96	
59.76/59.96	Lemma 62: converse(complement(X)) = complement(converse(X)).
59.76/59.96	Proof:
59.76/59.96	  converse(complement(X))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  converse(join(zero, complement(X)))
59.76/59.96	= { by lemma 61 }
59.76/59.96	  complement(converse(complement(join(zero, complement(X)))))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  complement(converse(meet(X, top)))
59.76/59.96	= { by lemma 40 }
59.76/59.96	  complement(converse(X))
59.76/59.96	
59.76/59.96	Lemma 63: converse(one) = one.
59.76/59.96	Proof:
59.76/59.96	  converse(one)
59.76/59.96	= { by axiom 2 (composition_identity) }
59.76/59.96	  composition(converse(one), one)
59.76/59.96	= { by lemma 19 }
59.76/59.96	  one
59.76/59.96	
59.76/59.96	Lemma 64: converse(composition(top, X)) = composition(converse(X), top).
59.76/59.96	Proof:
59.76/59.96	  converse(composition(top, X))
59.76/59.96	= { by axiom 3 (converse_multiplicativity) }
59.76/59.96	  composition(converse(X), converse(top))
59.76/59.96	= { by lemma 38 }
59.76/59.96	  composition(converse(X), top)
59.76/59.96	
59.76/59.96	Lemma 65: join(meet(Y, X), meet(complement(Y), X)) = X.
59.76/59.96	Proof:
59.76/59.96	  join(meet(Y, X), meet(complement(Y), X))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  join(meet(Y, X), meet(X, complement(Y)))
59.76/59.96	= { by lemma 33 }
59.76/59.96	  X
59.76/59.96	
59.76/59.96	Lemma 66: join(X, meet(Y, complement(X))) = join(X, Y).
59.76/59.96	Proof:
59.76/59.96	  join(X, meet(Y, complement(X)))
59.76/59.96	= { by lemma 57 }
59.76/59.96	  join(meet(X, join(Y, X)), meet(Y, complement(X)))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  join(meet(X, join(Y, complement(complement(X)))), meet(Y, complement(X)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  join(meet(X, join(Y, complement(complement(X)))), meet(complement(X), Y))
59.76/59.96	= { by lemma 59 }
59.76/59.96	  join(meet(X, join(Y, complement(complement(X)))), meet(complement(X), join(Y, complement(complement(X)))))
59.76/59.96	= { by lemma 65 }
59.76/59.96	  join(Y, complement(complement(X)))
59.76/59.96	= { by lemma 32 }
59.76/59.96	  join(Y, X)
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  join(X, Y)
59.76/59.96	
59.76/59.96	Lemma 67: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
59.76/59.96	Proof:
59.76/59.96	  converse(meet(X, converse(Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  converse(meet(converse(Y), X))
59.76/59.96	= { by lemma 61 }
59.76/59.96	  complement(converse(complement(meet(converse(Y), X))))
59.76/59.96	= { by lemma 47 }
59.76/59.96	  complement(converse(join(complement(converse(Y)), complement(X))))
59.76/59.96	= { by lemma 62 }
59.76/59.96	  converse(complement(join(complement(converse(Y)), complement(X))))
59.76/59.96	= { by lemma 54 }
59.76/59.96	  meet(converse(complement(join(complement(converse(Y)), complement(X)))), converse(join(complement(join(complement(converse(Y)), complement(X))), complement(join(complement(converse(Y)), X)))))
59.76/59.96	= { by lemma 62 }
59.76/59.96	  meet(complement(converse(join(complement(converse(Y)), complement(X)))), converse(join(complement(join(complement(converse(Y)), complement(X))), complement(join(complement(converse(Y)), X)))))
59.76/59.96	= { by lemma 47 }
59.76/59.96	  meet(complement(converse(complement(meet(converse(Y), X)))), converse(join(complement(join(complement(converse(Y)), complement(X))), complement(join(complement(converse(Y)), X)))))
59.76/59.96	= { by lemma 61 }
59.76/59.96	  meet(converse(meet(converse(Y), X)), converse(join(complement(join(complement(converse(Y)), complement(X))), complement(join(complement(converse(Y)), X)))))
59.76/59.96	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.76/59.96	  meet(converse(meet(converse(Y), X)), converse(converse(Y)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(converse(converse(Y)), converse(meet(converse(Y), X)))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  meet(Y, converse(meet(converse(Y), X)))
59.76/59.96	= { by lemma 15 }
59.76/59.96	  meet(Y, converse(meet(X, converse(Y))))
59.76/59.96	= { by lemma 61 }
59.76/59.96	  meet(Y, converse(meet(X, complement(converse(complement(Y))))))
59.76/59.96	= { by lemma 60 }
59.76/59.96	  meet(Y, join(complement(Y), converse(meet(X, complement(converse(complement(Y)))))))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  meet(Y, converse(join(converse(complement(Y)), meet(X, complement(converse(complement(Y)))))))
59.76/59.96	= { by lemma 66 }
59.76/59.96	  meet(Y, converse(join(converse(complement(Y)), X)))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  meet(Y, join(complement(Y), converse(X)))
59.76/59.96	= { by lemma 60 }
59.76/59.96	  meet(Y, converse(X))
59.76/59.96	
59.76/59.96	Lemma 68: meet(one, converse(X)) = converse(meet(X, one)).
59.76/59.96	Proof:
59.76/59.96	  meet(one, converse(X))
59.76/59.96	= { by lemma 67 }
59.76/59.96	  converse(meet(X, converse(one)))
59.76/59.96	= { by lemma 63 }
59.76/59.96	  converse(meet(X, one))
59.76/59.96	
59.76/59.96	Lemma 69: composition(join(X, one), Y) = join(Y, composition(X, Y)).
59.76/59.96	Proof:
59.76/59.96	  composition(join(X, one), Y)
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  composition(join(one, X), Y)
59.76/59.96	= { by axiom 9 (composition_distributivity) }
59.76/59.96	  join(composition(one, Y), composition(X, Y))
59.76/59.96	= { by lemma 20 }
59.76/59.96	  join(Y, composition(X, Y))
59.76/59.96	
59.76/59.96	Lemma 70: join(complement(one), composition(converse(X), complement(X))) = complement(one).
59.76/59.96	Proof:
59.76/59.96	  join(complement(one), composition(converse(X), complement(X)))
59.76/59.96	= { by axiom 2 (composition_identity) }
59.76/59.96	  join(complement(one), composition(converse(X), complement(composition(X, one))))
59.76/59.96	= { by lemma 21 }
59.76/59.96	  complement(one)
59.76/59.96	
59.76/59.96	Lemma 71: join(complement(one), composition(X, complement(converse(X)))) = complement(one).
59.76/59.96	Proof:
59.76/59.96	  join(complement(one), composition(X, complement(converse(X))))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  join(complement(one), composition(converse(converse(X)), complement(converse(X))))
59.76/59.96	= { by lemma 70 }
59.76/59.96	  complement(one)
59.76/59.96	
59.76/59.96	Lemma 72: meet(complement(X), complement(Y)) = complement(join(X, Y)).
59.76/59.96	Proof:
59.76/59.96	  meet(complement(X), complement(Y))
59.76/59.96	= { by lemma 31 }
59.76/59.96	  meet(join(zero, complement(X)), complement(Y))
59.76/59.96	= { by lemma 44 }
59.76/59.96	  complement(join(Y, complement(join(zero, complement(X)))))
59.76/59.96	= { by lemma 17 }
59.76/59.96	  complement(join(Y, meet(X, top)))
59.76/59.96	= { by lemma 40 }
59.76/59.96	  complement(join(Y, X))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  complement(join(X, Y))
59.76/59.96	
59.76/59.96	Lemma 73: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
59.76/59.96	Proof:
59.76/59.96	  converse(composition(X, converse(Y)))
59.76/59.96	= { by axiom 3 (converse_multiplicativity) }
59.76/59.96	  composition(converse(converse(Y)), converse(X))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.96	  composition(Y, converse(X))
59.76/59.96	
59.76/59.96	Lemma 74: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
59.76/59.96	Proof:
59.76/59.96	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
59.76/59.96	= { by axiom 3 (converse_multiplicativity) }
59.76/59.96	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
59.76/59.96	= { by axiom 9 (composition_distributivity) }
59.76/59.96	  composition(join(converse(X), Y), converse(Z))
59.76/59.96	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.96	  composition(join(Y, converse(X)), converse(Z))
59.76/59.96	
59.76/59.96	Lemma 75: composition(converse(X), complement(composition(X, top))) = zero.
59.76/59.96	Proof:
59.76/59.96	  composition(converse(X), complement(composition(X, top)))
59.76/59.96	= { by lemma 41 }
59.76/59.96	  join(zero, composition(converse(X), complement(composition(X, top))))
59.76/59.96	= { by lemma 16 }
59.76/59.96	  join(complement(top), composition(converse(X), complement(composition(X, top))))
59.76/59.96	= { by lemma 21 }
59.76/59.96	  complement(top)
59.76/59.96	= { by lemma 16 }
59.76/59.96	  zero
59.76/59.96	
59.76/59.96	Lemma 76: join(X, converse(zero)) = X.
59.76/59.96	Proof:
59.76/59.96	  join(X, converse(zero))
59.76/59.96	= { by lemma 36 }
59.76/59.96	  converse(join(converse(X), zero))
59.76/59.96	= { by lemma 34 }
59.76/59.96	  converse(converse(X))
59.76/59.96	= { by axiom 11 (converse_idempotence) }
59.76/59.97	  X
59.76/59.97	
59.76/59.97	Lemma 77: composition(join(X, complement(composition(top, converse(Y)))), Y) = composition(X, Y).
59.76/59.97	Proof:
59.76/59.97	  composition(join(X, complement(composition(top, converse(Y)))), Y)
59.76/59.97	= { by lemma 39 }
59.76/59.97	  composition(join(X, complement(converse(composition(Y, top)))), Y)
59.76/59.97	= { by lemma 62 }
59.76/59.97	  composition(join(X, converse(complement(composition(Y, top)))), Y)
59.76/59.97	= { by axiom 11 (converse_idempotence) }
59.76/59.97	  composition(join(X, converse(complement(composition(Y, top)))), converse(converse(Y)))
59.76/59.97	= { by lemma 74 }
59.76/59.97	  join(composition(X, converse(converse(Y))), converse(composition(converse(Y), complement(composition(Y, top)))))
59.76/59.97	= { by lemma 75 }
59.76/59.97	  join(composition(X, converse(converse(Y))), converse(zero))
59.76/59.97	= { by lemma 76 }
59.76/59.97	  composition(X, converse(converse(Y)))
59.76/59.97	= { by axiom 11 (converse_idempotence) }
59.76/59.97	  composition(X, Y)
59.76/59.97	
59.76/59.97	Lemma 78: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
59.76/59.97	Proof:
59.76/59.97	  join(complement(X), meet(X, Y))
59.76/59.97	= { by axiom 10 (maddux1_join_commutativity) }
59.76/59.97	  join(meet(X, Y), complement(X))
59.76/59.97	= { by lemma 45 }
59.76/59.97	  complement(meet(X, complement(meet(X, Y))))
59.76/59.97	= { by lemma 58 }
59.76/59.97	  complement(meet(X, complement(Y)))
59.76/59.97	= { by lemma 45 }
59.81/60.01	  join(Y, complement(X))
59.81/60.01	
59.81/60.01	Lemma 79: composition(meet(X, one), converse(meet(X, one))) = meet(X, one).
59.81/60.01	Proof:
59.81/60.01	  composition(meet(X, one), converse(meet(X, one)))
59.81/60.01	= { by lemma 32 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, complement(complement(one)))))
59.81/60.01	= { by lemma 22 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, complement(join(complement(one), complement(one))))))
59.81/60.01	= { by axiom 5 (maddux4_definiton_of_meet) }
59.81/60.01	  composition(meet(X, one), converse(meet(X, meet(one, one))))
59.81/60.01	= { by lemma 40 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, meet(one, meet(one, top)))))
59.81/60.01	= { by lemma 17 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, meet(one, complement(join(zero, complement(one)))))))
59.81/60.01	= { by lemma 51 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, complement(join(complement(one), join(zero, complement(one)))))))
59.81/60.01	= { by lemma 51 }
59.81/60.01	  composition(meet(X, one), converse(complement(join(complement(X), join(complement(one), join(zero, complement(one)))))))
59.81/60.01	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.01	  composition(meet(X, one), converse(complement(join(join(complement(X), complement(one)), join(zero, complement(one))))))
59.81/60.01	= { by lemma 72 }
59.81/60.01	  composition(meet(X, one), converse(meet(complement(join(complement(X), complement(one))), complement(join(zero, complement(one))))))
59.81/60.01	= { by axiom 5 (maddux4_definiton_of_meet) }
59.81/60.01	  composition(meet(X, one), converse(meet(meet(X, one), complement(join(zero, complement(one))))))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  composition(meet(X, one), converse(meet(complement(join(zero, complement(one))), meet(X, one))))
59.81/60.01	= { by lemma 17 }
59.81/60.01	  composition(meet(X, one), converse(meet(meet(one, top), meet(X, one))))
59.81/60.01	= { by lemma 40 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, meet(X, one))))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  composition(meet(X, one), converse(meet(meet(X, one), one)))
59.81/60.01	= { by axiom 11 (converse_idempotence) }
59.81/60.01	  composition(meet(X, one), converse(meet(converse(converse(meet(X, one))), one)))
59.81/60.01	= { by lemma 32 }
59.81/60.01	  composition(meet(X, one), converse(meet(complement(complement(converse(converse(meet(X, one))))), one)))
59.81/60.01	= { by lemma 62 }
59.81/60.01	  composition(meet(X, one), converse(meet(complement(converse(complement(converse(meet(X, one))))), one)))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, complement(converse(complement(converse(meet(X, one))))))))
59.81/60.01	= { by lemma 59 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(complement(converse(complement(converse(meet(X, one))))), complement(one)))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(complement(one), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 71 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(join(complement(one), composition(complement(converse(meet(X, one))), complement(converse(complement(converse(meet(X, one))))))), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(complement(one), join(composition(complement(converse(meet(X, one))), complement(converse(complement(converse(meet(X, one)))))), complement(converse(complement(converse(meet(X, one))))))))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(complement(one), join(complement(converse(complement(converse(meet(X, one))))), composition(complement(converse(meet(X, one))), complement(converse(complement(converse(meet(X, one)))))))))))
59.81/60.01	= { by lemma 60 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, join(complement(converse(complement(converse(meet(X, one))))), composition(complement(converse(meet(X, one))), complement(converse(complement(converse(meet(X, one))))))))))
59.81/60.01	= { by lemma 69 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(complement(converse(meet(X, one))), one), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(one, complement(converse(meet(X, one)))), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 63 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(converse(one), complement(converse(meet(X, one)))), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(converse(one), complement(converse(meet(one, X)))), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(complement(converse(meet(one, X))), converse(one)), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by axiom 5 (maddux4_definiton_of_meet) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(complement(converse(complement(join(complement(one), complement(X))))), converse(one)), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(join(complement(converse(complement(join(complement(one), complement(X))))), converse(join(complement(join(complement(one), complement(X))), complement(join(complement(one), X))))), complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 53 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(top, complement(converse(complement(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 62 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(top, complement(complement(converse(converse(meet(X, one)))))))))
59.81/60.01	= { by lemma 32 }
59.81/60.01	  composition(meet(X, one), converse(meet(one, composition(top, converse(converse(meet(X, one)))))))
59.81/60.01	= { by lemma 73 }
59.81/60.01	  converse(composition(meet(one, composition(top, converse(converse(meet(X, one))))), converse(meet(X, one))))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  converse(composition(meet(composition(top, converse(converse(meet(X, one)))), one), converse(meet(X, one))))
59.81/60.01	= { by lemma 77 }
59.81/60.01	  converse(composition(join(meet(composition(top, converse(converse(meet(X, one)))), one), complement(composition(top, converse(converse(meet(X, one)))))), converse(meet(X, one))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  converse(composition(join(complement(composition(top, converse(converse(meet(X, one))))), meet(composition(top, converse(converse(meet(X, one)))), one)), converse(meet(X, one))))
59.81/60.01	= { by lemma 78 }
59.81/60.01	  converse(composition(join(one, complement(composition(top, converse(converse(meet(X, one)))))), converse(meet(X, one))))
59.81/60.01	= { by lemma 77 }
59.81/60.01	  converse(composition(one, converse(meet(X, one))))
59.81/60.01	= { by lemma 73 }
59.81/60.01	  composition(meet(X, one), converse(one))
59.81/60.01	= { by lemma 63 }
59.81/60.01	  composition(meet(X, one), one)
59.81/60.01	= { by axiom 2 (composition_identity) }
59.81/60.01	  meet(X, one)
59.81/60.01	
59.81/60.01	Lemma 80: converse(meet(X, one)) = meet(X, one).
59.81/60.01	Proof:
59.81/60.01	  converse(meet(X, one))
59.81/60.01	= { by lemma 79 }
59.81/60.01	  converse(composition(meet(X, one), converse(meet(X, one))))
59.81/60.01	= { by lemma 73 }
59.81/60.01	  composition(meet(X, one), converse(meet(X, one)))
59.81/60.01	= { by lemma 79 }
59.81/60.01	  meet(X, one)
59.81/60.01	
59.81/60.01	Lemma 81: converse(meet(one, X)) = meet(X, one).
59.81/60.01	Proof:
59.81/60.01	  converse(meet(one, X))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  converse(meet(X, one))
59.81/60.01	= { by lemma 80 }
59.81/60.01	  meet(X, one)
59.81/60.01	
59.81/60.01	Lemma 82: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
59.81/60.01	Proof:
59.81/60.01	  join(composition(X, Y), join(Z, composition(W, Y)))
59.81/60.01	= { by lemma 26 }
59.81/60.01	  join(Z, join(composition(W, Y), composition(X, Y)))
59.81/60.01	= { by axiom 9 (composition_distributivity) }
59.81/60.01	  join(Z, composition(join(W, X), Y))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  join(Z, composition(join(X, W), Y))
59.81/60.01	
59.81/60.01	Lemma 83: meet(one, composition(complement(converse(X)), join(X, Y))) = meet(one, composition(converse(Y), complement(X))).
59.81/60.01	Proof:
59.81/60.01	  meet(one, composition(complement(converse(X)), join(X, Y)))
59.81/60.01	= { by lemma 63 }
59.81/60.01	  meet(converse(one), composition(complement(converse(X)), join(X, Y)))
59.81/60.01	= { by lemma 62 }
59.81/60.01	  meet(converse(one), composition(converse(complement(X)), join(X, Y)))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  meet(converse(one), composition(converse(complement(X)), join(Y, X)))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  meet(composition(converse(complement(X)), join(Y, X)), converse(one))
59.81/60.01	= { by lemma 67 }
59.81/60.01	  converse(meet(one, converse(composition(converse(complement(X)), join(Y, X)))))
59.81/60.01	= { by lemma 18 }
59.81/60.01	  converse(meet(one, composition(converse(join(Y, X)), complement(X))))
59.81/60.01	= { by lemma 81 }
59.81/60.01	  meet(composition(converse(join(Y, X)), complement(X)), one)
59.81/60.01	= { by lemma 15 }
59.81/60.01	  meet(one, composition(converse(join(Y, X)), complement(X)))
59.81/60.01	= { by lemma 60 }
59.81/60.01	  meet(one, join(complement(one), composition(converse(join(Y, X)), complement(X))))
59.81/60.01	= { by axiom 6 (converse_additivity) }
59.81/60.01	  meet(one, join(complement(one), composition(join(converse(Y), converse(X)), complement(X))))
59.81/60.01	= { by lemma 82 }
59.81/60.01	  meet(one, join(composition(converse(Y), complement(X)), join(complement(one), composition(converse(X), complement(X)))))
59.81/60.01	= { by lemma 70 }
59.81/60.01	  meet(one, join(composition(converse(Y), complement(X)), complement(one)))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  meet(one, join(complement(one), composition(converse(Y), complement(X))))
59.81/60.01	= { by lemma 60 }
59.81/60.01	  meet(one, composition(converse(Y), complement(X)))
59.81/60.01	
59.81/60.01	Lemma 84: join(complement(one), composition(join(X, Y), complement(converse(X)))) = join(complement(one), composition(Y, complement(converse(X)))).
59.81/60.01	Proof:
59.81/60.01	  join(complement(one), composition(join(X, Y), complement(converse(X))))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  join(complement(one), composition(join(Y, X), complement(converse(X))))
59.81/60.01	= { by lemma 82 }
59.81/60.01	  join(composition(Y, complement(converse(X))), join(complement(one), composition(X, complement(converse(X)))))
59.81/60.01	= { by lemma 71 }
59.81/60.01	  join(composition(Y, complement(converse(X))), complement(one))
59.81/60.01	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.01	  join(complement(one), composition(Y, complement(converse(X))))
59.81/60.01	
59.81/60.01	Lemma 85: join(X, complement(meet(X, Y))) = top.
59.81/60.01	Proof:
59.81/60.01	  join(X, complement(meet(X, Y)))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  join(X, complement(meet(Y, X)))
59.81/60.01	= { by lemma 47 }
59.81/60.01	  join(X, join(complement(Y), complement(X)))
59.81/60.01	= { by lemma 28 }
59.81/60.01	  join(complement(Y), top)
59.81/60.01	= { by lemma 29 }
59.81/60.01	  top
59.81/60.01	
59.81/60.01	Lemma 86: join(X, complement(meet(Y, X))) = top.
59.81/60.01	Proof:
59.81/60.01	  join(X, complement(meet(Y, X)))
59.81/60.01	= { by lemma 15 }
59.81/60.01	  join(X, complement(meet(X, Y)))
59.81/60.01	= { by lemma 85 }
59.81/60.02	  top
59.81/60.02	
59.81/60.02	Lemma 87: join(complement(one), converse(complement(one))) = converse(complement(one)).
59.81/60.02	Proof:
59.81/60.02	  join(complement(one), converse(complement(one)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  join(converse(complement(one)), complement(one))
59.81/60.02	= { by lemma 52 }
59.81/60.02	  complement(meet(complement(converse(complement(one))), one))
59.81/60.02	= { by lemma 65 }
59.81/60.02	  join(meet(converse(complement(one)), complement(meet(complement(converse(complement(one))), one))), meet(complement(converse(complement(one))), complement(meet(complement(converse(complement(one))), one))))
59.81/60.02	= { by lemma 52 }
59.81/60.02	  join(meet(converse(complement(one)), join(converse(complement(one)), complement(one))), meet(complement(converse(complement(one))), complement(meet(complement(converse(complement(one))), one))))
59.81/60.02	= { by lemma 55 }
59.81/60.02	  join(converse(complement(one)), meet(complement(converse(complement(one))), complement(meet(complement(converse(complement(one))), one))))
59.81/60.02	= { by lemma 58 }
59.81/60.02	  join(converse(complement(one)), meet(complement(converse(complement(one))), complement(one)))
59.81/60.02	= { by lemma 72 }
59.81/60.02	  join(converse(complement(one)), complement(join(converse(complement(one)), one)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  join(converse(complement(one)), complement(join(one, converse(complement(one)))))
59.81/60.02	= { by lemma 63 }
59.81/60.02	  join(converse(complement(one)), complement(join(one, converse(complement(converse(one))))))
59.81/60.02	= { by lemma 50 }
59.81/60.02	  join(converse(complement(one)), complement(top))
59.81/60.02	= { by lemma 16 }
59.81/60.02	  join(converse(complement(one)), zero)
59.81/60.02	= { by lemma 34 }
59.81/60.02	  converse(complement(one))
59.81/60.02	
59.81/60.02	Lemma 88: converse(complement(one)) = complement(one).
59.81/60.02	Proof:
59.81/60.02	  converse(complement(one))
59.81/60.02	= { by lemma 87 }
59.81/60.02	  join(complement(one), converse(complement(one)))
59.81/60.02	= { by lemma 35 }
59.81/60.02	  converse(join(complement(one), converse(complement(one))))
59.81/60.02	= { by lemma 87 }
59.81/60.02	  converse(converse(complement(one)))
59.81/60.02	= { by axiom 11 (converse_idempotence) }
59.81/60.02	  complement(one)
59.81/60.02	
59.81/60.02	Lemma 89: join(complement(Y), meet(X, Y)) = join(X, complement(Y)).
59.81/60.02	Proof:
59.81/60.02	  join(complement(Y), meet(X, Y))
59.81/60.02	= { by lemma 15 }
59.81/60.02	  join(complement(Y), meet(Y, X))
59.81/60.02	= { by lemma 78 }
59.81/60.02	  join(X, complement(Y))
59.81/60.02	
59.81/60.02	Lemma 90: join(complement(one), converse(X)) = join(X, complement(one)).
59.81/60.02	Proof:
59.81/60.02	  join(complement(one), converse(X))
59.81/60.02	= { by lemma 88 }
59.81/60.02	  join(converse(complement(one)), converse(X))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  converse(join(complement(one), X))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(join(X, complement(one)))
59.81/60.02	= { by lemma 89 }
59.81/60.02	  converse(join(complement(one), meet(X, one)))
59.81/60.02	= { by lemma 80 }
59.81/60.02	  converse(join(complement(one), converse(meet(X, one))))
59.81/60.02	= { by lemma 63 }
59.81/60.02	  converse(join(complement(one), converse(meet(X, converse(one)))))
59.81/60.02	= { by lemma 67 }
59.81/60.02	  converse(join(complement(one), meet(one, converse(X))))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(join(meet(one, converse(X)), complement(one)))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  join(converse(meet(one, converse(X))), converse(complement(one)))
59.81/60.02	= { by lemma 67 }
59.81/60.02	  join(meet(X, converse(one)), converse(complement(one)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  join(converse(complement(one)), meet(X, converse(one)))
59.81/60.02	= { by lemma 62 }
59.81/60.02	  join(complement(converse(one)), meet(X, converse(one)))
59.81/60.02	= { by lemma 63 }
59.81/60.02	  join(complement(one), meet(X, converse(one)))
59.81/60.02	= { by lemma 63 }
59.81/60.02	  join(complement(one), meet(X, one))
59.81/60.02	= { by lemma 89 }
59.81/60.02	  join(X, complement(one))
59.81/60.02	
59.81/60.02	Lemma 91: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
59.81/60.02	Proof:
59.81/60.02	  join(meet(X, Y), complement(join(Y, complement(X))))
59.81/60.02	= { by axiom 5 (maddux4_definiton_of_meet) }
59.81/60.02	  join(complement(join(complement(X), complement(Y))), complement(join(Y, complement(X))))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
59.81/60.02	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.81/60.02	  X
59.81/60.02	
59.81/60.02	Lemma 92: composition(converse(X), complement(converse(Z))) = converse(composition(complement(Z), X)).
59.81/60.02	Proof:
59.81/60.02	  composition(converse(X), complement(converse(Z)))
59.81/60.02	= { by lemma 56 }
59.81/60.02	  composition(converse(X), complement(join(converse(Z), meet(converse(Z), complement(?)))))
59.81/60.02	= { by lemma 44 }
59.81/60.02	  composition(converse(X), complement(join(converse(Z), complement(join(?, complement(converse(Z)))))))
59.81/60.02	= { by lemma 44 }
59.81/60.02	  composition(converse(X), meet(join(?, complement(converse(Z))), complement(converse(Z))))
59.81/60.02	= { by lemma 62 }
59.81/60.02	  composition(converse(X), meet(join(?, complement(converse(Z))), converse(complement(Z))))
59.81/60.02	= { by lemma 67 }
59.81/60.02	  composition(converse(X), converse(meet(complement(Z), converse(join(?, complement(converse(Z)))))))
59.81/60.02	= { by axiom 3 (converse_multiplicativity) }
59.81/60.02	  converse(composition(meet(complement(Z), converse(join(?, complement(converse(Z))))), X))
59.81/60.02	= { by lemma 41 }
59.81/60.02	  converse(composition(join(zero, meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by lemma 16 }
59.81/60.02	  converse(composition(join(complement(top), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by lemma 29 }
59.81/60.02	  converse(composition(join(complement(join(converse(?), top)), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by lemma 50 }
59.81/60.02	  converse(composition(join(complement(join(converse(?), join(Z, converse(complement(converse(Z)))))), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by lemma 26 }
59.81/60.02	  converse(composition(join(complement(join(Z, join(converse(complement(converse(Z))), converse(?)))), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  converse(composition(join(complement(join(Z, converse(join(complement(converse(Z)), ?)))), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(composition(join(complement(join(Z, converse(join(?, complement(converse(Z)))))), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(composition(join(complement(join(converse(join(?, complement(converse(Z)))), Z)), meet(complement(Z), converse(join(?, complement(converse(Z)))))), X))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(composition(join(meet(complement(Z), converse(join(?, complement(converse(Z))))), complement(join(converse(join(?, complement(converse(Z)))), Z))), X))
59.81/60.02	= { by lemma 31 }
59.81/60.02	  converse(composition(join(meet(join(zero, complement(Z)), converse(join(?, complement(converse(Z))))), complement(join(converse(join(?, complement(converse(Z)))), Z))), X))
59.81/60.02	= { by lemma 40 }
59.81/60.02	  converse(composition(join(meet(join(zero, complement(Z)), converse(join(?, complement(converse(Z))))), complement(join(converse(join(?, complement(converse(Z)))), meet(Z, top)))), X))
59.81/60.02	= { by lemma 17 }
59.81/60.02	  converse(composition(join(meet(join(zero, complement(Z)), converse(join(?, complement(converse(Z))))), complement(join(converse(join(?, complement(converse(Z)))), complement(join(zero, complement(Z)))))), X))
59.81/60.02	= { by lemma 91 }
59.81/60.02	  converse(composition(join(zero, complement(Z)), X))
59.81/60.02	= { by lemma 31 }
59.81/60.02	  converse(composition(complement(Z), X))
59.81/60.02	
59.81/60.02	Lemma 93: meet(one, composition(X, converse(Y))) = meet(one, composition(Y, converse(X))).
59.81/60.02	Proof:
59.81/60.02	  meet(one, composition(X, converse(Y)))
59.81/60.02	= { by lemma 15 }
59.81/60.02	  meet(composition(X, converse(Y)), one)
59.81/60.02	= { by lemma 81 }
59.81/60.02	  converse(meet(one, composition(X, converse(Y))))
59.81/60.02	= { by lemma 73 }
59.81/60.02	  converse(meet(one, converse(composition(Y, converse(X)))))
59.81/60.02	= { by lemma 67 }
59.81/60.02	  meet(composition(Y, converse(X)), converse(one))
59.81/60.02	= { by lemma 15 }
59.81/60.02	  meet(converse(one), composition(Y, converse(X)))
59.81/60.02	= { by lemma 63 }
59.81/60.02	  meet(one, composition(Y, converse(X)))
59.81/60.02	
59.81/60.02	Lemma 94: converse(join(X, composition(converse(Z), Y))) = join(converse(X), composition(converse(Y), Z)).
59.81/60.02	Proof:
59.81/60.02	  converse(join(X, composition(converse(Z), Y)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(join(composition(converse(Z), Y), X))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  join(converse(composition(converse(Z), Y)), converse(X))
59.81/60.02	= { by lemma 18 }
59.81/60.02	  join(composition(converse(Y), Z), converse(X))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  join(converse(X), composition(converse(Y), Z))
59.81/60.02	
59.81/60.02	Lemma 95: meet(one, composition(join(X, complement(converse(Y))), Y)) = meet(one, composition(X, Y)).
59.81/60.02	Proof:
59.81/60.02	  meet(one, composition(join(X, complement(converse(Y))), Y))
59.81/60.02	= { by lemma 60 }
59.81/60.02	  meet(one, join(complement(one), composition(join(X, complement(converse(Y))), Y)))
59.81/60.02	= { by lemma 82 }
59.81/60.02	  meet(one, join(composition(X, Y), join(complement(one), composition(complement(converse(Y)), Y))))
59.81/60.02	= { by lemma 88 }
59.81/60.02	  meet(one, join(composition(X, Y), join(converse(complement(one)), composition(complement(converse(Y)), Y))))
59.81/60.02	= { by lemma 62 }
59.81/60.02	  meet(one, join(composition(X, Y), join(converse(complement(one)), composition(converse(complement(Y)), Y))))
59.81/60.02	= { by lemma 94 }
59.81/60.02	  meet(one, join(composition(X, Y), converse(join(complement(one), composition(converse(Y), complement(Y))))))
59.81/60.02	= { by lemma 70 }
59.81/60.02	  meet(one, join(composition(X, Y), converse(complement(one))))
59.81/60.02	= { by lemma 88 }
59.81/60.02	  meet(one, join(composition(X, Y), complement(one)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  meet(one, join(complement(one), composition(X, Y)))
59.81/60.02	= { by lemma 60 }
59.81/60.02	  meet(one, composition(X, Y))
59.81/60.02	
59.81/60.02	Lemma 96: converse(join(composition(X, Y), composition(X, Z))) = converse(composition(X, join(Y, Z))).
59.81/60.02	Proof:
59.81/60.02	  converse(join(composition(X, Y), composition(X, Z)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(join(composition(X, Z), composition(X, Y)))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  join(converse(composition(X, Z)), converse(composition(X, Y)))
59.81/60.02	= { by axiom 3 (converse_multiplicativity) }
59.81/60.02	  join(composition(converse(Z), converse(X)), converse(composition(X, Y)))
59.81/60.02	= { by lemma 74 }
59.81/60.02	  composition(join(converse(Z), converse(Y)), converse(X))
59.81/60.02	= { by axiom 6 (converse_additivity) }
59.81/60.02	  composition(converse(join(Z, Y)), converse(X))
59.81/60.02	= { by axiom 3 (converse_multiplicativity) }
59.81/60.02	  converse(composition(X, join(Z, Y)))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  converse(composition(X, join(Y, Z)))
59.81/60.02	
59.81/60.02	Lemma 97: meet(one, composition(join(complement(converse(Y)), X), Y)) = meet(one, composition(X, Y)).
59.81/60.02	Proof:
59.81/60.02	  meet(one, composition(join(complement(converse(Y)), X), Y))
59.81/60.02	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.02	  meet(one, composition(join(X, complement(converse(Y))), Y))
59.81/60.02	= { by lemma 95 }
59.81/60.03	  meet(one, composition(X, Y))
59.81/60.03	
59.81/60.03	Lemma 98: meet(one, composition(top, meet(Y, converse(X)))) = meet(one, composition(X, Y)).
59.81/60.03	Proof:
59.81/60.03	  meet(one, composition(top, meet(Y, converse(X))))
59.81/60.03	= { by lemma 15 }
59.81/60.03	  meet(composition(top, meet(Y, converse(X))), one)
59.81/60.03	= { by lemma 81 }
59.81/60.03	  converse(meet(one, composition(top, meet(Y, converse(X)))))
59.81/60.03	= { by lemma 15 }
59.81/60.03	  converse(meet(composition(top, meet(Y, converse(X))), one))
59.81/60.03	= { by lemma 68 }
59.81/60.03	  meet(one, converse(composition(top, meet(Y, converse(X)))))
59.81/60.03	= { by lemma 64 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), top))
59.81/60.03	= { by lemma 38 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), converse(top)))
59.81/60.03	= { by lemma 85 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), converse(join(converse(Y), complement(meet(converse(Y), X))))))
59.81/60.03	= { by lemma 36 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), join(Y, converse(complement(meet(converse(Y), X))))))
59.81/60.03	= { by lemma 62 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), join(Y, complement(converse(meet(converse(Y), X))))))
59.81/60.03	= { by lemma 15 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), join(Y, complement(converse(meet(X, converse(Y)))))))
59.81/60.03	= { by lemma 67 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), join(Y, complement(meet(Y, converse(X))))))
59.81/60.03	= { by lemma 60 }
59.81/60.03	  meet(one, join(complement(one), composition(converse(meet(Y, converse(X))), join(Y, complement(meet(Y, converse(X)))))))
59.81/60.03	= { by axiom 11 (converse_idempotence) }
59.81/60.03	  meet(one, join(complement(one), converse(converse(composition(converse(meet(Y, converse(X))), join(Y, complement(meet(Y, converse(X)))))))))
59.81/60.03	= { by lemma 96 }
59.81/60.03	  meet(one, join(complement(one), converse(converse(join(composition(converse(meet(Y, converse(X))), Y), composition(converse(meet(Y, converse(X))), complement(meet(Y, converse(X)))))))))
59.81/60.03	= { by axiom 11 (converse_idempotence) }
59.81/60.03	  meet(one, join(complement(one), join(composition(converse(meet(Y, converse(X))), Y), composition(converse(meet(Y, converse(X))), complement(meet(Y, converse(X)))))))
59.81/60.03	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.03	  meet(one, join(join(complement(one), composition(converse(meet(Y, converse(X))), Y)), composition(converse(meet(Y, converse(X))), complement(meet(Y, converse(X))))))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  meet(one, join(join(composition(converse(meet(Y, converse(X))), Y), complement(one)), composition(converse(meet(Y, converse(X))), complement(meet(Y, converse(X))))))
59.81/60.03	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.03	  meet(one, join(composition(converse(meet(Y, converse(X))), Y), join(complement(one), composition(converse(meet(Y, converse(X))), complement(meet(Y, converse(X)))))))
59.81/60.03	= { by lemma 70 }
59.81/60.03	  meet(one, join(composition(converse(meet(Y, converse(X))), Y), complement(one)))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  meet(one, join(complement(one), composition(converse(meet(Y, converse(X))), Y)))
59.81/60.03	= { by lemma 60 }
59.81/60.03	  meet(one, composition(converse(meet(Y, converse(X))), Y))
59.81/60.03	= { by lemma 67 }
59.81/60.03	  meet(one, composition(meet(X, converse(Y)), Y))
59.81/60.03	= { by lemma 32 }
59.81/60.03	  meet(one, composition(meet(X, complement(complement(converse(Y)))), Y))
59.81/60.03	= { by lemma 97 }
59.81/60.03	  meet(one, composition(join(complement(converse(Y)), meet(X, complement(complement(converse(Y))))), Y))
59.81/60.03	= { by lemma 66 }
59.81/60.03	  meet(one, composition(join(complement(converse(Y)), X), Y))
59.81/60.03	= { by lemma 97 }
59.81/60.03	  meet(one, composition(X, Y))
59.81/60.03	
59.81/60.03	Lemma 99: composition(converse(join(X, complement(composition(Y, top)))), Y) = composition(converse(X), Y).
59.81/60.03	Proof:
59.81/60.03	  composition(converse(join(X, complement(composition(Y, top)))), Y)
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  composition(converse(join(complement(composition(Y, top)), X)), Y)
59.81/60.03	= { by lemma 18 }
59.81/60.03	  converse(composition(converse(Y), join(complement(composition(Y, top)), X)))
59.81/60.03	= { by lemma 96 }
59.81/60.03	  converse(join(composition(converse(Y), complement(composition(Y, top))), composition(converse(Y), X)))
59.81/60.03	= { by lemma 75 }
59.81/60.03	  converse(join(zero, composition(converse(Y), X)))
59.81/60.03	= { by lemma 94 }
59.81/60.03	  join(converse(zero), composition(converse(X), Y))
59.81/60.03	= { by lemma 41 }
59.81/60.03	  join(join(zero, converse(zero)), composition(converse(X), Y))
59.81/60.03	= { by lemma 76 }
59.81/60.03	  join(zero, composition(converse(X), Y))
59.81/60.03	= { by lemma 41 }
59.81/60.03	  composition(converse(X), Y)
59.81/60.03	
59.81/60.03	Lemma 100: join(composition(X, Z), complement(composition(meet(X, Y), Z))) = top.
59.81/60.03	Proof:
59.81/60.03	  join(composition(X, Z), complement(composition(meet(X, Y), Z)))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(complement(composition(meet(X, Y), Z)), composition(X, Z))
59.81/60.03	= { by axiom 5 (maddux4_definiton_of_meet) }
59.81/60.03	  join(complement(composition(complement(join(complement(X), complement(Y))), Z)), composition(X, Z))
59.81/60.03	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
59.81/60.03	  join(complement(composition(complement(join(complement(X), complement(Y))), Z)), composition(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), Z))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(complement(composition(complement(join(complement(X), complement(Y))), Z)), composition(join(complement(join(complement(X), Y)), complement(join(complement(X), complement(Y)))), Z))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(composition(join(complement(join(complement(X), Y)), complement(join(complement(X), complement(Y)))), Z), complement(composition(complement(join(complement(X), complement(Y))), Z)))
59.81/60.03	= { by axiom 9 (composition_distributivity) }
59.81/60.03	  join(join(composition(complement(join(complement(X), Y)), Z), composition(complement(join(complement(X), complement(Y))), Z)), complement(composition(complement(join(complement(X), complement(Y))), Z)))
59.81/60.03	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.03	  join(composition(complement(join(complement(X), Y)), Z), join(composition(complement(join(complement(X), complement(Y))), Z), complement(composition(complement(join(complement(X), complement(Y))), Z))))
59.81/60.03	= { by axiom 7 (def_top) }
59.81/60.03	  join(composition(complement(join(complement(X), Y)), Z), top)
59.81/60.03	= { by lemma 29 }
59.81/60.03	  top
59.81/60.03	
59.81/60.03	Lemma 101: join(top, X) = top.
59.81/60.03	Proof:
59.81/60.03	  join(top, X)
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(X, top)
59.81/60.03	= { by lemma 29 }
59.81/60.03	  top
59.81/60.03	
59.81/60.03	Lemma 102: composition(top, top) = top.
59.81/60.03	Proof:
59.81/60.03	  composition(top, top)
59.81/60.03	= { by lemma 41 }
59.81/60.03	  join(zero, composition(top, top))
59.81/60.03	= { by lemma 101 }
59.81/60.03	  join(zero, composition(join(top, one), top))
59.81/60.03	= { by lemma 69 }
59.81/60.03	  join(zero, join(top, composition(top, top)))
59.81/60.03	= { by axiom 12 (maddux2_join_associativity) }
59.81/60.03	  join(join(zero, top), composition(top, top))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(join(top, zero), composition(top, top))
59.81/60.03	= { by lemma 16 }
59.81/60.03	  join(join(top, complement(top)), composition(top, top))
59.81/60.03	= { by axiom 7 (def_top) }
59.81/60.03	  join(top, composition(top, top))
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  join(composition(top, top), top)
59.81/60.03	= { by lemma 29 }
59.81/60.03	  top
59.81/60.03	
59.81/60.03	Lemma 103: composition(converse(join(complement(composition(Y, top)), X)), Y) = composition(converse(X), Y).
59.81/60.03	Proof:
59.81/60.03	  composition(converse(join(complement(composition(Y, top)), X)), Y)
59.81/60.03	= { by axiom 10 (maddux1_join_commutativity) }
59.81/60.03	  composition(converse(join(X, complement(composition(Y, top)))), Y)
59.81/60.03	= { by lemma 99 }
59.81/60.03	  composition(converse(X), Y)
59.81/60.03	
59.81/60.03	Lemma 104: composition(top, zero) = zero.
59.81/60.03	Proof:
59.81/60.03	  composition(top, zero)
59.81/60.03	= { by lemma 41 }
59.81/60.03	  join(zero, composition(top, zero))
59.81/60.03	= { by lemma 16 }
59.81/60.03	  join(complement(top), composition(top, zero))
59.81/60.03	= { by lemma 38 }
59.81/60.03	  join(complement(top), composition(converse(top), zero))
59.81/60.03	= { by lemma 16 }
59.81/60.03	  join(complement(top), composition(converse(top), complement(top)))
59.81/60.03	= { by lemma 102 }
59.81/60.03	  join(complement(top), composition(converse(top), complement(composition(top, top))))
59.81/60.03	= { by lemma 21 }
59.81/60.03	  complement(top)
59.81/60.03	= { by lemma 16 }
60.34/60.55	  zero
60.34/60.55	
60.34/60.55	Lemma 105: composition(top, meet(composition(sK2_goals_X0, X), composition(sK2_goals_X0, complement(X)))) = zero.
60.34/60.55	Proof:
60.34/60.55	  composition(top, meet(composition(sK2_goals_X0, X), composition(sK2_goals_X0, complement(X))))
60.34/60.55	= { by lemma 102 }
60.34/60.55	  composition(composition(top, top), meet(composition(sK2_goals_X0, X), composition(sK2_goals_X0, complement(X))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(composition(top, top), meet(composition(sK2_goals_X0, complement(X)), composition(sK2_goals_X0, X)))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(composition(top, top), meet(composition(sK2_goals_X0, complement(X)), converse(converse(composition(sK2_goals_X0, X)))))
60.34/60.55	= { by lemma 67 }
60.34/60.55	  composition(composition(top, top), converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))
60.34/60.55	= { by axiom 2 (composition_identity) }
60.34/60.55	  composition(composition(top, top), composition(converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))), one))
60.34/60.55	= { by lemma 18 }
60.34/60.55	  composition(composition(top, top), converse(composition(converse(one), meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))))
60.34/60.55	= { by lemma 103 }
60.34/60.55	  composition(composition(top, top), converse(composition(converse(join(complement(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)), one)), meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))))
60.34/60.55	= { by lemma 66 }
60.34/60.55	  composition(composition(top, top), converse(composition(converse(join(complement(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)), meet(one, complement(complement(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)))))), meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))))
60.34/60.55	= { by lemma 103 }
60.34/60.55	  composition(composition(top, top), converse(composition(converse(meet(one, complement(complement(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))))), meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))))
60.34/60.55	= { by lemma 32 }
60.34/60.55	  composition(composition(top, top), converse(composition(converse(meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))), meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))))
60.34/60.55	= { by lemma 18 }
60.34/60.55	  composition(composition(top, top), composition(converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(top, composition(top, composition(converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)))))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(top, composition(composition(top, converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))))
60.34/60.55	= { by lemma 39 }
60.34/60.55	  composition(top, composition(converse(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(top, composition(converse(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one)))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(composition(top, converse(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top))), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 39 }
60.34/60.55	  composition(converse(composition(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), top)), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 99 }
60.34/60.55	  composition(converse(join(composition(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), top), complement(composition(meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one), top)))), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 100 }
60.34/60.55	  composition(converse(top), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 38 }
60.34/60.55	  composition(top, meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 29 }
60.34/60.55	  composition(join(?, top), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 38 }
60.34/60.55	  composition(join(?, converse(top)), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 100 }
60.34/60.55	  composition(join(?, converse(join(composition(one, top), complement(composition(meet(one, composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X)))), top))))), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(?, converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top))))), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top), one))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))), top)))
60.34/60.55	= { by lemma 38 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))), converse(top))))
60.34/60.55	= { by lemma 86 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))), converse(join(converse(composition(sK2_goals_X0, X)), complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))
60.34/60.55	= { by lemma 93 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(join(converse(composition(sK2_goals_X0, X)), complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(join(converse(composition(sK2_goals_X0, X)), complement(converse(converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))
60.34/60.55	= { by lemma 95 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))))
60.34/60.55	= { by lemma 93 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), converse(converse(composition(sK2_goals_X0, X))))))
60.34/60.55	= { by axiom 2 (composition_identity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), composition(converse(converse(composition(sK2_goals_X0, X))), one))))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), converse(converse(composition(sK2_goals_X0, X)))), one)))
60.34/60.55	= { by lemma 73 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))), one)))
60.34/60.55	= { by lemma 32 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))), complement(complement(one)))))
60.34/60.55	= { by lemma 83 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(composition(sK2_goals_X0, X)), converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))
60.34/60.55	= { by lemma 32 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(composition(sK2_goals_X0, X)), complement(complement(converse(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by lemma 62 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(composition(sK2_goals_X0, X)), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by lemma 84 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(join(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))), converse(composition(sK2_goals_X0, X))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(join(converse(composition(sK2_goals_X0, X)), complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by lemma 86 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(top, complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by axiom 7 (def_top) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(join(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))), complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(join(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))), complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))), complement(converse(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by lemma 84 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(converse(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(one)))))
60.34/60.55	= { by lemma 90 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), converse(composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))))))))
60.34/60.55	= { by lemma 92 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), complement(converse(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by lemma 84 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(join(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))), converse(complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))), complement(converse(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by axiom 6 (converse_additivity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(converse(join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))), complement(converse(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by lemma 92 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), converse(composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))))))))
60.34/60.55	= { by lemma 90 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X)))))))), complement(one)))))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(complement(converse(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))))))))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(complement(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))))))))
60.34/60.55	= { by lemma 32 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))))))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), join(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))), complement(converse(complement(meet(converse(composition(sK2_goals_X0, complement(X))), converse(composition(sK2_goals_X0, X))))))))))))
60.34/60.55	= { by axiom 7 (def_top) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(complement(converse(complement(one))), join(complement(one), composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)))))
60.34/60.55	= { by lemma 83 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))), top)), complement(complement(one)))))
60.34/60.55	= { by lemma 39 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(composition(top, converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))), complement(complement(one)))))
60.34/60.55	= { by lemma 32 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(composition(top, converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X)))))), one)))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(top, composition(converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))), one))))
60.34/60.55	= { by axiom 2 (composition_identity) }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(top, converse(meet(converse(composition(sK2_goals_X0, X)), converse(composition(sK2_goals_X0, complement(X))))))))
60.34/60.55	= { by lemma 67 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(top, meet(composition(sK2_goals_X0, complement(X)), converse(converse(composition(sK2_goals_X0, X)))))))
60.34/60.55	= { by lemma 98 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(one, composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X)))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), ?), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one))
60.34/60.55	= { by axiom 9 (composition_distributivity) }
60.34/60.55	  join(composition(converse(join(composition(one, top), complement(composition(meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one), top)))), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one)), composition(?, meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one)))
60.34/60.55	= { by lemma 99 }
60.34/60.55	  join(composition(converse(composition(one, top)), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one)), composition(?, meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one)))
60.34/60.55	= { by axiom 9 (composition_distributivity) }
60.34/60.55	  composition(join(converse(composition(one, top)), ?), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(?, converse(composition(one, top))), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one))
60.34/60.55	= { by lemma 39 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X))), one))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(composition(sK2_goals_X0, X)), composition(sK2_goals_X0, complement(X)))))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(composition(converse(composition(sK2_goals_X0, X)), sK2_goals_X0), complement(X))))
60.34/60.55	= { by lemma 18 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))), complement(X))))
60.34/60.55	= { by lemma 98 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(top, meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 38 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(top), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 100 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(composition(one, X), complement(composition(meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0)), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 20 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0)), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 34 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), zero), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 16 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(top)), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 29 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(join(one, top))), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 7 (def_top) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(join(one, join(composition(converse(sK2_goals_X0), sK2_goals_X0), complement(composition(converse(sK2_goals_X0), sK2_goals_X0)))))), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 12 (maddux2_join_associativity) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(join(join(one, composition(converse(sK2_goals_X0), sK2_goals_X0)), complement(composition(converse(sK2_goals_X0), sK2_goals_X0))))), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(join(join(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(composition(converse(sK2_goals_X0), sK2_goals_X0))))), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 14 (goals) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(join(meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one), complement(join(one, complement(composition(converse(sK2_goals_X0), sK2_goals_X0))))), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by lemma 91 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(composition(converse(sK2_goals_X0), sK2_goals_X0), X)))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 8 (composition_associativity) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))), meet(complement(X), converse(converse(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))))
60.34/60.55	= { by axiom 11 (converse_idempotence) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))), meet(complement(X), composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))))
60.34/60.55	= { by lemma 15 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, composition(converse(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))), meet(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X)), complement(X)))))
60.34/60.55	= { by lemma 60 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, join(complement(one), composition(converse(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))), meet(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X)), complement(X))))))
60.34/60.55	= { by lemma 44 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, join(complement(one), composition(converse(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X))))), complement(join(X, complement(composition(converse(sK2_goals_X0), composition(sK2_goals_X0, X)))))))))
60.34/60.55	= { by lemma 70 }
60.34/60.55	  composition(join(?, composition(top, converse(one))), meet(one, complement(one)))
60.34/60.55	= { by axiom 4 (def_zero) }
60.34/60.55	  composition(join(?, composition(top, converse(one))), zero)
60.34/60.55	= { by lemma 41 }
60.34/60.55	  join(zero, composition(join(?, composition(top, converse(one))), zero))
60.34/60.55	= { by lemma 104 }
60.34/60.55	  join(composition(top, zero), composition(join(?, composition(top, converse(one))), zero))
60.34/60.55	= { by axiom 9 (composition_distributivity) }
60.34/60.55	  composition(join(top, join(?, composition(top, converse(one)))), zero)
60.34/60.55	= { by lemma 101 }
60.34/60.55	  composition(top, zero)
60.34/60.55	= { by lemma 104 }
60.34/60.56	  zero
60.34/60.56	
60.34/60.56	Goal 1 (goals_1): zero = meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))).
60.34/60.56	Proof:
60.34/60.56	  zero
60.34/60.56	= { by lemma 105 }
60.34/60.56	  composition(top, meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))))
60.34/60.56	= { by lemma 101 }
60.34/60.56	  composition(join(top, one), meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))))
60.34/60.56	= { by axiom 10 (maddux1_join_commutativity) }
60.34/60.56	  composition(join(one, top), meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))))
60.34/60.56	= { by axiom 9 (composition_distributivity) }
60.34/60.56	  join(composition(one, meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1)))), composition(top, meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1)))))
60.34/60.56	= { by lemma 20 }
60.34/60.56	  join(meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))), composition(top, meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1)))))
60.34/60.56	= { by lemma 105 }
60.34/60.56	  join(meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1))), zero)
60.34/60.56	= { by lemma 34 }
60.34/60.56	  meet(composition(sK2_goals_X0, sK1_goals_X1), composition(sK2_goals_X0, complement(sK1_goals_X1)))
60.34/60.56	% SZS output end Proof
60.34/60.56	
60.34/60.56	RESULT: Theorem (the conjecture is true).
60.34/60.57	EOF