0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n024.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 180
0.12/0.34	% DateTime   : Thu Aug 29 09:52:35 EDT 2019
0.12/0.34	% CPUTime    : 
22.84/23.05	% SZS status Theorem
22.84/23.05	
22.84/23.05	% SZS output start Proof
22.84/23.05	Take the following subset of the input axioms:
24.65/24.82	  fof(composition_associativity, axiom, ![X0, X1, X2]: composition(composition(X0, X1), X2)=composition(X0, composition(X1, X2))).
24.65/24.82	  fof(composition_distributivity, axiom, ![X0, X1, X2]: composition(join(X0, X1), X2)=join(composition(X0, X2), composition(X1, X2))).
24.65/24.82	  fof(composition_identity, axiom, ![X0]: X0=composition(X0, one)).
24.65/24.82	  fof(converse_additivity, axiom, ![X0, X1]: converse(join(X0, X1))=join(converse(X0), converse(X1))).
24.65/24.82	  fof(converse_cancellativity, axiom, ![X0, X1]: complement(X1)=join(composition(converse(X0), complement(composition(X0, X1))), complement(X1))).
24.65/24.82	  fof(converse_idempotence, axiom, ![X0]: X0=converse(converse(X0))).
24.65/24.82	  fof(converse_multiplicativity, axiom, ![X0, X1]: converse(composition(X0, X1))=composition(converse(X1), converse(X0))).
24.65/24.82	  fof(dedekind_law, axiom, ![X0, X1, X2]: join(meet(composition(X0, X1), X2), composition(meet(X0, composition(X2, converse(X1))), meet(X1, composition(converse(X0), X2))))=composition(meet(X0, composition(X2, converse(X1))), meet(X1, composition(converse(X0), X2)))).
24.65/24.82	  fof(def_top, axiom, ![X0]: join(X0, complement(X0))=top).
24.65/24.82	  fof(def_zero, axiom, ![X0]: meet(X0, complement(X0))=zero).
24.65/24.82	  fof(goals, conjecture, ![X0, X1, X2]: (X0=join(composition(X2, converse(X1)), X0) <= join(composition(complement(X0), X1), complement(X2))=complement(X2))).
24.65/24.82	  fof(maddux1_join_commutativity, axiom, ![X0, X1]: join(X1, X0)=join(X0, X1)).
24.65/24.82	  fof(maddux2_join_associativity, axiom, ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
24.65/24.82	  fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0, X1]: X0=join(complement(join(complement(X0), complement(X1))), complement(join(complement(X0), X1)))).
24.65/24.82	  fof(maddux4_definiton_of_meet, axiom, ![X0, X1]: meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
24.65/24.82	
24.65/24.82	Now clausify the problem and encode Horn clauses using encoding 3 of
24.65/24.82	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
24.65/24.82	We repeatedly replace C & s=t => u=v by the two clauses:
24.65/24.82	  fresh(y, y, x1...xn) = u
24.65/24.82	  C => fresh(s, t, x1...xn) = v
24.65/24.82	where fresh is a fresh function symbol and x1..xn are the free
24.65/24.82	variables of u and v.
24.65/24.82	A predicate p(X) is encoded as p(X)=true (this is sound, because the
24.65/24.82	input problem has no model of domain size 1).
24.65/24.82	
24.65/24.82	The encoding turns the above axioms into the following unit equations and goals:
24.65/24.82	
24.65/24.82	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
24.65/24.82	Axiom 2 (composition_identity): X = composition(X, one).
24.65/24.82	Axiom 3 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
24.65/24.82	Axiom 4 (def_zero): meet(X, complement(X)) = zero.
24.65/24.82	Axiom 5 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
24.65/24.82	Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
24.65/24.82	Axiom 7 (def_top): join(X, complement(X)) = top.
24.65/24.82	Axiom 8 (composition_associativity): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
24.65/24.82	Axiom 9 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
24.65/24.82	Axiom 10 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
24.65/24.82	Axiom 11 (converse_idempotence): X = converse(converse(X)).
24.65/24.82	Axiom 12 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
24.65/24.82	Axiom 13 (converse_cancellativity): complement(X) = join(composition(converse(Y), complement(composition(Y, X))), complement(X)).
24.65/24.82	Axiom 14 (dedekind_law): join(meet(composition(X, Y), Z), composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z)))) = composition(meet(X, composition(Z, converse(Y))), meet(Y, composition(converse(X), Z))).
24.65/24.84	Axiom 15 (goals): join(composition(complement(sK3_goals_X0), sK2_goals_X1), complement(sK1_goals_X2)) = complement(sK1_goals_X2).
24.65/24.84	
24.65/24.84	Lemma 16: meet(X, Y) = meet(Y, X).
24.65/24.84	Proof:
24.65/24.84	  meet(X, Y)
24.65/24.84	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.84	  complement(join(complement(X), complement(Y)))
24.65/24.84	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.84	  complement(join(complement(Y), complement(X)))
24.65/24.84	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.84	  meet(Y, X)
24.65/24.84	
24.65/24.84	Lemma 17: complement(top) = zero.
24.65/24.84	Proof:
24.65/24.84	  complement(top)
24.65/24.84	= { by axiom 7 (def_top) }
24.65/24.84	  complement(join(complement(?), complement(complement(?))))
24.65/24.84	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.84	  meet(?, complement(?))
24.65/24.84	= { by axiom 4 (def_zero) }
24.65/24.84	  zero
24.65/24.84	
24.65/24.84	Lemma 18: complement(join(zero, complement(X))) = meet(X, top).
24.65/24.84	Proof:
24.65/24.84	  complement(join(zero, complement(X)))
24.65/24.84	= { by lemma 17 }
24.65/24.84	  complement(join(complement(top), complement(X)))
24.65/24.84	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.84	  meet(top, X)
24.65/24.84	= { by lemma 16 }
24.65/24.84	  meet(X, top)
24.65/24.84	
24.65/24.84	Lemma 19: composition(converse(one), X) = X.
24.65/24.84	Proof:
24.65/24.84	  composition(converse(one), X)
24.65/24.84	= { by axiom 11 (converse_idempotence) }
24.65/24.84	  composition(converse(one), converse(converse(X)))
24.65/24.84	= { by axiom 3 (converse_multiplicativity) }
24.65/24.84	  converse(composition(converse(X), one))
24.65/24.84	= { by axiom 2 (composition_identity) }
24.65/24.84	  converse(converse(X))
24.65/24.84	= { by axiom 11 (converse_idempotence) }
24.65/24.84	  X
24.65/24.84	
24.65/24.84	Lemma 20: composition(one, X) = X.
24.65/24.84	Proof:
24.65/24.84	  composition(one, X)
24.65/24.84	= { by lemma 19 }
24.65/24.84	  composition(converse(one), composition(one, X))
24.65/24.84	= { by axiom 8 (composition_associativity) }
24.65/24.84	  composition(composition(converse(one), one), X)
24.65/24.84	= { by axiom 2 (composition_identity) }
24.65/24.84	  composition(converse(one), X)
24.65/24.84	= { by lemma 19 }
24.65/24.84	  X
24.65/24.84	
24.65/24.84	Lemma 21: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
24.65/24.84	Proof:
24.65/24.84	  join(complement(X), composition(converse(Y), complement(composition(Y, X))))
24.65/24.84	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.84	  join(composition(converse(Y), complement(composition(Y, X))), complement(X))
24.65/24.84	= { by axiom 13 (converse_cancellativity) }
24.65/24.84	  complement(X)
24.65/24.84	
24.65/24.84	Lemma 22: join(complement(X), complement(X)) = complement(X).
24.65/24.84	Proof:
24.65/24.84	  join(complement(X), complement(X))
24.65/24.84	= { by lemma 19 }
24.65/24.84	  join(complement(X), composition(converse(one), complement(X)))
24.65/24.84	= { by lemma 20 }
24.65/24.84	  join(complement(X), composition(converse(one), complement(composition(one, X))))
24.65/24.84	= { by lemma 21 }
24.65/24.84	  complement(X)
24.65/24.84	
24.65/24.84	Lemma 23: join(X, join(Y, Z)) = join(Z, join(X, Y)).
24.65/24.84	Proof:
24.65/24.84	  join(X, join(Y, Z))
24.65/24.84	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.84	  join(join(X, Y), Z)
24.65/24.84	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.84	  join(Z, join(X, Y))
24.65/24.84	
24.65/24.84	Lemma 24: join(X, join(complement(X), Y)) = join(Y, top).
24.65/24.84	Proof:
24.65/24.84	  join(X, join(complement(X), Y))
24.65/24.84	= { by lemma 23 }
24.65/24.84	  join(complement(X), join(Y, X))
24.65/24.84	= { by lemma 23 }
24.65/24.84	  join(Y, join(X, complement(X)))
24.65/24.84	= { by axiom 7 (def_top) }
24.65/24.84	  join(Y, top)
24.65/24.84	
24.65/24.84	Lemma 25: join(X, join(Y, complement(X))) = join(Y, top).
24.65/24.84	Proof:
24.65/24.84	  join(X, join(Y, complement(X)))
24.65/24.84	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.84	  join(X, join(complement(X), Y))
24.65/24.85	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.85	  join(join(X, complement(X)), Y)
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  join(top, Y)
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(Y, top)
24.65/24.85	
24.65/24.85	Lemma 26: join(top, complement(X)) = top.
24.65/24.85	Proof:
24.65/24.85	  join(top, complement(X))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(complement(X), top)
24.65/24.85	= { by lemma 25 }
24.65/24.85	  join(X, join(complement(X), complement(X)))
24.65/24.85	= { by lemma 22 }
24.65/24.85	  join(X, complement(X))
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 27: join(X, top) = top.
24.65/24.85	Proof:
24.65/24.85	  join(X, top)
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  join(X, join(complement(X), complement(complement(X))))
24.65/24.85	= { by lemma 24 }
24.65/24.85	  join(complement(complement(X)), top)
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(top, complement(complement(X)))
24.65/24.85	= { by lemma 26 }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 28: join(zero, meet(X, top)) = X.
24.65/24.85	Proof:
24.65/24.85	  join(zero, meet(X, top))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(meet(X, top), zero)
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(complement(join(complement(X), complement(top))), zero)
24.65/24.85	= { by lemma 17 }
24.65/24.85	  join(complement(join(complement(X), complement(top))), complement(top))
24.65/24.85	= { by lemma 27 }
24.65/24.85	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 29: join(meet(X, Y), meet(X, complement(Y))) = X.
24.65/24.85	Proof:
24.65/24.85	  join(meet(X, Y), meet(X, complement(Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(meet(X, complement(Y)), meet(X, Y))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 30: complement(zero) = top.
24.65/24.85	Proof:
24.65/24.85	  complement(zero)
24.65/24.85	= { by lemma 17 }
24.65/24.85	  complement(complement(top))
24.65/24.85	= { by lemma 22 }
24.65/24.85	  complement(join(complement(top), complement(top)))
24.65/24.85	= { by lemma 17 }
24.65/24.85	  complement(join(zero, complement(top)))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  meet(top, top)
24.65/24.85	= { by lemma 28 }
24.65/24.85	  join(zero, meet(meet(top, top), top))
24.65/24.85	= { by axiom 4 (def_zero) }
24.65/24.85	  join(meet(top, complement(top)), meet(meet(top, top), top))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  join(meet(top, complement(top)), meet(top, meet(top, top)))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(meet(top, complement(top)), meet(top, complement(join(complement(top), complement(top)))))
24.65/24.85	= { by lemma 22 }
24.65/24.85	  join(meet(top, complement(top)), meet(top, complement(complement(top))))
24.65/24.85	= { by lemma 29 }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 31: join(zero, meet(X, X)) = X.
24.65/24.85	Proof:
24.65/24.85	  join(zero, meet(X, X))
24.65/24.85	= { by lemma 17 }
24.65/24.85	  join(complement(top), meet(X, X))
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  join(complement(join(complement(X), complement(complement(X)))), meet(X, X))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(complement(join(complement(X), complement(complement(X)))), complement(join(complement(X), complement(X))))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 32: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
24.65/24.85	Proof:
24.65/24.85	  join(X, meet(Y, Y))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(meet(Y, Y), X)
24.65/24.85	= { by lemma 31 }
24.65/24.85	  join(meet(Y, Y), join(zero, meet(X, X)))
24.65/24.85	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.85	  join(join(meet(Y, Y), zero), meet(X, X))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(join(zero, meet(Y, Y)), meet(X, X))
24.65/24.85	= { by lemma 31 }
24.65/24.85	  join(Y, meet(X, X))
24.65/24.85	
24.65/24.85	Lemma 33: join(X, zero) = X.
24.65/24.85	Proof:
24.65/24.85	  join(X, zero)
24.65/24.85	= { by lemma 17 }
24.65/24.85	  join(X, complement(top))
24.65/24.85	= { by lemma 26 }
24.65/24.85	  join(X, complement(join(top, complement(zero))))
24.65/24.85	= { by lemma 30 }
24.65/24.85	  join(X, complement(join(complement(zero), complement(zero))))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(X, meet(zero, zero))
24.65/24.85	= { by lemma 32 }
24.65/24.85	  join(zero, meet(X, X))
24.65/24.85	= { by lemma 31 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 34: join(zero, X) = X.
24.65/24.85	Proof:
24.65/24.85	  join(zero, X)
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(X, zero)
24.65/24.85	= { by lemma 33 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 35: meet(X, top) = X.
24.65/24.85	Proof:
24.65/24.85	  meet(X, top)
24.65/24.85	= { by lemma 34 }
24.65/24.85	  join(zero, meet(X, top))
24.65/24.85	= { by lemma 28 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 36: complement(complement(X)) = X.
24.65/24.85	Proof:
24.65/24.85	  complement(complement(X))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  complement(join(zero, complement(X)))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  meet(X, top)
24.65/24.85	= { by lemma 35 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 37: meet(top, X) = X.
24.65/24.85	Proof:
24.65/24.85	  meet(top, X)
24.65/24.85	= { by lemma 16 }
24.65/24.85	  meet(X, top)
24.65/24.85	= { by lemma 35 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 38: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
24.65/24.85	Proof:
24.65/24.85	  complement(join(complement(X), meet(Y, Z)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  complement(join(complement(X), meet(Z, Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  complement(join(meet(Z, Y), complement(X)))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  meet(join(complement(Z), complement(Y)), X)
24.65/24.85	= { by lemma 16 }
24.65/24.85	  meet(X, join(complement(Z), complement(Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(X, join(complement(Y), complement(Z)))
24.65/24.85	
24.65/24.85	Lemma 39: join(complement(X), complement(Y)) = complement(meet(X, Y)).
24.65/24.85	Proof:
24.65/24.85	  join(complement(X), complement(Y))
24.65/24.85	= { by lemma 37 }
24.65/24.85	  meet(top, join(complement(X), complement(Y)))
24.65/24.85	= { by lemma 38 }
24.65/24.85	  complement(join(complement(top), meet(X, Y)))
24.65/24.85	= { by lemma 17 }
24.65/24.85	  complement(join(zero, meet(X, Y)))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  complement(meet(X, Y))
24.65/24.85	
24.65/24.85	Lemma 40: complement(meet(X, complement(Y))) = join(Y, complement(X)).
24.65/24.85	Proof:
24.65/24.85	  complement(meet(X, complement(Y)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  complement(meet(complement(Y), X))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  complement(meet(join(zero, complement(Y)), X))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  join(complement(join(zero, complement(Y))), complement(X))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  join(meet(Y, top), complement(X))
24.65/24.85	= { by lemma 35 }
24.65/24.85	  join(Y, complement(X))
24.65/24.85	
24.65/24.85	Lemma 41: join(complement(converse(X)), converse(join(X, Y))) = top.
24.65/24.85	Proof:
24.65/24.85	  join(complement(converse(X)), converse(join(X, Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(complement(converse(X)), converse(join(Y, X)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(converse(join(Y, X)), complement(converse(X)))
24.65/24.85	= { by axiom 6 (converse_additivity) }
24.65/24.85	  join(join(converse(Y), converse(X)), complement(converse(X)))
24.65/24.85	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.85	  join(converse(Y), join(converse(X), complement(converse(X))))
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  join(converse(Y), top)
24.65/24.85	= { by lemma 27 }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 42: meet(converse(X), converse(join(X, Y))) = converse(X).
24.65/24.85	Proof:
24.65/24.85	  meet(converse(X), converse(join(X, Y)))
24.65/24.85	= { by lemma 33 }
24.65/24.85	  join(meet(converse(X), converse(join(X, Y))), zero)
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), zero)
24.65/24.85	= { by lemma 17 }
24.65/24.85	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(top))
24.65/24.85	= { by lemma 41 }
24.65/24.85	  join(complement(join(complement(converse(X)), complement(converse(join(X, Y))))), complement(join(complement(converse(X)), converse(join(X, Y)))))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  converse(X)
24.65/24.85	
24.65/24.85	Lemma 43: meet(X, join(X, Y)) = X.
24.65/24.85	Proof:
24.65/24.85	  meet(X, join(X, Y))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  meet(converse(converse(X)), join(X, Y))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  meet(converse(converse(X)), converse(converse(join(X, Y))))
24.65/24.85	= { by axiom 6 (converse_additivity) }
24.65/24.85	  meet(converse(converse(X)), converse(join(converse(X), converse(Y))))
24.65/24.85	= { by lemma 42 }
24.65/24.85	  converse(converse(X))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 44: complement(meet(complement(X), Y)) = join(X, complement(Y)).
24.65/24.85	Proof:
24.65/24.85	  complement(meet(complement(X), Y))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  complement(meet(Y, complement(X)))
24.65/24.85	= { by lemma 40 }
24.65/24.85	  join(X, complement(Y))
24.65/24.85	
24.65/24.85	Lemma 45: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
24.65/24.85	Proof:
24.65/24.85	  meet(join(X, complement(Y)), complement(meet(X, Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(join(complement(Y), X), complement(meet(X, Y)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  meet(join(complement(Y), X), complement(meet(Y, X)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  meet(complement(meet(Y, X)), join(complement(Y), X))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  complement(Y)
24.65/24.85	
24.65/24.85	Lemma 46: meet(X, X) = X.
24.65/24.85	Proof:
24.65/24.85	  meet(X, X)
24.65/24.85	= { by lemma 34 }
24.65/24.85	  join(zero, meet(X, X))
24.65/24.85	= { by lemma 31 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 47: complement(join(Y, complement(X))) = meet(X, complement(Y)).
24.65/24.85	Proof:
24.65/24.85	  complement(join(Y, complement(X)))
24.65/24.85	= { by lemma 46 }
24.65/24.85	  complement(join(Y, meet(complement(X), complement(X))))
24.65/24.85	= { by lemma 32 }
24.65/24.85	  complement(join(complement(X), meet(Y, Y)))
24.65/24.85	= { by lemma 38 }
24.65/24.85	  meet(X, join(complement(Y), complement(Y)))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  meet(X, complement(meet(Y, Y)))
24.65/24.85	= { by lemma 46 }
24.65/24.85	  meet(X, complement(Y))
24.65/24.85	
24.65/24.85	Lemma 48: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
24.65/24.85	Proof:
24.65/24.85	  meet(X, complement(meet(X, Y)))
24.65/24.85	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, Y)))
24.65/24.85	= { by lemma 43 }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, join(Y, complement(X))))))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, join(complement(X), Y)))))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), complement(meet(Y, join(complement(X), Y)))))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
24.65/24.85	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.85	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
24.65/24.85	= { by lemma 44 }
24.65/24.85	  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))))
24.65/24.85	= { by lemma 45 }
24.65/24.85	  complement(join(complement(X), Y))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  complement(join(Y, complement(X)))
24.65/24.85	= { by lemma 47 }
24.65/24.85	  meet(X, complement(Y))
24.65/24.85	
24.65/24.85	Lemma 49: join(complement(X), meet(X, Y)) = join(Y, complement(X)).
24.65/24.85	Proof:
24.65/24.85	  join(complement(X), meet(X, Y))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(meet(X, Y), complement(X))
24.65/24.85	= { by lemma 40 }
24.65/24.85	  complement(meet(X, complement(meet(X, Y))))
24.65/24.85	= { by lemma 48 }
24.65/24.85	  complement(meet(X, complement(Y)))
24.65/24.85	= { by lemma 40 }
24.65/24.85	  join(Y, complement(X))
24.65/24.85	
24.65/24.85	Lemma 50: meet(join(X, Y), join(X, complement(Y))) = X.
24.65/24.85	Proof:
24.65/24.85	  meet(join(X, Y), join(X, complement(Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(join(Y, X), join(X, complement(Y)))
24.65/24.85	= { by lemma 35 }
24.65/24.85	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
24.65/24.85	= { by lemma 40 }
24.65/24.85	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, complement(X))))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, join(zero, complement(X)))))
24.65/24.85	= { by lemma 45 }
24.65/24.85	  complement(join(zero, complement(X)))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  meet(X, top)
24.65/24.85	= { by lemma 35 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 51: converse(join(X, converse(Y))) = join(Y, converse(X)).
24.65/24.85	Proof:
24.65/24.85	  converse(join(X, converse(Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  converse(join(converse(Y), X))
24.65/24.85	= { by axiom 6 (converse_additivity) }
24.65/24.85	  join(converse(converse(Y)), converse(X))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  join(Y, converse(X))
24.65/24.85	
24.65/24.85	Lemma 52: converse(join(converse(X), Y)) = join(X, converse(Y)).
24.65/24.85	Proof:
24.65/24.85	  converse(join(converse(X), Y))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  converse(join(Y, converse(X)))
24.65/24.85	= { by lemma 51 }
24.65/24.85	  join(X, converse(Y))
24.65/24.85	
24.65/24.85	Lemma 53: join(X, converse(top)) = converse(top).
24.65/24.85	Proof:
24.65/24.85	  join(X, converse(top))
24.65/24.85	= { by lemma 52 }
24.65/24.85	  converse(join(converse(X), top))
24.65/24.85	= { by lemma 27 }
24.65/24.85	  converse(top)
24.65/24.85	
24.65/24.85	Lemma 54: converse(top) = top.
24.65/24.85	Proof:
24.65/24.85	  converse(top)
24.65/24.85	= { by lemma 53 }
24.65/24.85	  join(?, converse(top))
24.65/24.85	= { by lemma 53 }
24.65/24.85	  join(?, join(complement(?), converse(top)))
24.65/24.85	= { by lemma 24 }
24.65/24.85	  join(converse(top), top)
24.65/24.85	= { by lemma 27 }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 55: join(X, converse(complement(converse(X)))) = top.
24.65/24.85	Proof:
24.65/24.85	  join(X, converse(complement(converse(X))))
24.65/24.85	= { by lemma 52 }
24.65/24.85	  converse(join(converse(X), complement(converse(X))))
24.65/24.85	= { by axiom 7 (def_top) }
24.65/24.85	  converse(top)
24.65/24.85	= { by lemma 54 }
24.65/24.85	  top
24.65/24.85	
24.65/24.85	Lemma 56: join(X, complement(converse(complement(converse(X))))) = X.
24.65/24.85	Proof:
24.65/24.85	  join(X, complement(converse(complement(converse(X)))))
24.65/24.85	= { by lemma 37 }
24.65/24.85	  meet(top, join(X, complement(converse(complement(converse(X))))))
24.65/24.85	= { by lemma 55 }
24.65/24.85	  meet(join(X, converse(complement(converse(X)))), join(X, complement(converse(complement(converse(X))))))
24.65/24.85	= { by lemma 50 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 57: meet(X, join(Y, complement(X))) = meet(X, Y).
24.65/24.85	Proof:
24.65/24.85	  meet(X, join(Y, complement(X)))
24.65/24.85	= { by lemma 40 }
24.65/24.85	  meet(X, complement(meet(X, complement(Y))))
24.65/24.85	= { by lemma 48 }
24.65/24.85	  meet(X, complement(complement(Y)))
24.65/24.85	= { by lemma 36 }
24.65/24.85	  meet(X, Y)
24.65/24.85	
24.65/24.85	Lemma 58: converse(complement(converse(complement(X)))) = X.
24.65/24.85	Proof:
24.65/24.85	  converse(complement(converse(complement(X))))
24.65/24.85	= { by lemma 50 }
24.65/24.85	  meet(join(converse(complement(converse(complement(X)))), X), join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(join(X, converse(complement(converse(complement(X))))), join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  meet(join(X, converse(complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by lemma 52 }
24.65/24.85	  meet(converse(join(converse(X), complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by lemma 56 }
24.65/24.85	  meet(converse(converse(X)), join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  meet(X, join(converse(complement(converse(complement(X)))), complement(X)))
24.65/24.85	= { by lemma 57 }
24.65/24.85	  meet(X, converse(complement(converse(complement(X)))))
24.65/24.85	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.85	  complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))
24.65/24.85	= { by lemma 56 }
24.65/24.85	  complement(complement(X))
24.65/24.85	= { by lemma 36 }
24.65/24.85	  X
24.65/24.85	
24.65/24.85	Lemma 59: complement(converse(complement(X))) = converse(X).
24.65/24.85	Proof:
24.65/24.85	  complement(converse(complement(X)))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  converse(converse(complement(converse(complement(X)))))
24.65/24.85	= { by lemma 58 }
24.65/24.85	  converse(X)
24.65/24.85	
24.65/24.85	Lemma 60: meet(converse(X), converse(join(Y, complement(X)))) = converse(meet(X, Y)).
24.65/24.85	Proof:
24.65/24.85	  meet(converse(X), converse(join(Y, complement(X))))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  meet(converse(join(Y, complement(X))), converse(X))
24.65/24.85	= { by lemma 49 }
24.65/24.85	  meet(converse(join(complement(X), meet(X, Y))), converse(X))
24.65/24.85	= { by axiom 6 (converse_additivity) }
24.65/24.85	  meet(join(converse(complement(X)), converse(meet(X, Y))), converse(X))
24.65/24.85	= { by lemma 59 }
24.65/24.85	  meet(join(converse(complement(X)), complement(converse(complement(meet(X, Y))))), converse(X))
24.65/24.85	= { by lemma 59 }
24.65/24.85	  meet(join(converse(complement(X)), complement(converse(complement(meet(X, Y))))), complement(converse(complement(X))))
24.65/24.85	= { by lemma 42 }
24.65/24.85	  meet(join(converse(complement(X)), complement(converse(complement(meet(X, Y))))), complement(meet(converse(complement(X)), converse(join(complement(X), complement(Y))))))
24.65/24.85	= { by lemma 39 }
24.65/24.85	  meet(join(converse(complement(X)), complement(converse(complement(meet(X, Y))))), complement(meet(converse(complement(X)), converse(complement(meet(X, Y))))))
24.65/24.85	= { by lemma 45 }
24.65/24.85	  complement(converse(complement(meet(X, Y))))
24.65/24.85	= { by lemma 59 }
24.65/24.85	  converse(meet(X, Y))
24.65/24.85	
24.65/24.85	Lemma 61: complement(converse(X)) = converse(complement(X)).
24.65/24.85	Proof:
24.65/24.85	  complement(converse(X))
24.65/24.85	= { by lemma 35 }
24.65/24.85	  complement(converse(meet(X, top)))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  complement(converse(complement(join(zero, complement(X)))))
24.65/24.85	= { by lemma 59 }
24.65/24.85	  converse(join(zero, complement(X)))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  converse(complement(X))
24.65/24.85	
24.65/24.85	Lemma 62: meet(X, join(complement(X), Y)) = meet(X, Y).
24.65/24.85	Proof:
24.65/24.85	  meet(X, join(complement(X), Y))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  meet(X, join(Y, complement(X)))
24.65/24.85	= { by lemma 57 }
24.65/24.85	  meet(X, Y)
24.65/24.85	
24.65/24.85	Lemma 63: converse(meet(X, converse(Y))) = meet(Y, converse(X)).
24.65/24.85	Proof:
24.65/24.85	  converse(meet(X, converse(Y)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  converse(meet(converse(Y), X))
24.65/24.85	= { by lemma 60 }
24.65/24.85	  meet(converse(converse(Y)), converse(join(X, complement(converse(Y)))))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  meet(Y, converse(join(X, complement(converse(Y)))))
24.65/24.85	= { by lemma 61 }
24.65/24.85	  meet(Y, converse(join(X, converse(complement(Y)))))
24.65/24.85	= { by lemma 51 }
24.65/24.85	  meet(Y, join(complement(Y), converse(X)))
24.65/24.85	= { by lemma 62 }
24.65/24.85	  meet(Y, converse(X))
24.65/24.85	
24.65/24.85	Lemma 64: converse(composition(X, converse(Y))) = composition(Y, converse(X)).
24.65/24.85	Proof:
24.65/24.85	  converse(composition(X, converse(Y)))
24.65/24.85	= { by axiom 3 (converse_multiplicativity) }
24.65/24.85	  composition(converse(converse(Y)), converse(X))
24.65/24.85	= { by axiom 11 (converse_idempotence) }
24.65/24.85	  composition(Y, converse(X))
24.65/24.85	
24.65/24.85	Lemma 65: converse(one) = one.
24.65/24.85	Proof:
24.65/24.85	  converse(one)
24.65/24.85	= { by axiom 2 (composition_identity) }
24.65/24.85	  composition(converse(one), one)
24.65/24.85	= { by lemma 19 }
24.65/24.85	  one
24.65/24.85	
24.65/24.85	Lemma 66: join(X, complement(join(X, Y))) = join(X, complement(Y)).
24.65/24.85	Proof:
24.65/24.85	  join(X, complement(join(X, Y)))
24.65/24.85	= { by lemma 43 }
24.65/24.85	  join(meet(X, join(X, complement(Y))), complement(join(X, Y)))
24.65/24.85	= { by lemma 44 }
24.65/24.85	  join(meet(X, complement(meet(complement(X), Y))), complement(join(X, Y)))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), complement(join(X, Y)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), complement(join(Y, X)))
24.65/24.85	= { by lemma 35 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), complement(join(Y, meet(X, top))))
24.65/24.85	= { by lemma 18 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), complement(join(Y, complement(join(zero, complement(X))))))
24.65/24.85	= { by lemma 47 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), meet(join(zero, complement(X)), complement(Y)))
24.65/24.85	= { by lemma 34 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), meet(complement(X), complement(Y)))
24.65/24.85	= { by lemma 48 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), meet(complement(X), complement(meet(complement(X), Y))))
24.65/24.85	= { by lemma 16 }
24.65/24.85	  join(meet(complement(meet(complement(X), Y)), X), meet(complement(meet(complement(X), Y)), complement(X)))
24.65/24.85	= { by lemma 29 }
24.65/24.85	  complement(meet(complement(X), Y))
24.65/24.85	= { by lemma 44 }
24.65/24.85	  join(X, complement(Y))
24.65/24.85	
24.65/24.85	Lemma 67: join(complement(one), converse(complement(one))) = converse(complement(one)).
24.65/24.85	Proof:
24.65/24.85	  join(complement(one), converse(complement(one)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(converse(complement(one)), complement(one))
24.65/24.85	= { by lemma 66 }
24.65/24.85	  join(converse(complement(one)), complement(join(converse(complement(one)), one)))
24.65/24.85	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.85	  join(converse(complement(one)), complement(join(one, converse(complement(one)))))
24.65/24.85	= { by lemma 65 }
24.65/24.85	  join(converse(complement(one)), complement(join(one, converse(complement(converse(one))))))
24.65/24.85	= { by lemma 55 }
24.65/24.85	  join(converse(complement(one)), complement(top))
24.65/24.85	= { by lemma 17 }
24.65/24.85	  join(converse(complement(one)), zero)
24.65/24.85	= { by lemma 33 }
24.65/24.86	  converse(complement(one))
24.65/24.86	
24.65/24.86	Lemma 68: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
24.65/24.86	Proof:
24.65/24.86	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
24.65/24.86	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.86	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
24.65/24.86	= { by axiom 3 (converse_multiplicativity) }
24.65/24.86	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
24.65/24.86	= { by axiom 9 (composition_distributivity) }
24.65/24.86	  composition(join(converse(X), Y), converse(Z))
24.65/24.86	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.86	  composition(join(Y, converse(X)), converse(Z))
24.65/24.86	
24.65/24.86	Lemma 69: join(X, converse(zero)) = X.
24.65/24.86	Proof:
24.65/24.86	  join(X, converse(zero))
24.65/24.86	= { by lemma 52 }
24.65/24.86	  converse(join(converse(X), zero))
24.65/24.86	= { by lemma 33 }
24.65/24.86	  converse(converse(X))
24.65/24.86	= { by axiom 11 (converse_idempotence) }
24.65/24.86	  X
24.65/24.86	
24.65/24.86	Lemma 70: converse(zero) = zero.
24.65/24.86	Proof:
24.65/24.86	  converse(zero)
24.65/24.86	= { by lemma 34 }
24.65/24.86	  join(zero, converse(zero))
24.65/24.86	= { by lemma 69 }
24.65/24.86	  zero
24.65/24.86	
24.65/24.86	Lemma 71: join(converse(complement(X)), composition(Y, converse(complement(composition(X, Y))))) = converse(complement(X)).
24.65/24.86	Proof:
24.65/24.86	  join(converse(complement(X)), composition(Y, converse(complement(composition(X, Y)))))
24.65/24.86	= { by lemma 61 }
24.65/24.86	  join(complement(converse(X)), composition(Y, converse(complement(composition(X, Y)))))
24.65/24.86	= { by axiom 11 (converse_idempotence) }
24.65/24.86	  join(complement(converse(X)), composition(converse(converse(Y)), converse(complement(composition(X, Y)))))
24.65/24.86	= { by lemma 61 }
24.65/24.86	  join(complement(converse(X)), composition(converse(converse(Y)), complement(converse(composition(X, Y)))))
24.65/24.86	= { by axiom 3 (converse_multiplicativity) }
24.65/24.86	  join(complement(converse(X)), composition(converse(converse(Y)), complement(composition(converse(Y), converse(X)))))
24.65/24.86	= { by lemma 21 }
24.65/24.86	  complement(converse(X))
24.65/24.86	= { by lemma 61 }
24.65/24.86	  converse(complement(X))
24.65/24.86	
24.65/24.86	Lemma 72: composition(join(X, complement(composition(top, Y))), converse(Y)) = composition(X, converse(Y)).
24.65/24.86	Proof:
24.65/24.86	  composition(join(X, complement(composition(top, Y))), converse(Y))
24.65/24.86	= { by axiom 11 (converse_idempotence) }
24.65/24.86	  composition(join(X, converse(converse(complement(composition(top, Y))))), converse(Y))
24.65/24.86	= { by lemma 68 }
24.65/24.86	  join(composition(X, converse(Y)), converse(composition(Y, converse(complement(composition(top, Y))))))
24.65/24.86	= { by lemma 34 }
24.65/24.86	  join(composition(X, converse(Y)), converse(join(zero, composition(Y, converse(complement(composition(top, Y)))))))
24.65/24.86	= { by lemma 70 }
24.65/24.86	  join(composition(X, converse(Y)), converse(join(converse(zero), composition(Y, converse(complement(composition(top, Y)))))))
24.65/24.86	= { by lemma 17 }
24.65/24.86	  join(composition(X, converse(Y)), converse(join(converse(complement(top)), composition(Y, converse(complement(composition(top, Y)))))))
24.65/24.86	= { by lemma 71 }
24.65/24.86	  join(composition(X, converse(Y)), converse(converse(complement(top))))
24.65/24.86	= { by lemma 17 }
24.65/24.86	  join(composition(X, converse(Y)), converse(converse(zero)))
24.65/24.86	= { by lemma 70 }
24.65/24.86	  join(composition(X, converse(Y)), converse(zero))
24.65/24.86	= { by lemma 69 }
24.65/24.88	  composition(X, converse(Y))
24.65/24.88	
24.65/24.88	Lemma 73: composition(meet(X, one), converse(meet(X, one))) = meet(X, one).
24.65/24.88	Proof:
24.65/24.88	  composition(meet(X, one), converse(meet(X, one)))
24.65/24.88	= { by lemma 16 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, X)))
24.65/24.88	= { by lemma 57 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(X, complement(one)))))
24.65/24.88	= { by lemma 49 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), meet(one, X)))))
24.65/24.88	= { by lemma 16 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), meet(X, one)))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(meet(X, one), complement(one)))))
24.65/24.88	= { by lemma 40 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, complement(meet(one, complement(meet(X, one)))))))
24.65/24.88	= { by lemma 39 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), complement(complement(meet(X, one)))))))
24.65/24.88	= { by lemma 20 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(one, complement(complement(meet(X, one))))))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), complement(one)))))
24.65/24.88	= { by lemma 21 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(complement(one), composition(converse(complement(meet(X, one))), complement(composition(complement(meet(X, one)), one))))))))
24.65/24.88	= { by axiom 2 (composition_identity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(composition(one, complement(complement(meet(X, one)))), join(complement(one), composition(converse(complement(meet(X, one))), complement(complement(meet(X, one)))))))))
24.65/24.88	= { by lemma 23 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), join(composition(converse(complement(meet(X, one))), complement(complement(meet(X, one)))), composition(one, complement(complement(meet(X, one)))))))))
24.65/24.88	= { by axiom 9 (composition_distributivity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(converse(complement(meet(X, one))), one), complement(complement(meet(X, one))))))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(one, converse(complement(meet(X, one)))), complement(complement(meet(X, one))))))))
24.65/24.88	= { by lemma 65 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(converse(one), converse(complement(meet(X, one)))), complement(complement(meet(X, one))))))))
24.65/24.88	= { by lemma 16 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(converse(one), converse(complement(meet(one, X)))), complement(complement(meet(X, one))))))))
24.65/24.88	= { by lemma 61 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(converse(one), complement(converse(meet(one, X)))), complement(complement(meet(X, one))))))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(complement(converse(meet(one, X))), converse(one)), complement(complement(meet(X, one))))))))
24.65/24.88	= { by axiom 5 (maddux4_definiton_of_meet) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(join(complement(converse(complement(join(complement(one), complement(X))))), converse(one)), complement(complement(meet(X, one))))))))
24.65/24.88	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(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(complement(meet(X, one))))))))
24.65/24.88	= { by lemma 41 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(top, complement(complement(meet(X, one))))))))
24.65/24.88	= { by lemma 36 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, join(complement(one), composition(top, meet(X, one))))))
24.65/24.88	= { by lemma 62 }
24.65/24.88	  composition(meet(X, one), converse(meet(one, composition(top, meet(X, one)))))
24.65/24.88	= { by lemma 64 }
24.65/24.88	  converse(composition(meet(one, composition(top, meet(X, one))), converse(meet(X, one))))
24.65/24.88	= { by lemma 16 }
24.65/24.88	  converse(composition(meet(composition(top, meet(X, one)), one), converse(meet(X, one))))
24.65/24.88	= { by lemma 72 }
24.65/24.88	  converse(composition(join(meet(composition(top, meet(X, one)), one), complement(composition(top, meet(X, one)))), converse(meet(X, one))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  converse(composition(join(complement(composition(top, meet(X, one))), meet(composition(top, meet(X, one)), one)), converse(meet(X, one))))
24.65/24.88	= { by lemma 49 }
24.65/24.88	  converse(composition(join(one, complement(composition(top, meet(X, one)))), converse(meet(X, one))))
24.65/24.88	= { by lemma 72 }
24.65/24.88	  converse(composition(one, converse(meet(X, one))))
24.65/24.88	= { by lemma 64 }
24.65/24.88	  composition(meet(X, one), converse(one))
24.65/24.88	= { by lemma 65 }
24.65/24.88	  composition(meet(X, one), one)
24.65/24.88	= { by axiom 2 (composition_identity) }
24.65/24.88	  meet(X, one)
24.65/24.88	
24.65/24.88	Lemma 74: converse(complement(converse(X))) = complement(X).
24.65/24.88	Proof:
24.65/24.88	  converse(complement(converse(X)))
24.65/24.88	= { by lemma 35 }
24.65/24.88	  converse(complement(converse(meet(X, top))))
24.65/24.88	= { by lemma 18 }
24.65/24.88	  converse(complement(converse(complement(join(zero, complement(X))))))
24.65/24.88	= { by lemma 58 }
24.65/24.88	  join(zero, complement(X))
24.65/24.88	= { by lemma 34 }
24.65/24.88	  complement(X)
24.65/24.88	
24.65/24.88	Lemma 75: join(top, X) = top.
24.65/24.88	Proof:
24.65/24.88	  join(top, X)
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  join(X, top)
24.65/24.88	= { by lemma 27 }
24.65/24.88	  top
24.65/24.88	
24.65/24.88	Lemma 76: join(Y, composition(X, Y)) = composition(join(X, one), Y).
24.65/24.88	Proof:
24.65/24.88	  join(Y, composition(X, Y))
24.65/24.88	= { by lemma 20 }
24.65/24.88	  join(composition(one, Y), composition(X, Y))
24.65/24.88	= { by axiom 9 (composition_distributivity) }
24.65/24.88	  composition(join(one, X), Y)
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  composition(join(X, one), Y)
24.65/24.88	
24.65/24.88	Lemma 77: composition(top, zero) = zero.
24.65/24.88	Proof:
24.65/24.88	  composition(top, zero)
24.65/24.88	= { by lemma 75 }
24.65/24.88	  composition(join(top, one), zero)
24.65/24.88	= { by lemma 54 }
24.65/24.88	  composition(join(converse(top), one), zero)
24.65/24.88	= { by lemma 17 }
24.65/24.88	  composition(join(converse(top), one), complement(top))
24.65/24.88	= { by lemma 76 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(top)))
24.65/24.88	= { by lemma 27 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(composition(top, top), top))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(top, composition(top, top)))))
24.65/24.88	= { by axiom 7 (def_top) }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(join(top, complement(top)), composition(top, top)))))
24.65/24.88	= { by lemma 17 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(join(top, zero), composition(top, top)))))
24.65/24.88	= { by axiom 10 (maddux1_join_commutativity) }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(join(zero, top), composition(top, top)))))
24.65/24.88	= { by axiom 12 (maddux2_join_associativity) }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(zero, join(top, composition(top, top))))))
24.65/24.88	= { by lemma 76 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(join(zero, composition(join(top, one), top)))))
24.65/24.88	= { by lemma 34 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(composition(join(top, one), top))))
24.65/24.88	= { by lemma 75 }
24.65/24.88	  join(complement(top), composition(converse(top), complement(composition(top, top))))
24.65/24.88	= { by lemma 21 }
24.65/24.88	  complement(top)
24.65/24.88	= { by lemma 17 }
26.43/26.65	  zero
26.43/26.65	
26.43/26.65	Goal 1 (goals_1): sK3_goals_X0 = join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0).
26.43/26.65	Proof:
26.43/26.65	  sK3_goals_X0
26.43/26.65	= { by lemma 36 }
26.43/26.65	  complement(complement(sK3_goals_X0))
26.43/26.65	= { by lemma 45 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), complement(complement(sK3_goals_X0))), complement(meet(composition(sK1_goals_X2, converse(sK2_goals_X1)), complement(sK3_goals_X0))))
26.43/26.65	= { by lemma 36 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(meet(composition(sK1_goals_X2, converse(sK2_goals_X1)), complement(sK3_goals_X0))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(meet(complement(sK3_goals_X0), composition(sK1_goals_X2, converse(sK2_goals_X1)))))
26.43/26.65	= { by axiom 11 (converse_idempotence) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(meet(complement(sK3_goals_X0), composition(sK1_goals_X2, converse(sK2_goals_X1)))))))
26.43/26.65	= { by lemma 64 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(meet(complement(sK3_goals_X0), converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))))))
26.43/26.65	= { by lemma 63 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))))))
26.43/26.65	= { by axiom 2 (composition_identity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), one))))
26.43/26.65	= { by lemma 33 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, zero)))))
26.43/26.65	= { by lemma 17 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(top))))))
26.43/26.65	= { by lemma 54 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(top)))))))
26.43/26.65	= { by lemma 27 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(join(complement(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2)))))), top))))))))
26.43/26.65	= { by lemma 25 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(join(converse(one), join(complement(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2)))))), complement(converse(one)))))))))))
26.43/26.65	= { by lemma 39 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(join(converse(one), complement(meet(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))), converse(one)))))))))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(join(converse(one), complement(meet(converse(one), composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))))))))
26.43/26.65	= { by lemma 52 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(join(one, converse(complement(meet(converse(one), composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))))))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(join(one, converse(complement(meet(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))), converse(one)))))))))))
26.43/26.65	= { by lemma 66 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, complement(converse(complement(meet(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))), converse(one))))))))))
26.43/26.65	= { by lemma 59 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, converse(meet(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))), converse(one))))))))
26.43/26.65	= { by lemma 63 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, meet(one, converse(composition(converse(converse(complement(sK3_goals_X0))), converse(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 64 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(converse(converse(complement(sK3_goals_X0)))))))))))
26.43/26.65	= { by axiom 11 (converse_idempotence) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))), one)))))
26.43/26.65	= { by axiom 11 (converse_idempotence) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))), one)))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), join(one, meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))))
26.43/26.65	= { by axiom 3 (converse_multiplicativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(composition(converse(join(one, meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))), converse(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by axiom 6 (converse_additivity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(composition(join(converse(one), converse(meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))), converse(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by lemma 68 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(join(composition(converse(one), converse(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))))), converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))))
26.43/26.65	= { by lemma 19 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(join(converse(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0)))), converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))))
26.43/26.65	= { by axiom 6 (converse_additivity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(converse(converse(join(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))))
26.43/26.65	= { by axiom 11 (converse_idempotence) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by axiom 2 (composition_identity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(meet(composition(composition(sK2_goals_X1, converse(sK1_goals_X2)), one), converse(complement(sK3_goals_X0))), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by axiom 2 (composition_identity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(meet(composition(composition(sK2_goals_X1, converse(sK1_goals_X2)), one), converse(complement(sK3_goals_X0))), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), composition(converse(complement(sK3_goals_X0)), one)), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by lemma 65 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(meet(composition(composition(sK2_goals_X1, converse(sK1_goals_X2)), one), converse(complement(sK3_goals_X0))), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), composition(converse(complement(sK3_goals_X0)), converse(one))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0)))))))))
26.43/26.65	= { by axiom 14 (dedekind_law) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), composition(converse(complement(sK3_goals_X0)), converse(one))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0))))))))
26.43/26.65	= { by lemma 65 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), composition(converse(complement(sK3_goals_X0)), one)), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0))))))))
26.43/26.65	= { by axiom 2 (composition_identity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, composition(converse(composition(sK2_goals_X1, converse(sK1_goals_X2))), converse(complement(sK3_goals_X0))))))))
26.43/26.65	= { by axiom 3 (converse_multiplicativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))
26.43/26.65	= { by lemma 57 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(one, join(converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), complement(one)))))))
26.43/26.65	= { by lemma 65 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), complement(one)))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(complement(one), converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by axiom 11 (converse_idempotence) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(converse(converse(complement(one))), converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by lemma 67 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(converse(join(complement(one), converse(complement(one)))), converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by lemma 51 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(join(complement(one), converse(complement(one))), converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by lemma 67 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), join(converse(complement(one)), converse(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by axiom 6 (converse_additivity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), converse(join(complement(one), composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(converse(one), converse(join(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), complement(one))))))))
26.43/26.65	= { by lemma 60 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), converse(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), converse(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one))))))
26.43/26.65	= { by lemma 73 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), converse(composition(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one), converse(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one))))))))
26.43/26.65	= { by lemma 64 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), composition(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one), converse(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one)))))))
26.43/26.65	= { by lemma 73 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one)))))
26.43/26.65	= { by lemma 37 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), meet(top, one))))))
26.43/26.65	= { by lemma 35 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), meet(top, one)), top)))))
26.43/26.65	= { by lemma 18 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, complement(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), meet(top, one)))))))))
26.43/26.65	= { by axiom 5 (maddux4_definiton_of_meet) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, complement(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), complement(join(complement(top), complement(one)))))))))))
26.43/26.65	= { by lemma 40 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, join(join(complement(top), complement(one)), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))))
26.43/26.65	= { by axiom 12 (maddux2_join_associativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, join(complement(top), join(complement(one), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 39 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, join(complement(top), complement(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 39 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), complement(join(zero, complement(meet(top, meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 18 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(top, meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))), top)))))
26.43/26.65	= { by lemma 35 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(top, meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))))))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), top)))))
26.43/26.65	= { by lemma 27 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), top))))))
26.43/26.65	= { by lemma 27 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2)))), top)))))))
26.43/26.65	= { by axiom 7 (def_top) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2)))), join(composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2)))), complement(composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2)))))))))))))
26.43/26.65	= { by axiom 12 (maddux2_join_associativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(join(composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2)))), composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2))))), complement(composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by axiom 9 (composition_distributivity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(composition(join(complement(sK1_goals_X2), composition(complement(sK3_goals_X0), sK2_goals_X1)), converse(complement(complement(sK1_goals_X2)))), complement(composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(complement(composition(composition(complement(sK3_goals_X0), sK2_goals_X1), converse(complement(complement(sK1_goals_X2))))), composition(join(complement(sK1_goals_X2), composition(complement(sK3_goals_X0), sK2_goals_X1)), converse(complement(complement(sK1_goals_X2)))))))))))
26.43/26.65	= { by axiom 8 (composition_associativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2)))))), composition(join(complement(sK1_goals_X2), composition(complement(sK3_goals_X0), sK2_goals_X1)), converse(complement(complement(sK1_goals_X2)))))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2)))))), composition(join(composition(complement(sK3_goals_X0), sK2_goals_X1), complement(sK1_goals_X2)), converse(complement(complement(sK1_goals_X2)))))))))))
26.43/26.65	= { by axiom 15 (goals) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2)))))), composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2)))))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), join(composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2)))), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2)))))))))))))
26.43/26.65	= { by axiom 12 (maddux2_join_associativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(join(complement(one), composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2))))), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 74 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(join(converse(complement(converse(one))), composition(complement(sK1_goals_X2), converse(complement(complement(sK1_goals_X2))))), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 19 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(join(converse(complement(converse(one))), composition(complement(sK1_goals_X2), converse(complement(composition(converse(one), complement(sK1_goals_X2)))))), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 71 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(converse(complement(converse(one))), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by lemma 74 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(one), complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))))))))))
26.43/26.65	= { by axiom 10 (maddux1_join_commutativity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), join(complement(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2)))))), complement(one)))))))
26.43/26.65	= { by lemma 39 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), complement(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(complement(complement(sK1_goals_X2))))), one)))))))
26.43/26.65	= { by lemma 36 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), complement(meet(composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))), one)))))))
26.43/26.65	= { by lemma 16 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), meet(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2)))), complement(meet(one, composition(complement(sK3_goals_X0), composition(sK2_goals_X1, converse(sK1_goals_X2))))))))))
26.43/26.65	= { by axiom 4 (def_zero) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), zero))))
26.43/26.65	= { by lemma 34 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(zero, composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), zero)))))
26.43/26.65	= { by lemma 77 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(join(composition(top, zero), composition(meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0))), zero)))))
26.43/26.65	= { by axiom 9 (composition_distributivity) }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(join(top, meet(composition(sK2_goals_X1, converse(sK1_goals_X2)), converse(complement(sK3_goals_X0)))), zero))))
26.43/26.65	= { by lemma 75 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(composition(top, zero))))
26.43/26.65	= { by lemma 77 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(converse(zero)))
26.43/26.65	= { by lemma 70 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), complement(zero))
26.43/26.65	= { by lemma 30 }
26.43/26.65	  meet(join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0), top)
26.43/26.65	= { by lemma 35 }
26.43/26.65	  join(composition(sK1_goals_X2, converse(sK2_goals_X1)), sK3_goals_X0)
26.43/26.65	% SZS output end Proof
26.43/26.65	
26.43/26.65	RESULT: Theorem (the conjecture is true).
26.43/26.66	EOF