0.03/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.34	% Computer   : n026.cluster.edu
0.14/0.34	% Model      : x86_64 x86_64
0.14/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.34	% Memory     : 8042.1875MB
0.14/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.34	% CPULimit   : 180
0.14/0.34	% DateTime   : Thu Aug 29 14:26:20 EDT 2019
0.14/0.34	% CPUTime    : 
8.08/8.32	% SZS status Theorem
8.08/8.32	
8.16/8.32	% SZS output start Proof
8.16/8.32	Take the following subset of the input axioms:
8.32/8.49	  fof(composition_associativity, axiom, ![X0, X1, X2]: composition(composition(X0, X1), X2)=composition(X0, composition(X1, X2))).
8.32/8.49	  fof(composition_distributivity, axiom, ![X0, X1, X2]: composition(join(X0, X1), X2)=join(composition(X0, X2), composition(X1, X2))).
8.32/8.49	  fof(composition_identity, axiom, ![X0]: X0=composition(X0, one)).
8.32/8.49	  fof(converse_additivity, axiom, ![X0, X1]: converse(join(X0, X1))=join(converse(X0), converse(X1))).
8.32/8.49	  fof(converse_cancellativity, axiom, ![X0, X1]: complement(X1)=join(composition(converse(X0), complement(composition(X0, X1))), complement(X1))).
8.32/8.49	  fof(converse_idempotence, axiom, ![X0]: X0=converse(converse(X0))).
8.32/8.49	  fof(converse_multiplicativity, axiom, ![X0, X1]: converse(composition(X0, X1))=composition(converse(X1), converse(X0))).
8.32/8.49	  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)))).
8.32/8.49	  fof(def_top, axiom, ![X0]: join(X0, complement(X0))=top).
8.32/8.49	  fof(def_zero, axiom, ![X0]: meet(X0, complement(X0))=zero).
8.32/8.49	  fof(goals, conjecture, ![X0, X1, X2]: ((one=join(X0, one) & join(X1, one)=one) => composition(meet(X0, X1), X2)=meet(composition(X0, X2), composition(X1, X2)))).
8.32/8.49	  fof(maddux1_join_commutativity, axiom, ![X0, X1]: join(X1, X0)=join(X0, X1)).
8.32/8.49	  fof(maddux2_join_associativity, axiom, ![X0, X1, X2]: join(X0, join(X1, X2))=join(join(X0, X1), X2)).
8.32/8.49	  fof(maddux3_a_kind_of_de_Morgan, axiom, ![X0, X1]: X0=join(complement(join(complement(X0), complement(X1))), complement(join(complement(X0), X1)))).
8.32/8.49	  fof(maddux4_definiton_of_meet, axiom, ![X0, X1]: meet(X0, X1)=complement(join(complement(X0), complement(X1)))).
8.32/8.49	
8.32/8.49	Now clausify the problem and encode Horn clauses using encoding 3 of
8.32/8.49	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
8.32/8.49	We repeatedly replace C & s=t => u=v by the two clauses:
8.32/8.49	  fresh(y, y, x1...xn) = u
8.32/8.49	  C => fresh(s, t, x1...xn) = v
8.32/8.49	where fresh is a fresh function symbol and x1..xn are the free
8.32/8.49	variables of u and v.
8.32/8.49	A predicate p(X) is encoded as p(X)=true (this is sound, because the
8.32/8.49	input problem has no model of domain size 1).
8.32/8.49	
8.32/8.49	The encoding turns the above axioms into the following unit equations and goals:
8.32/8.49	
8.32/8.49	Axiom 1 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
8.32/8.49	Axiom 2 (composition_identity): X = composition(X, one).
8.32/8.49	Axiom 3 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
8.32/8.49	Axiom 4 (def_zero): meet(X, complement(X)) = zero.
8.32/8.49	Axiom 5 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
8.32/8.49	Axiom 6 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
8.32/8.49	Axiom 7 (def_top): join(X, complement(X)) = top.
8.32/8.49	Axiom 8 (composition_associativity): composition(composition(X, Y), Z) = composition(X, composition(Y, Z)).
8.32/8.49	Axiom 9 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
8.32/8.49	Axiom 10 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
8.32/8.49	Axiom 11 (converse_idempotence): X = converse(converse(X)).
8.32/8.49	Axiom 12 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
8.32/8.49	Axiom 13 (converse_cancellativity): complement(X) = join(composition(converse(Y), complement(composition(Y, X))), complement(X)).
8.32/8.49	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))).
8.32/8.49	Axiom 15 (goals): join(sK3_goals_X1, one) = one.
8.32/8.52	Axiom 16 (goals_1): one = join(sK2_goals_X0, one).
8.32/8.52	
8.32/8.52	Lemma 17: meet(X, Y) = meet(Y, X).
8.32/8.52	Proof:
8.32/8.52	  meet(X, Y)
8.32/8.52	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.52	  complement(join(complement(X), complement(Y)))
8.32/8.52	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.52	  complement(join(complement(Y), complement(X)))
8.32/8.52	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.52	  meet(Y, X)
8.32/8.52	
8.32/8.52	Lemma 18: complement(top) = zero.
8.32/8.52	Proof:
8.32/8.52	  complement(top)
8.32/8.52	= { by axiom 7 (def_top) }
8.32/8.52	  complement(join(complement(?), complement(complement(?))))
8.32/8.52	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.52	  meet(?, complement(?))
8.32/8.52	= { by axiom 4 (def_zero) }
8.32/8.52	  zero
8.32/8.52	
8.32/8.52	Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
8.32/8.52	Proof:
8.32/8.52	  complement(join(zero, complement(X)))
8.32/8.52	= { by lemma 18 }
8.32/8.52	  complement(join(complement(top), complement(X)))
8.32/8.52	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.52	  meet(top, X)
8.32/8.52	= { by lemma 17 }
8.32/8.53	  meet(X, top)
8.32/8.53	
8.32/8.53	Lemma 20: converse(composition(converse(X), Y)) = composition(converse(Y), X).
8.32/8.53	Proof:
8.32/8.53	  converse(composition(converse(X), Y))
8.32/8.53	= { by axiom 3 (converse_multiplicativity) }
8.32/8.53	  composition(converse(Y), converse(converse(X)))
8.32/8.53	= { by axiom 11 (converse_idempotence) }
8.32/8.53	  composition(converse(Y), X)
8.32/8.53	
8.32/8.53	Lemma 21: composition(converse(one), X) = X.
8.32/8.53	Proof:
8.32/8.53	  composition(converse(one), X)
8.32/8.53	= { by lemma 20 }
8.32/8.53	  converse(composition(converse(X), one))
8.32/8.53	= { by axiom 2 (composition_identity) }
8.32/8.53	  converse(converse(X))
8.32/8.53	= { by axiom 11 (converse_idempotence) }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 22: composition(one, X) = X.
8.32/8.53	Proof:
8.32/8.53	  composition(one, X)
8.32/8.53	= { by lemma 21 }
8.32/8.53	  composition(converse(one), composition(one, X))
8.32/8.53	= { by axiom 8 (composition_associativity) }
8.32/8.53	  composition(composition(converse(one), one), X)
8.32/8.53	= { by axiom 2 (composition_identity) }
8.32/8.53	  composition(converse(one), X)
8.32/8.53	= { by lemma 21 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 23: join(complement(X), composition(converse(Y), complement(composition(Y, X)))) = complement(X).
8.32/8.53	Proof:
8.32/8.53	  join(complement(X), composition(converse(Y), complement(composition(Y, X))))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(composition(converse(Y), complement(composition(Y, X))), complement(X))
8.32/8.53	= { by axiom 13 (converse_cancellativity) }
8.32/8.53	  complement(X)
8.32/8.53	
8.32/8.53	Lemma 24: join(complement(X), complement(X)) = complement(X).
8.32/8.53	Proof:
8.32/8.53	  join(complement(X), complement(X))
8.32/8.53	= { by lemma 21 }
8.32/8.53	  join(complement(X), composition(converse(one), complement(X)))
8.32/8.53	= { by lemma 22 }
8.32/8.53	  join(complement(X), composition(converse(one), complement(composition(one, X))))
8.32/8.53	= { by lemma 23 }
8.32/8.53	  complement(X)
8.32/8.53	
8.32/8.53	Lemma 25: join(X, join(Y, Z)) = join(Z, join(X, Y)).
8.32/8.53	Proof:
8.32/8.53	  join(X, join(Y, Z))
8.32/8.53	= { by axiom 12 (maddux2_join_associativity) }
8.32/8.53	  join(join(X, Y), Z)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(Z, join(X, Y))
8.32/8.53	
8.32/8.53	Lemma 26: join(X, join(complement(X), Y)) = join(Y, top).
8.32/8.53	Proof:
8.32/8.53	  join(X, join(complement(X), Y))
8.32/8.53	= { by lemma 25 }
8.32/8.53	  join(complement(X), join(Y, X))
8.32/8.53	= { by lemma 25 }
8.32/8.53	  join(Y, join(X, complement(X)))
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  join(Y, top)
8.32/8.53	
8.32/8.53	Lemma 27: join(X, join(Y, complement(X))) = join(Y, top).
8.32/8.53	Proof:
8.32/8.53	  join(X, join(Y, complement(X)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(X, join(complement(X), Y))
8.32/8.53	= { by axiom 12 (maddux2_join_associativity) }
8.32/8.53	  join(join(X, complement(X)), Y)
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  join(top, Y)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(Y, top)
8.32/8.53	
8.32/8.53	Lemma 28: join(top, complement(X)) = top.
8.32/8.53	Proof:
8.32/8.53	  join(top, complement(X))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(complement(X), top)
8.32/8.53	= { by lemma 27 }
8.32/8.53	  join(X, join(complement(X), complement(X)))
8.32/8.53	= { by lemma 24 }
8.32/8.53	  join(X, complement(X))
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  top
8.32/8.53	
8.32/8.53	Lemma 29: join(X, top) = top.
8.32/8.53	Proof:
8.32/8.53	  join(X, top)
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  join(X, join(complement(X), complement(complement(X))))
8.32/8.53	= { by lemma 26 }
8.32/8.53	  join(complement(complement(X)), top)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(top, complement(complement(X)))
8.32/8.53	= { by lemma 28 }
8.32/8.53	  top
8.32/8.53	
8.32/8.53	Lemma 30: join(zero, meet(X, top)) = X.
8.32/8.53	Proof:
8.32/8.53	  join(zero, meet(X, top))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(meet(X, top), zero)
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(complement(join(complement(X), complement(top))), zero)
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(complement(join(complement(X), complement(top))), complement(top))
8.32/8.53	= { by lemma 29 }
8.32/8.53	  join(complement(join(complement(X), complement(top))), complement(join(complement(X), top)))
8.32/8.53	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 31: join(meet(X, Y), meet(X, complement(Y))) = X.
8.32/8.53	Proof:
8.32/8.53	  join(meet(X, Y), meet(X, complement(Y)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(meet(X, complement(Y)), meet(X, Y))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(complement(join(complement(X), complement(complement(Y)))), meet(X, Y))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(complement(join(complement(X), complement(complement(Y)))), complement(join(complement(X), complement(Y))))
8.32/8.53	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 32: join(zero, meet(X, X)) = X.
8.32/8.53	Proof:
8.32/8.53	  join(zero, meet(X, X))
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(complement(top), meet(X, X))
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  join(complement(join(complement(X), complement(complement(X)))), meet(X, X))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(complement(join(complement(X), complement(complement(X)))), complement(join(complement(X), complement(X))))
8.32/8.53	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 33: join(X, meet(Y, Y)) = join(Y, meet(X, X)).
8.32/8.53	Proof:
8.32/8.53	  join(X, meet(Y, Y))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(meet(Y, Y), X)
8.32/8.53	= { by lemma 32 }
8.32/8.53	  join(meet(Y, Y), join(zero, meet(X, X)))
8.32/8.53	= { by axiom 12 (maddux2_join_associativity) }
8.32/8.53	  join(join(meet(Y, Y), zero), meet(X, X))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(join(zero, meet(Y, Y)), meet(X, X))
8.32/8.53	= { by lemma 32 }
8.32/8.53	  join(Y, meet(X, X))
8.32/8.53	
8.32/8.53	Lemma 34: join(X, zero) = X.
8.32/8.53	Proof:
8.32/8.53	  join(X, zero)
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(X, complement(top))
8.32/8.53	= { by lemma 28 }
8.32/8.53	  join(X, complement(join(top, complement(zero))))
8.32/8.53	= { by lemma 31 }
8.32/8.53	  join(X, complement(join(join(meet(top, complement(top)), meet(top, complement(complement(top)))), complement(zero))))
8.32/8.53	= { by axiom 4 (def_zero) }
8.32/8.53	  join(X, complement(join(join(zero, meet(top, complement(complement(top)))), complement(zero))))
8.32/8.53	= { by lemma 24 }
8.32/8.53	  join(X, complement(join(join(zero, meet(top, complement(join(complement(top), complement(top))))), complement(zero))))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(X, complement(join(join(zero, meet(top, meet(top, top))), complement(zero))))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  join(X, complement(join(join(zero, meet(meet(top, top), top)), complement(zero))))
8.32/8.53	= { by lemma 30 }
8.32/8.53	  join(X, complement(join(meet(top, top), complement(zero))))
8.32/8.53	= { by lemma 19 }
8.32/8.53	  join(X, complement(join(complement(join(zero, complement(top))), complement(zero))))
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(X, complement(join(complement(join(complement(top), complement(top))), complement(zero))))
8.32/8.53	= { by lemma 24 }
8.32/8.53	  join(X, complement(join(complement(complement(top)), complement(zero))))
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(X, complement(join(complement(zero), complement(zero))))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(X, meet(zero, zero))
8.32/8.53	= { by lemma 33 }
8.32/8.53	  join(zero, meet(X, X))
8.32/8.53	= { by lemma 32 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 35: join(zero, X) = X.
8.32/8.53	Proof:
8.32/8.53	  join(zero, X)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(X, zero)
8.32/8.53	= { by lemma 34 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 36: meet(X, top) = X.
8.32/8.53	Proof:
8.32/8.53	  meet(X, top)
8.32/8.53	= { by lemma 35 }
8.32/8.53	  join(zero, meet(X, top))
8.32/8.53	= { by lemma 30 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 37: complement(complement(X)) = X.
8.32/8.53	Proof:
8.32/8.53	  complement(complement(X))
8.32/8.53	= { by lemma 35 }
8.32/8.53	  complement(join(zero, complement(X)))
8.32/8.53	= { by lemma 19 }
8.32/8.53	  meet(X, top)
8.32/8.53	= { by lemma 36 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 38: meet(X, X) = X.
8.32/8.53	Proof:
8.32/8.53	  meet(X, X)
8.32/8.53	= { by lemma 35 }
8.32/8.53	  join(zero, meet(X, X))
8.32/8.53	= { by lemma 32 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 39: complement(join(complement(X), meet(Y, Z))) = meet(X, join(complement(Y), complement(Z))).
8.32/8.53	Proof:
8.32/8.53	  complement(join(complement(X), meet(Y, Z)))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  complement(join(complement(X), meet(Z, Y)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  complement(join(meet(Z, Y), complement(X)))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  complement(join(complement(join(complement(Z), complement(Y))), complement(X)))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  meet(join(complement(Z), complement(Y)), X)
8.32/8.53	= { by lemma 17 }
8.32/8.53	  meet(X, join(complement(Z), complement(Y)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  meet(X, join(complement(Y), complement(Z)))
8.32/8.53	
8.32/8.53	Lemma 40: meet(top, X) = X.
8.32/8.53	Proof:
8.32/8.53	  meet(top, X)
8.32/8.53	= { by lemma 17 }
8.32/8.53	  meet(X, top)
8.32/8.53	= { by lemma 36 }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 41: join(complement(X), complement(Y)) = complement(meet(X, Y)).
8.32/8.53	Proof:
8.32/8.53	  join(complement(X), complement(Y))
8.32/8.53	= { by lemma 40 }
8.32/8.53	  meet(top, join(complement(X), complement(Y)))
8.32/8.53	= { by lemma 39 }
8.32/8.53	  complement(join(complement(top), meet(X, Y)))
8.32/8.53	= { by lemma 18 }
8.32/8.53	  complement(join(zero, meet(X, Y)))
8.32/8.53	= { by lemma 35 }
8.32/8.53	  complement(meet(X, Y))
8.32/8.53	
8.32/8.53	Lemma 42: complement(join(Y, complement(X))) = meet(X, complement(Y)).
8.32/8.53	Proof:
8.32/8.53	  complement(join(Y, complement(X)))
8.32/8.53	= { by lemma 38 }
8.32/8.53	  complement(join(Y, meet(complement(X), complement(X))))
8.32/8.53	= { by lemma 33 }
8.32/8.53	  complement(join(complement(X), meet(Y, Y)))
8.32/8.53	= { by lemma 39 }
8.32/8.53	  meet(X, join(complement(Y), complement(Y)))
8.32/8.53	= { by lemma 41 }
8.32/8.53	  meet(X, complement(meet(Y, Y)))
8.32/8.53	= { by lemma 38 }
8.32/8.53	  meet(X, complement(Y))
8.32/8.53	
8.32/8.53	Lemma 43: complement(join(complement(X), Y)) = meet(X, complement(Y)).
8.32/8.53	Proof:
8.32/8.53	  complement(join(complement(X), Y))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  complement(join(Y, complement(X)))
8.32/8.53	= { by lemma 42 }
8.32/8.53	  meet(X, complement(Y))
8.32/8.53	
8.32/8.53	Lemma 44: join(one, sK2_goals_X0) = one.
8.32/8.53	Proof:
8.32/8.53	  join(one, sK2_goals_X0)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(sK2_goals_X0, one)
8.32/8.53	= { by axiom 16 (goals_1) }
8.32/8.53	  one
8.32/8.53	
8.32/8.53	Lemma 45: join(one, join(sK2_goals_X0, X)) = join(X, one).
8.32/8.53	Proof:
8.32/8.53	  join(one, join(sK2_goals_X0, X))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(one, join(X, sK2_goals_X0))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(one, join(sK2_goals_X0, X))
8.32/8.53	= { by axiom 12 (maddux2_join_associativity) }
8.32/8.53	  join(join(one, sK2_goals_X0), X)
8.32/8.53	= { by lemma 44 }
8.32/8.53	  join(one, X)
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(X, one)
8.32/8.53	
8.32/8.53	Lemma 46: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
8.32/8.53	Proof:
8.32/8.53	  join(meet(X, Y), complement(join(Y, complement(X))))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  join(complement(join(complement(X), complement(Y))), complement(join(Y, complement(X))))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
8.32/8.53	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.53	  X
8.32/8.53	
8.32/8.53	Lemma 47: meet(one, sK2_goals_X0) = sK2_goals_X0.
8.32/8.53	Proof:
8.32/8.53	  meet(one, sK2_goals_X0)
8.32/8.53	= { by lemma 17 }
8.32/8.53	  meet(sK2_goals_X0, one)
8.32/8.53	= { by lemma 34 }
8.32/8.53	  join(meet(sK2_goals_X0, one), zero)
8.32/8.53	= { by lemma 18 }
8.32/8.53	  join(meet(sK2_goals_X0, one), complement(top))
8.32/8.53	= { by lemma 29 }
8.32/8.53	  join(meet(sK2_goals_X0, one), complement(join(one, top)))
8.32/8.53	= { by axiom 7 (def_top) }
8.32/8.53	  join(meet(sK2_goals_X0, one), complement(join(one, join(sK2_goals_X0, complement(sK2_goals_X0)))))
8.32/8.53	= { by lemma 45 }
8.32/8.53	  join(meet(sK2_goals_X0, one), complement(join(complement(sK2_goals_X0), one)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  join(meet(sK2_goals_X0, one), complement(join(one, complement(sK2_goals_X0))))
8.32/8.53	= { by lemma 46 }
8.32/8.53	  sK2_goals_X0
8.32/8.53	
8.32/8.53	Lemma 48: complement(meet(X, complement(Y))) = join(Y, complement(X)).
8.32/8.53	Proof:
8.32/8.53	  complement(meet(X, complement(Y)))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  complement(meet(complement(Y), X))
8.32/8.53	= { by lemma 35 }
8.32/8.53	  complement(meet(join(zero, complement(Y)), X))
8.32/8.53	= { by lemma 41 }
8.32/8.53	  join(complement(join(zero, complement(Y))), complement(X))
8.32/8.53	= { by lemma 19 }
8.32/8.53	  join(meet(Y, top), complement(X))
8.32/8.53	= { by lemma 36 }
8.32/8.53	  join(Y, complement(X))
8.32/8.53	
8.32/8.53	Lemma 49: complement(meet(X, meet(Y, complement(Z)))) = join(Z, complement(meet(X, Y))).
8.32/8.53	Proof:
8.32/8.53	  complement(meet(X, meet(Y, complement(Z))))
8.32/8.53	= { by lemma 41 }
8.32/8.53	  join(complement(X), complement(meet(Y, complement(Z))))
8.32/8.53	= { by lemma 48 }
8.32/8.53	  join(complement(X), join(Z, complement(Y)))
8.32/8.53	= { by lemma 25 }
8.32/8.53	  join(Z, join(complement(Y), complement(X)))
8.32/8.53	= { by lemma 41 }
8.32/8.53	  join(Z, complement(meet(Y, X)))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  join(Z, complement(meet(X, Y)))
8.32/8.53	
8.32/8.53	Lemma 50: complement(meet(complement(X), Y)) = join(X, complement(Y)).
8.32/8.53	Proof:
8.32/8.53	  complement(meet(complement(X), Y))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  complement(meet(Y, complement(X)))
8.32/8.53	= { by lemma 48 }
8.32/8.53	  join(X, complement(Y))
8.32/8.53	
8.32/8.53	Lemma 51: meet(join(X, complement(Y)), complement(meet(X, Y))) = complement(Y).
8.32/8.53	Proof:
8.32/8.53	  meet(join(X, complement(Y)), complement(meet(X, Y)))
8.32/8.53	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.53	  meet(join(complement(Y), X), complement(meet(X, Y)))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  meet(join(complement(Y), X), complement(meet(Y, X)))
8.32/8.53	= { by lemma 17 }
8.32/8.53	  meet(complement(meet(Y, X)), join(complement(Y), X))
8.32/8.53	= { by lemma 41 }
8.32/8.53	  meet(join(complement(Y), complement(X)), join(complement(Y), X))
8.32/8.53	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.53	  complement(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), X))))
8.32/8.53	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.54	  complement(Y)
8.32/8.54	
8.32/8.54	Lemma 52: meet(X, complement(meet(X, Y))) = meet(X, complement(Y)).
8.32/8.54	Proof:
8.32/8.54	  meet(X, complement(meet(X, Y)))
8.32/8.54	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, Y)))
8.32/8.54	= { by axiom 11 (converse_idempotence) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, converse(converse(Y)))))
8.32/8.54	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(complement(join(complement(converse(converse(Y))), complement(converse(join(converse(Y), converse(complement(X))))))), complement(join(complement(converse(converse(Y))), converse(join(converse(Y), converse(complement(X))))))))))
8.32/8.54	= { by axiom 5 (maddux4_definiton_of_meet) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(complement(converse(converse(Y))), converse(join(converse(Y), converse(complement(X))))))))))
8.32/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(complement(converse(converse(Y))), converse(join(converse(complement(X)), converse(Y)))))))))
8.32/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(converse(join(converse(complement(X)), converse(Y))), complement(converse(converse(Y)))))))))
8.32/8.54	= { by axiom 6 (converse_additivity) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(join(converse(converse(complement(X))), converse(converse(Y))), complement(converse(converse(Y)))))))))
8.32/8.54	= { by axiom 12 (maddux2_join_associativity) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(converse(converse(complement(X))), join(converse(converse(Y)), complement(converse(converse(Y))))))))))
8.32/8.54	= { by axiom 7 (def_top) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(join(converse(converse(complement(X))), top))))))
8.32/8.54	= { by lemma 29 }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), complement(top)))))
8.32/8.54	= { by lemma 18 }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, join(meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))), zero))))
8.32/8.54	= { by lemma 34 }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(converse(converse(Y)), converse(join(converse(Y), converse(complement(X))))))))
8.32/8.54	= { by axiom 11 (converse_idempotence) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, converse(join(converse(Y), converse(complement(X))))))))
8.32/8.54	= { by axiom 6 (converse_additivity) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, converse(converse(join(Y, complement(X))))))))
8.32/8.54	= { by axiom 11 (converse_idempotence) }
8.32/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, join(Y, complement(X))))))
8.32/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), complement(meet(X, meet(Y, join(complement(X), Y)))))
8.38/8.54	= { by lemma 41 }
8.38/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), complement(meet(Y, join(complement(X), Y)))))
8.38/8.54	= { by lemma 41 }
8.38/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(complement(X), join(complement(Y), complement(join(complement(X), Y)))))
8.38/8.54	= { by axiom 12 (maddux2_join_associativity) }
8.38/8.54	  meet(join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))), join(join(complement(X), complement(Y)), complement(join(complement(X), Y))))
8.38/8.54	= { by lemma 50 }
8.38/8.54	  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))))
8.38/8.54	= { by lemma 51 }
8.38/8.54	  complement(join(complement(X), Y))
8.38/8.54	= { by lemma 43 }
8.38/8.54	  meet(X, complement(Y))
8.38/8.54	
8.38/8.54	Lemma 53: meet(X, complement(meet(Y, X))) = meet(X, complement(Y)).
8.38/8.54	Proof:
8.38/8.54	  meet(X, complement(meet(Y, X)))
8.38/8.54	= { by lemma 17 }
8.38/8.54	  meet(X, complement(meet(X, Y)))
8.38/8.54	= { by lemma 52 }
8.38/8.54	  meet(X, complement(Y))
8.38/8.54	
8.38/8.54	Lemma 54: meet(sK2_goals_X0, complement(meet(one, X))) = meet(sK2_goals_X0, complement(X)).
8.38/8.54	Proof:
8.38/8.54	  meet(sK2_goals_X0, complement(meet(one, X)))
8.38/8.54	= { by lemma 17 }
8.38/8.54	  meet(sK2_goals_X0, complement(meet(X, one)))
8.38/8.54	= { by lemma 43 }
8.38/8.54	  complement(join(complement(sK2_goals_X0), meet(X, one)))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  complement(join(meet(X, one), complement(sK2_goals_X0)))
8.38/8.54	= { by lemma 47 }
8.38/8.54	  complement(join(meet(X, one), complement(meet(one, sK2_goals_X0))))
8.38/8.54	= { by lemma 17 }
8.38/8.54	  complement(join(meet(X, one), complement(meet(sK2_goals_X0, one))))
8.38/8.54	= { by lemma 49 }
8.38/8.54	  complement(complement(meet(sK2_goals_X0, meet(one, complement(meet(X, one))))))
8.38/8.54	= { by lemma 53 }
8.38/8.54	  complement(complement(meet(sK2_goals_X0, meet(one, complement(X)))))
8.38/8.54	= { by lemma 49 }
8.38/8.54	  complement(join(X, complement(meet(sK2_goals_X0, one))))
8.38/8.54	= { by lemma 17 }
8.38/8.54	  complement(join(X, complement(meet(one, sK2_goals_X0))))
8.38/8.54	= { by lemma 47 }
8.38/8.54	  complement(join(X, complement(sK2_goals_X0)))
8.38/8.54	= { by lemma 42 }
8.38/8.54	  meet(sK2_goals_X0, complement(X))
8.38/8.54	
8.38/8.54	Lemma 55: converse(one) = one.
8.38/8.54	Proof:
8.38/8.54	  converse(one)
8.38/8.54	= { by axiom 2 (composition_identity) }
8.38/8.54	  composition(converse(one), one)
8.38/8.54	= { by lemma 21 }
8.38/8.54	  one
8.38/8.54	
8.38/8.54	Lemma 56: converse(join(one, X)) = join(one, converse(X)).
8.38/8.54	Proof:
8.38/8.54	  converse(join(one, X))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  converse(join(X, one))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  converse(join(one, X))
8.38/8.54	= { by axiom 6 (converse_additivity) }
8.38/8.54	  join(converse(one), converse(X))
8.38/8.54	= { by lemma 55 }
8.38/8.54	  join(one, converse(X))
8.38/8.54	
8.38/8.54	Lemma 57: join(one, sK3_goals_X1) = one.
8.38/8.54	Proof:
8.38/8.54	  join(one, sK3_goals_X1)
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  join(sK3_goals_X1, one)
8.38/8.54	= { by axiom 15 (goals) }
8.38/8.54	  one
8.38/8.54	
8.38/8.54	Lemma 58: join(one, complement(sK3_goals_X1)) = join(one, top).
8.38/8.54	Proof:
8.38/8.54	  join(one, complement(sK3_goals_X1))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  join(complement(sK3_goals_X1), one)
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  join(one, complement(sK3_goals_X1))
8.38/8.54	= { by lemma 57 }
8.38/8.54	  join(join(one, sK3_goals_X1), complement(sK3_goals_X1))
8.38/8.54	= { by axiom 12 (maddux2_join_associativity) }
8.38/8.54	  join(one, join(sK3_goals_X1, complement(sK3_goals_X1)))
8.38/8.54	= { by axiom 7 (def_top) }
8.38/8.54	  join(one, top)
8.38/8.54	
8.38/8.54	Lemma 59: converse(join(X, converse(Y))) = join(Y, converse(X)).
8.38/8.54	Proof:
8.38/8.54	  converse(join(X, converse(Y)))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  converse(join(converse(Y), X))
8.38/8.54	= { by axiom 6 (converse_additivity) }
8.38/8.54	  join(converse(converse(Y)), converse(X))
8.38/8.54	= { by axiom 11 (converse_idempotence) }
8.38/8.54	  join(Y, converse(X))
8.38/8.54	
8.38/8.54	Lemma 60: converse(join(converse(X), Y)) = join(X, converse(Y)).
8.38/8.54	Proof:
8.38/8.54	  converse(join(converse(X), Y))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  converse(join(Y, converse(X)))
8.38/8.54	= { by lemma 59 }
8.38/8.54	  join(X, converse(Y))
8.38/8.54	
8.38/8.54	Lemma 61: join(X, converse(top)) = converse(top).
8.38/8.54	Proof:
8.38/8.54	  join(X, converse(top))
8.38/8.54	= { by lemma 60 }
8.38/8.54	  converse(join(converse(X), top))
8.38/8.54	= { by lemma 29 }
8.38/8.54	  converse(top)
8.38/8.54	
8.38/8.54	Lemma 62: converse(top) = top.
8.38/8.54	Proof:
8.38/8.54	  converse(top)
8.38/8.54	= { by lemma 61 }
8.38/8.54	  join(?, converse(top))
8.38/8.54	= { by lemma 61 }
8.38/8.54	  join(?, join(complement(?), converse(top)))
8.38/8.54	= { by lemma 26 }
8.38/8.54	  join(converse(top), top)
8.38/8.54	= { by lemma 29 }
8.38/8.54	  top
8.38/8.54	
8.38/8.54	Lemma 63: join(composition(X, Y), join(Z, composition(W, Y))) = join(Z, composition(join(X, W), Y)).
8.38/8.54	Proof:
8.38/8.54	  join(composition(X, Y), join(Z, composition(W, Y)))
8.38/8.54	= { by lemma 25 }
8.38/8.54	  join(Z, join(composition(W, Y), composition(X, Y)))
8.38/8.54	= { by axiom 9 (composition_distributivity) }
8.38/8.54	  join(Z, composition(join(W, X), Y))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  join(Z, composition(join(X, W), Y))
8.38/8.54	
8.38/8.54	Lemma 64: meet(X, join(Y, complement(X))) = meet(X, Y).
8.38/8.54	Proof:
8.38/8.54	  meet(X, join(Y, complement(X)))
8.38/8.54	= { by lemma 48 }
8.38/8.54	  meet(X, complement(meet(X, complement(Y))))
8.38/8.54	= { by lemma 52 }
8.38/8.54	  meet(X, complement(complement(Y)))
8.38/8.54	= { by lemma 37 }
8.38/8.54	  meet(X, Y)
8.38/8.54	
8.38/8.54	Lemma 65: meet(X, join(complement(X), Y)) = meet(X, Y).
8.38/8.54	Proof:
8.38/8.54	  meet(X, join(complement(X), Y))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  meet(X, join(Y, complement(X)))
8.38/8.54	= { by lemma 64 }
8.38/8.54	  meet(X, Y)
8.38/8.54	
8.38/8.54	Lemma 66: join(composition(Y, converse(Z)), converse(composition(Z, X))) = composition(join(Y, converse(X)), converse(Z)).
8.38/8.54	Proof:
8.38/8.54	  join(composition(Y, converse(Z)), converse(composition(Z, X)))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.54	  join(converse(composition(Z, X)), composition(Y, converse(Z)))
8.38/8.54	= { by axiom 3 (converse_multiplicativity) }
8.38/8.54	  join(composition(converse(X), converse(Z)), composition(Y, converse(Z)))
8.38/8.54	= { by axiom 9 (composition_distributivity) }
8.38/8.54	  composition(join(converse(X), Y), converse(Z))
8.38/8.54	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  composition(join(Y, converse(X)), converse(Z))
8.38/8.55	
8.38/8.55	Lemma 67: join(composition(Z, W), composition(X, join(Y, W))) = join(composition(X, Y), composition(join(X, Z), W)).
8.38/8.55	Proof:
8.38/8.55	  join(composition(Z, W), composition(X, join(Y, W)))
8.38/8.55	= { by axiom 11 (converse_idempotence) }
8.38/8.55	  join(composition(Z, W), converse(converse(composition(X, join(Y, W)))))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  join(composition(Z, W), converse(converse(composition(X, join(W, Y)))))
8.38/8.55	= { by axiom 3 (converse_multiplicativity) }
8.38/8.55	  join(composition(Z, W), converse(composition(converse(join(W, Y)), converse(X))))
8.38/8.55	= { by axiom 6 (converse_additivity) }
8.38/8.55	  join(composition(Z, W), converse(composition(join(converse(W), converse(Y)), converse(X))))
8.38/8.55	= { by lemma 66 }
8.38/8.55	  join(composition(Z, W), converse(join(composition(converse(W), converse(X)), converse(composition(X, Y)))))
8.38/8.55	= { by axiom 3 (converse_multiplicativity) }
8.38/8.55	  join(composition(Z, W), converse(join(converse(composition(X, W)), converse(composition(X, Y)))))
8.38/8.55	= { by axiom 6 (converse_additivity) }
8.38/8.55	  join(composition(Z, W), converse(converse(join(composition(X, W), composition(X, Y)))))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  join(composition(Z, W), converse(converse(join(composition(X, Y), composition(X, W)))))
8.38/8.55	= { by axiom 11 (converse_idempotence) }
8.38/8.55	  join(composition(Z, W), join(composition(X, Y), composition(X, W)))
8.38/8.55	= { by lemma 63 }
8.38/8.55	  join(composition(X, Y), composition(join(Z, X), W))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  join(composition(X, Y), composition(join(X, Z), W))
8.38/8.55	
8.38/8.55	Lemma 68: meet(join(X, Y), join(X, complement(Y))) = X.
8.38/8.55	Proof:
8.38/8.55	  meet(join(X, Y), join(X, complement(Y)))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  meet(join(Y, X), join(X, complement(Y)))
8.38/8.55	= { by lemma 36 }
8.38/8.55	  meet(join(Y, meet(X, top)), join(X, complement(Y)))
8.38/8.55	= { by lemma 19 }
8.38/8.55	  meet(join(Y, complement(join(zero, complement(X)))), join(X, complement(Y)))
8.38/8.55	= { by lemma 48 }
8.38/8.55	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, complement(X))))
8.38/8.55	= { by lemma 35 }
8.38/8.55	  meet(join(Y, complement(join(zero, complement(X)))), complement(meet(Y, join(zero, complement(X)))))
8.38/8.55	= { by lemma 51 }
8.38/8.55	  complement(join(zero, complement(X)))
8.38/8.55	= { by lemma 19 }
8.38/8.55	  meet(X, top)
8.38/8.55	= { by lemma 36 }
8.38/8.55	  X
8.38/8.55	
8.38/8.55	Lemma 69: join(X, converse(complement(converse(X)))) = top.
8.38/8.55	Proof:
8.38/8.55	  join(X, converse(complement(converse(X))))
8.38/8.55	= { by lemma 60 }
8.38/8.55	  converse(join(converse(X), complement(converse(X))))
8.38/8.55	= { by axiom 7 (def_top) }
8.38/8.55	  converse(top)
8.38/8.55	= { by lemma 62 }
8.38/8.55	  top
8.38/8.55	
8.38/8.55	Lemma 70: join(X, complement(converse(complement(converse(X))))) = X.
8.38/8.55	Proof:
8.38/8.55	  join(X, complement(converse(complement(converse(X)))))
8.38/8.55	= { by lemma 40 }
8.38/8.55	  meet(top, join(X, complement(converse(complement(converse(X))))))
8.38/8.55	= { by lemma 69 }
8.38/8.55	  meet(join(X, converse(complement(converse(X)))), join(X, complement(converse(complement(converse(X))))))
8.38/8.55	= { by lemma 68 }
8.38/8.55	  X
8.38/8.55	
8.38/8.55	Lemma 71: converse(complement(converse(complement(X)))) = X.
8.38/8.55	Proof:
8.38/8.55	  converse(complement(converse(complement(X))))
8.38/8.55	= { by lemma 68 }
8.38/8.55	  meet(join(converse(complement(converse(complement(X)))), X), join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  meet(join(X, converse(complement(converse(complement(X))))), join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by axiom 11 (converse_idempotence) }
8.38/8.55	  meet(join(X, converse(complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by lemma 60 }
8.38/8.55	  meet(converse(join(converse(X), complement(converse(complement(converse(converse(X))))))), join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by lemma 70 }
8.38/8.55	  meet(converse(converse(X)), join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by axiom 11 (converse_idempotence) }
8.38/8.55	  meet(X, join(converse(complement(converse(complement(X)))), complement(X)))
8.38/8.55	= { by lemma 64 }
8.38/8.55	  meet(X, converse(complement(converse(complement(X)))))
8.38/8.55	= { by axiom 5 (maddux4_definiton_of_meet) }
8.38/8.55	  complement(join(complement(X), complement(converse(complement(converse(complement(X)))))))
8.38/8.55	= { by lemma 70 }
8.38/8.55	  complement(complement(X))
8.38/8.55	= { by lemma 37 }
8.38/8.55	  X
8.38/8.55	
8.38/8.55	Lemma 72: join(complement(composition(X, Y)), composition(join(X, Z), Y)) = top.
8.38/8.55	Proof:
8.38/8.55	  join(complement(composition(X, Y)), composition(join(X, Z), Y))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  join(complement(composition(X, Y)), composition(join(Z, X), Y))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  join(composition(join(Z, X), Y), complement(composition(X, Y)))
8.38/8.55	= { by axiom 9 (composition_distributivity) }
8.38/8.55	  join(join(composition(Z, Y), composition(X, Y)), complement(composition(X, Y)))
8.38/8.55	= { by axiom 12 (maddux2_join_associativity) }
8.38/8.55	  join(composition(Z, Y), join(composition(X, Y), complement(composition(X, Y))))
8.38/8.55	= { by axiom 7 (def_top) }
8.38/8.55	  join(composition(Z, Y), top)
8.38/8.55	= { by lemma 29 }
8.38/8.55	  top
8.38/8.55	
8.38/8.55	Lemma 73: meet(composition(X, Y), composition(join(Z, X), Y)) = composition(X, Y).
8.38/8.55	Proof:
8.38/8.55	  meet(composition(X, Y), composition(join(Z, X), Y))
8.38/8.55	= { by axiom 10 (maddux1_join_commutativity) }
8.38/8.55	  meet(composition(X, Y), composition(join(X, Z), Y))
8.38/8.55	= { by axiom 5 (maddux4_definiton_of_meet) }
8.38/8.55	  complement(join(complement(composition(X, Y)), complement(composition(join(X, Z), Y))))
8.38/8.55	= { by lemma 36 }
8.38/8.55	  complement(meet(join(complement(composition(X, Y)), complement(composition(join(X, Z), Y))), top))
8.38/8.55	= { by lemma 41 }
8.38/8.55	  join(complement(join(complement(composition(X, Y)), complement(composition(join(X, Z), Y)))), complement(top))
8.38/8.55	= { by lemma 72 }
8.38/8.55	  join(complement(join(complement(composition(X, Y)), complement(composition(join(X, Z), Y)))), complement(join(complement(composition(X, Y)), composition(join(X, Z), Y))))
8.38/8.55	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.38/8.55	  composition(X, Y)
8.38/8.55	
8.38/8.55	Lemma 74: meet(X, composition(converse(sK2_goals_X0), X)) = composition(converse(sK2_goals_X0), X).
8.38/8.55	Proof:
8.38/8.55	  meet(X, composition(converse(sK2_goals_X0), X))
8.38/8.55	= { by lemma 17 }
8.38/8.55	  meet(composition(converse(sK2_goals_X0), X), X)
8.38/8.55	= { by lemma 22 }
8.38/8.55	  meet(composition(converse(sK2_goals_X0), X), composition(one, X))
8.38/8.55	= { by lemma 55 }
8.38/8.55	  meet(composition(converse(sK2_goals_X0), X), composition(converse(one), X))
8.38/8.55	= { by lemma 44 }
8.38/8.55	  meet(composition(converse(sK2_goals_X0), X), composition(converse(join(one, sK2_goals_X0)), X))
8.38/8.55	= { by lemma 56 }
8.38/8.55	  meet(composition(converse(sK2_goals_X0), X), composition(join(one, converse(sK2_goals_X0)), X))
8.38/8.55	= { by lemma 73 }
8.39/8.56	  composition(converse(sK2_goals_X0), X)
8.39/8.56	
8.39/8.56	Lemma 75: meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0)) = composition(converse(sK2_goals_X0), sK2_goals_X0).
8.39/8.56	Proof:
8.39/8.56	  meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.56	= { by lemma 17 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), one)
8.39/8.56	= { by lemma 64 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.56	= { by lemma 46 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(join(one, complement(converse(sK2_goals_X0))))), sK2_goals_X0))))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(join(complement(converse(sK2_goals_X0)), one))), sK2_goals_X0))))
8.39/8.56	= { by lemma 55 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(join(complement(converse(sK2_goals_X0)), converse(one)))), sK2_goals_X0))))
8.39/8.56	= { by lemma 59 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(converse(join(one, converse(complement(converse(sK2_goals_X0))))))), sK2_goals_X0))))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(converse(join(converse(complement(converse(sK2_goals_X0))), one)))), sK2_goals_X0))))
8.39/8.56	= { by lemma 45 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(converse(join(one, join(sK2_goals_X0, converse(complement(converse(sK2_goals_X0)))))))), sK2_goals_X0))))
8.39/8.56	= { by lemma 69 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(converse(join(one, top)))), sK2_goals_X0))))
8.39/8.56	= { by lemma 29 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(converse(top))), sK2_goals_X0))))
8.39/8.56	= { by lemma 62 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), complement(top)), sK2_goals_X0))))
8.39/8.56	= { by lemma 18 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(join(meet(converse(sK2_goals_X0), one), zero), sK2_goals_X0))))
8.39/8.56	= { by lemma 34 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, complement(composition(meet(converse(sK2_goals_X0), one), sK2_goals_X0))))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(complement(composition(meet(converse(sK2_goals_X0), one), sK2_goals_X0)), one))
8.39/8.56	= { by lemma 45 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(sK2_goals_X0, complement(composition(meet(converse(sK2_goals_X0), one), sK2_goals_X0)))))
8.39/8.56	= { by lemma 21 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(composition(converse(one), sK2_goals_X0), complement(composition(meet(converse(sK2_goals_X0), one), sK2_goals_X0)))))
8.39/8.56	= { by lemma 17 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(composition(converse(one), sK2_goals_X0), complement(composition(meet(one, converse(sK2_goals_X0)), sK2_goals_X0)))))
8.39/8.56	= { by lemma 55 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(composition(converse(one), sK2_goals_X0), complement(composition(meet(converse(one), converse(sK2_goals_X0)), sK2_goals_X0)))))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(complement(composition(meet(converse(one), converse(sK2_goals_X0)), sK2_goals_X0)), composition(converse(one), sK2_goals_X0))))
8.39/8.56	= { by axiom 5 (maddux4_definiton_of_meet) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(complement(composition(complement(join(complement(converse(one)), complement(converse(sK2_goals_X0)))), sK2_goals_X0)), composition(converse(one), sK2_goals_X0))))
8.39/8.56	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, join(complement(composition(complement(join(complement(converse(one)), complement(converse(sK2_goals_X0)))), sK2_goals_X0)), composition(join(complement(join(complement(converse(one)), complement(converse(sK2_goals_X0)))), complement(join(complement(converse(one)), converse(sK2_goals_X0)))), sK2_goals_X0))))
8.39/8.56	= { by lemma 72 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), join(one, top))
8.39/8.56	= { by lemma 29 }
8.39/8.56	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), top)
8.39/8.56	= { by lemma 36 }
8.39/8.56	  composition(converse(sK2_goals_X0), sK2_goals_X0)
8.39/8.56	
8.39/8.56	Lemma 76: join(Z, composition(X, composition(Y, Z))) = composition(join(one, composition(X, Y)), Z).
8.39/8.56	Proof:
8.39/8.56	  join(Z, composition(X, composition(Y, Z)))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  join(composition(X, composition(Y, Z)), Z)
8.39/8.56	= { by axiom 8 (composition_associativity) }
8.39/8.56	  join(composition(composition(X, Y), Z), Z)
8.39/8.56	= { by lemma 21 }
8.39/8.56	  join(composition(composition(X, Y), Z), composition(converse(one), Z))
8.39/8.56	= { by axiom 9 (composition_distributivity) }
8.39/8.56	  composition(join(composition(X, Y), converse(one)), Z)
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  composition(join(converse(one), composition(X, Y)), Z)
8.39/8.56	= { by lemma 55 }
8.39/8.56	  composition(join(one, composition(X, Y)), Z)
8.39/8.56	
8.39/8.56	Lemma 77: meet(complement(X), complement(Y)) = complement(join(X, Y)).
8.39/8.56	Proof:
8.39/8.56	  meet(complement(X), complement(Y))
8.39/8.56	= { by lemma 35 }
8.39/8.56	  meet(join(zero, complement(X)), complement(Y))
8.39/8.56	= { by lemma 42 }
8.39/8.56	  complement(join(Y, complement(join(zero, complement(X)))))
8.39/8.56	= { by lemma 19 }
8.39/8.56	  complement(join(Y, meet(X, top)))
8.39/8.56	= { by lemma 36 }
8.39/8.56	  complement(join(Y, X))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  complement(join(X, Y))
8.39/8.56	
8.39/8.56	Lemma 78: complement(join(X, meet(X, Y))) = complement(X).
8.39/8.56	Proof:
8.39/8.56	  complement(join(X, meet(X, Y)))
8.39/8.56	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.56	  complement(join(meet(X, Y), X))
8.39/8.56	= { by lemma 77 }
8.39/8.56	  meet(complement(meet(X, Y)), complement(X))
8.39/8.56	= { by lemma 53 }
8.39/8.56	  meet(complement(meet(X, Y)), complement(meet(X, complement(meet(X, Y)))))
8.39/8.56	= { by lemma 52 }
8.39/8.56	  meet(complement(meet(X, Y)), complement(meet(X, complement(Y))))
8.39/8.56	= { by lemma 77 }
8.39/8.56	  complement(join(meet(X, Y), meet(X, complement(Y))))
8.39/8.56	= { by lemma 31 }
8.39/8.57	  complement(X)
8.39/8.57	
8.39/8.57	Lemma 79: composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)) = sK2_goals_X0.
8.39/8.57	Proof:
8.39/8.57	  composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 38 }
8.39/8.57	  composition(meet(sK2_goals_X0, sK2_goals_X0), composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by axiom 2 (composition_identity) }
8.39/8.57	  composition(meet(sK2_goals_X0, composition(sK2_goals_X0, one)), composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 55 }
8.39/8.57	  composition(meet(sK2_goals_X0, composition(sK2_goals_X0, converse(one))), composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 75 }
8.39/8.57	  composition(meet(sK2_goals_X0, composition(sK2_goals_X0, converse(one))), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0)))
8.39/8.57	= { by axiom 14 (dedekind_law) }
8.39/8.57	  join(meet(composition(sK2_goals_X0, one), sK2_goals_X0), composition(meet(sK2_goals_X0, composition(sK2_goals_X0, converse(one))), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by axiom 2 (composition_identity) }
8.39/8.57	  join(meet(sK2_goals_X0, sK2_goals_X0), composition(meet(sK2_goals_X0, composition(sK2_goals_X0, converse(one))), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by lemma 38 }
8.39/8.57	  join(sK2_goals_X0, composition(meet(sK2_goals_X0, composition(sK2_goals_X0, converse(one))), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by lemma 55 }
8.39/8.57	  join(sK2_goals_X0, composition(meet(sK2_goals_X0, composition(sK2_goals_X0, one)), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by axiom 2 (composition_identity) }
8.39/8.57	  join(sK2_goals_X0, composition(meet(sK2_goals_X0, sK2_goals_X0), meet(one, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by lemma 75 }
8.39/8.57	  join(sK2_goals_X0, composition(meet(sK2_goals_X0, sK2_goals_X0), composition(converse(sK2_goals_X0), sK2_goals_X0)))
8.39/8.57	= { by lemma 76 }
8.39/8.57	  composition(join(one, composition(meet(sK2_goals_X0, sK2_goals_X0), converse(sK2_goals_X0))), sK2_goals_X0)
8.39/8.57	= { by lemma 38 }
8.39/8.57	  composition(join(one, composition(sK2_goals_X0, converse(sK2_goals_X0))), sK2_goals_X0)
8.39/8.57	= { by lemma 76 }
8.39/8.57	  join(sK2_goals_X0, composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))
8.39/8.57	= { by axiom 11 (converse_idempotence) }
8.39/8.57	  converse(converse(join(sK2_goals_X0, composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))))
8.39/8.57	= { by axiom 6 (converse_additivity) }
8.39/8.57	  converse(join(converse(sK2_goals_X0), converse(composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))))
8.39/8.57	= { by lemma 21 }
8.39/8.57	  converse(join(composition(converse(one), converse(sK2_goals_X0)), converse(composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))))
8.39/8.57	= { by lemma 66 }
8.39/8.57	  converse(composition(join(converse(one), converse(composition(converse(sK2_goals_X0), sK2_goals_X0))), converse(sK2_goals_X0)))
8.39/8.57	= { by axiom 6 (converse_additivity) }
8.39/8.57	  converse(composition(converse(join(one, composition(converse(sK2_goals_X0), sK2_goals_X0))), converse(sK2_goals_X0)))
8.39/8.57	= { by axiom 3 (converse_multiplicativity) }
8.39/8.57	  converse(converse(composition(sK2_goals_X0, join(one, composition(converse(sK2_goals_X0), sK2_goals_X0)))))
8.39/8.57	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.57	  converse(converse(composition(sK2_goals_X0, join(composition(converse(sK2_goals_X0), sK2_goals_X0), one))))
8.39/8.57	= { by axiom 11 (converse_idempotence) }
8.39/8.57	  composition(sK2_goals_X0, join(composition(converse(sK2_goals_X0), sK2_goals_X0), one))
8.39/8.57	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.57	  composition(sK2_goals_X0, join(one, composition(converse(sK2_goals_X0), sK2_goals_X0)))
8.39/8.57	= { by lemma 74 }
8.39/8.57	  composition(sK2_goals_X0, join(one, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0))))
8.39/8.57	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.57	  composition(sK2_goals_X0, join(meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)), one))
8.39/8.57	= { by lemma 45 }
8.39/8.57	  composition(sK2_goals_X0, join(one, join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))))
8.39/8.57	= { by axiom 1 (maddux3_a_kind_of_de_Morgan) }
8.39/8.57	  composition(sK2_goals_X0, join(one, join(complement(join(complement(join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))), complement(?))), complement(join(complement(join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))), ?)))))
8.39/8.57	= { by lemma 78 }
8.39/8.57	  composition(sK2_goals_X0, join(one, join(complement(join(complement(sK2_goals_X0), complement(?))), complement(join(complement(join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))), ?)))))
8.39/8.57	= { by lemma 41 }
8.39/8.57	  composition(sK2_goals_X0, join(one, complement(meet(join(complement(sK2_goals_X0), complement(?)), join(complement(join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))), ?)))))
8.39/8.57	= { by lemma 41 }
8.39/8.57	  composition(sK2_goals_X0, join(one, complement(meet(complement(meet(sK2_goals_X0, ?)), join(complement(join(sK2_goals_X0, meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))), ?)))))
8.39/8.57	= { by lemma 78 }
8.39/8.57	  composition(sK2_goals_X0, join(one, complement(meet(complement(meet(sK2_goals_X0, ?)), join(complement(sK2_goals_X0), ?)))))
8.39/8.57	= { by lemma 50 }
8.39/8.57	  composition(sK2_goals_X0, join(one, join(meet(sK2_goals_X0, ?), complement(join(complement(sK2_goals_X0), ?)))))
8.39/8.57	= { by lemma 43 }
8.39/8.57	  composition(sK2_goals_X0, join(one, join(meet(sK2_goals_X0, ?), meet(sK2_goals_X0, complement(?)))))
8.39/8.57	= { by lemma 31 }
8.39/8.57	  composition(sK2_goals_X0, join(one, sK2_goals_X0))
8.39/8.57	= { by lemma 44 }
8.39/8.57	  composition(sK2_goals_X0, one)
8.39/8.57	= { by axiom 2 (composition_identity) }
8.39/8.57	  sK2_goals_X0
8.39/8.57	
8.39/8.57	Lemma 80: meet(X, composition(sK2_goals_X0, X)) = composition(sK2_goals_X0, X).
8.39/8.57	Proof:
8.39/8.57	  meet(X, composition(sK2_goals_X0, X))
8.39/8.57	= { by lemma 17 }
8.39/8.57	  meet(composition(sK2_goals_X0, X), X)
8.39/8.57	= { by lemma 34 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), zero)
8.39/8.57	= { by lemma 18 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(top))
8.39/8.57	= { by lemma 72 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(join(complement(composition(sK2_goals_X0, X)), composition(join(sK2_goals_X0, meet(one, complement(sK2_goals_X0))), X))))
8.39/8.57	= { by lemma 47 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(join(complement(composition(sK2_goals_X0, X)), composition(join(meet(one, sK2_goals_X0), meet(one, complement(sK2_goals_X0))), X))))
8.39/8.57	= { by lemma 31 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(join(complement(composition(sK2_goals_X0, X)), composition(one, X))))
8.39/8.57	= { by lemma 22 }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(join(complement(composition(sK2_goals_X0, X)), X)))
8.39/8.57	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.57	  join(meet(composition(sK2_goals_X0, X), X), complement(join(X, complement(composition(sK2_goals_X0, X)))))
8.39/8.57	= { by lemma 46 }
8.39/8.57	  composition(sK2_goals_X0, X)
8.39/8.57	
8.39/8.57	Lemma 81: composition(converse(sK2_goals_X0), sK2_goals_X0) = sK2_goals_X0.
8.39/8.57	Proof:
8.39/8.57	  composition(converse(sK2_goals_X0), sK2_goals_X0)
8.39/8.57	= { by lemma 74 }
8.39/8.57	  meet(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 17 }
8.39/8.57	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), sK2_goals_X0)
8.39/8.57	= { by lemma 79 }
8.39/8.57	  meet(composition(converse(sK2_goals_X0), sK2_goals_X0), composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0)))
8.39/8.57	= { by lemma 80 }
8.39/8.57	  composition(sK2_goals_X0, composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 79 }
8.39/8.57	  sK2_goals_X0
8.39/8.57	
8.39/8.57	Lemma 82: converse(sK2_goals_X0) = sK2_goals_X0.
8.39/8.57	Proof:
8.39/8.57	  converse(sK2_goals_X0)
8.39/8.57	= { by lemma 81 }
8.39/8.57	  converse(composition(converse(sK2_goals_X0), sK2_goals_X0))
8.39/8.57	= { by lemma 20 }
8.39/8.57	  composition(converse(sK2_goals_X0), sK2_goals_X0)
8.39/8.57	= { by lemma 81 }
8.39/8.58	  sK2_goals_X0
8.39/8.58	
8.39/8.58	Lemma 83: meet(X, composition(sK2_goals_X0, top)) = composition(sK2_goals_X0, X).
8.39/8.58	Proof:
8.39/8.58	  meet(X, composition(sK2_goals_X0, top))
8.39/8.58	= { by lemma 65 }
8.39/8.58	  meet(X, join(complement(X), composition(sK2_goals_X0, top)))
8.39/8.58	= { by lemma 82 }
8.39/8.58	  meet(X, join(complement(X), composition(converse(sK2_goals_X0), top)))
8.39/8.58	= { by lemma 62 }
8.39/8.58	  meet(X, join(complement(X), composition(converse(sK2_goals_X0), converse(top))))
8.39/8.58	= { by axiom 3 (converse_multiplicativity) }
8.39/8.58	  meet(X, join(complement(X), converse(composition(top, sK2_goals_X0))))
8.39/8.58	= { by lemma 60 }
8.39/8.58	  meet(X, converse(join(converse(complement(X)), composition(top, sK2_goals_X0))))
8.39/8.58	= { by axiom 2 (composition_identity) }
8.39/8.58	  meet(X, converse(join(composition(converse(complement(X)), one), composition(top, sK2_goals_X0))))
8.39/8.58	= { by axiom 7 (def_top) }
8.39/8.58	  meet(X, converse(join(composition(converse(complement(X)), one), composition(join(converse(complement(X)), complement(converse(complement(X)))), sK2_goals_X0))))
8.39/8.58	= { by lemma 67 }
8.39/8.58	  meet(X, converse(join(composition(complement(converse(complement(X))), sK2_goals_X0), composition(converse(complement(X)), join(one, sK2_goals_X0)))))
8.39/8.58	= { by lemma 44 }
8.39/8.58	  meet(X, converse(join(composition(complement(converse(complement(X))), sK2_goals_X0), composition(converse(complement(X)), one))))
8.39/8.58	= { by axiom 2 (composition_identity) }
8.39/8.58	  meet(X, converse(join(composition(complement(converse(complement(X))), sK2_goals_X0), converse(complement(X)))))
8.39/8.58	= { by axiom 10 (maddux1_join_commutativity) }
8.39/8.58	  meet(X, converse(join(converse(complement(X)), composition(complement(converse(complement(X))), sK2_goals_X0))))
8.39/8.58	= { by lemma 60 }
8.39/8.58	  meet(X, join(complement(X), converse(composition(complement(converse(complement(X))), sK2_goals_X0))))
8.39/8.58	= { by lemma 36 }
8.39/8.58	  meet(X, join(complement(X), converse(composition(complement(converse(meet(complement(X), top))), sK2_goals_X0))))
8.39/8.58	= { by lemma 19 }
8.39/8.58	  meet(X, join(complement(X), converse(composition(complement(converse(complement(join(zero, complement(complement(X)))))), sK2_goals_X0))))
8.39/8.58	= { by axiom 11 (converse_idempotence) }
8.39/8.58	  meet(X, join(complement(X), converse(composition(converse(converse(complement(converse(complement(join(zero, complement(complement(X)))))))), sK2_goals_X0))))
8.39/8.58	= { by lemma 71 }
8.39/8.58	  meet(X, join(complement(X), converse(composition(converse(join(zero, complement(complement(X)))), sK2_goals_X0))))
8.39/8.58	= { by lemma 35 }
8.39/8.58	  meet(X, join(complement(X), converse(composition(converse(complement(complement(X))), sK2_goals_X0))))
8.39/8.58	= { by lemma 20 }
8.39/8.58	  meet(X, join(complement(X), composition(converse(sK2_goals_X0), complement(complement(X)))))
8.39/8.58	= { by lemma 82 }
8.39/8.58	  meet(X, join(complement(X), composition(sK2_goals_X0, complement(complement(X)))))
8.39/8.58	= { by lemma 65 }
8.39/8.58	  meet(X, composition(sK2_goals_X0, complement(complement(X))))
8.39/8.58	= { by lemma 37 }
8.39/8.58	  meet(X, composition(sK2_goals_X0, X))
8.39/8.58	= { by lemma 80 }
8.39/8.58	  composition(sK2_goals_X0, X)
8.39/8.58	
8.39/8.58	Lemma 84: meet(composition(sK2_goals_X0, top), X) = composition(sK2_goals_X0, X).
8.39/8.58	Proof:
8.39/8.58	  meet(composition(sK2_goals_X0, top), X)
8.39/8.58	= { by lemma 17 }
8.39/8.58	  meet(X, composition(sK2_goals_X0, top))
8.39/8.58	= { by lemma 83 }
8.39/8.58	  composition(sK2_goals_X0, X)
8.39/8.58	
8.39/8.58	Lemma 85: meet(X, meet(Y, Z)) = meet(Y, meet(X, Z)).
8.39/8.58	Proof:
8.39/8.58	  meet(X, meet(Y, Z))
8.39/8.58	= { by lemma 36 }
8.39/8.58	  meet(meet(X, meet(Y, Z)), top)
8.39/8.58	= { by lemma 19 }
8.39/8.58	  complement(join(zero, complement(meet(X, meet(Y, Z)))))
8.39/8.58	= { by axiom 5 (maddux4_definiton_of_meet) }
8.39/8.58	  complement(join(zero, complement(meet(X, complement(join(complement(Y), complement(Z)))))))
8.39/8.58	= { by lemma 48 }
8.39/8.58	  complement(join(zero, join(join(complement(Y), complement(Z)), complement(X))))
8.39/8.58	= { by axiom 12 (maddux2_join_associativity) }
8.39/8.58	  complement(join(zero, join(complement(Y), join(complement(Z), complement(X)))))
8.39/8.58	= { by lemma 41 }
8.39/8.58	  complement(join(zero, join(complement(Y), complement(meet(Z, X)))))
8.39/8.58	= { by lemma 41 }
8.39/8.58	  complement(join(zero, complement(meet(Y, meet(Z, X)))))
8.39/8.58	= { by lemma 19 }
8.39/8.58	  meet(meet(Y, meet(Z, X)), top)
8.39/8.58	= { by lemma 36 }
8.39/8.58	  meet(Y, meet(Z, X))
8.39/8.58	= { by lemma 17 }
8.58/8.74	  meet(Y, meet(X, Z))
8.58/8.74	
8.58/8.74	Goal 1 (goals_2): composition(meet(sK2_goals_X0, sK3_goals_X1), sK1_goals_X2) = meet(composition(sK2_goals_X0, sK1_goals_X2), composition(sK3_goals_X1, sK1_goals_X2)).
8.58/8.74	Proof:
8.58/8.74	  composition(meet(sK2_goals_X0, sK3_goals_X1), sK1_goals_X2)
8.58/8.74	= { by lemma 17 }
8.58/8.74	  composition(meet(sK3_goals_X1, sK2_goals_X0), sK1_goals_X2)
8.58/8.74	= { by lemma 17 }
8.58/8.74	  composition(meet(sK2_goals_X0, sK3_goals_X1), sK1_goals_X2)
8.58/8.74	= { by lemma 37 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(sK3_goals_X1))), sK1_goals_X2)
8.58/8.74	= { by lemma 54 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(meet(one, complement(sK3_goals_X1)))), sK1_goals_X2)
8.58/8.74	= { by lemma 42 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(sK3_goals_X1, complement(one))))), sK1_goals_X2)
8.58/8.74	= { by lemma 48 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(complement(meet(one, complement(sK3_goals_X1)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 41 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), complement(complement(sK3_goals_X1)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 22 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(one, complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by axiom 10 (maddux1_join_commutativity) }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(composition(one, complement(complement(sK3_goals_X1))), complement(one))))), sK1_goals_X2)
8.58/8.74	= { by lemma 23 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(composition(one, complement(complement(sK3_goals_X1))), join(complement(one), composition(converse(complement(sK3_goals_X1)), complement(composition(complement(sK3_goals_X1), one)))))))), sK1_goals_X2)
8.58/8.74	= { by axiom 2 (composition_identity) }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(composition(one, complement(complement(sK3_goals_X1))), join(complement(one), composition(converse(complement(sK3_goals_X1)), complement(complement(sK3_goals_X1)))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 63 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(join(one, converse(complement(sK3_goals_X1))), complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 56 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(converse(join(one, complement(sK3_goals_X1))), complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 58 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(converse(join(one, top)), complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 56 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(join(one, converse(top)), complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 61 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(converse(top), complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 62 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(top, complement(complement(sK3_goals_X1))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 37 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(join(complement(one), composition(top, sK3_goals_X1))))), sK1_goals_X2)
8.58/8.74	= { by lemma 43 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(meet(one, complement(composition(top, sK3_goals_X1))))), sK1_goals_X2)
8.58/8.74	= { by lemma 54 }
8.58/8.74	  composition(meet(sK2_goals_X0, complement(complement(composition(top, sK3_goals_X1)))), sK1_goals_X2)
8.58/8.74	= { by lemma 37 }
8.58/8.74	  composition(meet(sK2_goals_X0, composition(top, sK3_goals_X1)), sK1_goals_X2)
8.58/8.74	= { by lemma 65 }
8.58/8.74	  composition(meet(sK2_goals_X0, join(complement(sK2_goals_X0), composition(top, sK3_goals_X1))), sK1_goals_X2)
8.58/8.74	= { by axiom 2 (composition_identity) }
8.58/8.74	  composition(meet(sK2_goals_X0, join(composition(complement(sK2_goals_X0), one), composition(top, sK3_goals_X1))), sK1_goals_X2)
8.58/8.74	= { by axiom 7 (def_top) }
8.58/8.74	  composition(meet(sK2_goals_X0, join(composition(complement(sK2_goals_X0), one), composition(join(complement(sK2_goals_X0), complement(complement(sK2_goals_X0))), sK3_goals_X1))), sK1_goals_X2)
8.58/8.74	= { by lemma 67 }
8.58/8.74	  composition(meet(sK2_goals_X0, join(composition(complement(complement(sK2_goals_X0)), sK3_goals_X1), composition(complement(sK2_goals_X0), join(one, sK3_goals_X1)))), sK1_goals_X2)
8.58/8.74	= { by lemma 57 }
8.58/8.74	  composition(meet(sK2_goals_X0, join(composition(complement(complement(sK2_goals_X0)), sK3_goals_X1), composition(complement(sK2_goals_X0), one))), sK1_goals_X2)
8.58/8.74	= { by axiom 2 (composition_identity) }
8.58/8.74	  composition(meet(sK2_goals_X0, join(composition(complement(complement(sK2_goals_X0)), sK3_goals_X1), complement(sK2_goals_X0))), sK1_goals_X2)
8.58/8.74	= { by axiom 10 (maddux1_join_commutativity) }
8.58/8.74	  composition(meet(sK2_goals_X0, join(complement(sK2_goals_X0), composition(complement(complement(sK2_goals_X0)), sK3_goals_X1))), sK1_goals_X2)
8.58/8.74	= { by lemma 65 }
8.58/8.74	  composition(meet(sK2_goals_X0, composition(complement(complement(sK2_goals_X0)), sK3_goals_X1)), sK1_goals_X2)
8.58/8.74	= { by lemma 37 }
8.58/8.74	  composition(meet(sK2_goals_X0, composition(sK2_goals_X0, sK3_goals_X1)), sK1_goals_X2)
8.58/8.74	= { by lemma 17 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), sK2_goals_X0), sK1_goals_X2)
8.58/8.74	= { by lemma 64 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, complement(composition(sK2_goals_X0, sK3_goals_X1)))), sK1_goals_X2)
8.58/8.74	= { by lemma 35 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, join(zero, complement(composition(sK2_goals_X0, sK3_goals_X1))))), sK1_goals_X2)
8.58/8.74	= { by lemma 71 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(converse(complement(join(zero, complement(composition(sK2_goals_X0, sK3_goals_X1))))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 19 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(converse(meet(composition(sK2_goals_X0, sK3_goals_X1), top)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 36 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(converse(composition(sK2_goals_X0, sK3_goals_X1)))))), sK1_goals_X2)
8.58/8.74	= { by axiom 3 (converse_multiplicativity) }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 73 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), composition(join(one, converse(sK3_goals_X1)), converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 56 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), composition(converse(join(one, sK3_goals_X1)), converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 57 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), composition(converse(one), converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 55 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), composition(one, converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 22 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), converse(sK2_goals_X0)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 17 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), join(sK2_goals_X0, converse(complement(meet(converse(sK2_goals_X0), composition(converse(sK3_goals_X1), converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 60 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), converse(join(converse(sK2_goals_X0), complement(meet(converse(sK2_goals_X0), composition(converse(sK3_goals_X1), converse(sK2_goals_X0))))))), sK1_goals_X2)
8.58/8.74	= { by lemma 17 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), converse(join(converse(sK2_goals_X0), complement(meet(composition(converse(sK3_goals_X1), converse(sK2_goals_X0)), converse(sK2_goals_X0)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 41 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), converse(join(converse(sK2_goals_X0), join(complement(composition(converse(sK3_goals_X1), converse(sK2_goals_X0))), complement(converse(sK2_goals_X0)))))), sK1_goals_X2)
8.58/8.74	= { by lemma 27 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), converse(join(complement(composition(converse(sK3_goals_X1), converse(sK2_goals_X0))), top))), sK1_goals_X2)
8.58/8.74	= { by lemma 29 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), converse(top)), sK1_goals_X2)
8.58/8.74	= { by lemma 62 }
8.58/8.74	  composition(meet(composition(sK2_goals_X0, sK3_goals_X1), top), sK1_goals_X2)
8.58/8.74	= { by lemma 36 }
8.58/8.74	  composition(composition(sK2_goals_X0, sK3_goals_X1), sK1_goals_X2)
8.58/8.74	= { by axiom 8 (composition_associativity) }
8.58/8.74	  composition(sK2_goals_X0, composition(sK3_goals_X1, sK1_goals_X2))
8.58/8.74	= { by lemma 46 }
8.58/8.74	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(sK1_goals_X2, complement(composition(sK3_goals_X1, sK1_goals_X2))))))
8.58/8.74	= { by axiom 10 (maddux1_join_commutativity) }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), sK1_goals_X2))))
8.58/8.75	= { by lemma 22 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(one, sK1_goals_X2)))))
8.58/8.75	= { by lemma 31 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(meet(one, sK3_goals_X1), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 17 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(meet(sK3_goals_X1, one), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 34 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(join(meet(sK3_goals_X1, one), zero), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 18 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(join(meet(sK3_goals_X1, one), complement(top)), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 29 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(join(meet(sK3_goals_X1, one), complement(join(one, top))), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 58 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(join(meet(sK3_goals_X1, one), complement(join(one, complement(sK3_goals_X1)))), meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 46 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(join(complement(composition(sK3_goals_X1, sK1_goals_X2)), composition(join(sK3_goals_X1, meet(one, complement(sK3_goals_X1))), sK1_goals_X2)))))
8.58/8.75	= { by lemma 72 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), complement(top)))
8.58/8.75	= { by lemma 18 }
8.58/8.75	  composition(sK2_goals_X0, join(meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2), zero))
8.58/8.75	= { by lemma 34 }
8.58/8.75	  composition(sK2_goals_X0, meet(composition(sK3_goals_X1, sK1_goals_X2), sK1_goals_X2))
8.58/8.75	= { by lemma 17 }
8.58/8.75	  composition(sK2_goals_X0, meet(sK1_goals_X2, composition(sK3_goals_X1, sK1_goals_X2)))
8.58/8.75	= { by lemma 84 }
8.58/8.75	  meet(composition(sK2_goals_X0, top), meet(sK1_goals_X2, composition(sK3_goals_X1, sK1_goals_X2)))
8.58/8.75	= { by lemma 85 }
8.58/8.75	  meet(sK1_goals_X2, meet(composition(sK2_goals_X0, top), composition(sK3_goals_X1, sK1_goals_X2)))
8.58/8.75	= { by lemma 84 }
8.58/8.75	  meet(sK1_goals_X2, composition(sK2_goals_X0, composition(sK3_goals_X1, sK1_goals_X2)))
8.58/8.75	= { by lemma 83 }
8.58/8.75	  meet(sK1_goals_X2, meet(composition(sK3_goals_X1, sK1_goals_X2), composition(sK2_goals_X0, top)))
8.58/8.75	= { by lemma 85 }
8.58/8.75	  meet(composition(sK3_goals_X1, sK1_goals_X2), meet(sK1_goals_X2, composition(sK2_goals_X0, top)))
8.58/8.75	= { by lemma 83 }
8.58/8.75	  meet(composition(sK3_goals_X1, sK1_goals_X2), composition(sK2_goals_X0, sK1_goals_X2))
8.58/8.75	= { by lemma 17 }
8.58/8.75	  meet(composition(sK2_goals_X0, sK1_goals_X2), composition(sK3_goals_X1, sK1_goals_X2))
8.58/8.75	% SZS output end Proof
8.58/8.75	
8.58/8.75	RESULT: Theorem (the conjecture is true).
8.58/8.75	EOF