
theorem :: THEOREM 4.16. (1) iff (4)
  for L be lower-bounded LATTICE holds ( L is continuous implies ex X be
  non empty set, p be projection Function of BoolePoset X,BoolePoset X st p is
  directed-sups-preserving & L,Image p are_isomorphic ) & (( ex X be set, p be
  projection Function of BoolePoset X,BoolePoset X st p is
directed-sups-preserving & L,Image p are_isomorphic ) implies L is continuous )
proof
  let L be lower-bounded LATTICE;
  thus L is continuous implies ex X be non empty set, p be projection Function
  of BoolePoset X,BoolePoset X st p is directed-sups-preserving & L,Image p
  are_isomorphic
  proof
    assume L is continuous;
    then
    ex A be arithmetic lower-bounded LATTICE, g be Function of A,L st g is
    onto & g is infs-preserving & g is directed-sups-preserving by Lm2;
    hence thesis by Lm3;
  end;
  thus thesis by Lm4;
end;
