
theorem :: THEOREM 4.16. (1) iff (3)
  for L be lower-bounded LATTICE holds L is continuous iff ex A be
  algebraic lower-bounded LATTICE, g be Function of A,L st g is onto & g is
  infs-preserving & g is directed-sups-preserving
proof
  let L be lower-bounded LATTICE;
  thus L is continuous implies ex A be algebraic lower-bounded LATTICE, g be
  Function of A,L st g is onto & g is infs-preserving & g is
  directed-sups-preserving
  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;
  end;
  thus (ex A be algebraic lower-bounded LATTICE, g be Function of A,L st g is
  onto & g is infs-preserving & g is directed-sups-preserving) implies L is
  continuous
  proof
    assume ex A be algebraic lower-bounded LATTICE, g be Function of A,L st g
    is onto & g is infs-preserving & g is directed-sups-preserving;
    then 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
by Lm3;
    hence thesis by Lm4;
  end;
end;
