reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;

theorem Th20:
  x is Ideal of L iff x is Filter of L.:
proof
  thus x is Ideal of L implies x is Filter of L.:
  proof
    assume x is Ideal of L;
    then reconsider I = x as Ideal of L;
    reconsider I as non empty Subset of L.:;
    I is Filter of L.:
    proof
      now let p9,q9;
      p9"/\"q9 = .:p9"\/".:q9;
      hence  p9 in I & q9 in I iff p9"/\"q9 in I by Lm1;
      end;
      hence thesis by FILTER_0:8;
    end;
    hence thesis;
  end;
  assume x is Filter of L.:;
  then reconsider F = x as Filter of L.:;
  reconsider F as non empty Subset of L;
  now
    let p,q;
    p.:"/\"q.: = p"\/"q;
    hence p in F & q in F iff p"\/"q in F by FILTER_0:8;
  end;
  hence thesis by Lm1;
end;
