reserve L for Lattice,
  p,q,r for Element of L,
  p9,q9,r9 for Element of L.:,
  x, y for set;
reserve I,J for Ideal of L,
  F for Filter of L;

theorem
  L is lower-bounded implies Bottom L in I
proof
  assume L is lower-bounded;
  then Top (L.:) = Bottom L & L.: is upper-bounded by LATTICE2:48,61;
  then Bottom L in I.: by FILTER_0:11;
  hence thesis;
end;
