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;
reserve D for non empty Subset of L,
  D9 for non empty Subset of L.:;
reserve D1,D2 for non empty Subset of L,
  D19,D29 for non empty Subset of L.:;

theorem Th41:
  L is upper-bounded & Top L in D implies (.D.> = (.L.> & (.D.> =
  the carrier of L
proof
  assume L is upper-bounded;
  then
A1: L.: is lower-bounded & Bottom (L.:) = Top L by LATTICE2:49,62;
  assume Top L in D;
  then <.D.:.) = <.L.:.) by A1,FILTER_0:25;
  hence thesis by Th36;
end;
