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 upper-bounded implies (.L.> = (.Top L.>
proof
  assume
A1: L is upper-bounded;
  then L.: is lower-bounded by LATTICE2:49;
  then
A2: <.L.:.) = <.Bottom (L.:).) by FILTER_0:17;
  Bottom (L.:) = (Top L).: by A1,LATTICE2:62;
  hence thesis by A2,Th29;
end;
