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
  L is upper-bounded & Top L in I implies I = (.L.> & I = the carrier of L
proof
  (.I.> = I by Th37;
  hence thesis by Th41;
end;
