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 for I st I <> the carrier of L ex J st I c=
  J & J is max-ideal
proof
  assume L is upper-bounded;
  then
A1: L.: is lower-bounded by LATTICE2:49;
  let I;
  assume I <> carr(L);
  then consider F being Filter of L.: such that
A2: I.: c= F & F is being_ultrafilter by A1,FILTER_0:18;
  take .:F;
  ( .:F).: = .:F;
  hence thesis by A2,Th32;
end;
