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 & p <> Top L implies ex I st p in I & I is max-ideal
proof
  assume L is upper-bounded;
  then
A1: L.: is lower-bounded & Bottom (L.: ) = Top L by LATTICE2:49,62;
  assume p <> Top L;
  then consider F being Filter of L.: such that
A2: p.: in F & F is being_ultrafilter by A1,FILTER_0:20;
  take .:F;
  ( .:F).: = .:F;
  hence thesis by A2,Th32;
end;
