
theorem
  for L being Lattice,
      A being Filter of L
    st L = BooleLatt {{}} & A is being_ultrafilter holds
      A = {Top L}
  proof
    let L be Lattice,
        A be Filter of L;
    assume
A0: L = BooleLatt {{}};
    assume
Z1: A is being_ultrafilter;
A1: Top L = {{}} by A0,LATTICE3:4; then
    reconsider B = {{{}}} as Filter of L by FILTER_0:12,A0;
Z2: not {} in A
    proof
      assume {} in A; then
      Bottom L in A by A0,LATTICE3:3;
      hence contradiction by Z1,LOPCLSET:29,A0;
    end;
    A <> {{},{{}}} by Z2,TARSKI:def 2;
    hence thesis by A1,lemma2,A0;
  end;
