
theorem
  for L being Lattice,
      A being Filter of L
    st L = BooleLatt {{}} holds
      A = {Top L} or A = <.L.)
  proof
    let L be Lattice,
        A be Filter of L;
    assume
A0: L = BooleLatt {{}};
    Top L = {{}} by A0,LATTICE3:4; then
    reconsider B = {{{}}} as Filter of L by FILTER_0:12,A0;
    per cases by lemma2,A0;
    suppose
      A = {};
      hence thesis;
    end;
    suppose A = {{},{{}}};
      hence thesis by lemma3,A0;
    end;
    suppose A = {{{}}};
      hence thesis by A0,LATTICE3:4;
    end;
  end;
