
theorem
  for L being non trivial Boolean Lattice,
      A being Filter of L
    st L = BooleLatt {{}} & A = {Top L} holds
      A is prime
  proof
    let L be non trivial Boolean Lattice,
        A be Filter of L;
    assume
A1: L = BooleLatt {{}} & A = {Top L};
    for H being Filter of L st
      A c= H & H <> the carrier of L holds A = H
    proof
      let H be Filter of L;
      assume A c= H & H <> the carrier of L; then
      H = {} or H = {{{}}} by lemma3,A1,lemma2;
      hence thesis by A1,LATTICE3:4;
    end; then
    A is being_ultrafilter by A1,FILTER_0:def 3;
    hence thesis by FILTER_0:45;
  end;
