
theorem lemma2:
  for L being Lattice,
      A being Filter of L
    st L = BooleLatt {{}} holds
      A = {} or A = {{},{{}}} or A = {{{}}}
  proof
    let L be Lattice,
        A be Filter of L;
    assume
A0: L = BooleLatt {{}};
    A <> {{}}
    proof
      assume
Z0:   A = {{}};
      reconsider b = {{}}, a = {} as Element of L by TARSKI:def 2,A0,lemma3;
z1:   {{}} /\ {} = b "/\" a by A0,LATTICE3:1;
      b in A & a in A iff b "/\" a in A by FILTER_0:8;
      hence thesis by z1,Z0,TARSKI:def 1;
    end;
    hence thesis by A0,lemma1;
  end;
