
theorem Lem1:
  for L being Lattice,
      F being Filter of L st F is not prime holds
    ex a, b being Element of L st
    a "\/" b in F & not a in F & not b in F
  proof
    let L be Lattice,
        F be Filter of L;
    assume F is not prime; then
    consider a, b being Element of L such that
Z1: not (a "\/" b in F iff a in F or b in F) by FILTER_0:def 5;
    now assume not a "\/" b in F & (a in F or b in F); then
      per cases;
      suppose not a "\/" b in F & a in F;
        hence contradiction by FILTER_0:10;
      end;
      suppose not a "\/" b in F & b in F;
        hence contradiction by FILTER_0:10;
      end;
    end;
    hence thesis by Z1;
  end;
