
theorem
  for L being Lattice,
      a being Element of L,
      F being Filter of L holds
    F in PrimeFilters L.a iff F is prime & a in F
  proof
    let L be Lattice,
        a be Element of L,
        F be Filter of L;
    hereby
      assume F in PrimeFilters L.a;
      then ex F0 being Filter of L st
        F0 = F & F0 is prime & a in F0 by Th17;
      hence F is prime & a in F;
    end;
    thus thesis by Th17;
  end;
