
theorem Th17:
  for L being Lattice,
      a being Element of L,
      x being set holds
  x in PrimeFilters L.a iff ex F being Filter of L st
    F = x & F is prime & a in F
  proof
    let L be Lattice,
        a be Element of L, x be set;
    thus x in PrimeFilters L.a implies ex F being Filter of L st
      F = x & F is prime & a in F
    proof
      assume x in PrimeFilters L.a;
      then x in { F1 where F1 is Filter of L : F1 is prime & a in F1 }
        by Def6;
      then consider F being Filter of L such that
  A2: F = x and
  A3: F is prime and
  A4: a in F;
      take F;
      thus thesis by A2,A3,A4;
    end;
    assume ex F being Filter of L st F = x & F is prime & a in F;
    then x in { F where F is Filter of L : F is prime & a in F };
    hence thesis by Def6;
  end;
