
theorem Th23:
  for L being distributive Lattice,
      x being set holds
    x in StoneR L iff ex a being Element of L st (PrimeFilters L).a = x
  proof
    let L be distributive Lattice,
        x be set;
    thus x in StoneR L implies ex a being Element of L st
      (PrimeFilters L).a = x
    proof
      assume x in StoneR L;
      then consider y being object such that
A2:   y in dom PrimeFilters L and
A3:   x = PrimeFilters L.y by FUNCT_1:def 3;
      reconsider a = y as Element of L by A2;
      take a;
      thus thesis by A3;
    end;
    given a being Element of L such that
A4: x = PrimeFilters L.a;
    a in the carrier of L;
    then a in dom PrimeFilters L by Def6;
    hence thesis by A4,FUNCT_1:def 3;
  end;
