reserve T for TopSpace;
reserve A,B for Subset of T;
reserve T for non empty TopSpace;
reserve P,Q for Element of Topology_of T;
reserve p,q for Element of Open_setLatt(T);
reserve L for D_Lattice;
reserve F for Filter of L;
reserve a,b for Element of L;
reserve x,X,X1,X2,Y,Z for set;

theorem Th13:
  x in StoneS(L) iff ex a st x=StoneH(L).a
proof
  hereby
    assume x in StoneS(L);
    then consider y be object such that
A1: y in dom StoneH(L) and
A2: x = StoneH(L).y by FUNCT_1:def 3;
    reconsider y as Element of L by A1,Def6;
    take y;
    thus x=StoneH(L).y by A2;
  end;
  given b such that
A3: x=StoneH(L).b;
  b in the carrier of L;
  then b in dom StoneH(L) by Def6;
  hence thesis by A3,FUNCT_1:def 3;
end;
