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;
reserve p,q for Element of StoneLatt(L);
reserve H for non trivial H_Lattice;
reserve p9,q9 for Element of H;

theorem Th30:
  StoneS(H) c= bool F_primeSet(H)
proof
  let x be object;
   reconsider xx=x as set by TARSKI:1;
  assume x in StoneS(H);
  then consider p9 such that
A1: x=StoneH(H).p9 by Th13;
A2: x={F where F is Filter of H:F in F_primeSet(H) & p9 in F} by A1,Def6;
  xx c= F_primeSet(H)
  proof
    let y be object;
    assume y in xx;
    then
    ex F being Filter of H st y=F & F in F_primeSet(H) & p9 in F by A2;
    hence thesis;
  end;
  hence thesis;
end;
