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 Th32:
  StoneS(H) c= the carrier of Open_setLatt(HTopSpace H)
proof
  let x be object;
  set carrO = the carrier of Open_setLatt(HTopSpace H);
  assume x in StoneS(H);
  then reconsider z={x} as Subset of StoneS(H) by ZFMISC_1:31;
A1: union z = x by ZFMISC_1:25;
  carrO = the set of all union A where A is Subset of StoneS(H) by Def12;
  hence thesis by A1;
end;
