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 Th28:
  StoneH(H).(Top H) = F_primeSet(H)
proof
  hereby
    let x be object;
    assume x in StoneH(H).(Top H);
    then
    ex F being Filter of H st F=x & F <> the carrier of H & F is prime
    & Top H in F by Th12;
    hence x in F_primeSet(H);
  end;
  let x be object;
  assume x in F_primeSet(H);
  then consider F being Filter of H such that
A1: F=x and
A2: F <> the carrier of H and
A3: F is prime;
  Top H in F by FILTER_0:11;
  hence thesis by A1,A2,A3,Th12;
end;
