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 Th29:
  StoneH(H).(Bottom H) = {}
proof
  set x = the Element of StoneH(H).(Bottom H);
  assume StoneH(H).(Bottom H) <> {};
  then
  ex F being Filter of H st F=x & F <> the carrier of H & F is prime &
  Bottom H in F by Th12;
  hence contradiction by FILTER_0:26;
end;
