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 Th22:
  a <> b implies StoneH(L).a <> StoneH(L).b
proof
  assume a <> b;
  then not a [= b or not b [= a by LATTICES:8;
  then (ex F st F in F_primeSet(L) & not b in F & a in F) or ex F st F in
  F_primeSet(L) & not a in F & b in F by Th20;
  then consider F such that
A1: F in F_primeSet(L) and
A2: b in F & not a in F or a in F & not b in F;
  F in StoneH(L).a & not F in StoneH(L).b or F in StoneH(L).b & not F in
  StoneH(L).a by A1,A2,Th11;
  hence thesis;
end;
