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 Th21:
  a <> b implies ex F st F in F_primeSet(L)
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;
  hence thesis;
end;
