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 Th19:
  not b [= a implies <.b.) in SF_have b \ SF_have a
proof
  assume not b [= a;
  then not a in <.b.) by FILTER_0:15;
  then
A1: not <.b.) in SF_have a by Th16;
  b in <.b.);
  then <.b.) in SF_have b;
  hence thesis by A1,XBOOLE_0:def 5;
end;
