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 Th17:
  x in SF_have b \ SF_have a implies x is Filter of L & b in x & not a in x
proof
  assume
A1: x in SF_have b \ SF_have a;
  then
A2: not x in SF_have a by XBOOLE_0:def 5;
A3: x in SF_have b by A1,XBOOLE_0:def 5;
  then x is Filter of L by Th16;
  hence thesis by A3,A2,Th16;
end;
