reserve Y for TopStruct;

theorem Th22:
  for A, B being Subset of Y st B c= A holds A is discrete implies
  B is discrete
proof
  let A, B be Subset of Y;
  assume
A1: B c= A;
  assume
A2: A is discrete;
  now
    let D be Subset of Y;
    assume
A3: D c= B;
    then D c= A by A1,XBOOLE_1:1;
    then consider G being Subset of Y such that
A4: G is open and
A5: A /\ G = D by A2;
    hereby
      take G;
      D c= G by A5,XBOOLE_1:17;
      then
A6:   D c= B /\ G by A3,XBOOLE_1:19;
      B /\ G c= A /\ G by A1,XBOOLE_1:26;
      hence G is open & B /\ G = D by A4,A5,A6;
    end;
  end;
  hence thesis;
end;
