reserve Y for TopStruct;

theorem Th21:
  for Y being non empty TopStruct, A being Subset of Y st A = the
  carrier of Y holds A is discrete iff Y is discrete
proof
  let Y be non empty TopStruct, A be Subset of Y;
  assume
A1: A = the carrier of Y;
  hereby
    assume
A2: A is discrete;
    now
      let C be object;
      assume C in bool the carrier of Y;
      then reconsider B = C as Subset of Y;
      consider G being Subset of Y such that
A3:   G is open and
A4:   A /\ G = B by A1,A2;
      G = B by A1,A4,XBOOLE_1:28;
      hence C in the topology of Y by A3;
    end;
    then bool the carrier of Y c= the topology of Y by TARSKI:def 3;
    then the topology of Y = bool the carrier of Y;
    hence Y is discrete by TDLAT_3:def 1;
  end;
  hereby
    assume Y is discrete;
    then reconsider Y as discrete non empty TopStruct;
    now
      let D be Subset of Y;
      assume
A5:   D c= A;
      reconsider G = D as Subset of Y;
      take G;
      thus G is open by TDLAT_3:15;
      thus A /\ G = D by A5,XBOOLE_1:28;
    end;
    hence A is discrete;
  end;
end;
