reserve Y for TopStruct;

theorem Th20:
  for Y being non empty TopStruct, Y0 being non empty SubSpace of
  Y, A being Subset of Y st A = the carrier of Y0 holds A is discrete iff Y0 is
  discrete
proof
  let Y be non empty TopStruct, Y0 be non empty SubSpace of Y, A be Subset of
  Y;
  assume
A1: A = the carrier of Y0;
A2: [#]Y = the carrier of Y;
  [#]Y0 = the carrier of Y0;
  then
A3: the carrier of Y0 c= the carrier of Y by A2,PRE_TOPC:def 4;
  hereby
    assume
A4: A is discrete;
    now
      let C be object;
      assume C in bool the carrier of Y0;
      then reconsider B = C as Subset of Y0;
      reconsider D = B as Subset of Y by A3,XBOOLE_1:1;
      consider G being Subset of Y such that
A5:   G is open and
A6:   A /\ G = D by A1,A4;
        G in the topology of Y & B = G /\ [#]Y0 by A1,A6,A5;
      hence C in the topology of Y0 by PRE_TOPC:def 4;
    end;
    then bool the carrier of Y0 c= the topology of Y0 by TARSKI:def 3;
    then the topology of Y0 = bool the carrier of Y0;
    hence Y0 is discrete by TDLAT_3:def 1;
  end;
  hereby
    assume
A7: Y0 is discrete;
    now
      let D be Subset of Y;
      assume D c= A;
      then reconsider B = D as Subset of Y0 by A1;
      B is open by A7,TDLAT_3:15;
      then B in the topology of Y0;
      then consider G being Subset of Y such that
A8:   G in the topology of Y and
A9:   B = G /\ [#]Y0 by PRE_TOPC:def 4;
      reconsider G as Subset of Y;
      take G;
      thus G is open by A8;
      thus A /\ G = D by A1,A9;
    end;
    hence A is discrete;
  end;
end;
