reserve Y for TopStruct;

theorem
  for Y being non empty TopStruct, Y0 being non proper SubSpace of Y
  holds the TopStruct of Y0 = the TopStruct of Y
proof
  let Y be non empty TopStruct;
  let Y0 be non proper SubSpace of Y;
A1: ex A being Subset of Y st A = the carrier of Y0 & A is non proper by Def1;
  now
    let D be object;
    assume
A2: D in the topology of Y;
    then reconsider E = D as Subset of Y;
    reconsider C = E as Subset of Y0 by A1;
    now
      take E;
      thus E in the topology of Y & C = E /\ [#]Y0 by A2,XBOOLE_1:28;
    end;
    hence D in the topology of Y0 by PRE_TOPC:def 4;
  end;
  then
A3: the topology of Y c= the topology of Y0 by TARSKI:def 3;
A4: the carrier of Y0 = the carrier of Y by A1;
  now
    let D be object;
    assume
A5: D in the topology of Y0;
    then reconsider C = D as Subset of Y0;
    ex E being Subset of Y st E in the topology of Y & C = E /\ [#]Y0
    by A5,PRE_TOPC:def 4;
    hence D in the topology of Y by A4,XBOOLE_1:28;
  end;
  then the topology of Y0 c= the topology of Y by TARSKI:def 3;
  then the topology of Y0 = the topology of Y by A3;
  hence thesis by A1;
end;
