reserve X for TopSpace;

theorem
  for X1 being closed SubSpace of X, X2 being closed SubSpace of X1
  holds X2 is closed SubSpace of X
proof
  let X1 be closed SubSpace of X, X2 be closed SubSpace of X1;
  now
    reconsider C = [#]X1 as Subset of X by BORSUK_1:1;
    let B be Subset of X;
    assume
A1: B = the carrier of X2;
    then reconsider BB = B as Subset of X1 by BORSUK_1:1;
    BB is closed by A1,BORSUK_1:def 11;
    then
A2: ex A being Subset of X st A is closed & A /\ [#]X1 = BB by PRE_TOPC:13;
    C is closed by BORSUK_1:def 11;
    hence B is closed by A2;
  end;
  hence thesis by Th7,BORSUK_1:def 11;
end;
