reserve X for TopSpace;

theorem
  for X being non empty TopSpace, X1 being closed non empty SubSpace of
  X, X2 being non empty SubSpace of X st the carrier of X1 c= the carrier of X2
  holds X1 is closed non empty SubSpace of X2
proof
  let X be non empty TopSpace, X1 be closed non empty SubSpace of X, X2 be non
  empty SubSpace of X;
  assume the carrier of X1 c= the carrier of X2;
  then reconsider B = the carrier of X1 as Subset of X2;
  now
    let C be Subset of X2;
    assume
A1: C = the carrier of X1;
    then reconsider A = C as Subset of X by BORSUK_1:1;
A2: A /\ [#]X2 = C by XBOOLE_1:28;
    A is closed by A1,Th11;
    hence C is closed by A2,PRE_TOPC:13;
  end;
  then B is closed;
  hence thesis by Th4,Th11;
end;
