reserve X for TopSpace;

theorem
  for X1, X2 being SubSpace of X st X1 is SubSpace of X2 & X2 is
  SubSpace of X1 holds the TopStruct of X1 = the TopStruct of X2
proof
  let X1,X2 be SubSpace of X;
  set A1 = the carrier of X1, A2 = the carrier of X2;
  assume X1 is SubSpace of X2 & X2 is SubSpace of X1;
  then A1 c= A2 & A2 c= A1 by Th4;
  then A1 = A2 by XBOOLE_0:def 10;
  hence thesis by Th5;
end;
