reserve a, b, r, s for Real;

theorem
  r <= s implies for A being Subset of Closed-Interval-TSpace(r,s) holds
  A is real-bounded Subset of REAL
proof
  assume r <= s;
  then
A1: the carrier of Closed-Interval-TSpace(r,s) = [.r,s.] by TOPMETR:18;
  let A be Subset of Closed-Interval-TSpace(r,s);
  A is bounded_above bounded_below by A1,XXREAL_2:43,44;
  hence thesis by A1,XBOOLE_1:1;
end;
