reserve i for Integer,
  a, b, r, s for Real;

theorem
  a <= r & s <= b implies [.r,s.] is closed Subset of
  Closed-Interval-TSpace(a,b)
proof
  set T = Closed-Interval-TSpace(a,b);
  set A = [.r,s.];
  assume that
A1: a <= r and
A2: s <= b;
  per cases;
  suppose
    r > s;
    then A = {}T by XXREAL_1:29;
    hence thesis;
  end;
  suppose
    r <= s;
    then a <= s by A1,XXREAL_0:2;
    then the carrier of T = [.a,b.] by A2,TOPMETR:18,XXREAL_0:2;
    then reconsider A as Subset of T by A1,A2,XXREAL_1:34;
    reconsider C = A as Subset of R^1 by TOPMETR:17;
    C is closed & C /\ [#]T = A by TREAL_1:1,XBOOLE_1:28;
    hence thesis by PRE_TOPC:13;
  end;
end;
