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

theorem
  a <= r & s <= b implies ].r,s.[ is open 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_1:28;
    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_1:37;
    reconsider C = A as Subset of R^1 by TOPMETR:17;
    C is open & C /\ [#]T = A by JORDAN6:35,XBOOLE_1:28;
    hence thesis by TOPS_2:24;
  end;
end;
