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

theorem
  a <= b & r <= b implies [.a,r.[ is open Subset of Closed-Interval-TSpace(a,b)
proof
  set T = Closed-Interval-TSpace(a,b);
  assume that
A1: a <= b and
A2: r <= b;
A3: the carrier of T = [.a,b.] by A1,TOPMETR:18;
  then reconsider A = [.a,r.[ as Subset of T by A2,XXREAL_1:35;
  reconsider C = ].a-1,r.[ as Subset of R^1 by TOPMETR:17;
A4: C /\ [#]T c= A
  proof
    let x be object;
    assume
A5: x in C /\ [#]T;
    then
A6: x in C by XBOOLE_0:def 4;
    then reconsider x as Real;
A7: x < r by A6,XXREAL_1:4;
    a <= x by A3,A5,XXREAL_1:1;
    hence thesis by A7;
  end;
  a-1 < a-0 by XREAL_1:15;
  then A c= C by XXREAL_1:48;
  then A c= C/\[#]T by XBOOLE_1:19;
  then C is open & C /\ [#]T = A by A4,JORDAN6:35;
  hence thesis by TOPS_2:24;
end;
