reserve r for Real,
  X for set,
  f, g, h for real-valued Function;
reserve T for non empty TopSpace,
  A for closed Subset of T;

theorem Th21:
  for T being non empty TopSpace for f being Function of T, R^1
  for r being positive Real holds f, the carrier of T
is_absolutely_bounded_by r iff f is Function of T, Closed-Interval-TSpace(-r,r)
proof
  let T be non empty TopSpace;
  let f be Function of T, R^1;
  let r be positive Real;
A1: dom f = the carrier of T by FUNCT_2:def 1;
  thus f, the carrier of T is_absolutely_bounded_by r implies f is Function of
  T, Closed-Interval-TSpace(-r,r)
  proof
    assume
A2: f, the carrier of T is_absolutely_bounded_by r;
    now
      let x be object;
      assume x in the carrier of T;
      then x in (the carrier of T) /\ dom f by A1;
      then |.f.x.|<=r by A2;
      then 2*|.f.x.|<=2*r by XREAL_1:64;
      then |.2.|*|.f.x.|<=2*r by ABSVALUE:def 1;
      then |.-2.|*|.f.x.|<=2*r by COMPLEX1:52;
      then |.(-2)*f.x.|<=r--r by COMPLEX1:65;
      then |.-r+r-2*f.x.|<=r--r;
      then f.x in [.-r,r.] by RCOMP_1:2;
      hence f.x in the carrier of Closed-Interval-TSpace(-r,r) by TOPMETR:18;
    end;
    hence thesis by A1,FUNCT_2:3;
  end;
  assume
A3: f is Function of T, Closed-Interval-TSpace(-r,r);
  let x be set;
  assume x in (the carrier of T) /\ dom f;
  then f.x in the carrier of Closed-Interval-TSpace(-r,r) by A3,FUNCT_2:5;
  then f.x in [.-r,r.] by TOPMETR:18;
  then |.-r+r-2*f.x.|<=r--r by RCOMP_1:2;
  then |.(-2)*f.x.|<=r--r;
  then |.-2.|*|.f.x.|<=2*r by COMPLEX1:65;
  then |.2.|*|.f.x.|<=2*r by COMPLEX1:52;
  then 2*|.f.x.|<=2*r by ABSVALUE:def 1;
  then (2*|.f.x.|)/2<=(2*r)/2 by XREAL_1:72;
  hence thesis;
end;
