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 Th22:
  f-g, X is_absolutely_bounded_by r implies g-f, X is_absolutely_bounded_by r
proof
  assume
A1: f-g, X is_absolutely_bounded_by r;
  let x be set;
  assume
A2: x in X /\ dom (g-f);
  then
A3: x in dom (g-f) by XBOOLE_0:def 4;
A4: dom (f-g)= dom f /\ dom g by VALUED_1:12
    .= dom (g-f) by VALUED_1:12;
  then |.(f-g).x.| <= r by A1,A2;
  then |.f.x-g.x.| <= r by A4,A3,VALUED_1:13;
  then |.-(f.x-g.x).| <= r by COMPLEX1:52;
  then |.g.x-f.x.| <= r;
  hence thesis by A3,VALUED_1:13;
end;
