reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem Th15:
  f c= g & g,X is_absolutely_bounded_by r implies f,X
  is_absolutely_bounded_by r
proof
  assume that
A1: f c= g and
A2: g,X is_absolutely_bounded_by r;
  let x be set;
  assume
A3: x in X /\ dom f;
  then
A4: x in dom f by XBOOLE_0:def 4;
A5: x in X by A3,XBOOLE_0:def 4;
  dom f c= dom g by A1,GRFUNC_1:2;
  then x in X /\ dom g by A4,A5,XBOOLE_0:def 4;
  then |.g.x.| <= r by A2;
  hence thesis by A1,A4,GRFUNC_1:2;
end;
