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

theorem Th16:
  (X c= dom f or dom g c= dom f) & f|X = g|X & f,X
  is_absolutely_bounded_by r implies g,X is_absolutely_bounded_by r
proof
  assume that
A1: X c= dom f or dom g c= dom f and
A2: f|X = g|X and
A3: f,X is_absolutely_bounded_by r;
  let x be set;
  assume
A4: x in X /\ dom g;
  then
A5: x in X by XBOOLE_0:def 4;
  then
A6: f.x = (f|X).x by FUNCT_1:49
    .= g.x by A2,A5,FUNCT_1:49;
  x in dom g by A4,XBOOLE_0:def 4;
  then x in X /\ dom f by A1,A5,XBOOLE_0:def 4;
  hence thesis by A3,A6;
end;
