
theorem Th63:
  for S being RelStr, T being non empty RelStr, f being Function of S,T
  for S9 being SubRelStr of S for T9 being SubRelStr of T
  st f.:the carrier of S9 c= the carrier of T9
  holds f|the carrier of S9 is Function of S9, T9
proof
  let S be RelStr, T be non empty RelStr, f be Function of S,T;
  let S9 be SubRelStr of S, T9 be SubRelStr of T such that
A1: f.:the carrier of S9 c= the carrier of T9;
  set g = f|the carrier of S9;
A2: dom f = the carrier of S by FUNCT_2:def 1;
  the carrier of S9 c= the carrier of S by YELLOW_0:def 13;
  then
A3: dom g = the carrier of S9 by A2,RELAT_1:62;
  rng g = f.:the carrier of S9 by RELAT_1:115;
  hence thesis by A1,A3,FUNCT_2:2;
end;
