
theorem Th6:
  for S being RelStr, T being non empty RelStr, F being Subset of (
  T |^ the carrier of S) holds sup F is Function of S, T
proof
  let S be RelStr, T be non empty RelStr;
  let F be Subset of (T |^ the carrier of S);
  set f = sup F;
  f in the carrier of (T |^ the carrier of S);
  then f in Funcs (the carrier of S, the carrier of T) by YELLOW_1:28;
  then ex f9 being Function st f9 = f & dom f9 = the carrier of S & rng f9 c=
  the carrier of T by FUNCT_2:def 2;
  hence thesis by FUNCT_2:def 1,RELSET_1:4;
end;
