reserve X1, X2, Y for non empty RelStr,
  f for Function of [:X1, X2:], Y,
  x for Element of X1,
  y for Element of X2;

theorem Th19:
  for S being set, T being non empty RelStr, f be set holds f is
  Element of T |^ S iff f is Function of S, the carrier of T
proof
  let S be set, T be non empty RelStr, f be set;
  hereby
    assume f is Element of (T |^ S);
    then f in the carrier of (T |^ S);
    then f in Funcs (S, the carrier of T) by YELLOW_1:28;
    then
    ex a being Function st a = f & dom a = S & rng a c= the carrier of T by
FUNCT_2:def 2;
    hence f is Function of S, the carrier of T by FUNCT_2:def 1,RELSET_1:4;
  end;
  assume f is Function of S, the carrier of T;
  then f in Funcs (S, the carrier of T) by FUNCT_2:8;
  hence thesis by YELLOW_1:28;
end;
