reserve a, I for set,
  S for non empty non void ManySortedSign;

theorem Th2:
  for X being non empty set, Y being set, f being Function of X, {Y
  } holds rng f = {Y}
proof
  let X be non empty set, Y be set, f be Function of X, {Y};
  thus rng f c= {Y};
  let q be object;
  consider x being object such that
A1: x in X by XBOOLE_0:def 1;
  assume q in {Y};
  then
A2: dom f = X & q = Y by FUNCT_2:def 1,TARSKI:def 1;
  f.x = Y by A1,FUNCT_2:50;
  hence thesis by A2,A1,FUNCT_1:def 3;
end;
