reserve X for set,
  Y for non empty set;

theorem Th3:
  for f being Function of X,Y, A being Subset of X st f is onto
  holds (f.:A)` c= f.:A`
proof
  let f be Function of X,Y, A be Subset of X such that
A1: f is onto;
  let e be object;
  assume
A2: e in (f.:A)`;
  then reconsider y = e as Element of Y;
  consider u being object such that
A3: u in X and
A4: y = f.u by A1,Th1;
  reconsider x=u as Element of X by A3;
  now
    assume x in A;
    then y in f.:A by A4,FUNCT_2:35;
    hence contradiction by A2,XBOOLE_0:def 5;
  end;
  then x in A` by A3,SUBSET_1:29;
  hence thesis by A4,FUNCT_2:35;
end;
