reserve X for set,
  Y for non empty set;

theorem
  for f being Function of X,Y st f is onto for y being Element of Y ex x
  being Element of X st y = f.x
proof
  let f be Function of X,Y such that
A1: f is onto;
  let y be Element of Y;
  ex x being object st x in X & f.x = y by A1,Th1;
  hence thesis;
end;
