theorem Th42:
  f is onto implies for c being Element of X9 ex x st c = f.x
proof
A1: for c being set holds c in rng f iff ex x st c = f.x
  proof
    let c be set;
    thus c in rng f implies ex x st c = f.x
    proof
      assume c in rng f;
      then consider y being object such that
A2:   y in dom f and
A3:   f.y = c by FUNCT_1:def 3;
      reconsider y as Element of X by A2;
      take y;
      thus thesis by A3;
    end;
    given x such that
A4: c = f.x;
    the carrier of X = dom f by FUNCT_2:def 1;
    hence thesis by A4,FUNCT_1:def 3;
  end;
  assume f is onto;
  then rng f = the carrier of X9 by FUNCT_2:def 3;
  hence thesis by A1;
end;
