theorem Th4:
  for Y being non empty set for f being sequence of Y holds x
  in rng f iff ex n st x = f.n
proof
  let Y be non empty set;
  let f be sequence of Y;
  thus x in rng f implies ex n st x = f.n
  proof
    assume x in rng f;
    then consider y being object such that
A1: y in dom f and
A2: x = f.y by FUNCT_1:def 3;
    reconsider m=y as Element of NAT by A1;
    take m;
    thus thesis by A2;
  end;
  given n such that
A3: x = f.n;
A4: n in NAT by ORDINAL1:def 12;
  dom f = NAT by FUNCT_2:def 1;
  hence thesis by A3,FUNCT_1:def 3,A4;
end;
