reserve a, b, k, n, m for Nat,
  i for Integer,
  r for Real,
  p for Rational,
  c for Complex,
  x for object,
  f for Function;

theorem
  f is sequence of NAT iff dom f=NAT & for x st x in NAT holds f.x is natural
proof
  thus f is sequence of NAT implies dom f=NAT & for x st x in NAT holds f.x is
  natural by FUNCT_2:def 1;
  assume that
A1: dom f= NAT and
A2: for x st x in NAT holds f.x is natural;
  rng f c= NAT
  proof
    let x be object;
    assume x in rng f;
    then ex y being object st y in NAT & x = f.y by A1,FUNCT_1:def 3;
    then x is natural by A2;
    hence thesis by ORDINAL1:def 12;
  end;
  hence thesis by A1,FUNCT_2:def 1,RELSET_1:4;
end;
