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 Th8:
  f is Integer_Sequence iff dom f=NAT & for x st x in NAT holds f.x is integer
proof
  thus f is Integer_Sequence implies dom f=NAT & for x st x in NAT holds f.x
  is integer by SEQ_1:1;
  assume that
A1: dom f= NAT and
A2: for x st x in NAT holds f.x is integer;
  now
    let y be object;
    assume y in rng f;
    then consider x being object such that
A3: x in dom f and
A4: y=f.x by FUNCT_1:def 3;
    f.x is integer by A1,A2,A3;
    hence y in INT by A4;
  end;
  then
A5: rng f c=INT;
  for x st x in NAT holds f.x is real
  proof
    let x;
    assume x in NAT;
    then f.x is integer by A2;
    hence thesis;
  end;
  hence thesis by A1,A5,RELAT_1:def 19,SEQ_1:1;
end;
