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;
reserve l, n1, n2 for Nat;
reserve s1, s2 for Real_Sequence;

theorem Th50:
  c_d(r).n in NAT
proof
  set s=scf(r);
  set s2=c_d(r);
  defpred P[Nat] means s2.$1 in NAT;
  s2.0 = 1 by Def6;
  then
A1: P[0];
A2: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume that
A3: s2.n in NAT and
A4: s2.(n+1) in NAT;
    reconsider n2=s2.(n+1) as Element of NAT by A4;
    reconsider n1=s2.n as Element of NAT by A3;
    n+2>=0+1 by XREAL_1:7;
    then reconsider n3=s.(n+2) as Element of NAT by Th38,INT_1:3;
    n3*n2+n1 in NAT;
    hence thesis by Def6;
  end;
  s2.1=s.1 by Def6;
  then
A5: P[1] by Th38,INT_1:3;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A1,A5,A2);
  hence thesis;
end;
