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 Th44:
  scf(r).0 > 0 implies for n holds c_n(r).n in NAT
proof
  set s1=c_n(r);
  set s=scf(r);
  defpred P[Nat] means s1.$1 in NAT;
A1: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume that
A2: s1.n in NAT and
A3: s1.(n+1) in NAT;
    reconsider n2=s1.(n+1) as Element of NAT by A3;
    reconsider n1=s1.n as Element of NAT by A2;
    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 Def5;
  end;
  assume
A4: scf(r).0 > 0;
A5: P[1]
  proof
    reconsider n2=s.0 as Element of NAT by A4,INT_1:3;
    reconsider n1=s.1 as Element of NAT by Th38,INT_1:3;
    n1*n2+1 in NAT;
    hence thesis by Def5;
  end;
  let n;
  s.0 in NAT by A4,INT_1:3;
  then
A6: P[0] by Def5;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A6,A5,A1);
  hence thesis;
end;
