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 Th51:
  c_d(r).n >= 0
proof
  set s=scf(r);
  set s1=c_d(r);
  defpred P[Nat] means s1.$1>=0;
A1: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume
A2: s1.n>=0;
A3: s1.(n+2)=s.(n+2) * s1.(n+1) + s1.n by Def6;
    n+2>1+0 by XREAL_1:8;
    then
A4: s.(n+2) >=0 by Th38;
    assume s1.(n+1)>=0;
    hence thesis by A4,A3,A2;
  end;
  s.1>=0 by Th38;
  then
A5: P[1] by Def6;
A6: P[0] by Def6;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A6,A5,A1);
  hence thesis;
end;
