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 Th45:
  scf(r).0 > 0 implies for n holds c_n(r).n > 0
proof
  assume
A1: scf(r).0>0;
  set s1=c_n(r);
  set s=scf(r);
  defpred P[Nat] means s1.$1>0;
  s1.1 = s.1 * s.0 +1 & s.1 >=0 by Def5,Th38;
  then
A2: P[1] by A1;
  let n;
A3: for n being Nat st P[n] & P[n+1] holds P[n+2]
  proof
    let n be Nat;
    assume
A4: s1.n>0 & s1.(n+1)>0;
    n+2>1+0 by XREAL_1:8;
    then
A5: s.(n+2) >=0 by Th38;
    s1.(n+2)=s.(n+2) * s1.(n+1) + s1.n by Def5;
    hence thesis by A5,A4;
  end;
A6: P[0] by A1,Def5;
  for n being Nat holds P[n] from FIB_NUM:sch 1(A6,A2,A3);
  hence thesis;
end;
