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
  scf(r).0 > 0 implies for n holds bcf(r).(n+1)=c_n(r).(n+1)/c_n(r).n
proof
  set s1=c_n(r);
  set s=scf(r);
  defpred X[Nat] means bcf(r).($1+1)=s1.($1+1)/s1.$1;
  set s3=bcf(r);
  assume
A1: scf(r).0 > 0;
A2: for n st X[n] holds X[n+1]
  proof
    let n;
    assume
A3: bcf(r).(n+1)=s1.(n+1)/s1.n;
A4: s1.(n+1)>0 by A1,Th45;
    bcf(r).(n+1+1) = 1/s3.(n+1) + s.(n+1+1) by Def8
      .=s1.n/s1.(n+1)+ s.(n+1+1) by A3,XCMPLX_1:57
      .=(s1.n+ s.(n+2)*s1.(n+1))/s1.(n+1) by A4,XCMPLX_1:113
      .=s1.(n+2)/s1.(n+1) by Def5;
    hence thesis;
  end;
  bcf(r).(0+1) = 1/s3.0 + s.(0+1) by Def8
    .=1/s.0 + s.1 by Def8
    .=(1+s.0 * s.1)/s.0 by A1,XCMPLX_1:113
    .=s1.1/s.0 by Def5
    .=s1.(0+1)/s1.0 by Def5;
  then
A5: X[0];
  for n holds X[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
end;
