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
  (for n holds scf(r).n<>0) implies for n st n >= 1 holds c_n(r).n / c_d
  (r).n = (c_n(r).(n+1)-c_n(r).(n-1)) / (c_d(r).(n+1)-c_d(r).(n-1))
proof
  set s1=c_n(r), s2=c_d(r);
  set s=scf(r);
  defpred P[Nat] means s1.$1/s2.$1=(s1.($1+1)-s1.($1-1))/(s2.($1+1)-s2.($1-1));
  assume
A1: for n holds scf(r).n<>0;
A2: for n being Nat st n>=1 holds P[n] implies P[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
    s1.n/s2.n=(s1.(n+1)-s1.(n-1))/(s2.(n+1)-s2.(n-1));
    s.(n+2) * s1.(n+1) + s1.n - s1.n = s1.(n+2) - s1.n & s2.(n+2) - s2.n =
    s.(n+ 2) * s2.(n+1) + s2.n - s2.n by Def5,Def6;
    hence thesis by A1,XCMPLX_1:91;
  end;
  let n;
  (s1.(1+1)-s1.(1-1))/(s2.(1+1)-s2.(1-1)) =(s.(2+0) * s1.(0+1) + s1.0-s1.0
  )/(s2.(2+0)-s2.0) by Def5
    .=(s.2 * s1.1 + s1.0-s1.0)/(s.2 * s2.1 + s2.0-s2.0) by Def6
    .=s1.1/s2.1 by A1,XCMPLX_1:91;
  then
A3: P[1];
  for n being Nat st n>=1 holds P[n] from NAT_1:sch 8(A3,A2);
  hence thesis;
end;
