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 Th65:
  (for n holds c_d(r).n<>0) implies c_n(r).(n+1)/c_d(r).(n+1) -
  c_n(r).n/c_d(r).n = (-1)|^n / (c_d(r).(n+1)*c_d(r).n)
proof
  set s1=c_n(r), s2=c_d(r);
  assume for n holds s2.n<>0;
  then s2.n<>0 & s2.(n+1)<>0;
  then
  s1.(n+1)/s2.(n+1)-s1.n/s2.n =(s1.(n+1)*s2.n-s1.n*s2.(n+1))/(s2.(n+1)*s2.
  n) by XCMPLX_1:130
    .=(-1)|^n /(s2.(n+1)*s2.n) by Th64;
  hence thesis;
end;
