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 c_d(r).n<>0) implies for n holds |.c_n(r).(n+1)/c_d(r)
  .(n+1) - c_n(r).n/c_d(r).n.| = 1 / |.c_d(r).(n+1)*c_d(r).n.|
proof
  set s1=c_n(r), s2=c_d(r);
  assume
A1: for n holds s2.n<>0;
  let n;
  reconsider n as Element of NAT by ORDINAL1:def 12;
  |.s1.(n+1)/s2.(n+1)-s1.n/s2.n.| =|. (-1) |^n / (s2.(n+1)*s2.n).| by A1
,Th65
    .=|. (-1) |^n .| / |.s2.(n+1)*s2.n.| by COMPLEX1:67
    .=|. (-1) to_power n .| / |.s2.(n+1)*s2.n.| by POWER:41
    .=(|.-1.| to_power n) / |.s2.(n+1)*s2.n.| by SERIES_1:2
    .=((-(-1)) to_power n) / |.s2.(n+1)*s2.n.| by ABSVALUE:def 1
    .=1 / |.s2.(n+1)*s2.n.| by POWER:26;
  hence thesis;
end;
