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 holds 1/(scf(r).(n+1)*(c_d(r).
  n)^2) <= 1/(c_d(r).n)^2
proof
  assume
A1: for n holds scf(r).n <>0;
  let n;
  set s=scf(r);
  set s2=c_d(r);
  s.1 <>0 by A1;
  then s.1>0 by Th38;
  then
A2: s2.n>0 by Th52;
  n+1>=1+0 & s.(n+1) <>0 by A1,XREAL_1:7;
  then s.(n+1)*(s2.n)^2 >= 1*(s2.n)^2 by A2,Th40,XREAL_1:64;
  hence thesis by A2,SQUARE_1:12,XREAL_1:118;
end;
