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 c_d(r).(n+2) <= 2*scf(r
  ).(n+2) * c_d(r).(n+1)
proof
  assume
A1: for n holds scf(r).n > 0;
  let n;
  set s=scf(r);
  set s1=c_d(r);
A2: s.(n+2) > 0 by A1;
  s.1 > 0 by A1;
  then
A3: s1.n > 0 by Th52;
A4: s1.(n+1)/s1.n >= 1/s.(n+2) by A1,Th58;
  s1.(n+1)/s1.n * s1.n =s1.n / s1.n * s1.(n+1) .=s1.(n+1) by A3,XCMPLX_1:88;
  then
A5: s1.(n+1) >= s1.n / s.(n+2) by A4,A3,XREAL_1:64;
  s1.n / s.(n+2) * s.(n+2) =s.(n+2) /s.(n+2) * s1.n .= s1.n by A2,XCMPLX_1:88;
  then s1.(n+1) * s.(n+2) >= s1.n by A5,A2,XREAL_1:64;
  then s1.(n+1) * s.(n+2) + s1.(n+1) * s.(n+2)>= s1.n+s1.(n+1) * s.(n+2) by
XREAL_1:6;
  hence thesis by Def6;
end;
