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 1 / (c_d(r).n*c_d
  (r).(n+1)) < 1 / (scf(r).(n+1)*(c_d(r).n)^2)
proof
  set s=scf(r),s2=c_d(r);
  defpred X[Nat] means 1/(s2.$1*s2.($1+1))<1/(s.($1+1)*(s2.$1)^2);
  assume
A1: for n holds scf(r).n>0;
  then
A2: s.2>0;
A3: scf(r).1>0 by A1;
A4: for n being Nat st n>=1 & X[n] holds X[n+1]
  proof
    let n be Nat;
    assume that
    n>=1 and
    1/(s2.n*s2.(n+1))<1/(scf(r).(n+1)*(s2.n)^2);
A5: s2.(n+1)>0 by A3,Th52;
    then
A6: (s2.(n+1))^2>0 by SQUARE_1:12;
    s.(n+2)>0 by A1;
    then
A7: s.(n+2)*(s2.(n+1))^2>0 by A6,XREAL_1:129;
    s2.n>0 by A3,Th52;
    then
A8: s2.(n+1)*s2.n>0 by A5,XREAL_1:129;
    s2.(n+1)*s2.(n+1+1) =s2.(n+1)*(s.(n+2)*s2.(n+1)+s2.n) by Def6
      .=s.(n+2)*(s2.(n+1))^2+s2.(n+1)*s2.n;
    hence thesis by A8,A7,XREAL_1:29,76;
  end;
A9: s.1>0 by A1;
  then (s.1)^2>0 by SQUARE_1:12;
  then s.2*(s.1)^2>0 by A2,XREAL_1:129;
  then
A10: 1/(s.2*(s.1)^2+s.1)<1/(s.2*(s.1)^2) by A9,XREAL_1:29,76;
  let n;
  1/(s2.1*s2.(1+1)) =1/(s2.1*(s.(0+2)*s2.(0+1)+s2.0)) by Def6
    .=1/(s2.1*(s.2*s.1+s2.0)) by Def6
    .=1/(s.1*(s.2*s.1+s2.0)) by Def6
    .=1/(s.1*(s.2*s.1+1)) by Def6
    .=1/(s.2*(s.1)^2+s.1);
  then
A11: X[1] by A10,Def6;
  for n being Nat st n>=1 holds X[n] from NAT_1:sch 8(A11,A4);
  hence thesis;
end;
