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 Th58:
  (for n holds scf(r).n > 0) implies for n holds c_d(r).(n+1)/c_d(
  r).n >= 1/scf(r).(n+2)
proof
  set s=scf(r);
  set s1=c_d(r);
  defpred X[Nat] means s1.($1+1)/s1.$1 >= 1/s.($1+2);
  assume
A1: for n holds scf(r).n > 0;
  then
A2: scf(r).1<>0;
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    assume s1.(n+1)/s1.n >= 1/s.(n+2);
    set r=s1.(n+1);
A4: s.1 > 0 by A1;
    then
A5: s1.(n+1)>0 by Th52;
    n+3>=0+1 & s.(n+3) <>0 by A1,XREAL_1:7;
    then s.(n+3)>=1 by Th40;
    then
A6: 1/s.(n+3)<=1/1 by XREAL_1:118;
    n+2>=0+1 & s.(n+2) <>0 by A1,XREAL_1:7;
    then
A7: s.(n+2)>=1 by Th40;
A8: s1.n>0 by A4,Th52;
    s1.(n+2)/s1.(n+1)=(s.(n+2) * r + s1.n)/r by Def6
      .=(s.(n+2) * r)/r + s1.n/r
      .=s.(n+2)* (r/r) + s1.n/r
      .=s.(n+2)+ s1.n/r by A5,XCMPLX_1:88;
    then s1.(n+2)/s1.(n+1)>=1+0 by A5,A8,A7,XREAL_1:7;
    hence thesis by A6,XXREAL_0:2;
  end;
  scf(r).2<>0 by A1;
  then s.(0+2)>=1 by Th40;
  then
A9: 1/s.(0+2)<=1/1 by XREAL_1:118;
  s1.0 = 1 & s1.1 = s.1 by Def6;
  then s1.(0+1)/s1.0>=1 by A2,Th40;
  then
A10: X[0] by A9,XXREAL_0:2;
  for n holds X[n] from NAT_1:sch 2(A10,A3);
  hence thesis;
end;
