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 c_n(r).(2*n) /
  c_d(r).(2*n) > c_n(r).(2*n-2) / c_d(r).(2*n-2)
proof
  set s=scf(r), s1=c_n(r), s2=c_d(r);
  defpred X[Nat] means s1.(2*$1)/s2.(2*$1)>s1.(2*$1-2)/s2.(2*$1-2);
  assume
A1: for n holds scf(r).n>0;
  then
A2: s.1>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
    s1.(2*n)/s2.(2*n)>s1.(2*n-2)/s2.(2*n-2);
    s1.(2*(n+1))*s2.(2*(n+1)-2)-s1.(2*(n+1)-2)*s2.(2*(n+1)) = (s.(2*n+2)
    * s1.(2*n+1) + s1.(2*n))*s2.(2*n)-s1.(2*n)*s2.(2*n+2) by Def5
      .= (s.(2*n+2) * s1.(2*n+1) + s1.(2*n))*s2.(2*n) -s1.(2*n)*(s.(2*n+2) *
    s2.(2*n+1) + s2.(2*n)) by Def6
      .= s.(2*n+2) * (s1.(2*n+1) *s2.(2*n)-s1.(2*n)* s2.(2*n+1))
      .= s.(2*n+2) * (-1)|^(2*n) by Th64
      .= s.(2*n+2) * (1|^(2*n)) by WSIERP_1:2
      .= s.(2*n+2) * 1
      .= s.(2*n+2);
    then
A5: s1.(2*(n+1))*s2.(2*(n+1)-2)>s1.(2*(n+1)-2)*s2.(2*(n+1)) by A1,XREAL_1:47;
    s2.(2*n)>0 & s2.(2*n+2)>0 by A3,Th52;
    hence thesis by A5,XREAL_1:106;
  end;
  let n;
A6: s1.(2*1-2)/s2.(2*1-2) = s.0/s2.0 by Def5
    .=s.0/1 by Def6
    .=s.0;
A7: s.2>0 by A1;
  cocf(r).2 = c_n(r).2 * ((c_d(r))").2 by SEQ_1:8
    .= c_n(r).2 * (c_d(r).2)" by VALUED_1:10
    .= c_n(r).2 *(1/c_d(r).2)
    .= c_n(r).2 /c_d(r).2;
  then s1.(2*1)/s2.(2*1) = s.0 + 1/(s.1 + 1/s.2) by A1,Th73
    .= s.0+1/((s.1 * s.2 +1)/ s.2) by A7,XCMPLX_1:113
    .= s.0 + s.2/(s.1 * s.2 +1) by XCMPLX_1:57;
  then
A8: X[1] by A2,A7,A6,XREAL_1:29,139;
  for n being Nat st n>=1 holds X[n] from NAT_1:sch 8(A8,A4);
  hence thesis;
end;
