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+1) /
  c_d(r).(2*n+1) < c_n(r).(2*n-1) / c_d(r).(2*n-1)
proof
  set s=scf(r), s1=c_n(r), s2=c_d(r);
  defpred X[Nat] means s1.(2*$1+1)/s2.(2*$1+1)<s1.(2*$1-1)/s2.(2*$1-1);
  assume
A1: for n holds scf(r).n>0;
  then
A2: s.3>0;
  cocf(r).3 = c_n(r).3 * ((c_d(r))").3 by SEQ_1:8
    .= c_n(r).3 * (c_d(r).3)" by VALUED_1:10
    .= c_n(r).3 *(1/c_d(r).3)
    .= c_n(r).3 /c_d(r).3;
  then
A3: s1.(2*1+1)/s2.(2*1+1) = s.0 + 1/(s.1 + 1/(s.2 + 1/s.3)) by A1,Th74
    .= s.0 + 1/(s.1 + 1/((s.2*s.3 + 1)/s.3)) by A2,XCMPLX_1:113
    .= s.0 + 1/(s.1 + s.3/(s.2*s.3 + 1)) by XCMPLX_1:57;
  let n;
A4: s.1>0 by A1;
A5: scf(r).1 > 0 by A1;
A6: 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+1)/s2.(2*n+1)<s1.(2*n-1)/s2.(2*n-1);
    s1.(2*(n+1)+1)*s2.(2*(n+1)-1)-s1.(2*(n+1)-1)*s2.(2*(n+1)+1) =(s.(2*n+
    1+2) * s1.(2*n+1+1) + s1.(2*n+1))*s2.(2*n+1)-s1.(2*n+1)*s2.(2*n+3) by Def5
      .=(s.(2*n+1+2) * s1.(2*n+1+1) + s1.(2*n+1))*s2.(2*n+1) -s1.(2*n+1)*(s.
    (2*n+1+2) * s2.(2*n+1+1) + s2.(2*n+1)) by Def6
      .=s.(2*n+1+2)*(s1.(2*n+1+1)*s2.(2*n+1)-s1.(2*n+1)*s2.(2*n+1+1))
      .=s.(2*n+1+2) * (-1)|^(2*n+1) by Th64
      .=s.(2*n+1+2) * (-1|^(2*n+1)) by WSIERP_1:2
      .=s.(2*n+1+2) * (-1);
    then s1.(2*(n+1)+1)*s2.(2*(n+1)-1)-s1.(2*(n+1)-1)*s2.(2*(n+1)+1)<0 by A1,
XREAL_1:132;
    then
A7: s1.(2*(n+1)+1)*s2.(2*(n+1)-1)<s1.(2*(n+1)-1)*s2.(2*(n+1)+1) by XREAL_1:48;
    s2.(2*n+1)>0 & s2.(2*n+3)>0 by A5,Th52;
    hence thesis by A7,XREAL_1:106;
  end;
  s.2>0 by A1;
  then s.3/(s.2*s.3 + 1)>0 by A2,XREAL_1:139;
  then
A8: 1/(s.1 +s.3/(s.2*s.3 + 1))<1/s.1 by A4,XREAL_1:29,76;
  s1.(2*1-1)/s2.(2*1-1) = (s.1 * s.0 +1)/s2.1 by Def5
    .= (s.1 * s.0 +1)/s.1 by Def6
    .= s.0 +1 / s.1 by A4,XCMPLX_1:113;
  then
A9: X[1] by A8,A3,XREAL_1:8;
  for n being Nat st n>=1 holds X[n] from NAT_1:sch 8(A9,A6);
  hence thesis;
end;
