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
  scf(r).1 > 0 implies for n holds c_n(r).(2*n+1) / c_d(r).(2*n+1) > c_n
  (r).(2*n) / c_d(r).(2*n)
proof
  set s1=c_n(r), s2=c_d(r), s=scf(r);
  defpred X[Nat] means s1.(2*$1+1)/s2.(2*$1+1)>s1.(2*$1)/s2.(2*$1);
A1: s1.(2*0)/s2.(2*0) =s.0/s2.0 by Def5
    .=s.0/1 by Def6
    .=s.0;
  assume
A2: scf(r).1>0;
A3: for n st X[n] holds X[n+1]
  proof
    let n;
    assume s1.(2*n+1)/s2.(2*n+1)>s1.(2*n)/s2.(2*n);
    reconsider n as Element of NAT by ORDINAL1:def 12;
    s1.(2*(n+1)+1)*s2.(2*(n+1))-s1.(2*(n+1))*s2.(2*(n+1)+1) =(-1)|^(2*(n+1
    )) by Th64
      .=(1|^(2*(n+1))) by WSIERP_1:2
      .=1;
    then
A4: s1.(2*(n+1)+1)*s2.(2*(n+1))>s1.(2*(n+1))*s2.(2*(n+1)+1) by XREAL_1:47;
    s2.(2*(n+1)+1)>0 & s2.(2*(n+1))>0 by A2,Th52;
    hence thesis by A4,XREAL_1:106;
  end;
  s1.(2*0+1)/s2.(2*0+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 A2,XCMPLX_1:113;
  then
A5: X[0] by A2,A1,XREAL_1:29;
  for n holds X[n] from NAT_1:sch 2(A5,A3);
  hence thesis;
end;
