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-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)/s2.(2*$1)<s1.(2*$1-1)/s2.(2*$1-1);
  assume
A1: for n holds scf(r).n>0;
  then s.1>0 & s.2>0;
  then
A2: 1/(s.1 + 1/s.2) < 1/s.1 by XREAL_1:29,76;
  let n;
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-1)/s2.(2*n-1);
    s1.(2*(n+1))*s2.(2*(n+1)-1)-s1.(2*(n+1)-1)*s2.(2*(n+1)) =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
A5: s1.(2*(n+1))*s2.(2*(n+1)-1)<s1.(2*(n+1)-1)*s2.(2*(n+1)) by XREAL_1:48;
    s2.(2*n+1)>0 & s2.(2*n+2)>0 by A3,Th52;
    hence thesis by A5,XREAL_1:106;
  end;
  cocf(r).1 = c_n(r).1 * ((c_d(r))").1 by SEQ_1:8
    .= c_n(r).1 * (c_d(r).1)" by VALUED_1:10
    .= c_n(r).1 *(1/c_d(r).1)
    .= s1.(2*1-1)/s2.(2*1-1);
  then
A6: s1.(2*1-1)/s2.(2*1-1) = s.0 + 1/s.1 by A3,Th72;
  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)
    .= s1.(2*1)/s2.(2*1);
  then s1.(2*1)/s2.(2*1) = s.0 + 1/(s.1 + 1/s.2) by A1,Th73;
  then
A7: X[1] by A6,A2,XREAL_1:8;
  for n being Nat st n>=1 holds X[n] from NAT_1:sch 8(A7,A4);
  hence thesis;
end;
