 reserve i,j,k,m,n,m1,n1 for Nat;
 reserve a,r,r1,r2 for Real;
 reserve m0,cn,cd for Integer;
 reserve x1,x2,o for object;

theorem Th17:
  r is irrational & n is odd implies
    r < c_n(r).n/c_d(r).n
  proof
    assume
A1: r is irrational;
    assume n is odd; then
    reconsider m = n -1 as even natural number;
    (c_d(r).(m+1)*(c_d(r).(m+1)*rfs(r).(m+2) + c_d(r).m)) > 0
      by A1,Th13; then
    (-1)|^m /(c_d(r).(m+1)*(c_d(r).(m+1)*rfs(r).(m+2) + c_d(r).m)) > 0; then
    c_n(r).(m+1)/c_d(r).(m+1) - r > 0 by A1,Th15; then
    c_n(r).(m+1)/c_d(r).(m+1) - r + r > 0 + r by XREAL_1:8;
    hence thesis;
  end;
