 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 Th21:
  r is irrational implies
    |. r - c_n(r).(n+1)/c_d(r).(n+1) .| < 1/(c_d(r).(n+1)*c_d(r).(n+2))
  proof
    assume
A1: r is irrational; then
A2: c_d(r).(n+1) >= 1 by Th8;
A3: c_d(r).(n+1+1) >= 1 by A1,Th8;
    scf(r).(n+2)*c_d(r).(n+1) < rfs(r).(n+2)*c_d(r).(n+1)
      by A1,Th4,A2,XREAL_1:68; then
    scf(r).(n+2)*c_d(r).(n+1)+c_d(r).n < rfs(r).(n+2)*c_d(r).(n+1)+c_d(r).n
      by XREAL_1:8; then
A5: c_d(r).(n+2)<c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n by REAL_3:def 6; then
A6: c_d(r).(n+1)*c_d(r).(n+2) <
    c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n) by A2,XREAL_1:68;
    |. r - c_n(r).(n+1)/c_d(r).(n+1) .|
      = |. (-1)*(c_n(r).(n+1)/c_d(r).(n+1) - r) .|
    .= |. (-1)*((-1)|^n /(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n))).|
       by A1,Th15
    .= |. (-1)*(-1)|^n /(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)).|
    .= |. (-1)|^(n+1) /(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n)).|
       by NEWTON:6
    .= |.(-1)|^(n+1).|/|.(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)).|
       by COMPLEX1:67
    .= 1/|.(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)).| by SERIES_2:1
    .= 1/(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n))
      by A2,A3,A5,COMPLEX1:43;
    hence thesis by A6,A2,A3,XREAL_1:88;
  end;
