 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 Th22:
  r is irrational implies
    |. r*c_d(r).(n+1) - c_n(r).(n+1) .| < |. r*c_d(r).n - c_n(r).n .|
  & |. r - c_n(r).(n+1)/c_d(r).(n+1) .| < |. r - c_n(r).n/c_d(r).n .|
  proof
    assume
A1: r is irrational; then
A2: r=(c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n)/(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n)
      by Th14 .=(c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n)*
    (1/(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n));
A3: c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n <> 0 by A1,Th12;
A4: r*c_d(r).(n+1)*rfs(r).(n+2) + r*c_d(r).n =
     (c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n)
      *((1/(c_d(r).(n+1)*rfs(r).(n+2)+c_d(r).n))
      *(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n)) by A2
   .= (c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n) * 1 by A3,XCMPLX_1:106
   .= c_n(r).(n+1)*rfs(r).(n+2)+c_n(r).n;
A6:  -(r*c_d(r).n - c_n(r).n)
     = r*c_d(r).(n+1)*rfs(r).(n+2) - c_n(r).(n+1)*rfs(r).(n+2) by A4
    .= rfs(r).(n+2)*(r*c_d(r).(n+1) - c_n(r).(n+1));
A7:  1 < rfs(r).(n+1+1) by A1,Th4;
A8:   |. r*c_d(r).n-c_n(r).n .| = |.-(r*c_d(r).n-c_n(r).n).| by COMPLEX1:52
   .= |. rfs(r).(n+2)*(r*c_d(r).(n+1) - c_n(r).(n+1)) .| by A6
   .= |. rfs(r).(n+2) .|*|.(r*c_d(r).(n+1) - c_n(r).(n+1)) .| by COMPLEX1:65
   .= rfs(r).(n+2) *|. r*c_d(r).(n+1) - c_n(r).(n+1) .| by A7,COMPLEX1:43;
A9:  c_d(r).(n+1) >= 1 & c_d(r).n >=1 by A1,Th8;
     |. r - c_n(r).(n+1)/c_d(r).(n+1) .| > 0 by A1,Th19; then
A11: r-c_n(r).(n+1)/c_d(r).(n+1) <> 0 by COMPLEX1:47;
A12: (r-c_n(r).(n+1)/c_d(r).(n+1)) * c_d(r).(n+1) =
     r*c_d(r).(n+1) - c_n(r).(n+1)/c_d(r).(n+1) * c_d(r).(n+1)
     .= r*c_d(r).(n+1) - c_n(r).(n+1) by A9,XCMPLX_1:87; then
A13: |. r*c_d(r).n-c_n(r).n .| > |. r*c_d(r).(n+1) - c_n(r).(n+1) .|
     by A7,A8,A9,A11,COMPLEX1:47,XREAL_1:155;
A14:   |. r - c_n(r).(n+1)/c_d(r).(n+1) .|
     = |. r*c_d(r).(n+1)/c_d(r).(n+1)-c_n(r).(n+1)/c_d(r).(n+1) .|
       by A9,XCMPLX_1:89
    .= |. (r*c_d(r).(n+1)-c_n(r).(n+1))/c_d(r).(n+1) .|
    .= |. (r*c_d(r).(n+1)-c_n(r).(n+1)).|/|.c_d(r).(n+1) .| by COMPLEX1:67
    .= |. r*c_d(r).(n+1)-c_n(r).(n+1).|/c_d(r).(n+1) by A9,COMPLEX1:43;
A15: |. r - c_n(r).n/c_d(r).n .|
     = |. r*c_d(r).n/c_d(r).n-c_n(r).n/c_d(r).n .| by A9,XCMPLX_1:89
    .= |. (r*c_d(r).n-c_n(r).n)/c_d(r).n .|
    .= |. (r*c_d(r).n-c_n(r).n).|/|.c_d(r).n .| by COMPLEX1:67
    .= |. r*c_d(r).n-c_n(r).n.|/c_d(r).n by A9,COMPLEX1:43;
A16: |. r*c_d(r).(n+1)-c_n(r).(n+1).|/c_d(r).(n+1) <
     |. r*c_d(r).n - c_n(r).n .| / c_d(r).(n+1) by A9,A13,XREAL_1:74;
     |. r*c_d(r).n - c_n(r).n .| >= 0 by COMPLEX1:46; then
     |. r*c_d(r).n - c_n(r).n .| / c_d(r).(n+1) <=
     |. r*c_d(r).n - c_n(r).n .| / c_d(r).n by A1,Th7,A9,XREAL_1:118;
     hence thesis by A7,A8,A9,A11,A12,A14,A15,A16,COMPLEX1:47,
       XREAL_1:155,XXREAL_0:2;
end;
