 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 Th24:
  r is irrational implies
    |. r - c_n(r).n/c_d(r).n .| < 1/c_d(r).n|^2
  proof
    assume
A1: r is irrational;
    |. r - c_n(r).n/c_d(r).n .| < 1/c_d(r).n|^2
    proof
      per cases;
      suppose
A3:     n = 0; then
A6:     |. r - c_n(r).n/c_d(r).n .| = |. r - c_n(r).0/1 .| by REAL_3:def 6
         .= |. r - scf(r).0 .| by REAL_3:def 5 .= |.r-[\ rfs(r).0 /].|
           by REAL_3:def 4
         .= |.r-[\ r /].| by REAL_3:def 3 .= |. frac r .| by INT_1:def 8
         .= frac r by ABSVALUE:def 1,INT_1:43;
        1/c_d(r).n|^2 = 1/c_d(r).0|^2 by A3 .= 1/1|^2 by REAL_3:def 6
           .= 1;
        hence thesis by A6,INT_1:43;
      end;
      suppose
A8:     n > 0;
        set m = n - 1;
        reconsider m as Nat by A8;
A10:    c_d(r).(m+1) >= 1 by A1,Th8; then
        c_d(r).(m+1)*c_d(r).(m+1) >= 1 by XREAL_1:159; then
A12:    c_d(r).(m+1)|^2 >=1 by WSIERP_1:1;
        c_d(r).(m+1+1)*c_d(r).(m+1)>=c_d(r).(m+1)*c_d(r).(m+1)
          by A1,A10,Th7,XREAL_1:64; then
        c_d(r).(m+1)*c_d(r).(m+2)>=c_d(r).(m+1)|^2 by WSIERP_1:1; then
        1/(c_d(r).(m+1)*c_d(r).(m+2)) <= 1/c_d(r).(m+1)|^2
          by A12,XREAL_1:85;
        hence thesis by A1,XXREAL_0:2,Th21;
      end;
    end;
    hence thesis;
  end;
