 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 Th15:
  r is irrational implies
    c_n(r).(n+1)/c_d(r).(n+1) - r
  = (-1)|^n /(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n))
  proof
    assume
A1: r is irrational; then
A3: c_d(r).(n+1) <> 0 by Th8;
A4: c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n <> 0 by A1,Th12;
A5: c_n(r).(n+1)/c_d(r).(n+1) - r
     = c_n(r).(n+1)/c_d(r).(n+1) - (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 A1,Th14
    .= (c_n(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n)
     - (c_n(r).(n+1)*rfs(r).(n+2) + c_n(r).n)*c_d(r).(n+1) )
       /(c_d(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n))
       by A3,A4,XCMPLX_1:130;
    (c_n(r).(n+1)*(c_d(r).(n+1)*rfs(r).(n+2) + c_d(r).n)
     - (c_n(r).(n+1)*rfs(r).(n+2) + c_n(r).n)*c_d(r).(n+1))
     = c_n(r).(n+1)*c_d(r).(n+1)*rfs(r).(n+2) + c_n(r).(n+1)*c_d(r).n
     - c_n(r).(n+1)*c_d(r).(n+1)*rfs(r).(n+2) - c_n(r).n*c_d(r).(n+1)
    .= (-1)|^n by REAL_3:64;
    hence thesis by A5;
  end;
