 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
  r is irrational & n > 0 implies
    |. r - c_n(r).n/c_d(r).n .| + |. r - c_n(r).(n+1)/c_d(r).(n+1) .|
       = |. c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1) .|
  proof
    assume that
A1: r is irrational and
A2: n > 0;
    per cases;
    suppose
A3:   n is even;
      r > c_n(r).n/c_d(r).n by A1,A2,A3,Th16; then
A4:   r - c_n(r).n/c_d(r).n > c_n(r).n/c_d(r).n - c_n(r).n/c_d(r).n
      by XREAL_1:14;
      r - c_n(r).n/c_d(r).n <> 0 by A1,A2,A3,Th16; then
      |. r - c_n(r).n/c_d(r).n.| <> 0 by ABSVALUE:2; then
A5:   |. r - c_n(r).n/c_d(r).n .| > 0 by COMPLEX1:46;
A8:   r - c_n(r).(n+1)/c_d(r).(n+1) <
      c_n(r).(n+1)/c_d(r).(n+1) -c_n(r).(n+1)/c_d(r).(n+1)
        by A1,A3,Th17,XREAL_1:14;then
      |. r - c_n(r).(n+1)/c_d(r).(n+1).| <> 0 by ABSVALUE:2; then
A9:   |. r - c_n(r).(n+1)/c_d(r).(n+1) .| > 0 by COMPLEX1:46;
A10:  |. r - c_n(r).n/c_d(r).n .| + |. r - c_n(r).(n+1)/c_d(r).(n+1) .| =
      r - c_n(r).n/c_d(r).n + |. r - c_n(r).(n+1)/c_d(r).(n+1) .|
      by A4,ABSVALUE:def 1  .= r - c_n(r).n/c_d(r).n +
      -(r - c_n(r).(n+1)/c_d(r).(n+1)) by A8,ABSVALUE:def 1
      .= - (c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1));
      c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1) < 0 by A5,A9,A10;
      hence thesis by A10,ABSVALUE:def 1;
    end;
    suppose
A11:  n is odd; then
A13:  r - c_n(r).n/c_d(r).n < c_n(r).n/c_d(r).n - c_n(r).n/c_d(r).n
       by XREAL_1:14,A1,Th17;
      r - c_n(r).n/c_d(r).n <> 0 by A1,A11,Th17; then
      |. r - c_n(r).n/c_d(r).n.| <> 0 by ABSVALUE:2; then
A14:  |. r - c_n(r).n/c_d(r).n .| > 0 by COMPLEX1:46;
      r > c_n(r).(n+1)/c_d(r).(n+1) by A1,A11,Th16; then
A18:  r - c_n(r).(n+1)/c_d(r).(n+1) >
      c_n(r).(n+1)/c_d(r).(n+1)-c_n(r).(n+1)/c_d(r).(n+1) by XREAL_1:14;
      |. r - c_n(r).(n+1)/c_d(r).(n+1).| <> 0 by A18,ABSVALUE:2; then
A19:  |. r - c_n(r).(n+1)/c_d(r).(n+1) .| > 0 by COMPLEX1:46;
      |. r - c_n(r).n/c_d(r).n .| + |. r - c_n(r).(n+1)/c_d(r).(n+1) .| =
      -(r - c_n(r).n/c_d(r).n) + |. r - c_n(r).(n+1)/c_d(r).(n+1) .|
      by A13,ABSVALUE:def 1 .=
      -r + c_n(r).n/c_d(r).n + (r - c_n(r).(n+1)/c_d(r).(n+1))
       by A18,ABSVALUE:def 1
      .= c_n(r).n/c_d(r).n - c_n(r).(n+1)/c_d(r).(n+1);
      hence thesis by A14,A19,ABSVALUE:def 1;
    end;
end;
