 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 Th23:
  r is irrational & m > n implies
    |. r - c_n(r).n/c_d(r).n .| > |. r - c_n(r).m/c_d(r).m .|
  proof
    assume that
A1: r is irrational and
A3: m > n;
    defpred P[Nat] means
      |. r - c_n(r).n/c_d(r).n .| > |. r - c_n(r).(n+1+$1)/c_d(r).(n+1+$1) .|;
A4: P[0] by A1,Th22;
A5: for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P[k]; then
      |.r-c_n(r).n/c_d(r).n.|
        > |.r-c_n(r).(n+1+k+1)/c_d(r).(n+1+k+1).| by A1,Th22,XXREAL_0:2;
      hence thesis;
    end;
A8: for k be Nat holds P[k] from NAT_1:sch 2(A4,A5);
    m - n > n-n by A3,XREAL_1:14; then
    m - n >= 0+1 by INT_1:7; then
A9: (m - n) - 1 >= 1 - 1 by XREAL_1:9;
    reconsider i = m - n - 1 as Integer;
    i in NAT by A9,INT_1:3; then
    reconsider i as Nat;
    |. r - c_n(r).n/c_d(r).n .| > |. r - c_n(r).(n+1+i)/c_d(r).(n+1+i) .|
      by A8;
    hence thesis;
  end;
