 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 implies c_d(r).(n+2) > c_d(r).(n+1)
  proof
    assume
A1: r is irrational; then
A2: scf(r).(n+1+1) > 0 by Th5;
A3: n+2 >= 0+1 by XREAL_1:8;
A4: c_d(r).(n+2) = scf(r).(n+2) * c_d(r).(n+1) + c_d(r).n by REAL_3:def 6;
    c_d(r).n > 0 by A1,Th8; then
A5: c_d(r).(n+2) -0 > scf(r).(n+2) * c_d(r).(n+1) + c_d(r).n - c_d(r).n
      by A4,XREAL_1:15;
    c_d(r).(n+1) >= 1 by A1,Th8; then
    scf(r).(n+2)*c_d(r).(n+1) >= c_d(r).(n+1) by A2,A3,REAL_3:40,XREAL_1:151;
    hence thesis by A5,XXREAL_0:2;
  end;
