 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) >= 2*c_d(r).n
  proof
    assume
A1: r is irrational; then
    scf(r).(n+1+1) > 0 by Th5; then
A4: scf(r).(n+2) >=0+1 by INT_1:7;
A5: 0 < c_d(r).n by A1,Th8;
    c_d(r).n <= c_d(r).(n+1) by A1,Th7; then
    scf(r).(n+2) * c_d(r).(n+1) >= 1*c_d(r).n by A4,A5,XREAL_1:66; then
    scf(r).(n+2)*c_d(r).(n+1) + c_d(r).n >= 1*c_d(r).n + c_d(r).n by XREAL_1:6;
    hence thesis by REAL_3:def 6;
  end;
