 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 Th8:
  r is irrational implies c_d(r).n >=1
  proof
    assume
A1: r is irrational;
    defpred P[Nat] means c_d(r).$1 >=1;
A2: P[0] by REAL_3:def 6;
A3: for n be Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume
A4:   P[n];
      c_d(r).(n+1) >= c_d(r).n by A1,Th7;
      hence thesis by A4,XXREAL_0:2;
    end;
    for n be Nat holds P[n] from NAT_1:sch 2(A2,A3);
    hence thesis;
  end;
