 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 Th10:
  r is irrational implies c_d(r).n >= n
  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];
      set m = n -1;
      per cases;
        suppose n=0;
          hence thesis by A1,Th8;
        end;
        suppose n > 0; then
          reconsider m as Nat;
A7:       scf(r).(m+1+1) > 0 by A1,Th5;
A8:       m+2 >= 0+1 by XREAL_1:8;
          c_d(r).(m+1) >= 1 by A1,Th8; then
A9:       scf(r).(m+2)*c_d(r).(m+1) >= c_d(r).(m+1)
            by A7,A8,REAL_3:40,XREAL_1:151;
          scf(r).(m+2)*c_d(r).(m+1) + c_d(r).m >= c_d(r).(m+1) +c_d(r).m
            by A9,XREAL_1:6; then
A12:      c_d(r).(m+2) >= c_d(r).(m+1) + c_d(r).m by REAL_3:def 6;
A13:      c_d(r).(m+1) + c_d(r).m >= n+c_d(r).m by A4,XREAL_1:6;
          n+c_d(r).m >= n + 1 by A1,Th8,XREAL_1:6; then
          c_d(r).(m+1) + c_d(r).m >= n + 1 by A13,XXREAL_0:2;
          hence thesis by A12,XXREAL_0:2;
        end;
      end;
      for n be Nat holds P[n] from NAT_1:sch 2(A2,A3);
      hence thesis;
    end;
