reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem
  for a holds ex b st b|^(n+1) <= a & a < (b+1)|^(n+1)
  proof
    defpred P[Nat] means ex b st b|^(n+1) <= $1 & $1 < (b+1)|^(n+1);
    A1: P[0]
    proof
      0|^(n+1) <= 0 & 0 < (0+1)|^(n+1);
      hence thesis;
    end;
    A2: P[k] implies P[k+1]
    proof
      assume P[k]; then consider b such that
      A3: b|^(n+1) <= k & k < (b+1)|^(n+1);
      A4: b + 1 > b + 0 & (b+1)+1 > (b+1) + 0 by XREAL_1:6;
      k+1 <= (b+1)|^(n+1) by A3,NAT_1:13; then
      per cases by XXREAL_0:1;
      suppose
        A5: k+1 < (b+1)|^(n+1);
        k+1 > b|^(n+1) by A3,NAT_1:13;
        hence thesis by A5;
      end;
      suppose
        A5: k+1 = (b+1)|^(n+1); then
        k+1 < (b+1+1)|^(n+1) by A4,Th40;
        hence thesis by A5;
      end;
    end;
    for x holds P[x] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;
