reserve i,j,n,n1,n2,m,k,u for Nat,
        r,r1,r2 for Real,
        x,y for Integer,
        a,b for non trivial Nat;

theorem Th16:
  Py(a,n) >= n
proof
  defpred P[Nat] means Py(a,$1) >= $1;
A1:P[0];
A2:P[k] implies P[k+1]
  proof
    k+1 > k by NAT_1:13;
    then
A3:   Py(a,k)+1 <= Py(a,k+1) by Th14,NAT_1:13;
    assume P[k];
    then k+1 <= Py(a,k)+1 by XREAL_1:6;
    hence thesis by A3,XXREAL_0:2;
  end;
  P[k] from NAT_1:sch 2(A1,A2);
  hence thesis;
end;
