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 Th14:
  k < n implies Py(a,k) < Py(a,n)
proof
  assume
A1: k <n;
  then reconsider nk=n-k as Nat by NAT_1:21;
A2: nk <>0 by A1;
  defpred P[Nat] means $1 > 0  implies Py(a,k) < Py(a,k+$1);
A3: P[0];
A4: for i be Nat holds P[i] implies P[i+1]
  proof
    let i be Nat;
    assume
A5:   P[i];
A6:   Py(a,k+i+1) = Py(a,k+i)*a+Px(a,k+i) by Th9;
    a >=2 by NAT_2:29;
    then a >= 1 by XXREAL_0:2;
    then Py(a,k+i)*a >= Py(a,k+i)*1 by XREAL_1:64;
    then
A7:   Py(a,k+i+1) > Py(a,k+i)*1+0 by A6,XREAL_1:8;
    i = 0 or i >0;
    hence thesis by A7,A5,XXREAL_0:2;
  end;
  P[n1] from NAT_1:sch 2(A3,A4);
  then P[nk];
  hence thesis by A2;
end;
