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 Th13:
  k <= n implies Px(a,k) <= Px(a,n)
proof
  assume k <=n;
  then reconsider nk=n-k as Nat by NAT_1:21;
  defpred P[Nat] means Px(a,k) <= Px(a,k+$1);
A1: P[0];
A2: for i be Nat holds P[i] implies P[i+1]
  proof
    let i be Nat;
    assume
A3:   P[i];
A4:   Px(a,k+i+1) = Px(a,k+i)*a+Py(a,k+i)*(a^2-'1) by Th9;
    a >=2 by NAT_2:29;
    then a >= 1 by XXREAL_0:2;
    then Px(a,k+i)*a >= Px(a,k+i)*1 by XREAL_1:64;
    then Px(a,k+i+1) >= Px(a,k+i)*1+0 by A4,XREAL_1:7;
    hence thesis by A3,XXREAL_0:2;
  end;
  P[n1] from NAT_1:sch 2(A1,A2);
  then P[nk];
  hence thesis;
end;
