Step in proof of Structure theorem over PID - Jacobson, Basic Algebra I

Question

I'm trying to do exercise 8.6 in chapter 3 of Jacobson's Basic algebra I, which provides a proof of the structure theorem that does not require the Smith Normal form.

Let $x_1,x_2,...,x_n$ be a set of generators for $M$, a left $D$-module finitely generated over a PID, such that:

1. n is minimal;
2. $l(x_1)$ is minimal for all sets of n generators for $M$

Here $l(x) $ is defined as $l(x)=l(d)$ for $d\in{D}$ such that $ann x=(d)$, $l$ is the length of the element of $D$ and we put $l(0)=\infty$.

Show that $M=Dx_1\oplus{N}$ where $N=\sum_{i=2}^n Dx_i$ and that $ann y\subseteq{ann x_1} \forall{y}\in{N}$. From this the structure theorem for finitely generated module over a PID will follow by induction.

Following the sketch of proof given in the book, I managed to prove the first part and also to prove that $ann x_j\subseteq{ann x_1}$ for all j.

What I don't understand is how this implies the result for all $y\in{N}$; for example I tried to prove it for an element of the form $ax_2$, $a\in{D}$ but I couldn't reach the conclusion even in this simpler case.

Any suggestions?