In ***[Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$][1]***, 

$\lambda\in\mathfrak{h}^*$ is
antidominant if $\langle \lambda + \rho, \alpha^\lor\rangle \not\in \mathbb{Z}^{>0}$ for all $\alpha\in \Phi^+$

Then the theorem 4.4 states:
Let $\lambda\in\mathfrak{h}^*$. Then $M(\lambda) = L(\lambda)$ if and only if $\lambda$ is antidominant.

In the proof of theorem 4.4, for integral case:

"Conversely, suppose $\lambda$ is antidominant. Thanks to 3.5, $\lambda \le w \cdot \lambda$ for
all $w \in W$. Since all composition factors of $M(\lambda)$ are of the form $L(w \cdot \lambda)$ with $w \cdot \lambda \le \lambda$, it follows that only $L(\lambda)$ can occur as a composition factor.
**But it occurs just once**, so $M(\lambda) = L(\lambda)$."

1. I would like to know how to prove the claim: **But it occurs just once**?
2. Does "**But it occurs just once**" implies $\{0\}\subseteq M(\lambda)$ is the composition series of $M(\lambda)$ and then $M(\lambda)\cong M(\lambda)/\{0\}\cong L(\lambda)$?




  [1]: https://bookstore.ams.org/gsm-94/