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

$\lambda\in\mathfrak{h}^*$ 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 see 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 $M(\lambda)\cong M(\lambda)/\{0\}\cong L(\lambda)$?

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

$\lambda\in\mathfrak{h}^*$ 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 $M(\lambda)\cong M(\lambda)/\{0\}\cong L(\lambda)$?