I am trying to solve the question 6.12 in Arora-Barak (Computational Complexity: A modern approach). The question asks you to show that the $\mathsf{PATH}$ problem (decide whether a graph $G$ has a path from a given node $s$ to another given node $t$) which is complete for $\mathbf{NL}$ is also contained in $\mathbf{NC}$ (this is easy). The question then also makes a remark that this implies that $\mathbf{NL} \subseteq \mathbf{NC}$ which is not obvious to me.
I think in order to show this, one has to show that $\mathbf{NC}$ is closed under logspace reductions, i.e
$$(1): B \in \mathbf{NC} \hbox{ and } A \le_l B \Longrightarrow A \in \mathbf{NC}$$
where $\le_l$ is the logspace reduction defined as
$$A \le_l B :\Longleftrightarrow (\exists M \hbox{ TM}, \forall x)[x \in A \Longleftrightarrow M(x) \in B]$$
($M$ is a TM which runs in logarithmic space).
I would appreciate if someone could give a tip for proving the statement $(1)$.