Derivative of a map into R^n by components

Statement

Let $U \subseteq \mathbb{R}^n$ be open, let $f = (f_1, \dots, f_m) \colon U \to \mathbb{R}^m$, let $a \in U$, and let $L \colon \mathbb{R}^n \to \mathbb{R}^m$ be linear with components $L_1, \dots, L_m$. Then $f$ is differentiable at $a$ with $Df(a) = L$ if and only if, for each $j \in \{1, \dots, m\}$, the component $f_j$ is differentiable at $a$ with $Df_j(a) = L_j$.