A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$

Let $n$, $m$ and $p$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ and let $V$ be an open subset of $\mathbb{R}^m$. Let $F=(F_1,\dots,F_m):U\to\mathbb{R}^m$ satisfy $F(x)\in V$ for every $x\in U$, let $G=(G_1,\dots,G_p):V\to\mathbb{R}^p$, and let $G\circ F:U\to\mathbb{R}^p$ be the map given by $(G\circ F)(x)=G(F(x))$, with coordinate functions $(G\circ F)_j$.

1. (Chain rule for partial derivatives) Suppose $F$ is of class $C^1$ on $U$ and $G$ is of class $C^1$ on $V$. Then for all $i\in\{1,\dots,n\}$ and $j\in\{1,\dots,p\}$ the partial derivative of $(G\circ F)_j$ with respect to the $i$th variable exists at every $x\in U$, and

$$\partial_i (G\circ F)_j(x)=\sum_{l=1}^{m}\partial_l G_j\bigl(F(x)\bigr)\,\partial_i F_l(x),$$

the sum being the finite sum in the field of real numbers.

2. (Finite order) Let $k$ be a natural number. If $F$ is of class $C^k$ on $U$ and $G$ is of class $C^k$ on $V$, then $G\circ F$ is of class $C^k$ on $U$.

3. (Smoothness) If $F$ is smooth on $U$ and $G$ is smooth on $V$, then $G\circ F$ is smooth on $U$.