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

We use throughout the following continuity toolkit. By claim 1 of Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions, a real-valued function on a subset of a Euclidean space is continuous at a point exactly when it is continuous there relative to that subset as a map into the real line with the metric of The Absolute Value Metric on the Real Line; so by claims 2, 3 and 4 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, sums, scalar multiples and products of real-valued functions continuous at every point of a common domain are continuous at every point of that domain, and by induction on the number of summands, using the recursion of Finite Sum Notation in a Field, the same holds for finite sums.

Claim 1. Let $x\in U$. Since $F$ is of class $C^1$ on $U$, A C^1 Map into a Euclidean Space is Differentiable at Every Point shows that the Jacobian matrix $DF(x)$ is defined and that $F$ is differentiable at $x$ with derivative matrix $DF(x)$; since $G$ is of class $C^1$ on $V$ and $F(x)\in V$, the same corollary shows that $G$ is differentiable at $F(x)$ with derivative matrix $DG(F(x))$. By Chain Rule for Differentiable Maps Between Euclidean Spaces, $G\circ F$ is differentiable at $x$ with derivative matrix the product of real matrices $DG(F(x))\,DF(x)$, a matrix with $p$ rows and $n$ columns. By claim 1 of A Derivative Matrix is the Jacobian Matrix, and is Unique, the partial derivative of $(G\circ F)_j$ with respect to the $i$th variable exists at $x$ and equals the entry of that product in row $j$ and column $i$, which by Product of Real Matrices is

By Jacobian Matrix of a Map Between Euclidean Spaces these entries are $\partial_l G_j(F(x))$ and $\partial_i F_l(x)$, which gives the asserted formula.

Claim 2. We argue by induction on $k$, the assertion being proved simultaneously for all $n$, $m$, $p$, $U$, $V$, $F$ and $G$ of the stated form.

Base case $k=1$. Suppose $F$ is of class $C^1$ on $U$ and $G$ is of class $C^1$ on $V$. By clause 1 of C^k Maps on a Euclidean Open Set every $F_l$ is continuous at every point of $U$ and every $G_j$ is continuous at every point of $V$, so by Coordinatewise Criterion for Continuity of a Map Between Euclidean Spaces the map $F$ is continuous at every point of $U$. Since $(G\circ F)_j=G_j\circ F$ and $F$ takes its values in $V$, Composition of Continuous Euclidean Maps (with $p=1$) shows that every $(G\circ F)_j$ is continuous at every point of $U$.

Fix $i$ and $j$. By claim 1 the partial derivative of $(G\circ F)_j$ with respect to the $i$th variable exists at every point of $U$, and as a function on $U$,

By clause 1 of C^k Maps on a Euclidean Open Set each $\partial_l G_j:V\to\mathbb{R}$ is continuous at every point of $V$ and each $\partial_i F_l:U\to\mathbb{R}$ is continuous at every point of $U$; since $F$ is continuous at every point of $U$ with values in $V$, Composition of Continuous Euclidean Maps shows that each $(\partial_l G_j)\circ F$ is continuous at every point of $U$, and the continuity toolkit then shows that $\partial_i (G\circ F)_j$ is continuous at every point of $U$. As $i$ and $j$ were arbitrary, clause 1 of C^k Maps on a Euclidean Open Set gives that $G\circ F$ is of class $C^1$ on $U$.

Induction step. Let $k$ be a natural number and assume the assertion for $k$, for all data of the stated form. Suppose $F$ is of class $C^{k+1}$ on $U$ and $G$ is of class $C^{k+1}$ on $V$. By clause 2 of C^k Maps on a Euclidean Open Set, $F$ and $G$ are of class $C^1$ on $U$ and $V$, so the base case gives that $G\circ F$ is of class $C^1$ on $U$, with the partial derivatives displayed above; moreover, for all $i$, $l$ and $j$, the functions $\partial_i F_l:U\to\mathbb{R}$ and $\partial_l G_j:V\to\mathbb{R}$ are of class $C^k$ on $U$ and on $V$ respectively (clauses 2 and 3 of that definition). By claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, $F$ is also of class $C^k$ on $U$.

Fix $i$ and $j$. Applying the induction hypothesis to the map $F:U\to\mathbb{R}^m$, which is of class $C^k$ and takes values in $V$, and to the map $\partial_l G_j:V\to\mathbb{R}$, which is of class $C^k$ (read as a map into $\mathbb{R}^1$ by clause 3 of C^k Maps on a Euclidean Open Set), we get that $(\partial_l G_j)\circ F$ is of class $C^k$ on $U$ for every $l$. By claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set each product $\bigl((\partial_l G_j)\circ F\bigr)\cdot\partial_i F_l$ is of class $C^k$ on $U$, and, by that claim together with induction on the number of summands using the recursion of Finite Sum Notation in a Field, so is the finite sum $\partial_i (G\circ F)_j$.

Thus $G\circ F$ is of class $C^1$ on $U$ and every $\partial_i (G\circ F)_j$ is of class $C^k$ on $U$, so $G\circ F$ is of class $C^{k+1}$ on $U$ by clauses 2 and 3 of C^k Maps on a Euclidean Open Set.

Claim 3. If $F$ and $G$ are smooth then by Smooth Map on a Euclidean Open Set they are of class $C^k$ on their domains for every natural number $k$, so claim 2 gives that $G\circ F$ is of class $C^k$ on $U$ for every natural number $k$, that is, smooth on $U$. $\blacksquare$