Peeling the Innermost Variable from a Multi-Index Partial Derivative

Let $n$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, and let $f:U\to\mathbb{R}$. Let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index of length $n$ with $\alpha\ne 0$, with the zero multi-index $0$, the standard basis multi-indices $e_i$ and the difference $\alpha-e_i$ as in that definition; index ranges such as $1\le i\le n$ use the order on the natural numbers.

Let $i$ satisfy $1\le i\le n$, $1\le\alpha_i$, and $\alpha_l=0$ for every $l$ with $i<l\le n$; that is, $i$ is the largest index at which $\alpha$ is nonzero. Assume that the partial derivative of $f$ with respect to the $i$th variable exists at every point of $U$, and let $\partial_i f:U\to\mathbb{R}$ be the function whose value at $x\in U$ is that partial derivative at $x$. Assume further that the partial derivative of multi-index $\alpha-e_i$ of $\partial_i f$ exists on $U$.

Then $\partial^{\alpha}f$ exists on $U$ and

$$\partial^{\alpha}f=\partial^{\alpha-e_i}(\partial_i f).$$