Uniqueness of the Partial Derivative on a Euclidean Open Set

Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, let $\mathbb{R}$ be the real numbers, let $f:U\to\mathbb{R}$, let $a=(a_1,\dots,a_n)\in U$, let $i\in\{1,\dots,n\}$, and let $L,L'\in\mathbb{R}$. If the partial derivative of $f$ with respect to the $i$th variable exists at $a$ with value $L$ and also exists at $a$ with value $L'$, then $L=L'$.