Theorem: Let $F:X\subseteq \mathbb{R^n}\rightarrow \mathbb{R}$ be of class of $C^1$ and let $a$ be a point of the level set $S=\{x\in\mathbb{R^n}|F(x)=c\}$. If $F_{x_n}(a)\neq 0$ then there is a neighborhood $U$ of $(a_1,a_1\dots a_{n-1})\in\mathbb{R^{n-1}}$, a neighborhood $V$ for of $a_n\in\mathbb{R}$ and a function $f:U\subseteq\mathbb{R^{n-1}}\rightarrow V$ of class $C^1$ such that if $(x_1,x_2,...x_{n-1})\in U$ and $x_n \in V$ satisfy $F(x_1,x_2...x_n)=c$ ,then $x_n=f(x_1,x_2,...x_n)$.
I can't quite understand statement " If $F_{x_n}(a)\neq 0$ then there is a neighborhood $U$ of $(a_1,a_1\dots a_{n-1})\in\mathbb{R^{n-1}}$, a neighborhood $V$ for of $a_n\in\mathbb{R}$ and a function $f:U\subseteq\mathbb{R^{n-1}}\rightarrow V$ of class $C^1$ such that if $(x_1,x_2,...x_{n-1})\in U$ and $x_n \in V$ satisfy $F(x_1,x_2...x_n)=c$, then $x_n=f(x_1,x_2,...x_n)$."Why the statement is true and what idea it talks about?