general relativity - Can we always locally split the metric as $- N^2dt^2+g_{ij}dx^idx^j$?

In the book by Y. Choquet-Bruhat, General Relativity and the Einstein Equations, the following technical lemma is found on page 9:

A Lorentzian metric can always be written in a small enough neighborhood by a change of coordinates under the form

$$-N^2dt^2+g_{ij}dx^idx^j.$$

The proof (I think that's what it's supposed to be) she gives makes little sense:

Indeed, under a change of coordinates $(x'^\alpha)\mapsto (x^\beta)$ with $x^0=x'^0$ we have

$$g'_{i0}=\frac{\partial x^j}{\partial x'^i}\left(g_{j0}+g_{jh}\frac{\partial x^h}{\partial x'^0}\right),$$ we make $g_{i0}'=0$ by solving the linear first-order system $g_{j0}+g_{jh}\frac{\partial x^h}{\partial x'^0}=0$ for the functions $x^h(x'^i,x'^0)$.

The reason this is problematic is that we assumed $x^0=x'^0$, so that in fact $x^h$ cannot be a function of $x'^0$, and the linear system falls apart.

Am I misinterpreting what she's saying? Can the proof be salvaged or is this a bad typo? Is the result true?