Just a trouble understanding the notation in the proof.
Let $F : N \to M$ be a $C^{\infty}$ map of manifolds, with $\text{dim} \; N = n$ and $\text{dim} \; M = m$. Then a nonempty regular level set $F^{-1}(c)$ where $c \in M$, is a regular submanifold of $N$ of dimension equal to $n - m$
Proof: Choose a chart $(V,\psi) = (V,y^1,\ldots,y^m)$ of $M$ centered at $c$, i.e. such that $\psi(c) = 0$ in $\mathbb{R}^m$. Then $F^{-1}(V)$ is an open set in $N$ that contains $F^{-1}(c)$ is the zero set of $\psi \circ F$. If $F^i = y^i \circ F = r^i \circ (\psi \circ F)$, then $F^{-1}(c)$ is also the common zero set of the functions $F^1,\ldots,F^m$ on $F^{-1}(V)$. Because the regular level set is assumed nonempty, $n \geq m$. Fix a point $p \in F^{-1}(c)$ and let $(U,\phi) = (U,x^1,\ldots,x^n)$ be a coordinate neighborhood of $p$ in $N$ contained in $F^{-1}(V)$. Since $F^{-1}(c)$ is a regular level set, $p \in F^{-1}(c)$ is a regular point of $F$. Therefore, the $m \times n$ Jacobian matrix $\left[ \partial F^i/\partial x^j(p) \right]$ has rank $m$. By renumbering the $F^i$ and $x^j$'s we may assume the first $m \times m$ block $\left[ \partial F^i / \partial x^j (p)\right]_{1 \leq i, j\leq m}$ is non singular. Replace the first $m$ coordinates $x^1,\ldots,x^m$ of the chart $(U,\phi)$ by $F^1,\ldots,F^m$. We claim that there's a neighborhood $U_p$ of $p$ such that $(U_p,F^1,\ldots,F^m,x^{m+1},\ldots,x^n)$ is a chart in the atlas of $N$. It suffices to compute it's Jacobian matrix at $p$: $$ \begin{bmatrix} \frac{\partial F^i}{\partial x^j} & \frac{\partial F^i}{\partial x^\beta} \\ \frac{\partial x^{\alpha}}{\partial x^j} & \frac{\partial x^{\alpha}}{\partial x^\beta} \end{bmatrix} = \begin{bmatrix} \frac{\partial F^i}{\partial x^j} & * \\ 0 & I \end{bmatrix} $$ where $1 \leq i, j \leq m$ and $m + 1 \leq \alpha, \beta \leq n$ Since this matrix has determinant $$ \text{det} \left[ \frac{\partial F^i}{\partial x^j}(p)\right]_{1 \leq i,j \leq m} \neq 0 $$ The inverse function theorem in the form of Corollary 6.27 implies the claim.
First of all why we replace the first $m$ coordinates with the components of $F$, what exactly do we end up having? Also the jacobian notation is a bit weird, it seems to me that it's a 2x2 matrix, but then I see the $I$ (identity matrix) as a subblock.
I hope the question is clear.