I am reading Frank Warner's "Foundations of Differentiable Manifolds and Lie Groups" and my question concerns his proof of existence of slice coordinates for immersions. The proposition is:
Let $f:M^c\rightarrow N^d$ be an immersion and let $p\in M$. Then there exists a cubic centered coordinate system $(V,\varphi)$ about $f(p)$ and a neighborhood $U$ of $p$ such that $f\mid_U$ is injective and $f(U)$ is a slice of $(V,\varphi)$.
The proof utilizes many other results established before so I will try to fill those in directly:
Let $(W,\tau)$ be a coordinate system centered on $f(p)$ with coordinate functions $y_1,\ldots,y_d$. Since $f$ is an immersion, by reindexing, we may assume $\tilde{\tau} = \pi_c\circ\tau\circ f$ is a coordinate map on a neighborhood $V'$ of $p$ where $\pi_c:\mathbb{R}^d\rightarrow\mathbb{R}^c$ is projection on the first $c$ coordinates.
I am fine up to this point. Equivalently, the maps $y_1\circ f,\ldots,y_c\circ f$ form a coordinate map on a neighborhood $V'$ of $p$. Then it is this next step that I would like intuition:
Define functions $(x_i)$ on $(\pi_c\circ\tau)^{-1}(\tilde{\tau}(V'))$ by setting $x_i = \begin{cases} y_i & i = 1,\ldots,c\\ y_i-y_i\circ f\circ\tilde{\tau}^{-1}\circ\pi_c\circ\tau & i = c+1,\ldots,d \end{cases}$
Now these functions are independent at $f(p)$ since there, $dx_i = \begin{cases} dy_i & i = 1,\ldots,c\\ dy_i + \sum_{j = 1}^{c}a_{ij}dy_j & i = c+1,\ldots,d \end{cases}$
for constants $a_{ij}$...
For reference, a set of functions $f_1,\ldots,f_n$ defined on a manifold $M$ are independent at $p\in M$ if their differentials $d(f_1)_p, \ldots ,d(f_n)_p$ are linearly independent in $T^\ast_pM$. The rest of the proof uses the fact that independent functions form a coordinate neighborhood of $f(p)$ and then constructs the required neighborhood $U$ of $p$.
My question: What is going on with the functions $x_i$ where $i>c$? What goes wrong when the $x_i$ are just set as the $y_i$ for all $i$? Since the $y_i$ already form a coordinate system, their differentials are linearly independent in $T^\ast_pM$ so what are the extra stuff for?