Frobenius Theorem; Slices Good Night. I am studying the Frobenius theorem. I'm reading the book Foundations of differentiable manifolds and Lie Groups; Frank Warner. In the first third part of the statement is written, "is a slice S $ Y_ {1} = 0 $", where $ Y_ {1} $ is a function of a coordinated system of coordinates. I do not understand this! Also, I do not understand the phrase "The subspace S of M with the coordinate system {$ x_ {i} | S: j = 1, ..., c\} $" which is said in the definition of Slice. 
(Warner definition)
1.34 Slices Suppose that $(U,\varphi)$ is a coordinate system on $M$ with 
coordinate functions $x_{1} ..... x_{d}$, and that $c$ is an integer, $0\leq c \leq d$. Let
$a\in \varphi(U)$, and let
$ S =\{q\in U:x_{i}(q)=r_{i}(a), i = c +i .... ,d\}$.
The subspace $S$ of $M$ together with the coordinate system
$\{x_{j}\mid S: j= 1,..., d\}$
forms a manifold which is a submanifod of $M$ called a slice of the coordinate system $(U, \varphi)$.
 A: You start with a chart $(U, \varphi)$ and a point in $a \in U$. The subspace $S$ is simply the subset of $U$ of the elements that have some particular coordinates in common with $a$.
Let's see an example in $\mathbb{R}^2$ (with the standard manifold structures). Let's be $U = \{(u,v)\in R^2\mid -1\le u,v\le 1\}$ and $a = (0,0)$. The slice with $x_1 = 0$ is simply the line segment between the points $(0,-1)$ and $(0,1)$.
EDIT 1: “Let $S$ be the slice $y_1 = 0$” means that $S$ is the subset of $V$ of points that have the first coordinate equal to $0$. In the Warner's definition, the points of a slice have the last $d-c$ coordinates equal. Nevertheless, you can choose them: you simply need to permute the basis and use it as a new chart. The second part of the definition simply said that if you fix $d-c$ coordinates you do not really need them and so you can ignore them. The slice depends only on the other $c$ coordinates. In my previous example, for example, the line segment depends merely on the value on the $y$ and it is in fact locally diffeomorphic to the real line.
P.S: $\{y_i|S \colon j = 1,\dots,n\}$ literally means the set of the restrictions on $S$ of the first $c$ coordinates functions. However, you can read it as the set of the restrictions of the coordinates that are not equal in all the elements of the slice. 
