# Trying to understand why this local coordinates parametrizes a manifold

First of all, I would like to say that I think this question fits better on Math Overflow than on Math Stack Exchange, in view of the proposal of the two sites. However, if my analysis of the situation was not correct I can move this question to Math Stack Exchange.

I'm reading the paper "Vector Fields Near the Boundary of a Manifold - S.M. Vishik", and the author gives us the following definitions:

• $M$ is a smooth manifold and $Q \subset M$ is a regular submanifold of codimension 1 (for my purposes $M=\mathbb{S}^3$ and $Q= \frac{1}{\sqrt{2}}\mathbb{T}^2$ are enough),
• $\Gamma(TM, Q)$ is the space of the germs (the author calls "germ" as "shoot") on $Q$ of the vector fields in a neighborhood of $Q$.
• $J^r(TM, Q)$ is the space of r-jets of germs of vector fields at points of $Q$.
• We shall call a point $x\in Q$ a singularity of an r-jet of a vector field $h$ at $x$ if $h(x) =0$. The set of all r-jets of vector fields on $Q$ which have singularities will be denoted by $Z(r)$.

The author also defines the space $\Sigma^{1_k} (r)$ but this isn't important to my question.

Now, comes the weird proof:

My Doubts: 1) What does he mean by "$\{Y_1 =0 ,..., Y_m=0\}$ which are idependent at each point of $\pi^{-1}(Q \cap U)$"?

2) Why 1) implies the lemma, i.e. proves that $Z(r)$ is a submanifold of $J^r (TM,Q)$?