With the following lemma :
Lemma : Let $f_{+},f_{-} : U \longmapsto \mathbb{R}$ be $C^{1}$ maps with $f_{-} \leq 0 \leq f_{+}$ with $U \subset \mathbb{R}^{n}$ open and bounden with $C^{1}$ boundary. Let $\Omega : \lbrace (x,t) : x \in U \wedge f_{-}(x) < t < f_{+}(x) \rbrace$ and $u \in C^{1}(\overline{\Omega})$. Let $y = (x,t)$ then we have
$$\int_{\Omega} \partial_{t}u(y)dy = \int_{\Sigma_{+}\cup \Sigma_{-}}u(z) \cdot \langle e_{t},v_{ext}(z)\rangle d\sigma(z)$$
Where $\Sigma_{+} = graph(f_{+}),\Sigma_{-} = graph(f_{-})$ and $v_{ext}(z)$ is the normal tangent vector.
I'd like to prove the general case, which means the following theorem :
Theorem : Let $\Omega \subset \mathbb{R}^{m}$ open and bounded with $C^{1}$ boundary. Then given $u \in C^{1}{\overline{\Omega}}$ and said $v_{ext}(z)$ the normal tangent vector of the boundary in the point $z \in \partial \Omega$ in outgoing direction with respect to the "inside" of the manifold, we have $$\int_{\Omega}\partial_{i}u(y)dy = \int_{\partial \Omega}u(z)\cdot \langle e_{i},v_{ext}(z)\rangle d\sigma(z)$$
I do understand the proof of the lemma, what I don't understand is the general proof of the theorem which requires to an easy Sard's theorem version to affirm that, the image under the projection $p : \mathbb{R}^{n} \longmapsto \mathbb{R}^{n-1} \hspace{0.2cm} p(x_{1},\cdots,x_{n}) = (x_{1},\cdots,x_{n-1})$ of the point of the boundary where the restriction of $p$ to the tangent space $T_{x}\partial \Omega$ is not invertible, has null measure, in other words $p(\lbrace x \in \partial\Omega : \lvert p_{|_{T_{x}\partial\Omega}} \mbox{not invertible}\rbrace) \rvert = 0$. Thanks to this observation we can proceed as if $\partial\Omega$ didn't contain "critics" points where $v_{ext} \perp e_{i}$. Then called $U = p(\Omega)-p(\lbrace x \in \partial\Omega : \langle e_{i},v_{ext} \rangle = 0\rbrace)$, open in $\mathbb{R}^{n-1}$, and $V$ a connected component of $U$, I should able to prove that $\Omega \cap p^{-1}(V)$ is a finite union of open and disjoint set which are of the form of the lemma already proven and conclude by additivity of the integral.
Any solution and help on how to prove that $\Omega \cap p^{-1}(V)$ is a finite union of open and disjoint set which are of the form of the lemma already proven is well accepted. I know there are easier way to prove the theorem for istance using partition of unity, tool which I'm not aware of. I would like to keep the proof as linear as possible, possibly using just general topology and the analysis required.
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