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 bounded 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 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 measure $0$, 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 "critical" 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 be able to prove that $\Omega \cap p^{-1}(V)$ is a finite union of open and disjoint sets which are of the form of the lemma already proven and conclude the theorem 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 ways to prove the theorem for instance using partition of unity, a tool 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.
As general as possible solutions using just general topology and the analysis concerning the second year of math university are well accepted too.
Edit 1 :
The definition of manifold I'm currently using is :
Definition : A subset $M \subseteq \mathbb{R}^{n}$ is a differentiable manifold of class $C^{l}$ and dimension $k$ if $\forall \hspace{0.1cm} x \in M \hspace{0.1cm} \exists \hspace{0.1cm} U \in \mathbb{R}^{k},V \in \mathbb{R}^{n}$ open subsets, with $x \in V$ and a diffeomorphism $\phi : U \longmapsto V \cap M$ of class $C^{l}$.
While the "easy" version of Sard's Theorem for manifolds is the following :
Definition : Let $f : \Sigma \longmapsto \Gamma$ a function of class $C^{1}$ between manifolds of same dimension. A point $p \in \Sigma$ is said to be critical is $d_{p}f$ is not surjective. The immage of the critical point are said critical values.
Theorem : Let $f : \Sigma \longmapsto \Gamma$ a function of class $C^{1}$ between manifolds. The critical values have null measure.
