j_m}$. Then $x$ belongs to some $U_{ j_k}$, so $\lambda_{ j_k} ( x) = 1$. If $\lambda_{ j_k} ( y) \neq 1$, then $\Psi( y) \neq \Psi ( x)$ and we are done. If $\lambda_{ j_k} (y) = 1$, [*i.e.*]{} if $y \in W_{
j_k}$, then the $j_k$-th component of $\Psi$ distinguishes $x$ from $y$, since $\lambda_{j_k} \cdot \phi_{j_k} (y) = \phi_{ j_k} ( y)$ differs from $\phi_{ j_k} (x)$ because $\phi_{ j_k}$ embeds $U_{ j_k}
'$ into ${\mathbb{R}}^{ 2 N_{ j_k}}$. So $\Psi$ embeds into ${\mathbb{R}}^N$ the neighborhood $U_{j_1}
\cup \cdots \cup U_{ j_m}$ of $\overline{ \Omega}$.

We choose a regularized distance function ${\sf d}_{\Psi (K)}$ for $\Psi ( K)$ in ${\mathbb{R}}^N$. We stratify $X$ so that $X_{\rm reg}$ is the single largest stratum (remind it is connected) and then stratify $X_{\rm sing}$ by listing all connected components of $\big[ X_{\rm
sing} \big]_{\rm reg}$, then continuing with $\big[ X_{\rm sing}
\big]_{\rm sing}$, and so on inductively. By Sard’s theorem and the stratified transversality theorem ([@hi1976]), for almost every $\varepsilon >0$, the intersection $$\big\{ x \in {\mathbb{R}}^N : \ {\sf
d}_{\Psi(K)} (x) = \varepsilon \big\} \cap 
\Psi\big(\Omega_{\rm reg}\big)$$ is a $\mathcal{ C }^\infty$ real hypersurface of $\Psi ( \Omega_{\rm
reg})$ 