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I am working in the following setting.

$\newcommand{\R}{\mathbb{R}} \newcommand{\Vreg}{{V^{reg}}}$

I have two functions $F \colon \R^{n+1}\to \R^{n}$ and $L \colon \R^{n+1} \to \R$, such that $F$ is smooth and $L$ is a linear function.

I also know that $L_{|\Vreg(F)} \colon \Vreg(F) \to \R$ is a Morse function for the smooth manifold $\Vreg(F)$.

With a little of calculations it is possible to see that the critical points for $L_{|\Vreg(F)}$, that is regular zeros (since $L_{\Vreg(F)}$ is a Morse function) of $D(L_{|\Vreg(F)}$ are those point of $\Vreg(F)$ that are zeros for $$J(L,F)=det\begin{pmatrix} \nabla L \\ D(F) \end{pmatrix} $$ so $\Vreg(L_{|\Vreg(F)})=\Vreg(F)\cap V(J(L,F))$.

Apparently it should be $$\Vreg(L_{|\Vreg(F)})=\Vreg(J(L,F),F)$$ but I have no idea on how to go from here.

Thanks in advance to anyone would could provide any help, or a counter-example in case this statement is false.

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