# Singular point for Morse functions

I am working in the following setting.


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.