Conic singularities and immediate transversality

I introduced the notion of immediate transversality in a recent paper on $A_{\mathrm{\infty }}$-structures on Morse complexes. I the present chapter, I prove that immediate transversality holds for every compact submanifold with $C^1$ conic singularities, not only in the setup of Morse Theory. This is an occasion of speaking of this type of singularities.