*Reference request:* Firstly, I'm looking for a proof of the following well-known result about handle decompositions:

($\ast$) Given two handle decompositions of a smooth $n$-manifold $M$, there exists a sequence of handle pair creations, cancellations, and handle slides (as well as isotopies) that takes one decomposition to the other.

I think I broadly understand the conceptual idea behind the proof, namely that handle decompositions correspond to Morse functions on $M$, and homotopies of the Morse functions give such sequences of handle moves, but I'd like to be able to work through this more rigorously.
So far the only lead I've been able to find is Gompf and Stipsicz's *4-Manifolds and Kirby Calculus* (Theorem 4.2.12), where they attribute the result to Cerf and simply reference his paper *La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie*.
Unfortunately my French is extremely rudimentary so I haven't been able to locate the relevant part(s) of the paper.
Would anyone be able to point me in the right direction, or know where I can find a proof of this theorem?

**Main question:**

If an $n$-dimensional smooth manifold $M$ has handle decompositions $\mathscr{H}, \mathscr{H}^\prime$, both in terms of $0,1,\dots, k$-handles where $k <n$, does there exist a sequence of handle moves taking $\mathscr{H}$ to $\mathscr{H}^\prime$ which only involves handles of index at most $k$?

I expect the answer is "yes", and that it follows immediately from the proof of ($\ast$), but as I have not yet seen such a proof written down, I am not sure.