Where does Hironaka prove that (a_f) stratifications exist?

Question

In "Some remarks on relative monodromy" (MR0476739), Lê has an analytic function $f : X\to\mathbb{C}$ where $X$ is an analytic subset of an open subset of $\mathbb{C}^N.$ Lê claims that Hironaka proved that such an $X$ can be stratified in a way obeying Whitney's condition, Thom's condition (a_f), and such that $f^{-1}(0)$ is a union of stratum.

Massey seems to repeat the claim that Hironaka proved this possible in the third paragraph of the introduction to https://arxiv.org/pdf/math/0605369.pdf.

Both cite Hironaka's "Stratification and flatness" (MR0499286) as the reference. However, I cannot seem to find where in this paper Hironaka actually proves the stated fact!

The closest I can find is that, at the end of section 5, Hironaka gives a "Corollary 1" stating that you can find, for a proper map $f : X\to Y$ where $Y$ is a curve, a Whitney stratification of $X$ obeying Thom's condition $a_f.$ But it seems that Massey and Lê do not assume properness of there map, so I am not sure how this corollary can be applied.

Question. Where can I find a proof of the statement that Lê uses?

Answer 1

I do not know about Hironaka's proof, but I know how to find a proof for this. Also, as far as I understand, a common style of the time was to not give many details and attribution of results was more complicated. Luckily that is not the case any more: see the paper Thom condition and monodromy by R. Giménez Conejero, D. T. Lê and J.J. Nuño-Ballesteros (MR4508018, in particular Lemma 2.5 and the beginning of the proof of Theorem 2.6); and also Localisation de systèmes différentiels, stratifications de Whitney et condition de Thom by J. Briançon, P. Maisonobe and M. Merle .

First, you need to give a Whitney stratification to $X$, you can always do this (in semialgebraic sets) by, for example, Theorem 2.7 of the book of Gibson Wirthmuller Du Plesis Looijenga, Topological stability of smooth mappings. Then, you adapt the stratification to have $f^{-1}(0)$ a union of strata. This is a very particular case of Lemma 2.5 I was citing above. Then, Thom's condition is very easy in dimension one by applying its definition for strata that map to the same strata. For the rest, you have Briançon, Maisonobe and Merle's proof (Th. 4.2.1).