First of all thank you for cleaning up my messages. I am new to this and have already learnt a lot from you (and about you:-)).
Second, I reverted to "VAS is based on Budan's theorem", but I still disagree with you. Here are the reasons why: A. Sturm himself said 
"m'appuyant sur" = "by relying upon" and not "having been inspired by".  B.  In Greek, "to rely upon" is βασίζομαι, and there you can see the word βάση which means base. I also leave it up to you to decide the wording for Sturm's method and Fourier's theorem. I simply cannot accept the word inspiration; reminds me of religion.