One straightforward kind of proof analysis was designed to reveal what assumptions there are behind accepted ‘theorems’, and this is clearly pertinent in the case of Zermelo's Axiom of Choice (his sixth axiom) and the WOT. What Zermelo's work showed, in effect, is that the ‘choice’ principle behind the Axiom is a necessary and sufficient condition for WOT; and he shows this by furnishing a Hilbertian style proof for the theorem, i.e., a conclusion which follows from (fairly) clear assumptions by means of a finite number of inferential steps. Indeed, the Axiom is chosen so as to make the WOT provable, and it transpired subsequently that it also made provable a vast array of results, mainly (but not solely) in set theory and in set-theoretic algebra. To understand the importance of Zermelo's work, it's necessary to appreciate the centrality of the the WOT.