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The problem is, sometimes you can't reformulate the problem this way. If there are 1000 equations with 200 variables and this is it, then interval arithmetic is useless. (As well as anything else, I guess.)
Thank you. Siebenmann's article is quite a nice reading. By the way, one quote from it: "One can expect that mathematicians will consequently come to use freely the notions of homeomorphism and topological manifold untroubled by the frustrating difficulties that worried their early history". It was 50 years ago!
This may be a difficult part, so if you want to fill the details in, then you better ask an expect. (I am familiar with differential geometry, but this point is about analysis on manifolds, where I am lacking.) Simply put, the Weitzenböck identity is helpful because the Bochner Laplacian is better then the Hodge Laplacian in the following way: if it is zero, then the form is covariant constant and, in particular, does not vanish (which is far from the case for the Hodge one).
Edwin Moise in his well known paper "Affine structures ..." (Annals of Math,1952) for this sort of arguments makes a reference to "Zur Topologie der Mannigfaltigkeiten", G Nöbeling, Monatshefte für Mathematik und Physik, 1935. I can't say much more for I could not read it even if I had it (I am not very good at German). On the undergrad level, it is plausible at least for manifolds because continuous maps can be approximated by piecewise linear ones. Of course, today we know that this logic is wrong.
