I wonder - do you know of any example where the Woodbury formula (cf. http://en.wikipedia.org/wiki/Woodbury_matrix_identity) was crucially used to prove anything? 

It might be a useful computational procedure but I've tried many times to use it in proofs and always ended up exasperated - it just replaces a nasty-looking matrix expression with an even nastier-looking expression.

But I've not given up hope on it and therefore am asking for examples of successful theoretical usage.