I don't understand why all authors of this proof assume that Df_a = id_n, how doesn't this destroy generality? For example, see https://www.physicsforums.com/showthread.php?t=476508. The λ in his post (and the post he quotes) is always Df_a (its not stated in that post, but in the book and the post that is quoted in that post). It doesn't seem like the answer is ever made. My attempt at an answer: I FEEL like the assumption is valid because its only a computation and therefore doesn't change the "structure" of the problem itself (i.e. the spaces are preserved). But it at first glance does seem like a pretty big leap in a proof. The same (or similar) argument is made in every proof i've seen : Spivak, and MIT's opencourse http://ocw.mit.edu/courses/mathematics/18-101-analysis-ii-fall-2005/lecture-notes/lecture7.pdf (i.e. Df(0) = id), also the proof given in Jerry Shurman's "Multivariable Calculus", the online book. This is a fairly simple question, could I have a simple answer?