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I am studying the following proof for which an excerpt is provided below:

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Update: I have written out a fully-detailed proof of an argument that seeks verify the claim that $\partial \psi$ is invertible. (1) I am unclear on what is special about the point $(x_0, 0)$ as the proof seems to goes through irrespective of the value of the particular point and (2) The author's logic at the end seems reversed to me. It would be helpful if someone could critique the proof below and indicate what step, if any, is incorrect. Also, I apologize for not including the actual TEX; it was formulated locally and I made use of many macros that mathjax wouldn't understand:

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    @LeonidKovalev The question is still unanswered. Compare my proof with the excerpt given from the text; they can't both be right.2012-07-29

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(1) It is true that $\partial \psi(x_0,y)$ is invertible for all values of $y$, not just $0$. We only care about the invertibility of $\partial \psi(x_0,0)$ because we need a diffeomorphism onto a neighborhood of (a piece of) the given manifold, which corresponds to $y=0$.

(2) I agree that the logic of the last sentence in the first excerpt is backwards. Just ignore "Therefore" and replace "and thus" with "because".