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Let $f:R^n\to R^n$ be an analytic map such that $df_a$ is invertible. The "usual" local inversion theorem says that $f$ is locally invertible and that the inverse will be smooth.

I'm quite sure that, if $f$ is analytic, the inverse will be analytic too, but I cannot prove it.

I know a proof in the case $n=1$ trough complex analysis.

This is almost a duplicate of this question with two differences :

  • I don't mind passing trough complex analysis
  • I need a statement on an open set of $R^n$, not an interval of R.
  • I want a proof or a link, not a reference to a book (I do not have access to a math library)
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