Local projective transformations

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We know that is a map from a real projective space to itself, which sends real projective lines to real projective lines, is a real projective transformation. I am interested to know if there is a local version of this result. That is, given a local diffeomorphism of a real projective space, which sends projective lines to real projective lines, is it a real projective transformation of the space restricted to an open set?