change of parameters, surfaces

Hello I have a very simple question of a proof in the book of Docarmo Differential Geometry. In some page of the book , they said that using the proposition 1 , we can deduce that the parametrizations are also diffeomorphism. How can I prove that? I only know that $x^{-1} y$ it´s diffeomorphism for all parametrizations y. Using this how can I prove that the function $x^{-1}$ it´s also differentiable? Here are the definitions used in Docarmo of parametrization, and the proposition 1.

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