I am reading Silverman's "The Arithmetic of Elliptic Curves". On page 10 he defines the function field of a projective variety $V$ over a field $K$ to be the function field of $V\cap\mathbb{A}^n$, where $\mathbb{A}^n$ is one of the standard affine patches of $\mathbb{P}^n$ that intersects V. In this definition he claims that the definition is independent of the chosen affine patch, since the resulting function fields are isomorphic for each patch. In the answers on this question, this is also mentioned, but without a proof. I am unable to find an isomorphism between the resulting function fields. Could someone give me a hint?
Thank you in advance!