Proof in Shafarevich that local ring is normal

Question

In Chapter II 5.1 Theorem 1 of Basic Algebraic Geometry I, Shafarevich sets about proving that $\mathcal{O}_x$ is integrally closed. He introduces an element $\alpha\in k(X)$ in the form $\alpha=u/v$ where $u,v\in\mathcal{O}_x$. Should this be $u,v\in \mathcal{O}(X)$, the coordinate ring not the local ring? As far as I can judge, the proof then goes through fine.