Corollary 2 (Runge's Theorem)

For each $i\in I$, let $\lambda_i$ be a point of $O_i$. Let $f$ be a holomorphic complex function in an open neighbourhood of X. Then $f|X$ is the uniform limit of restrictions to X of rational fractions whose poles are some of the $\lambda_i$.

With $E =\{\lambda_i\}_{i\in I}$, the assumption is condition (i) of cor. 1. Hence $R_E(X) = R(X)$, and th. 3 of I, p. 69 shows that $R(X) =\mathscr{O}(X)$.