Let $D\subseteq \mathbb{C}$ be a domain. Let $f,g:D\to \mathbb{C}$ be holomorphic, and let $J\subseteq D$ possess an accumulation point in $D$. If $f(z)=g(z)$ for all 1;;$z\in J$, then $f(z)=g(z)$ for all 1;;$z\in D$.

Later:
Notice how this is similar to the fact that if $(X,{\mathcal{O}}_X)$ is an affine 8. Definition 4.5 Affine Variety, then if two sections agree on some open subset, then they agree on the larger set.