Let $f_1$ and $f_2$ be holomorphic functions defined respectively on neighborhoods $U_1$ and $U_2$ of $0$ on the complex plane. We say that $f_1\sim f_2$ if there exists a neighborhood $U$ of $0$ in both domains of definitions such that $f_1|U = f_2|U$.
Let $O_0$ be the set of equivalence classes. I need to show that linear combinations and multiplications of functions define a ring structure on $O_0$.
Any idea or suggestion is welcomed. Thank you!