### Slight generalization of the Local Normal Form

The Local Normal Form states that if $F:X\to Y$ is a holomorphic map at $p\in X$, which is not constant, then there is a unique integer $m\geq 1$ which satisfies the following property: for every chart $\phi_2: U_2\to V_2$ on $Y$ centered at $F(p)$, there exists a chart $\phi_1: U_i\to V_1$ on $X$ centered at $p$ such that $\phi_2(F(\phi_1^{-1}(z)))=z^m$.

The way I understand the proof, I think we can extend it to say that for any chart $U_1\to V_1$ on $X$ centered at $p$, then there exists a chart $\phi_2: U_2\to V_2$ on $Y$ centered at $F(p)$ such that $\phi_2(F(\phi_1^{-1}(z)))=z^m$. The only added condition is that $\phi_2: U_2\to V_2$ is non-zero except at $F(p)$.

Proving this would be quite easy, and is left as an exercise.