$e^{z+a}=e^ze^a$ where $a,z\in\mathbf{C}$ by using properties of analytic functions.

I am trying to show that $e^{z+a}=e^ze^a$ where $a,z\in\mathbf{C}$ by using the fact that if $f,g$ are analytic on a region of $\mathbf{C}$ then $f=g$ if and only if the set $Z=\{z\in G: f(z)=g(z)\}$ has a limit point in $G$.

I know this result can be shown by other methods, but I want it done this way. It is frustrating me because I thought there would be a quick obvious answer. I tried using the fact that it holds for real numbers, but since $a\in\mathbf{C}$, I didn't get very far.

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First fix $a$ real and use that it holds for all real $z$ to conclude that it holds for all complex $z$. Then fix complex $z$ and use the fact that it holds for all real $a$ (which you just showed) to show that it holds for all complex $a$. I.e., just apply the argument twice.