On a Refinement of the Schneider–Lang theorem

We consider some arithmetic properties of values of meromorphic functions $g_1(z)$, …, $g_m(z)$ such that each of $g'_i(z)$ is algebraically dependent over a field $K$ of algebraic numbers, $[K:\mathbb Q]<\infty$, with the functions $g_1(z),\dots,g_m(z)$. We show that if all $\{g_i(z)\}$ are meromorphic of finite order, then either they all are rational functions, or they all are rational functions of some exponential, or they all are elliptic functions, or there exists a discrete set $U$ such that the number of points $z\notin U$ such that all $\{g_i( z)\}$ lie in $K$ is finite.