Infinite system of ODE

[Klaus10]

Let $\Gamma$ be an arbitrary nonempty set. For each $\gamma \in \Gamma$, let $f_\gamma$ be a scalar-valued function depending on a finite number of variables:

$$ f_\gamma \in C([t_1, t_2] \times \mathbb{R}^{n_\gamma}), \quad f_\gamma = f_\gamma(t, x_{\sigma_1}, \dots, x_{\sigma_{n_\gamma}}), $$

where $\{\sigma_1, \dots, \sigma_{n_\gamma}\} \subset \Gamma$ is a finite subset of indices, and

$$ \sup_{[t_1, t_2] \times \mathbb{R}^{n_\gamma}} |f_\gamma| \le M_\gamma. $$

Theorem.
The Cauchy problem

$$ \dot{x}_\gamma = f_\gamma, \quad x_\gamma(t_1) = \hat{x}_\gamma, \quad \gamma \in \Gamma, $$

has a solution $\{x_\gamma(t)\}_{\gamma \in \Gamma}$ such that $x_\gamma \in C^1[t_1, t_2]$ for each $\gamma \in \Gamma$.


What is your take on this construction?