Suppose that $f: E_1 \times E_2 \times \dots \times E_m \to (-\infty, \infty]$ is defined as
$$f(x_1, x_2, \dots, x_m) = \sum_{i=1}^m f_i(x_i)$$
for any $x_i \in E_i, \forall i = 1 \dots m$.
Prove that, for any $x_1 \in E_1, x_2 \in E_2 \dots x_m \in E_m$
$$prox_{f, \lambda}(x_1, x_2, \dots, x_m) = prox_{f_1, \lambda}(x_1) \times prox_{f_2, \lambda}(x_2) \times \dots prox_{f_m, \lambda}(x_m)$$
where $\times$ represents the cartesian product between sets.
