Finite Direct Product of Unitary Modules is Unitary Module/Proof 2

Proof
From Finite Direct Product of Modules is Module we have that $G$ is a module.

It remains to be shown that:


 * $\forall x \in G: 1_R \circ x = x$

Let $x = \tuple {x_1, x_2, \ldots, x_n} \in G$.

Then:

Hence the result.