Finite Direct Product of Modules is Module/Proof 2

Let $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n} \in G$.

So $(1)$ holds.

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

So $(2)$ holds.

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

So $(3)$ holds.

Hence the result.