Finite Submodule of Function Space

Theorem
Let $$G$$ be a group whose identity is $$e$$.

Let $$R$$ be a ring.

Let $$\left({G: \circ}\right)_R$$ be an $R$-module.

Let $$S$$ be a set.

Let $$G^S$$ the set of all mappings $$f: S \to G$$.

Let $$G^{\left({S}\right)}$$ be the set of all mappings $$f: S \to G$$ such that $$f \left({x}\right) = e$$ for all but finitely many elements $$x$$ of $$S$$.

Then $$\left({G^{\left({S}\right)}: \circ}\right)_R$$ is a submodule of $$\left({G^S: \circ}\right)_R$$.