Free Module on Set is Free

Theorem
Let $R$ be a ring with unity.

Let $I$ be a set.

Let $R^{\paren I}$ be the free $R$-module on $I$.

Then $R^{\paren I}$ is a free $R$-module.

Proof
From Canonical Basis of Free Module on Set is Basis, $R^{\paren I}$ has a basis.