Unitary R-Modules with n-Element Bases are Isomorphic

Theorem
Let $G$ and $H$ be unitary $R$-modules for some ring with unity $R$.

Let $G$ and $G$ both have bases with $n$ elements.

Then $G$ and $H$ are isomorphic.

Proof
From Isomorphism from $R^n$ via $n$-Term Sequence, both $G$ and $H$ are isomorphic to the $R$-module $R^n$.