Free groups satisfy IBN

Statement
Free groups satisfy the invariant basis number (IBN) property: If $$S$$ and $$T$$ are two sets such that the free group on the set $$S$$ is isomorphic to the free group on the set $$T$$, then $$S$$ and $$T$$ have the same cardinality.

Facts used

 * Free Abelian groups satisfy IBN

Proof
We use the fact that the free Abelian group on a set is the Abelianization of the free group on the set. Thus, if the free group on $$S$$ is isomorphic to the free group on $$T$$, so are their Abelianizations, and hence the free Abelian group on $$S$$ is isomorphic to the free Abelian group on $$T$$. The problem thus reduces to showing that free Abelian groups satisfy IBN.