Exercise 23, § 7

Let $F$ be a free group with basic family $(x, y)$ with $x \neq y$. Now prove that the derived group of $F$ has as basic family the family of commutators $(x^i, y^j)$ for $i \in \mathbf{Z}, j \in \mathbf{Z}, i \neq 0, j \neq 0$. (Use Exercise 20 or I, p. 149, Exercise 32.)