Proposition 7

Let g be an element of the free group $F(X)$. There exist an integer $n \geq 0$ and a sequence $(x_\alpha, m(\alpha))_{1 \leq \alpha \leq n}$ determined uniquely by the relations $x_\alpha \in X$, $x_\alpha \neq x_{\alpha+1}$ for $1 \leq \alpha < n$, $m(\alpha) \in \mathbf{Z}$, $m(\alpha) \neq 0$ for $1 \leq \alpha \leq n$, and $g = \prod_{\alpha=1}^n x_\alpha^{m(\alpha)}$.

The free group $F(X)$ enjoys the following universal property: