Proposition 6

*Let $w$ be an element of $\mathrm{Mo}(X)$. (a) There exist an integer $n \geq 0$, elements $x_\alpha$ of $X$ and integers $m(\alpha) > 0$ (for

† Note that $G_1 * G_2$ is not the "product" of $G_1$ and $G_2$ in the sense of Set Theory, IV, § 2, no. 4 (nor in the sense of the "theory of categories"; in the context of this theory, $G_1 * G_2$ is the "sum" of $G_1$ and $G_2$).

$1 \leq \alpha \leq n$ such that $x_\alpha \neq x_{\alpha+1}$ for $1 \leq \alpha < n$ and $w = \prod_{\alpha=1}^n x_\alpha^{m(\alpha)}$. The sequence $(x_\alpha, m(\alpha))_{1 \leq \alpha \leq n}$ is determined uniquely by these conditions.

(b) Let $p$ be a positive integer, $x'_\beta$ in $X$ and $m'(\beta)$ in $\mathbf{N}$ for $1 \leq \beta \leq p$ such that $w = \prod_{\beta=1}^p {x'}^{m'(\beta)}$. Then $p \geq n$. If $p = n$, then $x'_\beta = x_\beta$ and $m'(\beta) = m(\beta)$ for $1 \leq \beta \leq p$.

In the notation of Lemma 2, $h^{-1}(\phi_x(n)) = x^n$ for $x \in X$ and $n \in \mathbf{N}$. Proposition 6 then follows from no. 3, Proposition 5.