Theorem 7 (decomposition into prime factors)

Let $a$ be a strictly positive integer. There exists one and only one family $(v_p(a))_{p \in \mathfrak{P}}$ of integers $> 0$ such that the set of $p \in \mathfrak{P}$ with $v_p(a) \neq 0$ is finite and

As the group $\mathbf{Z}/a\mathbf{Z}$ is finite, it admits a Jordan-Hölder series. Lemma 2 then implies that $a$ is a product of prime integers, whence the existence of the family $(v_p(a))$; further, for every family $(v_p(a))_{p \in \mathfrak{P}}$ satisfying the conditions of Theorem 7, the integer $v_p(a)$ is, for all $p \in \mathfrak{P}$, equal to the number of factors of a Jordan-Hölder series of $\mathbf{Z}/a\mathbf{Z}$ isomorphic to $\mathbf{Z}/p\mathbf{Z}$ (Lemma 2). The uniqueness of the family $(v_p(a))$ therefore follows from the Jordan-Hölder theorem (no. 7, Theorem 6).