Euclid's Lemma for Unique Factorization Domain/General Result

Lemma
Let $\left({D, +, \times}\right)$ be a unique factorization domain. Let $p$ be an irreducible element of $D$.

Let $n \in D$ such that:
 * $\displaystyle n = \prod_{i \mathop = 1}^r a_i$

where $a_i \in D$ for all $i: 1 \le i \le r$.

Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$.

That is:
 * $p \mathop \backslash a_1 a_2 \ldots a_n \implies p \mathop \backslash a_1 \lor p \mathop \backslash a_2 \lor \cdots \lor p \mathop \backslash a_n$

Proof
Identical to the proof of Euclid's Lemma for Irreducible Elements: General Result.