Euclid's Lemma for Prime Divisors/General Result

Lemma
Let $p$ be a prime number.

Let $\displaystyle n = \prod_{i \mathop = 1}^r a_i$.

Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$ such that $1 \le i \le r$.

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