Product of Commuting Elements in Monoid is Unit iff Each Element is Unit

Theorem
Let $A$ be a monoid.

Let $\map G A$ be the group of units of $A$.

Let $n \ge 2$ be an integer.

Let $x_1, \ldots, x_n$ be commuting elements in $A$.

Let:
 * $\ds x = \prod_{i \mathop = 1}^n x_i$

Then:
 * $x \in \map G A$ $x_i \in \map G A$ for each $1 \le i \le n$.