Unique Quotient in Natural Numbers

Definition
Let $\left({S, \circ, *, \preceq}\right)$ be a naturally ordered semigroup with product.

Let $n \in S$ and $m \in S_{\ne 0}$ such that:
 * $m \mathrel \backslash n$

where $m \mathrel \backslash n$ denotes that $m$ is a divisor of $n$.

Then there exists exactly one element $p \in S$ such that $m * p = n$.

Proof
Let $n = m * p$.

Suppose that $n = 0$.

Then from Multiplication in Naturally Ordered Semigroup has No Proper Zero Divisors it follows that $p = 0$.

Thus in this case the unique value of $p$ is zero.

Now suppose $n \ne 0$.

Let $n = m * p = m * q$ for $p, q \in S$.

Then from Cancellable Elements of Naturally Ordered Semigroup under Multiplication it follows that $p = q$.

Hence the result.