Difference in Naturally Ordered Semigroup is Unique

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Theorem

Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.

Let $n, m \in S$ such that $m \preceq n$.


Then there exists a unique difference $n - m$ of $m$ and $n$.


Proof

Since $m \preceq n$, by axiom $(NO3)$:

$\exists p \in S: m + p = n$


Now suppose that $p, q \in S$ are such that:

$m + p = m + q = n$


Then it follows from axiom $(NO2)$ that:

$p = q$


Hence the result.

$\blacksquare$


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