Finite Semigroup Equal Elements for Different Powers

Theorem
Let $$\left({S, \circ}\right)$$ be a finite semigroup.

Then $$\forall x \in S: \exists m, n \in \N: m \ne n: x^m = x^n$$.

Proof
List the positive powers $$x, x^2, x^3, \ldots$$ of any element $$x$$ of a finite semigroup $$\left({S, \circ}\right)$$.

Since all are elements of $$S$$, and the semigroup has a finite number of elements, it follows from the Pigeonhole Principle this list must contain repetitions.

So there must be at least one instance where $$x^m = x^n$$ for some $$m, n \in \N$$.