User:Dfeuer/CTR5

Theorem
Let $(S,\circ)$ be a semigroup.

Let $\prec$ be a transitive relation on $S$ which is compatible with $\circ$.

Let $x,i \in S$.

Suppose that $i$ is idempotent with respect to $\circ$.

Let $n \in \N_{>0}$.

Then the following statements hold:


 * $i \prec x \implies i \prec x^n$
 * $x \prec i \implies x^n \prec i$.

Proof
By User:Dfeuer/Operating Repeatedly on Transitive Relationship Compatible with Associative Operation,


 * $i \prec x \implies i^n \prec x^n$
 * $x \prec i \implies x^n \prec i^n$.

By the definition of an idempotent element, $i^n = i$, so the theorem holds.