Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements

Theorem
Let $(S,\circ)$ be a Definition:Closed Algebraic Structure.

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

Let $x,y \in S$, and suppose that $x \prec y$.

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

Then $x^n \prec y^n$.

Proof
The result follows inductively from repeated application of User:Dfeuer/Operating on Transitive Relationships Compatible with Operation.