Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements

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

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

Let $x,y \in S$, and suppose that $x \mathrel{\mathcal R} y$.

Let $n \in \N_{>0}$ be a strictly positive integer.

Then $x^n \mathrel{\mathcal R} y^n$.

Proof
We proceed by Mathematical Induction:

$x^1=x$ and $y^1 = y$, so by assumption, $x^1 \mathrel{\mathcal R} y^1$.

Suppose now that $(1): \quad x^n \mathrel{\mathcal R} y^n$.

We have assumed that $(2): \quad x \mathrel{\mathcal R} y$. (2)

Applying User:Dfeuer/Operating on Transitive Relationships Compatible with Operation to $(1)$ and $(2)$,

$x^n\circ x \mathrel{\mathcal R} y^n \circ y$.

but $x^{n+1} = x^n \circ x$ and $y^{n+1} =y^n \circ y$, so the theorem holds.