Homomorphism of Powers/Natural Numbers

Theorem
Let $\left({T_1, \odot_1}\right)$ and $\left({T_2, \odot_2}\right)$ be semigroups.

Let $\phi: \left({T_1, \odot_1}\right) \to \left({T_2, \odot_2}\right)$ be a (semigroup) homomorphism.

Let $n \in \N$.

Let $\odot_1^n$ and $\odot_2^n$ be as defined as in Index Laws for Semigroups.

Then:
 * $\forall a \in T_1: \forall n \in \N: \phi \left({\odot_1^n \left({a}\right)}\right) = \odot_2^n \left({\phi \left({a}\right)}\right)$

Proof
Follows directly from Homomorphism of Powers: Naturally Ordered Semigroup as the Natural Numbers are a Naturally Ordered Semigroup.