Homomorphism of Powers

Theorem
Let $$\left({S, \circ; \preceq}\right)$$ be a naturally ordered semigroup.

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 homomorphism.

Let $$\odot_1^n$$ and $$\odot_2^n$$ be as defined as in Naturally Ordered Semigroup Power Law.

Then $$\forall a \in T_1: \forall n \in \left({S^*, \circ; \preceq}\right): \phi \left({\odot_1^n a}\right) = \odot_2^n \phi \left({a}\right)$$,

Proof
Can be proved by the Principle of Finite Induction.