Distributivity is Preserved in Induced Structure

Theorem
Let $\struct {T, \oplus, \otimes}$ be an algebraic structure, and let $S$ be a set.

Let $T^S$ denote the set of all mappings from $S$ to $T$.

Let $\struct {T^S, \oplus}$ be the structure on $T^S$ induced by $\oplus$.

Let $\struct {T^S, \otimes}$ be the structure on $T^S$ induced by $\otimes$.

If $\otimes$ is distributive over $\oplus$, then the operation induced on $T^S$ by $\otimes$ is distributive over the operation induced by $\oplus$.

Proof
Let $\struct {T, \oplus, \otimes}$ be an algebraic structure in which $\otimes$ distributes over $\oplus$.

Let $f, g, h \in T^S$.

Then:

This shows left distributivity.

The proof for right distributivity is identical.