Pointwise Operation on Distributive Structure is Distributive

Theorem
Let $S$ be a set.

Let $\left({T, +, \circ}\right)$ be an algebraic structure with two operations $+$ and $\circ$.

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

Let $\left({T^S, +', \circ'}\right)$ be the structure induced on $T^S$ by $+$ and $\circ$.

Let $\circ$ be distributive over $+$ in $T$.

Then $\circ'$ is distributive over $+'$ in $T$.

Proof
Let $f, g, h: S \to T$ be elements of $T^S$.

Let $x \in S$.

Then:

Similarly:
 * $\left ({\left({g +' h}\right) \circ' f}\right) \left({x}\right) = \left ({\left ({g \circ' f}\right) +' \left ({h \circ' f}\right)}\right) \left({x}\right)$

Hence the result.