External Direct Product of Ringoids is Ringoid

Theorem
Let $\left({R_1, +_1, \circ_1}\right), \left({R_2, +_2, \circ_2}\right), \ldots, \left({R_n, +_n, \circ_n}\right)$ be ringoids.

Let their external direct product be $\displaystyle \left({R, +, \circ}\right) = \prod_{k=1}^n \left({R_k, +_k, \circ_k}\right)$.

Then the operation $\circ$ is distributive over $+$.

Proof
As all the $\left({R_k, +_k, \circ_k}\right)$ are ringoids, $\circ_k$ distributes over $+_k$ for all $k$.

Let $x, y, z \in R$.

Then:

In the same way:
 * $\left({y + z}\right) \circ x = \left({y \circ x}\right) + \left({z \circ x}\right)$