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 $$\left({R, +, \circ}\right) = \prod_{k=1}^n \left({R_k, +_k, \circ_k}\right)$$.

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