Let $k$ be a field, let $n,m\in\mathbb N$, and let $X\subseteq \mathbb A_k^n$ and $Y\subseteq \mathbb A_k^m$ be affine varieties over $k$, allowing the empty variety and using the convention that $k[X]$ and $k[Y]$ are the $k$-algebras of polynomial functions restricted to $X$ and $Y$. Regard $X\times Y\subseteq \mathbb A_k^{n+m}$ as the affine variety with coordinates $(x_1,\dots,x_n,y_1,\dots,y_m)$ whose coordinate ring $k[X\times Y]$ is the $k$-algebra of polynomial functions in these variables restricted to $X\times Y$. Then the $k$-bilinear map $k[X]\times k[Y]\to k[X\times Y]$ sending $(f,g)$ to the regular function $(p,q)\mapsto f(p)g(q)$ induces an isomorphism of $k$-algebras $k[X]\otimes_k k[Y]\cong k[X\times Y]$.