Union of Many-to-One Relations with Disjoint Domains is Many-to-One

Theorem
Let $S_1, S_2, T_1, T_2$ be sets or classes.

Let $\mathcal R_1$ be a many-to-one relation on $S_1 \times T_1$.

Let $\mathcal R_2$ be a many-to-one relation on $S_2 \times T_2$.

Suppose that the domains of $\mathcal R_1$ and $\mathcal R_2$ are disjoint.

Then $\mathcal R_1 \cup \mathcal R_2$ is a many-to-one relation on $(S_1 \cup S_2) \times (T_1 \cup T_2)$.