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

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

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

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

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

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