Union of Right-Total Relations is Right-Total

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

Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be right-total relations.

Then $\mathcal R_1 \cup \mathcal R_2$ is right-total.

Proof
Define the predicates $L$ and $R$ by:
 * $\map L X \iff \text {$X$ is left-total}$
 * $\map R X \iff \text {$X$ is right-total}$

Also see

 * Union of Left-Total Relations is Left-Total