Inverse of Left-Total Relation is Right-Total

Theorem
Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse of $\mathcal R$.

Then:
 * $\mathcal R$ is left-total $\mathcal R^{-1}$ is right-total.

Proof
From Inverse of Inverse Relation, the inverse of $\mathcal R^{-1}$ is $\mathcal R$.

From Inverse of Right-Total Relation is Left-Total:
 * $\mathcal R^{-1}$ is right-total $\mathcal R$ is left-total.

Hence the result.