Inverse of Right-Total Relation is Left-Total

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

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

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

Sufficient Condition
Let $\RR$ be right-total.

Then by definition:
 * $\forall t \in T: \exists s \in S: \tuple {s, t} \in \RR$

By definition of the inverse of $\RR$, it follows that:
 * $\forall t \in T: \exists s \in S: \tuple {t, s} \in \RR^{-1}$

So by definition $\RR^{-1}$ is left-total.

Necessary Condition
Let $\RR^{-1}$ is left-total.

Then by definition:
 * $\forall t \in T: \exists s \in S: \tuple {t, s} \in \RR^{-1}$

and so:
 * $\forall t \in T: \exists s \in S: \tuple {s, t} \in \RR$

So by definition $\RR$ is right-total.