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

Sufficient Condition
Let $\mathcal R$ be right-total.

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

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

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

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

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

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

So by definition $\mathcal R$ is right-total.