One-to-Many Relation Composite with Inverse is Coreflexive

Theorem
Let $$\mathcal{R} \subseteq S \times T$$ be a relation which is one-to-many.

Then the composite of $$\mathcal{R}$$ with its inverse is a subset of the diagonal relation:

$$\mathcal{R}^{-1} \circ \mathcal{R} \subseteq \Delta_X$$

That is, by this result, $$\mathcal{R}^{-1} \circ \mathcal{R}$$ is both symmetric and antisymmetric.