Image of Intersection under One-to-Many Relation

Theorem
Let $$\mathcal{R} \subseteq S \times T$$ be a relation which is one-to-many. Let $$A$$ and $$B$$ be subsets of $$S$$. Then:

$$\mathcal{R} \left({A \cap B}\right) = \mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right)$$

Proof
From Image of Intersection, we already have:

$$\mathcal{R} \left({A \cap B}\right) \subseteq \mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right)$$

So we just need to show:

$$\mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right) \subseteq \mathcal{R} \left({A \cap B}\right)$$

Let $$t \in \mathcal{R} \left({A}\right) \cap \mathcal{R} \left({B}\right)$$.