Intersection of Transitive Relations is Transitive

Theorem
The intersection of two transitive relations is also a transitive relation.

Proof
Let $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ be transitive relations (on what sets is immaterial for this argument).

Let $$\left({s_1, s_2}\right) \in \mathcal{R}_1 \cap \mathcal{R}_2$$ and $$\left({s_2, s_3}\right) \in \mathcal{R}_1 \cap \mathcal{R}_2$$.

Then by definition of intersection:
 * $$\left({s_1, s_2}\right) \in \mathcal{R}_1$$ and $$\left({s_1, s_2}\right) \in \mathcal{R}_2$$;
 * $$\left({s_2, s_3}\right) \in \mathcal{R}_1$$ and $$\left({s_2, s_3}\right) \in \mathcal{R}_2$$.

Then as $$\mathcal{R}_1$$ and $$\mathcal{R}_2$$ are both transitive:
 * $$\left({s_1, s_3}\right) \in \mathcal{R}_1$$ and $$\left({s_1, s_3}\right) \in \mathcal{R}_2$$

and by definition of intersection:
 * $$\left({s_1, s_3}\right) \in \mathcal{R}_1 \cap \mathcal{R}_2$$

hence $$\mathcal{R}_1 \cap \mathcal{R}_2$$ is transitive.