Intersection of Transitive Relations is Transitive

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

Proof
Let $\RR_1$ and $\RR_2$ be transitive relations (on what sets is immaterial for this argument).

Let $\tuple {s_1, s_2} \in \RR_1 \cap \RR_2$ and $\tuple {s_2, s_3} \in \RR_1 \cap \RR_2$.

Then by definition of intersection:
 * $\tuple {s_1, s_2} \in \RR_1$ and $\tuple {s_1, s_2} \in \RR_2$
 * $\tuple {s_2, s_3} \in \RR_1$ and $\tuple {s_2, s_3} \in \RR_2$

Then as $\RR_1$ and $\RR_2$ are both transitive:
 * $\tuple {s_1, s_3} \in \RR_1$ and $\tuple {s_1, s_3} \in \RR_2$

and by definition of intersection:
 * $\tuple {s_1, s_3} \in \RR_1 \cap \RR_2$

hence $\RR_1 \cap \RR_2$ is transitive.