Composition of Relations Preserves Subsets

Theorem
Let $A, B, S, T$ be relations as subsets of Cartesian products.

Let $A \subseteq B$ and $S \subset T$.

Then:
 * $A \circ S \subseteq B \circ T$

Proof
We have:

By definition of subset:
 * $A \circ S \subseteq B \circ T$