Intersection of Subgroups is Subgroup

Theorem
The intersection of two subgroups of a group is itself a subgroup of that group:


 * $\forall H_1, H_2 \le \left({G, \circ}\right): H_1 \cap H_2 \le G$

It also follows that $H_1 \cap H_2 \le H_1$ and $H_1 \cap H_2 \le H_2$.

Proof
Let $H = H_1 \cap H_2$ where $H_1, H_2 \le \left({G, \circ}\right)$. Then:

As $H \subseteq H_1$ and $H \subseteq H_2$, the other results follow directly.