Index in Subgroup

Theorem
Let $G$ be a group.

Let $H, K$ be subgroups of finite index of $G$.

Then:
 * $\left[{H : H \cap K}\right] \le \left[{G : K}\right]$

where $\left[{G : K}\right]$ denotes the index of $K$ in $G$.

Equality happens iff $G = H K$.