Index of Intersection of Subgroups

Theorem
Let $$G$$ be a group.

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

Then $$\left[{G : H \cap K}\right] \le \left[{G : H}\right] \left[{G : K}\right]$$ where $$\left[{G : H}\right]$$ is the index of $H$ in $G$.

Note that here the symbol $$\le$$ is being used with its meaning "less than or equal to".