Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$. Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
I need this theorem to prove something. 

We know that
$$HK=\bigcup_{h\in H} hK$$
and each $hK$ has the same cardinality $|hK|=|K|$. (See ProofWiki.)
We also know that for any $h,h'\in G$ either $hK\cap h'K=\emptyset$ or $hK=h'K$.
So the only problem is to find out how many of the cosets $hK$, $h\in H$, are distinct.
Since
$$hK=h'K \Leftrightarrow h^{-1}h'\in K$$
(see ProofWiki)
we see that for each $k\in K$, the elements $h'=hk$ represent the same set. (We have $k=h^{-1}h'$.)
We also see that if $k=h^{-1}h'$ then $k$ must belong to $H$. 
Since the number of elements that represent the same coset is $|H\cap K|$, we have $|H|/|H\cap K|$ distinct cosets and $\frac{|H||K|}{|H\cap K|}$ elements in the union. 
