number of distinct ways of writing each element of the set $HK$ in the form $hk$ Let $H$ and $K$ be subgroups of the group $G$. The number of distinct 
ways of writing each element of the set $HK$ in the form $hk$, for some $h \in H$ and $k \in K$ is $|H \cap K|$. 
My thoughts:-
Let $|H \cap K|=n$.  Let $h_i=k_i$ where $i=1,2,3,....,n$.then if $\alpha \in HK$ then $\alpha = hk$ and also $\alpha = (hh_i)(h_i^{-1}k)$.
but how can I able to show these are the only representation toprove the problem.  

If $hk=h'k'$ then $kk'^{-1}=h^{-1}h'\in H\cap K$ and...?
