A confusion in the proof of $|HK|=\displaystyle{\frac{|H||K|}{|H\cap K|}}$

A confusion in the proof of $|HK|=\displaystyle{\frac{|H||K|}{|H\cap K|}}$

I'm trying to understand the proof of
$|HK|=\displaystyle{\frac{|H||K|}{|H\cap K|}}$, where $H$ and $K$ are
separate groups.
Let $[k_1],[k_2],\dots,[k_n]$ be the equivalence classes of
$\frac{K}{H\cap K}$. The proof says $Hk_1,Hk_2,\dots, Hk_n$ are disjoint.
In other words, $Hk_i\cap Hk_j=\emptyset$ for $i\neq j$. I don't
understand why this is the case.
Let $h_1k_1=h_2k_2$ for $h_1,h_2\in H$. Note that $k_1\neq k_2$. This
implies $h_2^{-1}h_1=k_2k_1^{-1}$. We know that $h_2^{-1}h_1\in H$ and
$k_2k_1^{-1}\in K$, where one or none of $k_1^{-1},k_2\in H\cap K$. On a
little investigation, we find out that both $k_1^{-1},k_2\in K\setminus
(H\cap K)$.
Can't two elements, which belong to $K\setminus (H\cap K)$, multiply to
give an element in $H\cap K$?
Thanks in advance.