I'm trying to solve this question which I found very complex and difficult due to the amount of details.

> 
> Let $H, K, N$ be subgroups of a group G such that $H\leq K$, $H\cap N
> =K\cap N$, and $HN=KN$. Show that $H=K$.
> 
> 
> 

I'm trying to prove it using this:

![1](https://i.imgur.com/QAzAfL2.png)

and this:

![2](https://i.imgur.com/M61vqaC.png)

Am I in the right way? I really need a hand here, there are so many data in this question and I'm a little confused.