This example is a very special example. In the general case, we would use normality, but here we are not allowed to as H is not stated as being normal. It turns out, however, that H must be normal (in fact, this is what you are proving. If you prove that for all , as you are doing here, then , and so is normal).
Clearly, . We thus only require to show that . To do this, we must prove two things:
1) Prove that and .
2) Prove that (alternatively, prove that there can only exist two right cosets).
Can you see why this gives us the result?
The proofs of these two results are not hard, and this site exists merely to help you understand (not to provide answers), and so I will omit the proofs for these two results. They are, however, not too hard (although the second one is a bit tricky).