So there is a preposition that says;

If AB is invertible, then A is invertible and B is invertible too.

I uploaded the proof that I made, however I think it's not enough.

The problem with my proof, in my opinion, is that I showed that B and A is invertible from one side (either from left or right), so I think that in order to show that they are invertible I have to show that they have inverses from both side and they are the same, am I mistaken?

I would appreciate if you can verify, and tell me if my proof is correct, however if it's not don't tell me what's the right proof, I want to work on it. Thank you!