> I proved the fourth theorem. It seems one hypothesis is not needed.

That's impressive, it didn't cross my mind that one need not assume the 
second operation to be closed.

> This proof is way too long so probably I used a very inefficient approach.

I'm also surprised at the length, given that the proof I know (from 
wikipedia...) is quite short. In hindsight, I probably should have split 
the task into smaller subtasks anyway, maybe I will do so in a revised 
version of the problem set.

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