I'll try not to take the set theory approach...

Consider the following.

If and , then as you've observed, .

If I understand correctly, ^n=e_H\}" alt="\left<P\right>_H=\{p\in P^n=e_H\}" />. Since . So

This seems a bit hard to find a really formal proof since its a pretty straightforward concept...I'm not sure if this is what you're looking for.. xD