Proof check

[Boi]

I'll be using $o(g)$ to denote the order of an element $g$.
For the proof I'll have to prove 2 lemmas first (they're very obvious results, but they are not stated in the book, so I've felt the need to prove them).
Lemma 1: If $G$ is a group, $g \in G$, $o(g) = n$, $p, q \in \mathbb{Z}$ and $p \equiv q \; (\mathrm{mod} \; n)$, then $g^{p}=g^{q}$.

 

[khansaheb]

View attachment 39388
I'll be using $o(g)$ to denote the order of an element $g$.
For the proof I'll have to prove 2 lemmas first (they're very obvious results, but they are not stated in the book, so I've felt the need to prove them).
Lemma 1: If $G$ is a group, $g \in G$, $o(g) = n$, $p, q \in \mathbb{Z}$ and $p \equiv q \; (\mathrm{mod} \; n)$, then $g^{p}=g^{q}$.

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