Let be nonzero integers

Lemma 1: If , then

Lemma 2: if and , then

Theorem: If are positive integers such that any two are coprime, then

Proof:

By (1), and

By (2),

By (2),

By (1),

WLOG, this means

Therefore, by (2), so

Q.E.D.

Can someone verify these two lemmas? I am not completely well versed in number theory just yet. I believe these are very easily correct, but I do not know a formal proof.