Product of Coprime Numbers whose Divisor Sum is Square has Square Divisor Sum

Theorem
Let $m, n \in \Z_{>0}$ be a positive integer.

Let the $\sigma$ value of $m$ and $n$ both be square.

Let $m$ and $n$ be coprime.

Then the $\sigma$ value of $m n$ is square.

Proof
Let $\sigma \left({m}\right) = k^2$.

Let $\sigma \left({n}\right) = l^2$.

Thus: