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 divisor sum of $m$ and $n$ both be square.

Let $m$ and $n$ be coprime.

Then the divisor sum of $m n$ is square.

Proof
Let $\map {\sigma_1} m = k^2$.

Let $\map {\sigma_1} n = l^2$.

Thus: