Product of Integers of form 4n + 1

Theorem
Let $m, n \in \Z$ such that both $m$ and $n$ are of the form $4 k + 1$ where $k \in \Z$.

Then $m n$ is also of the form $4 k + 1$.

Proof
Let $m = 4 k_1 + 1, n = 4 k_2 + 1$.

Then: