I am currently learning direct proofs. I couldn't solve the following exercise.
Define an integer $m$ to be bisquare iff, $\exists a \in Z, \exists b \in Z, m = a^2 + b^2$.
Let $r$ and $s$ be fixed integers.
Prove: If $r$ and $s$ are bisquare, then $rs$ is bisquare.
Proof. Assume $r$ and $s$ are bisquare. I must prove that $rs$ is bisquare. I have assumed that $\exists a \in Z, \exists b \in Z, r = a^2 + b^2$ and $\exists x \in Z, \exists y \in Z, s = x^2 + y^2$. I must prove that $\exists m \in Z, \exists n \in Z, rs = m^2 + n^2$.
From my assumption, I can show that
$rs = (a^2+b^2)(x^2+y^2)$.
This is where I got stuck. I've tried a couple algebraic tricks but I didn't get anywhere.