# A product of two sums of four squares

I am dealing with a problem and I hope you can help me. I have already proved this:

Let us suppose that integers $m$ and $n$ can be written as sum of squares of two integers. Prove that m*n can also be written as sum of squares of two integers.

Now I am trying to prove this: Let us suppose that integers $m$ and $n$ can be written as the sum of squares of four integers, i.e., $m=m_1^2+m_2^2+m_3^2+m_4^2$ and $n=n_1^2+n_2^2+n_3^2+n_4^2$. Prove that $mn$ can also be written as sum of squares of four integers.

Can you help me on this? Thank You for reading my post.