Write $xy$ as the sum of the squares of two rational integers Show that if $x$ and $y$ may both be written as the sum of the squares of two rational integers, then their product $xy$ may also be written as the sum of the squares of two rational integers.

If $x = a^2 + b^2$ and $y = c^2 + d^2$ then:
$$xy = (a^2+b^2)(c^2+d^2) = a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 = (ac+bd)^2 + (ad-bc)^2$$
