2011/002) product of sum of squares is sum of squares

Prove that product of two numbers, each of which can be expressed as sum of two squares, can itself be expressed as sum of two squares.

Let 1st number be a^2+b^2 and second be c^2 + d^2
one way

(a^2+b^2)(c^2+d^2)
= (a^2c^2 + b^2 d^2 + a^2d^2 + b^2 c^2)
= (a^2c^2 + b^2 d^2 + 2 abcd + a^2d^2 + b^2 c^2- 2abcd)
= (ac+bd)^2 + ( ad - bc)^2

also
(a^2+b^2)(c^2+d^2)
= (a^2c^2 + b^2 d^2 + a^2d^2 + b^2 c^2)
= (a^2c^2 + b^2 d^2 - 2 abcd + a^2d^2 + b^2 c^2+ 2abcd)
= (ac-bd)^2 + ( ad + bc)^2

so we can do in two different ways

further we can prove using complex numbers

(a^2+b^2)(c^2+d^2) = | a + ib|^2 |c + id|^2
= | (a + ib)( c+ id)| ^ 2
= | (ac - bd) + (ad +bc) i | ^2
= (ac-bd)^2 + (ad + bc)^2

and taking (a^2+b^2)(c^2+d^2) = | a + ib|^2 |c - id|^2

we get (ac+bd)^2 + (ad - bc)^2