Sum of 2 Squares in 2 Distinct Ways

Theorem
Let $m, n \in \Z_{>0}$ be distinct positive integers that can be expressed as the sum of two distinct square numbers.

Then $m n$ can be expressed as the sum of two square numbers in at least two distinct ways.

Proof
Let:
 * $m = a^2 + b^2$
 * $n = c^2 + d^2$

Then:

It remains to be shown that $a \ne b$ and $c \ne d$, then $\paren {a c + b d}^2 + \paren {a d - b c}^2 \ne \paren {a c - b d}^2 + \paren {a d + b c}^2$.