Sign of Quotient of Factors of Difference of Squares/Corollary

Corollary to Sign of Quotient of Factors of Difference of Squares
Let $a, b \in \R$ such that $a \ne b$.

Then
 * $-\operatorname{sgn} \left({\dfrac {b - a} {b + a} }\right) = \operatorname{sgn} \left({a^2 - b^2}\right) = -\operatorname{sgn} \left({\dfrac {b + a} {b - a} }\right)$

where $\operatorname{sgn}$ denotes the signum of a real number.