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
 * $-\map \sgn {\dfrac {b - a} {b + a} } = \map \sgn {a^2 - b^2} = -\map \sgn {\dfrac {b + a} {b - a} }$

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