Sign of Quotient of Factors of Difference of Squares

Theorem
Let $a, b \in \R$ such that $a \ne b$.

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

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