Bisectors of Adjacent Angles between Straight Lines Meeting at Point are Perpendicular/Proof 2

Proof
Let $\LL_1$ and $\LL_2$ be straight lines embedded in a cartesian plane $\CC$, expressed in normal form as:

From Bisectors of Angles between Two Straight Lines, the angle bisectors of the angles formed at the point of intersection of $\LL_1$ and $\LL_2$ are given by:

These are in the form $l x + m y + n = 0$.

We use Condition for Straight Lines in Plane to be Perpendicular to prove that $l_1 l_2 + m_1 m_2 = 0$, where:

Hence

Hence the result.