Equation of Line in Complex Plane/Formulation 2

Theorem
Let $\C$ be the complex plane.

Let $L$ be the infinite straight line in $\C$ which is the locus of the equation:
 * $l x + m y = 1$

Then $L$ may be written as:
 * $\map \Re {a z} = 1$

where $a$ is the point in $\C$ defined as:
 * $a = l - i m$

Proof
Let $z = x + i y$.

Let $a = l - i m$.

Then: