Homogeneous Cartesian Coordinates represent Unique Finite Point

Theorem
Let $\CC$ denote the Cartesian plane.

Let $P$ be an arbitrary point in $\CC$ which is not the point at infinity.

Let $Z \in \R_{\ne 0}$ be fixed.

Then $P$ can be expressed in uniquely in homogeneous Cartesian coordinates as:
 * $P = \tuple {X, Y, Z}$

Proof
Let $P$ be expressed in (conventional) Cartesian coordinates as $\tuple {x, y}$.

As $Z$ is fixed and non-zero, there exists a unique $X \in \R$ and a unique $Y \in \R$ such that:

Then by definition of homogeneous Cartesian coordinates, $\tuple {X, Y, Z}$ uniquely represents $P$.