Multiples of Homogeneous Cartesian Coordinates represent Same Point

Theorem
Let $\CC$ denote the Cartesian plane.

Let $P$ be an arbitrary point in $\CC$.

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

Then $P$ can also be expressed as:
 * $P = \tuple {\rho X, \rho Y, \rho Z}$

where $\rho \in \R$ is an arbitrary real number such that $\rho \ne 0$.

Proof
By definition of homogeneous Cartesian coordinates, $P$ can be expressed in conventional Cartesian coordinates as:
 * $P = \tuple {x, y}$

where:

for arbitrary $Z$.

We have that:

The result follows.