Kernel of Projection in Plane between Lines passing through Origin

Theorem
Let $M$ and $N$ be distinct lines in the plane both of which pass through the origin $O$.

Let $\pr_{M, N}$ be the projection on $M$ along $N$:
 * $\forall x \in \R^2: \map {\pr_{M, N} } x =$ the intersection of $M$ with the line through $x$ parallel to $N$.

Then $N$ is the kernel of $\pr_{M, N}$.