Fixed Points of Projection in Plane

Theorem
Let $M$ and $N$ be distinct lines in the plane.


 * Projection-in-plane.png

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

Then $M$ is the set of fixed points of $\pr_{M, N}$ in the sense that:


 * $x \in M$ $\map {\pr_{M, N} } x = x$