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

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

Proof
Let $x \in \R^2$ be arbitrary.

By definition, the image of $x$ is the intersection of $M$ with the line through $x$ parallel to $N$.

Therefore $\map {\pr_{M, N} } x \in M$.

Hence:
 * $\Img {\pr_{M, N} } \subseteq M$.

Now consider $y \in M$.

By Playfair's axiom there exists exactly one straight line $L$ parallel to $N$ passing through $y$.

Hence $y$ is the image of every point on $L$.

As $y$ is arbitrary, it follows that each point on $M$ is the image of some straight line in $\R^2$ under $\pr_{M, N}$.

That is:
 * $M \subseteq \Img {\pr_{M, N} }$

The result follows by definition of set equality.