Projection in Plane on X-Axis along Y-Axis

Theorem
Let $\pr_{X, Y}$ denote the projection on the $x$-axis along the $y$-axis as the mapping:
 * $\forall P \in \R^2: \map {\pr_{X, Y} } P =$ the intersection of the $x$-axis with the line through $P$ parallel to the $y$-axis.

Let $P = \tuple {\lambda_1, \lambda_2}$ be an arbitrary point in $\R^2$.

Then:
 * $\map {\pr_{X, Y} } {\lambda_1, \lambda_2} = \tuple {\lambda_1, 0}$