Let $n\geq 2$ be a positive integer. We call a vertex every point in the coordinate plane, whose $x$ and $y$ coordinates both are from the set $\{1,2,3,...,n\}$. We call a segment between two vertices an edge, if its length if $1$. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge $f$ is vital for an edge $e$, if the path of red edges connecting the two endpoints of $e$ contain $f$. Prove that there is a red edge, such that it is vital for at least $n$ edges.