Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties: any vertical or horizontal line in the plane cuts $K$ in at most $2$ points, any point of $H\setminus K$ is contained by a segment with endpoints from $K$.
What I've thought so far: explicitly giving an algorithm to construct the set starting from the outside to the inside, but I can't seem to rigorously prove it.