A system of disjoint representatives of line segments with given $k$ directions

We prove that for all positive integers $n$ and $k$, there exists an integer $N = N(n,k)$ satisfying the following. If $U$ is a set of $k$ direction vectors in the plane and $mathcal{J}_U$ is the set of all line segments in direction $u$ for some $uin U$, then for every $N$ families $mathcal{F}_1, ldots, mathcal{F}_N$, each consisting of $n$ mutually disjoint segments in $mathcal{J}_U$, there is a set ${A_1, ldots, A_n}$ of $n$ disjoint segments in $bigcup_{1leq ileq N}mathcal{F}_i$ and distinct integers $p_1, ldots, p_nin {1, ldots, N}$ satisfying that $A_jin mathcal{F}_{p_j}$ for all $jin {1, ldots, n}$. We generalize this property for underlying lines on fixed $k$ directions to $k$ families of simple curves with certain conditions.