Could somebody help to solve this problem:
Prove or disprove: If $G$ is a k-edge-connected graph with nonempty disjoint subsets $S_{1}$ and $S_{2}$ of $V(G)$, then there exist $k$ edge-disjoint paths $P_{1},P_{2},...,P_{k}$ such that $P_{i}$ is a $u_{i}-v_{i}$ path, where $u_{i} \in S_{1}$ and $v_{i} \in S_{2}$, for $i=1,2...,k$, and $|S_{1} \cap V(P_{i})|=|S_{2} \cap V(P_{i})| = 1$.