solve the problem

Q4 25 Points Given a graph, suppose that we need to construct a minimum- weight spanning "network" (i.e., subgraph), that is somewhat resistant to broken links (edges). A tree is not a good solution, because damage to a single edge would disconnect the network. So we want something with better connectivity than a MST. We decide that between every pair of vertices in the network there should be at least two disjoint paths. (That way, if one path contains a damaged edge, we can still use the other path). This implies that every vertex in the network must have at least two incident edges. Let G represent a graph such that it is possible to create at least one spanning network according to the connectivity requirements. Prove or disprove that for every vertex v of every valid graph G the two lightest edges incident to v must be part of the minimum-weight spanning network of G