C. (10 points) Let G = (V, E) be a connected, undirected graph where each edge e in E has an associated integer weight w(e). Let E' be a subset of E that is contained in some minimum-weight spanning tree (MST) of G. (As in the lecture slides, we identify an MST (V,T) of G with its edge set T, since the vertex set of any spanning tree of G is understood to be V.) Thus |E| = |V] - 1. Let G' denote the subgraph (V, E') of G, and let k denote the number of connected components of G. It is not difficult to prove that k = |VI - |El. Assume that k > 1, and that the graph (V1, Ey) is a connected component of G'. Let E* denote the set of all edges in E with exactly one endpoint in Vi, and let e* be a minimum-weight edge in E*. Prove that some MST of G contains E' + e*. Remark: You proof should not assume that the edge weights are distinct