Please provide steps with an explanation for the 2 proofs. Do not copy-paste old answers or answers from chatGPT. I will give an up-vote for answers following directions. Thanks!

Problem Two: Minimum Spanning Trees Let e=(u,v) be a minimum-weight edge in a graph G. Prove that (u,v) belongs to every minimum spanning tree of G. It's OK to assume that the edge costs of G are distinct. (Four points) Graduate Students Suppose we are given an undirected graph G=(V,E), with nonnegative edge weights ce. If the edge weights are not distinct, then there can be many distinct MSTs. Suppose we are then given a spanning tree TE such that each edge eT belongs to some minimum spanning tree of G. Is it then the case that T itself is a minimum spanning tree? Prove or show a counterexample