Proof: Assume the contrary, i.e. that belongs to an MST . Then deleting will break into two subtrees with the two ends of in different subtrees. The remainder of reconnects the subtrees, hence there is an edge of with ends in different subtrees, i.e., it reconnects the subtrees into a tree with weight less than that of , because the weight of is less than the weight of .
Proof: Assume the contrary, i.e. that belongs to an MST . Then deleting will break into two subtrees with the two ends of in different subtrees. The remainder of reconnects the subtrees, hence .there is an edge of with ends in different subtrees, i.e. it reconnects the subtrees into a tree with weight less than that of , because the weight of is less than the weight of .