I came across this problem in Introduction to algorithms Third Edition exercise.
So, how I proceeded was trying to contradict the situation when A minimum spanning tree has the largest edge greater than the largest edge of a bottleneck tree by cut and paste argument.
But it is not necessary that if I remove the largest Edge there exists an edge which connects this disconnected pieces into one!
Then, I tried proving it using Kruksal's Algorithm as basis as the maximum edge limits all other edges to be lesser than or equal to this maximum edge, but couldn't succeed.
Any help?