Is a subtree of a minimum spanning tree a minimum spanning tree of the subgraph spanned by the subtree?
Let’s prove this by contradiction. Say $T_a$ is not an MST of $G_a$, that means there is some cheaper MST $T_a^*$. However, if that were true, then by reconnecting $T_a^*$ to $T_b$ we would obtain a tree that spans $G_a$ that has a lower weight than the MST $T$, which is a contradiction.