Proof that "local" minimum spanning tree is "global" minimum spanning tree
If we remove $e = (x,y)$ from $T_2$, then it breaks into two connected components, say $T_{2,x},T_{2,y}$. There is a unique path in $T_1$ from $x$ to $y$, which together with $e$ forms a fundamental cycle of $T_1$. The path starts at $T_{2,x}$ and ends at $T_{2,y}$, hence at some point it crosses between the two. Adding the corresponding edge to $T_2$ would result in a fundamental cycle containing $e$.