Let $G$ be an undirected and connected graph. Let $T$ be a spanning tree of $G$ with edges weights: $w_1 \le, w_2 \le ... \le w_{n-1}$ which are responing to the edges. $e_1,e_2,...,e_{n-1}$.
Now I assume that there exists a spanning tree of $G$, let's call it $T'$, that has a path from node $u$ to node $v$ with all edges in a path from weight $<w_i$ for some $1 \le i \le n-1$. And I assume that the path from $u$ to $v$ in $T$ has at least one edge with weight $\ge w_i$. I other words, the path from $u$ to $v$ in $T$ has an edge which is heavier than all the edges in the corresponding path in $T'$.
I need to prove formally that $T$ is not a MST.
What I've tried is to somehow find a tree which is of less wight than $T$ using Kruskal's algorithm, but I couldn't manage to do it.