I have some difficulty proving the correctness of my solution to the following exercise. Let $G = (V,E)$ undirected connected graph, $w \colon E \to \mathbb{R}$ weight function. Let $T$ a MST (minimum spanning tree) of $G$. Now, we add a new edge $e'$ to $E$ with weight $w(e')$. Find an algorithm that updates $T$ so it will be a MST of the new graph $G' = (V,E \cup \{e'\})$. The time complexity of the algorithm should be $O(V+E)$.
Intuitively, the algorithm is quite simple: add $e'$ to $T$, and then we got one cycle closed. Remove the edge with maximal weight from this cycle, and obtain a MST of $G'$.
However, I had some difficulties prove the correctness of this algorithm formally, and would appriciate some help.
BTW. I have read this post: If I have an MST, and I add any edge to create a cycle, will removing the heaviest edge from that cycle result in an MST? but unfortunately haven't fully got the proof there.
Update: I tried Steven's solution and it helped me a lot, thanks again. I also found this solution online (I add a picture), but didn't understand it well. I also don't understand why in this solution, $T'$ is the MST on $G'$ instead of a MST on $G'$. Can someone help me understand what is going on the proof there, please?