Turn MST of G to MST of G with one new edge

Given $T$, an MST of $G(V,E)$ connected and undirected. Assume we add an edge $e'$ with weight $w(e')$. Suggest an algorithm which takes $T$ as input, and outs $T'$ MST of $G'(V,E\cup\{e'\})$.So i thought about adding $e'$ to $T$ and removing the heaviest edge in the cycle that was formed. Can anyone suggest how to go about proving that $T'$ is indeed the MST of $G'$?

Before we add $e'$ with weight $w(e')$, let us add $e'$ with weight so large (or, for example, larger than the weight of every other existing edge) that $e'$ will not be in any $MST$. In particular, $T$ remains an MST with this addition.

Now let us change the weight of $e'$ to $w(e')$. Then this question becomes the same question in the linked answer above.