qerwn
Can you please explain the essence of the proof? I understood till the underlined statements. After that, I am not able to understand the proof. What does the author do here?
update on the question. Now I understood that I need to prove the following proposition.
The underlined text comprises an explanation of the fact that spanning trees of $G \circ e$ are in one-to-one correspondence with spanning trees of $G$ containing $e$ (with each of the former ones being $T \circ e$ for some unique $T$ from the latter ones - unique due to $T \neq T' \Rightarrow T \circ e \neq T' \circ e$).
If what you need is a formal proof of that, your way is to a more formal definition of $G \circ e$.