Matrix Tree Theorem Proof Question

Question

I was going through the following proof for the Matrix Tree Theorem (see a screenshot of part of the proof below).

$\textbf{Question:}$ I am confused about the last red rectangle in the proof. To me, property $4$ states the $v^{th}$ row and column should be deleted from $L(G-\{e\})_0$ to yield $L(G-\{e\})_e$. What does vertex $u$ have to do with anything at all? I feel like I am missing something..

Answer 1

They are deleting rows and columns corresponding to vertices $u$ and $v$ in the matrix $L(G-\{e\})$, which is basically the same then deleting $v$ in $L(G-\{e\})_0$ (up to 1 offset).