I believe that instead of the condition to prove being E=-\left|V\right|, it should be E=-\left|E\right| instead.
Hello @EmGiXIII, welcome to the forum!
Yes, I think you’re correct, which would make E an unfortunate choice of notation for the number of edges. I’ll think of how to fix it and will update it soon
Thank you for spotting this, you have great attention to detail!