Having trouble understanding a proof of Mahaney’s theorem


I was reading this blog post: https://blog.computationalcomplexity.org/2011/09/mahaneys-theorem.html?m=1, but I am not sure why a’ cannot be in between w_i and w_j in the case 1, and also why a’ cannot be in between w_0 and w_1 in the case 2. Can anyone give me some explanation?