Unclear proof in the Gauss-Markov theorem

Question

I don't understand a step in the proof of the Gauss-Markov theorem:

Mainly, why can we simply replace y with e, given that y is defined as:

Thanks in advance!

Answer 1

Just substitute the $y = W\beta + e$ into the $y-W[(W'W)^{-1}W'+C]y$ (expression ahead of the red line you draw) yield:

$e - W(W'W)^{-1}W'e - WCW\beta - WCe = e - W(W'W)^{-1}W'e - WCe$

thanks to $CW=0.$

I guess the last sign in the bracket should be $-$.