In between eq. $(2.F.2)$ and $(2.F.3)$ we read
...Walras's law tells us that $w' = p'\cdot x(p',w')$
Assume now that homogeneity of degree zero does not hold. Then we have, $a>0$
$$x(ap,aw) \neq x(p,w)$$
see definition $2.E.1$
Then we can set $p'=ap,w'=aw$ to examine the case $x(p',w') \neq x(p,w)$.
But then Walras' law would imply
$$w' = p'\cdot ...