Regularity of weak solutions with $C^2$ boundary - Evans

Question

I'm reading the proof of Theorem 4 from section 6.3 of Evans - Partial Differential Equations. I'm looking for an explanation about the following claim

Previously, I already proved that $u_k\in H^1(V)$ by Theorem 3 from section 5.8.2 but the last case claim that equation 55 implies that $u_{{x_n}{x_n}}$ is in $H^1(V)$ and moreover $u\in H^2(V)$. Any suggestion would be appreciated.