I tried to prove Corollary 3.4 by virtue of Theorem 3.3.
In order to get the conclusion of Cor 3.4, it is natural to verify the condition (ii) is satisfied.
Since $\psi$ is a faithful normal state on $N$, we have a faithful representation $(H_{\psi},\rho_{\psi})$ of $N$ on the separable Hilbert space $H_{\psi}$. How to find a faithful normal state $\sigma$ on $B(H_{\psi})$ with $\psi=\sigma\circ\rho_{\psi}$?
You take $\rho$ to be the identity representation, and $\sigma=\omega=\psi$. Normal states always extend as normal to all of $B(H)$. This is because normal states are always of the form $\psi(x)=\operatorname{Tr}(hx)$ for some positive $h$. See for instance Theorem 7.1.12 in Kadison-Ringrose.