I am following Peter Bickel's MS Volume one theorem 5.4.2. I understand the main proof but there is one particular computation I am stuck on.
We have $$ \left(\bar{\theta}_n-\theta(P)\right)\left(-E_P \frac{\partial \psi}{\partial \theta}\left(X_1, \theta(P)\right)+o_p(1)\right)=\frac{1}{n} \sum_{i=1}^n \psi\left(X_i, \theta(P)\right) . $$
By the central limit theorem, $$ \frac{1}{n} \sum_{i=1}^n \psi\left(X_i, \theta(P)\right)=O_p\left(n^{-1 / 2}\right) . $$
Dividing by the second factor of the LHS of the first equation, we finally obtain $$ \bar{\theta}_n-\theta(P)=\frac{1}{n} \sum_{i=1}^n \widetilde{\psi}\left(X_i, \theta(P)\right)+o_p\left(\frac{1}{n} \sum_{i=1}^n \psi\left(X_i, \theta(P)\right)\right) $$
where $$ \tilde{\psi}(x, P)=\psi \cdot(x, \theta(P)) /\left(-E_P \frac{\partial \psi}{\partial \theta}\left(X_1, \theta(P)\right)\right) . $$
I don't understand how the last equation about $\bar{\theta}_n-\theta(P)$ was derived. I think some computation was omitted. I main concern is that the when I move the second factor to the right, there will be a $o_p(1)$ in the denominator, how to deal with that?