TeX source:

\begin{split} |\psi''\rangle = \dfrac{\mathbf{A}_i|\psi'\rangle}{\sqrt{\langle\psi'|\mathbf{A}_i|\psi'\rangle}} = \dfrac{\mathbf{A}_i|\psi\rangle}{\sqrt{\langle\psi'|\mathbf{A}_i|\psi'\rangle\langle\psi|\mathbf{A}_i|\psi\rangle}} = \dfrac{\mathbf{A}_i|\psi\rangle}{\sqrt{\frac{\langle\psi|\mathbf{A}^\dagger_i}{\sqrt{p_i}}\mathbf{A}_i\frac{\mathbf{A}_i|\psi\rangle}{\sqrt{p_i}}p_i}} = |\psi'\rangle, \end{split}