The aim of this post is to prove the equivalence theorem. In the post Equivalence Theorem of the S-Matrix, they treat the subject but they do not prove the relation of the residues.
The LSZ reduction formula is given by $$ \left.\left\langle p_{1}, \ldots, p_{n} \text { out }\right| q_{1}, \ldots, q_{m} \text { in }\right\rangle=\prod_{i=1}^{m}\left\{-\frac{i\left(p_{i}^{2}-m^{2}\right)}{(2 \pi)^{\frac{3}{2}} Z^{\frac{1}{2}}}\right\} \prod_{j=1}^{n}\left\{-\frac{i\left(q_{j}^{2}-m^{2}\right)}{(2 \pi)^{\frac{3}{2}} Z^{\frac{1}{2}}}\right\} \Gamma\left(p_{1}, \ldots, p_{n} ;-q_{1}, \ldots,-q_{m}\right), $$
where $\Gamma\left(p_{1}, \ldots, p_{n}\right)$ is the Fourier transform of the correlation functions $$ \Gamma\left(p_{1}, \ldots, p_{n}\right)=\int \prod_{i=1}^{n}\left\{\mathrm{~d}^{4} x_{i} e^{i p_{i} \cdot x_{i}}\right\}\left\langle 0\left|\mathrm{~T} \varphi\left(x_{1}\right) \ldots \varphi\left(x_{n}\right)\right| 0\right\rangle $$
and $Z$ is Wave function renormalization
Suppose we make a field redefinition of the form $ \phi=\psi +F(\psi)$. The equivalence theorem says that the S-matrix obtained using the field $\phi$ is the same as the one obtained using $\psi$.
Suppose that where $G\left(p_{1}, \ldots, p_{n}\right)$ is the Fourier transform of the correlation functions $$ G\left(p_{1}, \ldots, p_{n}\right)=\int \prod_{i=1}^{n}\left\{\mathrm{~d}^{4} x_{i} e^{i p_{i} \cdot x_{i}}\right\}\left\langle 0\left|\mathrm{~T} \psi\left(x_{1}\right) \ldots \psi\left(x_{n}\right)\right| 0\right\rangle $$ and let $Q_i$ be one of residues of $G\left(p_{1}, \ldots, p_{n}\right)$ and let $Z'$ be the Wave function renormalization associated to the filed $\psi$.
For two point function for example we have $$ \begin{aligned} \langle 0|T(\varphi(x) \varphi(y))| 0\rangle &=\langle 0|T(\psi(x) \psi(y))| 0\rangle+\langle 0|T(F(\psi(x)) \psi(y))| 0\rangle \\ &+\langle 0|T(\psi(x) F(\psi(y)))| 0\rangle+\langle 0|T(F(\psi(x)) F(\psi(y)))| 0\rangle \end{aligned} $$ How do we prove that $$\frac{Q_i}{Z'}=\frac{R_i}{Z},$$
where $R_i$ is one of residues of $\Gamma\left(p_{1}, \ldots, p_{n}\right)$?
