I am trying to prove equation 97 in Chapter I of Zavialov's "Renormalized Quantum Field Theory" book, the statement claims that the Green functions of Wightman fields $\check{\phi}$, defined by $$\langle0 | T\left\{\check{\phi}(x_1)\cdots \check{\phi}(x_n)\right\} | 0\rangle$$ can be calculated as $$\langle0 | T\left\{\phi(x_1)\cdots \phi(x_n)S(\phi)\right\} | 0\rangle.$$
The definition I have for the Wightman field $\check{\phi}$ is $$\check{\phi}(x) = S^\dagger \otimes T\{\phi(x)S(\phi)\}$$ where $S$ is the S-matrix. From where we can prove $$\check{\phi}(x) = \phi(x) -i \int D^{ret}(x-y)\left(S^\dagger \otimes \frac{\delta S}{\delta \phi(y)}\right)dy.$$
The definition of the $\otimes$ product is $$A(\phi)\otimes B(\phi) = \left.\exp\left(-i\int D^-(x-y)\frac{\delta}{\delta \phi_1(x)}\frac{\delta}{\delta \phi(y)}\mathrm{d}x\mathrm{d}y\right)A(\phi_1)B(\phi_2)\right|_{\phi_1=\phi_2=\phi}$$
And the definition of the $T$-product is the same but with the $D^c(x-y)$ function.
Can anyone point me in the correct direction for the proof?
I tried to do the simpler case, with only one field, then the relation is $$\langle0 | \check{\phi}(x) | 0\rangle=\langle0 | T\left\{\phi(x)S(\phi)\right\} | 0\rangle.$$ The time-order can be calculated as $$T\left\{\phi(x)S(\phi)\right\} = :\phi(x) S(\phi): -i\int D^c(x-y)\frac{\delta S}{\delta \phi(y)}\mathrm{d}y$$ And the vev is given by the second term $$\langle0 | T\left\{\phi(x)S(\phi)\right\} | 0\rangle = -i\int D^c(x-y)S_1(y)\mathrm{d}y.$$ with $S_1(y) = \langle0 | \frac{\delta S}{\delta \phi(y)} | 0\rangle$. But I couldn't calculate $\langle0 | \check{\phi}(x) | 0\rangle$.
