I have seen two versions of the Central sets theorem. The first one, which is due to H. Furstenberg, is the following.
Central sets theorem, version 1: Let $C$ be a central set in $\mathbb{N}$, and for any $m>1$, let $\{v_{\alpha}\}$ be any IP-system in $\mathbb{Z}^{m}$. We can find an IP-subsystem $\{v_{\phi(\alpha)}\}$ and an IP-system $\{h_{\alpha}\}\subseteq\mathbb{N}$ such that the vector $\overline{h}_{\alpha}^{(m)}+v_{\phi(\alpha)}\in C^{m}$ for each $\alpha\in\mathcal{F}$. ($\overline{h}_{\alpha}^{(m)}:=(h_\alpha,h_\alpha,...,h_\alpha).$)
The second version, which is due to N. Hindman and D. Strauss, is the following.
Central sets theorem, version 2: Let $C$ be a central subset of $\mathbb{N}$. Let $m\in\mathbb{N}$ and for each $i\in\{1,2,...,m\}$, let $f_{i}$ be a sequence in $\mathbb{Z}$. Then there exist sequences $\langle a_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ and $\langle H_{n}\rangle_{n=1}^{\infty}$ in $\mathcal{P}_{f}(\mathbb{N})$ such that
- for all $n,\,\max H_n <\min H_{n+1}$ and
- for all $F\in\mathcal{P}_f(\mathbb{N})$ and all $i\in\{1,2,...,m\}$, $$\sum_{n\in F}\big(a_n+\sum_{t\in H_n}f_i(t)\big)\in C$$.
Problem: Prove that the Version $1$ of Central sets theorem implies Version $2$.
My attempt: I have deduced that Version $2$ implies Version $1$ as follows. Let $C$ be a central set in $\mathbb{N}$, and for any $m>1$, let $\{v_{\alpha}\}_{\alpha\in\mathcal{P}_{f}(\mathbb{N})}$ be any IP-system in $\mathbb{Z}^{m}$. For each $i\in\{1,2,...,m\}$, let us define the sequence $f_{i}$ by $f_{i}(t):=\pi_{i}(v_{t})$, where $\pi_{i}$ is the projection of $\mathbb{Z}^{m}$ onto the $i^{\text{{th}}}$ component. Then there exists sequences $\langle a_{n}\rangle_{n=1}^{\infty}$ in $\mathbb{N}$ and $\langle H_{n}\rangle_{n=1}^{\infty}$ in $\mathcal{P}_{f}(\mathbb{N})$ such that
for all $n$, $\max H_n < \min H_{n+1}$ and
for all $\alpha\in\mathcal{P}_f(\mathbb{N})$ and all $i\in\{1,2,...,m\},\,\sum_{n\in\alpha}\big(a_n +\sum_{t\in H_n}\pi_i(v_t)\big)\in C$.
Let us consider the homomorphism $\phi:\mathcal{P}_{f}(\mathbb{N})\rightarrow\mathcal{P}_{f}(\mathbb{N})$ such that $\phi(n)=H_{n}$. Also consider the IP-system $\{h_{\alpha}\}_{\alpha\in\mathcal{P}_{f}(\mathbb{N})}$ where $h_{n}=a_{n}$ for each $n\in\mathbb{N}$. Then
$\begin{equation*} \begin{split} & h_\alpha + \sum_{n\in\alpha}\sum_{t\in\phi (n)}\pi_i(v_t)\in C \text{ for each } i\in\{1,2,...,m\}\\ \implies & (h_\alpha,h_\alpha,...,h_\alpha) + \sum_{n\in\alpha}\sum_{t\in\phi (n)}(\pi_1(v_t),\pi_2(v_t),...,\pi_m(v_t))\in C^m\\ \implies & \overline{h}_\alpha^{(m)}+\sum_{n\in\alpha}\sum_{t\in\phi (n)}v_t\in C^m\\ \implies & \overline{h}_\alpha^{(m)}+\sum_{n\in\alpha}v_{\phi (n)}\in C^m\\ \implies & \overline{h}_\alpha^{(m)}+v_{\phi (\alpha)}\in C^m. \end{split} \end{equation*}$.
Thus we have deduced that Version $2$ implies Version $1$. Now I wanted to revert this implication in the opposite direction. But I am not getting how to deduce the condition "$\max H_n<\min H_{n+1}$" of Version $2$.
Definition 1: A subset $C\subseteq \mathbb{N}$ is a central subset if there exists a dynamical system $(X,T)$ (i.e. $X$ is compact and $T:X\rightarrow X$ is continuous map), a point $x\in X$ and a uniformly recurrent point $y$ proximal to $x$, and a neighborhood $U_y$ of $y$ such that $C=\{n:T^nx\in U_y\}$.
Definition 2: A homomorphism $\phi:\mathcal{P}_{f}(\mathbb{N})\rightarrow\mathcal{P}_{f}(\mathbb{N})$ is a map such that $\alpha\cap\beta=\emptyset\implies\phi(\alpha)\cap\phi(\beta)=\emptyset$ and $\phi(\alpha\cup\beta)=\phi(\alpha)\cup\phi(\beta)$ for $\alpha,\beta\in\mathcal{P}_{f}(\mathbb{N})$. ($\mathcal{P}_f(\mathbb{N})$ is the set of all finite subsets of $\mathbb{N}$)
Definition 3: A $\mathcal{P}_{f}(\mathbb{N})$-sequence of elements in an arbitrary space $X$ is a sequence $\{x_{\alpha}\}_{\alpha\in\mathcal{P}_{f}(\mathbb{N})}$. If $X$ is a semigroup we say that a $\mathcal{P}_{f}(\mathbb{N})$-sequence defines an IP-system if $x_{\{i_{1},i_{2},...,i_{k}\}}=x_{i_{1}}\cdot x_{i_{2}}\cdot...\cdot x_{i_{k}}$ where $i_{1}<i_{2}<...<i_{k}$. The set $\{x_{\alpha}:\alpha\in\mathcal{P}_{f}(\mathbb{N})\}$ is then called an IP-set. An IP-subsystem of an IP-system $\{x_{\alpha}\}_{\alpha\in\mathcal{P}_{f}(\mathbb{N})}$ is the $\mathcal{P}_{f}(\mathbb{N})$-sequence $\{x_{\phi(\alpha)}\}_{\alpha\in\mathcal{P}_{f}(\mathbb{N})}$ for a homomorphism $\phi:\mathcal{P}_{f}(\mathbb{N})\rightarrow\mathcal{P}_{f}(\mathbb{N})$.
Thanks in advance for any help or suggestion.