I would like to prove that stochastics process $\left\{X_t\right\}_{t\in T}$ holds Markov property if and only if for each $t,s_1,s_2\in T$ such as $s_1<t$ \begin{align} \mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_t\right)\right)= \mathrm{E}\left(X_{t-s_1}|\sigma\left(X_t\right)\right)\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)\quad a.s. \end{align}
I can prove that if stochastics process $\left\{X_t\right\}_{t\in T}$ holds Markov property then \begin{align} \mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_t\right)\right)= \mathrm{E}\left(X_{t-s_1}|\sigma\left(X_t\right)\right)\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)\quad a.s. \end{align}
The reason is that if the stochastic process holds Markov property then for each Borel mesurable function we have \begin{align} \mathrm{E}\left(f(X_{t+s_2})|\sigma\left(X_l; l\leq t\right)\right)=\mathrm{E}\left(f(X_{t+s_2})|\sigma\left(X_t\right)\right)\quad a.s. \end{align}
So if we take $f=id$, we have
\begin{align} \mathrm{E}\left(X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)=\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)\quad a.s. \end{align} Then we can write down that
\begin{equation} \label{pod1} \begin{aligned} \textstyle\mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)&=\textstyle X_{t-s_1}\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)=\\ \textstyle &=X_{t-s_1}\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)\quad a.s. \end{aligned} \end{equation} Finally we get
$\begin{aligned} \textstyle\mathrm{E}\left[\mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)|\sigma\left(X_t\right)\right]&=\textstyle\mathrm{E}\left[X_{t-s_1}\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)|\sigma\left(X_t\right)\right]=\\ &=\textstyle \mathrm{E}\left[X_{t-s_1}\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)|\sigma\left(X_t\right)\right]=\\ &=\textstyle \mathrm{E}\left(X_{t-s_1}|\sigma\left(X_t\right)\right)\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right). \end{aligned}$
On the other hand if we use tower property, we have \begin{align*} \textstyle\mathrm{E}\left[\mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_l; l\leq t\right)\right)|\sigma\left(X_t\right)\right]&=\mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_t\right)\right). \end{align*}
So if we put these two expression together we have \begin{align*} \mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_t\right)\right)=\mathrm{E}\left(X_{t-s_1}|\sigma\left(X_t\right)\right)\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right) \quad a.s. \end{align*}
But I cannot deal with the other implication. It means I do not know how to prove that if the \begin{align} \mathrm{E}\left(X_{t-s_1}X_{t+s_2}|\sigma\left(X_t\right)\right)= \mathrm{E}\left(X_{t-s_1}|\sigma\left(X_t\right)\right)\mathrm{E}\left(X_{t+s_2}|\sigma\left(X_t\right)\right)\quad a.s. \end{align} holds then stochastic process $\left\{X_t\right\}_{t\in T}$ is Markov. I wolud like to show that for each Borel measurable function holds equation \begin{align} \mathrm{E}\left(f(X_{t+s_2})|\sigma\left(X_l; l\leq t\right)\right)=\mathrm{E}\left(f(X_{t+s_2})|\sigma\left(X_t\right)\right)\quad a.s. \end{align} Or I would like to show that \begin{align} \label{MR vlastnosť} P\left(X_{t+s}\in B|\sigma(X_l; l\leq t )\right)=P\left(X_{t+s}\in B|\sigma(X_t)\right)\quad a.s. \end{align} holds for each Borel set $B$ and each $t,s\in T$.
Any help will be appreciated. Thank you very much.