This comes from page 2 of General Theory of Markov Processes by Michael Sharpe, with some changes in notation.
Suppose $P(A_1\cap A_2\,|\, {\cal F}_{=t} )= P(A_1 \,|\, {\cal F}_{=t} )P( A_2\,|\, {\cal F}_{=t} )$ for all $A_1\in {\cal F}_{\leq t}$ and $A_2\in {\cal F}_{\geq t}$. Using well known properties of conditional expectations, \begin{eqnarray*} P(A_1\cap A_2) &=&P(P(A_1\cap A_2\ |\ {\cal F}_{=t}))\cr &=&P\left( P(A_1\ |\ {\cal F}_{=t})\ P(A_2 \ | \ {\cal F}_{=t}) \right)\cr &=&P(P(A_2\ |\ {\cal F}_{=t}) ; A_1). \end{eqnarray*}
My question is by what property of conditional expectations, we can have $ P\left( P(A_1\ |\ {\cal F}_{=t})\ P(A_2 \ | \ {\cal F}_{=t}) \right) = P(P(A_2\ |\ {\cal F}_{=t}) ; A_1)?$
What is the general form for the property of conditional expectation?
Any reference?
Thanks in advance!