Let $\sigma$ and $\tau$ be markov moments regarding filtering $\mathbb{F} = (F_t, t \in T), T \subset \mathbb{R}$. I need to prove that $\forall A \in F_{\tau}$, where $F_{\tau} = \left\{ A \in F: A \cap \left\{\tau \leq t \right\} \in F_t \right\}$, $F$ is a $\sigma$-algebra on probability space $(\Omega, F, P)$: $$ A \cap \left\{\tau \leq \sigma \right\} \in F_{\tau} \cap F_{\sigma} $$ My approach was to prove that: $$ (A \cap \left\{\tau \leq \sigma \right\}) \cap \left\{\tau \leq t \right\} \in F_{t}\\ (A \cap \left\{\tau \leq \sigma \right\}) \cap \left\{\sigma \leq t \right\} \in F_{t} $$
for the second one I could write: $ (A \cap \left\{\tau \leq \sigma \right\}) \cap \left\{\sigma \leq t \right\} \subset A \cap \left\{\tau \leq t \right\} \in F_{t} $, since $A \in F_{\tau}$. But I have problems with proving $(A \cap \left\{\tau \leq \sigma \right\}) \cap \left\{\tau \leq t \right\} \in F_{t}$. Thanks for any help!