I have the following from a book:
Assume that $ P_x(\tau_C \circ \theta_{(k-1)N} > N|F_{(k-1)N}) = P_{X_{(k-1)N}}(\tau_C > N). $ Integrating over $\{ \tau_C > (k-1)N\}$ using the definition of conditional probability we have $ P_x(\tau_C > kN) = E_x\left(\mathbf{1}\{\tau_C \circ \theta_{(k-1)N} > N\} \cdot\mathbf{1}\{\tau_C > (k-1)N\}\right) $
I'm a bit unsure how from that equality he gets the second equality we see. I can see that the LHS of the first equality multiplied by $\mathbf{1}\{\tau_C > (k-1)N\}$ and then taking the expectation wrt. x yields the RHS of the second equality, but how does the RHS of the first equality being "integrated over" as claimed produce the LHS of the second equality?
Here, $\tau_C$ is the first hitting time of some set $C$.
Thanks.