Let $T_{1}, T_{2}, ...$ be a sequence of independent, exponentially distributed random variables with parameter $\lambda > 0.$ i.e. $\Pr(T_{i} \geq t) = e^{-\lambda t} $
Let $S_{0}=0 $ and $S_{n}=\sum_{i=1}^{n}T_{i}$ and $$ N_{t}=max \{ n \geq 0:S_{n} \leq t \} ; 0\leq t \leq \infty $$ , which is a continuous time, integer-valued, right continuous with left limits process. Define $\mathcal{F}_{t}^{N}= \sigma(\{N_{s}: s \leq t \})$
I want to show that for all $0 \leq s < t $, $$ \Pr(S_{N_{s}+1} > t | \mathcal{F}_{s}^{N})=e^{-\lambda(t-s)}, a.s. $$
So I chose $E \in \mathcal{F}_{s}^{N} $ and $n \geq 0 $ and tried to prove the equality $$\int_{E \bigcap \{ N_{s}=n \}} E[1_{A}|\mathcal{F}_{s}^{N}]=Pr(A)Pr({E \bigcap \{ N_{s}=n \}})$$ where $A=\{ S_{N_{s}+1} > t \}= \{N_{t}-N_{s}<1 \} = \{N_t-N_s=0 \}. $
But I am stuck at the use of the independency (and considering about measurable sets?) and computing $Pr(A)$. I'd be thankful for any help.