$N(t)$ - inhomogenous poisson process with intensity $\lambda(t)$
Number of events $N(t)-N(s)$ in the time interval $(s, t]$ is poisson distributed with parameters $\int_{s}^{t}\lambda(u)\,du$
Number of events in disjoint time intervals are independent
$F_t$ is generated by $N(s)$, s$\leqslant$t
$M(t)=N(t)-\int_{0}^{t}\lambda(u)\,du$
PROVE THAT: $$E(M(t)\mid F_s)=M(s)\tag{1}$$
I have tried to put $M(t)-M(s)$ in the expected value, but I find it difficult to show that the above is true, which gives me (2):
$$E(M(t)-M(s)\mid F_s)=E(N(t)-N(s))-\int_{0}^{t}\lambda(u)\,du+\int_{s}^{t}\lambda(u)\,du\tag{2}$$
Is not even sure if the above is the correct way to think, but it was an attempt to prove statement (1).
Someone who can help me?