I am trying to prove that definition 17.1 + "certain time homogenity" imply definition 17.3iii. In this post the vice versa (17.3iii implies 17.1) was shown. So I want to show that for any $s,t\in[0,\infty)$ it holds
$$P_x[X_{t+s}\in A|\mathcal{F}_s]=\kappa_t(X_s,A)=P_{X_s}[X_t\in A]$$
by using $$P_x[X_{t+s}\in A|\mathcal{F}_s]=P_x[X_{t+s}\in A|X_s]$$
and a "certain time homogenity", which the author does not specify.
I tried following: Note that $P_x[X_0=x]=1$.
$$P_x[X_{t+s}\in A|\mathcal{F}_s]=P_x[X_{t+s}\in A|X_s]=P_x[X_t\in A|X_0]=P_{X_0}[X_t\in A]$$
But now, I do not know how to proceed. Also I do not know if, that's the right use of this "time homogenity", the author is writing about.
I would really appreciate some help on this one. Thanks in advance!