I'm reading about strong Markov property. In the text there is the next proposition which I'm, trying to prove:
If $X$ is a càdlag process and $z$ is a discrete stopping time, then $(X_{z+t}-X_{z})_{t\geq 0}$ satisfies:
a) $(X_{z+t}-X_{z})$ has the same distribution as $X_{t}$ for all $t\geq 0.$
b) $(X_{z+t}-X_{z})$ is independent of $\mathcal{F}_{z}.$
I'm stuck, because I can't see how to get the relation between right-continuity and the discrete stopping time. I thought that was necessary that $X$ be a Levy process, but in this case we have a kind of strong Markov property and my feeling is this proposition can be used to prove it.
How Could be proved this proposition?
Any kind of help is thanked in advanced.