Independence of increments with stopping times in Levy processes

Question

Let $X$ be a Levy Process and $S<T<U<V$ be stopping times. Let $F^X$ be the natural filtration of $X$. How can one show that

$X_V - X_U$ and $X_T - X_S$

are independent and

$X_V - X_U$ and $F^X_U$

are as well? Or is it not true in general?

Thank you a lot