Referring to Theorem 30.1 of Sato's book "Levy processes and Infinitely divisible distributions". In the proof, page 200, when he shows that the subordinated levy process has independent increments, is he covertly using the freezing lemma? I found the same proof in a thesis and it has a couple more passages involving the freezing lemma, conditioning to the whole vector of (Z(t{1}),...,Z(t_{n})). I'm not really getting why you should condition to get the result
