Poisson process

Suppose N1 (t) and N2(t) are Poisson processes defined on the same probability space (Ω,ℱ, ℙ) relative to the same filtration ℱ (t) , t ≥0. Assume that almost surely N1 (t) and N2(t) have no simultaneous jump. Show that, for each fixed t, the random variables N1 (t) and N2(t) are independent. (Hint: Adapt the proof of Corollary 11.5.3.) (In fact, the whole path of N1 is independent of the whole path of N2, although you are not being asked to prove this stronger statement.)