I am having a problem with a dirichlet distribution

For (Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5)

let Xk (k is a lower case) = [∑(from i=1 to k) Yi] / [∑(from i=1 to k+1) Yi] where k = 1,2,3

How can I prove X = (X1, X2, X3) is independent?

What I did was...

(Y1, Y2, Y3, Y4) ~ D4(1,2,3,4;5) = (Z1, Z2, Z3, Z4) / (Z1+Z2+Z3+Z4+Z5) where Z ~ N(0,1), Z IID G(1/2)

Now, we have

X1 = Z1 / (Z1+Z2)

X2 = (Z1+Z2) / (Z1+Z2+Z3)

X3 = (Z1+Z2+Z3) / (Z1+Z2+Z3+Z4)

I think if i can somehow show X1 and X2 are independent and X2 and X3 are independent then X1 and X3 are independent as well

BUt i don't know how