I am trying to prove decomposition property of conditional dependency but I'm kind of stuff in the middle ! This is ow I proceeded!
we want to prove :
$x\bot (y,w)| z $ implies $x\bot y | z$
That means x is independent ($\bot$) of y and w , given(|) z!
$x\bot (y,w)| z = x | (y,w),z= \frac{p(x,y,w,z)}{p(y,w,z)}=\frac{p(x,y,z|w) p(w)}{p(y,z|w)p(w)}$
now here I apply he marginalization on w(This is the part I dont know whether its correct or not!!)
$ \Rightarrow \frac{ \sum_w p(x,y,z|w)p(w)}{\sum_wp(y,z|w)p(w)}=\frac{p(x,y,z)}{p(y,z)}=p(x|y,z)$
so it has proven! But I am not sure about the marginalization part!Thanks in advance