I don't follow your computation of the LHS. It seems to me that you can choose sub_2 in C(n, r-k) ways, and then you can choose sub_1 in C(n-r+l, k) ways. This gives
C(n, k) C(n-k, r-k) = C(n, r-k) C(n-r+k, k),
which is true, but it's not what you set out to prove.