I have a vector $A$ ($l$) and three matrices $B$ ($l \times k$), $C$ ($i \times k$), and $D$ ($k \times j$). I wonder how to proof
$\sum_{l}A_{l}\sum_{k}B_{lk}C_{ik}D_{kj} = \sum_{k}(\sum_{l}A_lB_{lk})C_{ik}D_{kj}$
At first, I am not so positive it can be equal but I proofed by a simulation in programming and the result is almost equal (less than 1e8 different which probably comes from programming computation error). May someone suggest how to proof or just guide me what properties of matrix I should go look for to proof this?