How are these different?

$\displaystyle AB=[c_{ij}]$

where $\displaystyle c_{ij}=\sum_{k=1}a_{ik}b_{kj}$

and

$\displaystyle AB=[AB]_{ij}=\sum_{k=1}[A]_{ik}[b]_{kj}$

For instance,

My book proves the distributive property of multiplication as follows (im going to leave stuff out)

$\displaystyle A(B+C)=a_{i1}(b_{1j}+c_{1j})+...+a_{in}(b_{in}+c_{ nj})$

Then you expand AB+AC and do some regrouping to find A(B+C)=AB+AC

But, in another text I found the same proof done with the afore mentioned notation

$\displaystyle [A(B+C)]_{ij}=[A]_{*i}[(B+C)]_{*j}=\sum_{k}[A]_{ik}[B+C]_{kj}...$

I don't understand how to read the latter proof. What's being done in the proof makes sense to me, but I'm just not sure how to read the notation I guess.