divergence of the cross product of two vectors proof

You just need to write everything out neatly, so you can see the equating terms.

Let's deal with the left hand side first. $$\vec{A}\times\vec{B}=(A_2B_3-A_3B_2)\vec{i}+(A_3B_1-A_1B_3)\vec{j}+(A_1B_2-A_2B_1)\vec{k}$$ I wrote the component of $A$ always in front of the component of $B$ in order to see easily. Now the whole left hand side is the divergence of this: $$\frac{\partial(A_2B_3-A_3B_2)}{\partial x}+\frac{\partial(A_3B_1-A_1B_3)}{\partial y}+\frac{\partial(A_1B_2-A_2B_1)}{\partial z}$$

Let's wait for a while to do the product rule, and instead, look at the right hand side. $$\nabla \times \vec{A}=(\frac{\partial A_3}{\partial y}-\frac{\partial A_2}{\partial z})\vec{i}+(\frac{\partial A_1}{\partial z}-\frac{\partial A_3}{\partial x})\vec{j}+(\frac{\partial A_2}{\partial x}-\frac{\partial A_1}{\partial y})\vec{k}$$ The first term is the dot product of $B$ with it, so just replace the $i,j,k$ by $B_1,B_2,B_3$: $$\vec{B}\cdot \nabla \times \vec{A}=B_1(\frac{\partial A_3}{\partial y}-\frac{\partial A_2}{\partial z})+B_2(\frac{\partial A_1}{\partial z}-\frac{\partial A_3}{\partial x})+B_3(\frac{\partial A_2}{\partial x}-\frac{\partial A_1}{\partial y}) $$

Now the second term is similar. We just need to switch $A$ and $B$, and remember it is negative: $$\vec{A}\cdot \nabla \times \vec{B}=A_1(\frac{\partial B_3}{\partial y}-\frac{\partial B_2}{\partial z})+A_2(\frac{\partial B_1}{\partial z}-\frac{\partial B_3}{\partial x})+A_3(\frac{\partial B_2}{\partial x}-\frac{\partial B_1}{\partial y}) $$

Now you can compare each product in the left hand side with the corresponding terms in the right hand side. For example, the first product rule in the left hand side is $$\frac{\partial(A_2B_3)}{\partial x}=A_2\frac{\partial(B_3)}{\partial x}+\frac{\partial(A_2)}{\partial x}B_3$$

It is easy to see which terms in the right hand side equal to these two terms. I'll leave it to you to proceed.

In index notation

\begin{eqnarray*} (A \times B)_{i} = \epsilon_{ijk} A_j B_k \end{eqnarray*} (Einstein's convention of sum over repeated indices). Then if $A_{j_{,i}}=\partial A_j/\partial x_i$, and from $\nabla \times A=\epsilon_{ijk} A_{k_{,j}}$ (and so for the other symbols)

\begin{eqnarray*} \nabla \cdot (A \times B) &=& [\epsilon_{ijk} A_j B_k],_{i} \\ &=& \epsilon_{ijk} A_{j_{,i}} B_k + \epsilon_{ijk} A_j B_{k_{,i}} \\ &=& B_k ( \epsilon_{kij} A_{j_{,i}}) - A_j ( \epsilon_{jik} B_{k_{,i}}) \\ &=& B \cdot (\nabla \times A) - A \cdot ( \nabla \times B ) \end{eqnarray*} where the minus ``-'' sign appears since $\epsilon_{ijk}=-\epsilon_{jik}$.