Prove a variant of Tensor Product

If $A_{1}$, $A_{2}$, $B_{1}$, $B_{2}$ are matrices for which the matrix products $A_{1}A_{2}$ and $B_{1}B_{2}$ are defined, the following rule holds:

$(A_{1} \otimes B_{1})(A_{2}\otimes B_{2}) = (A_{1}A_{2})\otimes(B_{1}B_{2})$

Prove this. Explain that this result makes clear why the transposition in the definition of the Kronecker product is 'necessary'.