In the textbook A course in Probability Theory written by Kai Lai Chuang, he write that in the two-dimension Euclidean space, the Borel field $\mathcal{B^2}$ is generated by rectangles of the form$$\{(x,y):a<x\le b, c<y\le d\}.$$ It is also generated by the products set of the form $$B_1\times B_2=\{(x,y):x\in B_1, y\in B_2\}$$ where $B_1$ and $B_2$ belong to 1-dimension Borel field $\mathcal{B^1}$.
It is trivial that $\mathcal{B^2}$ is generated by the collection of rectangles of above form. My question is that how to show that $\mathcal{B^2}$ is also generated by the collection of the products set of the form $B_1\times B_2$.