This question comes from page 25, line 4, of Halmos' "Measure Theory", as the image below shows. This line says that $\displaystyle \bf C$ is a $\displaystyle \sigma$-ring which can be verified easily, but I just can not verify this easy statement because I can not make up the difference $\displaystyle [B_1\cup(C_1-A)]-[B_2\cup(C_2-A)]$ into a form of $\displaystyle B\cup(C-A)$. Could you please help me with this problem? Thanks!

NOTE: Here $\displaystyle \bf S(E)$ means the $\displaystyle \sigma$-ring generated by $\displaystyle \bf E$.