I have a set-theory equation in my book that I don't understand how to prove.
I have a disjoint collection of sets $A_1, A_2,\cdots\ , A_n$ and a disjoint collection $A_1\star, A_2\star, ..., A_n\star$ is constructed with the property that $(A_1\star)\cup(A_2\star)\cup(A_3\star)\cdots = (A_1)\cup(A_2)\cup(A_3)\cdots$
$A_i\star$ is defined by $$A_1\star = A_1,\quad A_i\star = A_i \diagdown [(A_1)\cup(A_2)\cup\cdots(A_{i-1})]$$ where $$A_i \diagdown [(A_1)\cup(A_2)\cup\cdots(A_{i-1})] = A_i\cap\overline{[(A_1)\cup(A_2)\cup...(A_{i-1})]}$$.
The book says it "should be easy to see that" $$(A_1\star)\cup(A_2\star)\cup(A_3\star)\cup... = (A_1)\cup(A_2)\cup(A_3)\cup\cdots$$
Should I use induction, and if so how?
Thanks.