Prove Theorem 2.2.2 on page 71. The only if direction is direct from the definition of independence

Question:

Prove Theorem 2.2.2 on page 71. The €œonly if€ direction is direct from the definition of independence on page 68. For the €œif€ direction, use induction on the value of j in the definition of independence. Let m = j ˆ’ 1 and let = 1 with j1 = ij.
In Theorem 2.2.2
Let A1, . . . , Ak be events such that Pr(A1 ˆ© . . . ˆ© Ak) > 0. Then A1, . . . , Ak are independent if and only if, for every two disjoint subsets {i1, . . . , im} and {j1, . . . , je} of {1, . . . , k}, we have

Theorem 2.2.2 says that k events are independent if and only if learning that some of the events occur does not change the probability that any combination of the other events occurs.