Let $A_1,\ldots,A_n$ be $n$ events in a discrete probability space. Can someone give me an example such that $P(A_1 \cap \cdots \cap A_n)=P(A_1)\cdot \ldots \cdot P(A_n)$ holds, but such that there is a subset $A_{i_1},\ldots,A_{i_k}$of the $A_1,\ldots,A_n$, such that $P(A_{i_1}\cap\ldots \cap A_{i_k})\neq P(A_{i_1})\cdot\ldots \cdot P(A_{i_k}) \ \ ?$
If that weren't such an example that would mean that the usual definition of mutual independence - as here, for example - is too strong, since the independence of the intersection of all events already implies the independence of all partial intersections of events. In that case I would like a proof of this implication.