(related to Proposition: Characterization of Independent Events II)
The proposition follows immediately from the other characterization of independent events, according to which given an event \(B\) with \(0 < p(B) < 1\), the event \(A\) is independent from \(B\) if and only if
\[p(A)=p(A|B).\]
By definition of conditional probability, it follows
\[p(A)=p(A|B)=\frac{p(A\cap B)}{p(B)}, \] which is equivalent to \[p(A)\cdot p(B)=p(A\cap B).\]
Assuming that \(A\) is independent from \(B\), like shown above, and in addition assuming \(0 < p(A) < 1\), we can conclude that \(B\) is independent from \(A\), for symmetry reasons.
