(related to Proposition: Probability of Event Difference)
We first note that
\[A=(A\cap B)\cup (A\cap \overline B)=(A\cap B)\cup (A\setminus B).\]
Because the events \((A\cap B)\) and \((A\setminus B)\) are mutually exclusive, it follows from the definition of probability that
\[p(A)=p(A\cap B)+p(A\setminus B),\]
which gives us the required equation
\[p(A\setminus B)=p(A)-p(A\cap B).\]
