Suppose that M$1$ and M$2$ are two DFA$$s with the same input alphabet. We$$ll refer to the

state set, start state, final state set, and transition function of M$1$ as S$1,\backslash$iota $1,$ F$1,$ and $\backslash$delta $1$

respectively, and similarly for M$2.$ We define the product DFA M$1\backslash$times M$2$ as follows. The

state set is the direct product S$1\backslash$times S$2,$ the set of ordered pairs $$s$1,$ s$2$ with s$1$ in S$1$ and

s$2$ in S$2.$ The start state is the pair $\backslash$iota $1,\backslash$iota $2$ and the final state set is F$1\backslash$times F$2.$ The new

transition function takes a state $$s$1,$ s$2$ and a letter a to $\backslash$delta $1($s$1,$ a$),\backslash$delta $2($s$2,$ a$).$ Prove that the

product DFA decides the language L$($M$1)\backslash$cap L$($M