Suppose that M1 and M2 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 M1 as S1, 1, F1, and 81 respectively, and similarly for M2. We define the product DFA M1 × M2 as follows. The state set is the direct product Si x S2, the set of ordered pairs (s1, s2) with s1 E S1 and s2 E S2. The start state is the pair (11, 12) and the final state set is F1 x F2. The new transition function takes a state (s1, s2) and a letter a to (81(s1, a), 82(s2, a)). Prove that the product DFA decides the language L(M1)n L(M2).