A Turing machine that halts on all inputs is called a halting Turing machine $($also known as Decider$).$ Prove the following:

$($a$)$ If $M_1$ and $M_2$ are two halting Turing machines, then there exists a halting Turing machine that recognizes $L(M_1)L(M_2).$

$($b$)$ If $M_1$ and $M_2$ are two $($not necessarily halting$)$ Turing machines, then there exists a Turing machine that recognizes $L(M_1)L(M_2).$