I know that for a universal Turing machine to emulate the operation of another Turing machine on input

*w*takes time at least proportional to the number of steps that the emulated Turing machine would take to halt on input

*w*.

What I believe is true, but cannot think of a way to prove, is that emulating multiple Turing machines on input

*w*, and returning an integer stating how many of those machines accepted on

*w*, must take time at least proportional to the time it takes the first TM to halt on

*w*+ the time it takes the second TM to halt on

*w*+ the time it takes the second TM to halt on

*w*, etc, all the way down to the last TM.

Can anyone demonstrate that this must be the case or point me to a good book or website that has the answer?