without loss of generality assume that M'

satisfies Lemma 1 given above. We now construct M from M' as follows.

                  

1. If M' pops its stock, then M pops its stock and remembers the symbol popped (in its finite control) by the move

            if, is in M'.

2. If M' changes its top symbol of the stock, then M remembers this without changing its top of the stock that is .

           if is in M'.