CASE II : PDA M accepts by empty stack.
We will construct 
from M in such a way that simulates M and detects when M empties its stack. enters its final state 
when and only when M empties its stack.Thus will accept a string 
iff M accepts.
Let 
where 
and X
and 
contains all the transition of 
, as well as the following two transitions.
and
Transitions 1 causes to enter the initial configuration of M except that will have its own bottom-of-stack marker X which is below the symbols of M's stack. From this point onward will simulate every move of M since all the transitions of M are also in .