a string, it empties its stock and at that point P0' would expose the bottom of stock marker Z0' and enters the final state P' by using rule 2. So, it is obvious that, an input string X is accepted by P iff it is accepted by P'.
The converse of lemma 2 is not necessarily true. But it can be shown that every language that has the prefix property and is accepted by a DPDA with final state is also accepted by some DPDA that accepts by empty stock, as given in the lemma 3.
Lemma 3 : If a language L has the prefix property and is accepted by a DPDAP by final state, then there is some DPDA P' that accepts by empty stock such that L=L(P').
Proof : Let 
for DPDA
that accepts by final state. We construct P' from P as follows.