, since 
.
So, while processing the string xy , the DPDA must arrive at the configuration given below because of its deterministic property.
From the point onward the DPDAP cannot move since it has already emptied its stock and 
. Hence, xy is not accepted by P as assumed.
The lemma 2 given below shows that every language accepted by a DPDA by some DPDA that accepts by final state.
Lemma 2 If L is accepted by some DPDA P that accepts by empty stock, than there is some DPDA P'that accepts by final state such that L=L(p').
Proof :
If 
accepts L by empty stack .