We can prove Theorem $6.19$ in three parts$:$
- Show that if $L=N(P)$ for some $DPDA$ $P,$ then $L$ has the prefix property.
- Show that if $L=N(P)$ for some $DPDA$ $P,$ then there exists a $DPDA$ $P'$ such that $L=L(P').$
- Show that if $L$ has the prefix property and is $L(P')$ for some $DPDA$ $P',$ then there exists a $DPDA$ $P$ such that $L=N(P).$