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We can prove Theorem $6.19$ in three parts$:$

  1. Show that if $L=N(P)$ for some $DPDA$  $P,$ then $L$ has the prefix property.
  2. Show that if $L=N(P)$ for some $DPDA$  $P,$ then there exists a $DPDA$ $P'$ such that $L=L(P').$
  3. Show that if $L$ has the prefix property and is $L(P')$ for some $DPDA$  $P',$ then there exists a $DPDA$ $P$ such that $L=N(P).$

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