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* If <math>w = \varepsilon ,</math> this follows from the definition of <math>F' .</math>
* Otherwise, let <math>w = va</math> with <math>v \in \Sigma^*</math> and <math>a \in \Sigma .</math>
:From <math>\delta'^*(q_0,w) = \delta^*(q_0,w)</math> and <math>F \subseteq F' ,</math> we have <math display=block>\delta'^*(q_0,w) \cap F' \neq \{\} \;\Leftarrow\; \delta^*(q_0,w) \cap F \neq \{\} ;</math> we still have to show the "<math>\Rightarrow</math>" direction.
:*If <math>\delta'^*(q_0,w)</math> contains a state in <math>F' \setminus \{ q_0 \} ,</math> then <math>\delta^*(q_0,w)</math> contains the same state, which lies in <math>F</math>.
:*If <math>\delta'^*(q_0,w)</math> contains <math>q_0 ,</math> and <math>q_0 \in F ,</math> then <math>\delta^*(q_0,w)</math> also contains a state in <math>F ,</math> viz. <math>q_0 .</math>
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