# Why $P$ cannot have NULL string in Arden's Theorem?

Arden's Theorem says that in the equation $$R=Q+RP$$, the $$P$$ cannot have NULL string. In this respect,the theorem will not be valid for the expression $$R=Q+R(NULL+01)$$. Am I correct? If so, then what will be the justification?

Here is the version of Arden's theorem on Wikipedia:

One solution of the language equation $$R = Q+RP$$ is $$R = QP^*$$.

If $$\epsilon \notin P$$, then this is the only solution.

When $$\epsilon \in P$$, there are more solutions. In fact, we can prove the following result:

The solutions of the language equation $$R = Q+RP$$, where $$\epsilon \in P$$, are $$R=SP^*$$ for all $$S \supseteq Q$$.

Proof. Let us first show that $$SP^*$$ is always a solution. Since $$\epsilon \in P$$, $$RP = SP^+ = SP^*$$. Since $$SP^* \supseteq S \supseteq Q$$, $$Q + RP = Q + SP^* = SP^* = R$$.

Let us now show that all solutions are of this form. Suppose that $$R$$ is a solution. Clearly $$R \supseteq Q$$. Since $$\epsilon \in P$$, this implies that $$RP \supseteq R \supseteq Q$$, and so $$RP = Q + RP = R$$. Induction shows that $$RP^n = R$$ for all $$n \in \mathbb{N}$$, and so $$RP^* = R$$. Since $$R \supseteq Q$$, this is a solution of the required form. $$\square$$