# Show that $R^{+} \equiv R \leftrightarrow L(RR) \subset L(R)$

Show that $$R^{+} \equiv R \leftrightarrow L(RR) \subset L(R)$$

sigma is any alphabet. R is a regular expression.

How can L(RR) even be a subset or equal to L(R)?

If $$\varepsilon\in L(R)$$, then $$RR$$ matches every string matched by $$R$$: you can match the first $$R$$ against $$\varepsilon$$ and the second $$R$$ against the rest of the string.