Prove the equivalence of regular expressions
Start with
$$
R^*R^* = \left(\bigcup_{i=0}^\infty R^i\right) \left(\bigcup_{j=0}^\infty R^j\right) = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^i R^j = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j}.
$$
As $i,j$ go over all pairs of non-negative integers, $i+j$ goes over all non-negative integers, hence
$$
R^*R^* = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j} = \bigcup_{k=0}^\infty R^k = R^*.
$$