Corollary 3

The right (resp. left) regular representation of G is isomorphic to the Hilbert sum

$\pi \bigoplus\in \widehat{G}\pi$dim($\pi$ ).

By the theorem and proposition 5, the restriction of the biregular representation of G to the subgroup $H =\{e\} \times G$ is isomorphic to the Hilbert sum of the restrictions to H of the representations $\overline{\pi}\boxtimes \pi$ for $\pi \in \widehat{G}$. These are isomorphic to the direct sum of dim($\pi$ ) copies of $\pi$ (lemma 8 of V, p. 384). This implies the result for the right regular representation, and the case of the left regular representation is similar.