Corollary 1

For every $\pi \in \widehat{G}$, the subrepresentation $\boldsymbol{\varrho }_G(\pi )$ is the $(\overline{\pi}\boxtimes \pi )$-isotypical component of $\boldsymbol{\varrho }_G$.

Let F be the $(\overline{\pi}\boxtimes \pi$)-isotypical component of $\boldsymbol{\varrho }_G$. The space F contains $\boldsymbol{\varrho }_G(\pi )$ (prop. 5). Moreover, for every $\tau \in \widehat{G}$ different from $\pi$, the intersection of F and $\boldsymbol{\varrho }_G(\tau )$ is zero (prop. 3 of V, p. 461). It follows that $F =\boldsymbol{\varrho }_G(\pi )$ by applying the theorem.