Corollary 1

Let G be a topological group and let $\varrho$ be a semisimple unitary representation of G in a complex Hilbert space E. Every subrepresentation of $\varrho ($resp. every quotient representation of $\varrho )$ is semisimple.

If $\varrho_1$ is a subrepresentation of $\varrho$, then every nonzero subrepresentation of $\varrho_1$ contains an irreducible subrepresentation (Prop. 7), hence $\varrho_1$ is semisimple (loc. cit.).

By Prop. 3 of V, p. 383, every quotient representation of $\varrho$ is isomorphic to a subrepresentation of $\varrho$, hence is semisimple.