Lemma 1

Let $W^G$ be the subspace of W consisting of the elements invariant under G. The endomorphism $\int_G\rho (g)dg$ of W is a projection with image $W^G$, compatible with the operations of G. In particular, dim $W^G=\int_G$ Tr $\rho (g)dg$.

Put $p=\int_G\rho (g)dg$; for $h\in G$,

and similarly $p\circ \rho (h) =p$. Thus, $p$ is compatible with the operations of G, and its image is contained in $W^G$. If $w\in W^G$, we have $p(w) =\int_G\rho (g)w dg=w$, hence the lemma.