Spectral measures

Question

Let $E:\Sigma\to\mathcal{L}(\mathcal{H})$ be a spectral measure on the Borel $\sigma$-algebra $\Sigma$ of $\mathbb{C}$. Assume also that $E$ is compactly supported in the sense that $E(K)=\operatorname{id}_\mathcal{H}$ for some compact $K\subset\mathbb{C}$, so that $$A:=\int\lambda dE$$ is a well-defined, bounded normal operator on $\mathcal{H}$. Do you know a nice proof for the fact, that $E(\operatorname{spec}A)=\operatorname{id}_\mathcal{H}$, which (of course) does not use the spectral theorem?