= Composition rule for holomorphic functional calculus
{title2=$g(f(x))=(g\circ f)(x)$}
Choose a cycle $\Gamma$ surrounding $\sigma(x)$ and a cycle $\Delta$ surrounding $f(\sigma(x))$, with $f(\Gamma)$ inside the region of winding number one for $\Delta$. The algebra-homomorphism property gives
$$
(\zeta1-f(x))^{-1}=\frac1{2\pi i}\int_\Gamma\frac{(z1-x)^{-1}}{\zeta-f(z)}\,dz.
$$
Substitute this into the contour formula for $g(f(x))$, interchange integrals, and apply the <Cauchy integral formula> to the inner $\zeta$ integral. This proves composition as an equality of algebra elements, not merely an equality on characters, which need not separate a nonsemisimple algebra.
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