 By a result from complex analysis, there exists a function "g" holomorphic on a neighborhood of "σ"("T") such that

According to the homomorphism property, "f"("T") − "f"("μ") = ("T" − "μ")"g"("T").Therefore, "μ" ∈ "σ"("T") implies "f"("μ") ∈ "σ"("f"("T")).For the other inclusion, if "μ" is not in "f"("σ"("T")), then the functional calculus is applicable to

So "g"("T")("f"("T") − "μ") = "I".