 So the vector-valued version of Cauchy's theorem gives

i.e.Hence the functional calculus is well-defined.Consequently, if "f"1 and "f"2 are two holomorphic functions defined on neighborhoods "D"1 and "D"2 of σ("T") and they are equal on an open set containing σ("T"), then "f"1("T") = "f"2("T").