Injectivity through a holomorphic functional calculus with nonvanishing derivative

Let $x_1,x_2$ belong to a commutative unital Banach algebra, have the same Gelfand transform, and have common spectrum $K$. If $f$ is holomorphic near $K$ and $f^{\prime }$ has no zero on $K$, then $f(x_1)=f(x_2)$ implies $x_1=x_2$. Apply the two-variable holomorphic functional calculus to the divided difference of $f$; its Gelfand transform never vanishes, so it is invertible.