Proposition 8

Let $f\in \mathscr{C}$(Sp$_A(x)$). Denoting again by $f$ the restriction of $f$ to Sp$_{A'}(\pi (x))$, one has the equality $\pi (f(x)) =f(\pi (x))$. In particular, for every $\chi \in \mathsf{X}(A)$, one has $\chi (f(x)) =f(\chi (x))$.

Let $z$ be the identical mapping on Sp$_A(x)$. The mappings defined by $f\mapsto \pi (f(x))$ and $f\mapsto f(\pi (x))$ are continuous unital morphisms of involutive algebras of $\mathscr{C}$(Sp$_A(x)$) into B which map $z$ onto $\pi (x)$. These morphisms therefore coincide on the unital involutive subalgebra of $\mathscr{C}$ (Sp$_A(x)$) generated by $z$. Since the latter is dense in $\mathscr{C}$(Sp$_A(x)$) (TG, X, p. 40, Cor. 1), these morphisms are equal.