Theorem 4

Let A be a commutative complex unital Banach algebra, let $n\geqslant 1$ be an integer, and let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. Let $\boldsymbol{f}=$ $(f_1, . . . , f_p)$, where $f_1, . . . , f_p$ are elements of $\mathscr{O}$(Sp$^n(\boldsymbol{a})$). The image of Sp$^n(\boldsymbol{a})$ under the mapping $\boldsymbol{z}\mapsto \boldsymbol{f}(\boldsymbol{z}) = (f_1(\boldsymbol{z}), . . . , f_p(\boldsymbol{z}))$ is equal to Sp$^p(\boldsymbol{f}(\boldsymbol{a}))$.

For every $g\in \mathscr{O}$(Sp$^p(\boldsymbol{f}(\boldsymbol{a})); A$), the composite germ $g\circ \boldsymbol{f}$ is an element of $\mathscr{O}$(Sp$^n(\boldsymbol{a}); A$) and one has $g(\boldsymbol{f}(\boldsymbol{a})) = (g\circ \boldsymbol{f})(\boldsymbol{a})$.

The first assertion concerning the image of Sp$^n(\boldsymbol{a})$ follows from cor. 2 of I, p. 67. To prove the second, we shall use the following lemma.