Lemma 2

Let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$. There exist infinitely differentiable mappings $v_1, . . . , v_n$ of $\mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$ into A such that

for every $\boldsymbol{z}= (z_1, . . . , z_n)\in \mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$.

Let $\boldsymbol{w}= (w_1, . . . , w_n)\in \mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$. By definition of the simultaneous spectrum, there exist $b_1, . . . , b_n$ in A such that

(I, p. 41, n$^o6$). There exists an open neighbourhood $W_{\boldsymbol{w}}$ of $\boldsymbol{w}$ such that the element $(z_1-a_1)b_1+\cdots + (z_n-a_n)b_n$ of A is invertible if $\boldsymbol{z}= (z_1, . . . , z_n)$ belongs to $W_{\boldsymbol{w}}$. For every integer $j$ such that $1\leqslant j\leqslant n$ and every $\boldsymbol{z}$ in $W_{\boldsymbol{w}}$, let

The functions $u_1, u_2, . . . , u_n$ from $W_{\boldsymbol{w}}$ into A thus defined are infinitely differentiable in $W_{\boldsymbol{w}}$, and we have

for every $\boldsymbol{z}$ in $W_{\boldsymbol{w}}$.

Since the family $(W_{\boldsymbol{w}})_{\boldsymbol{w}\in\mathbf{C}^{n-}Sp^n(\boldsymbol{a})}$ is an open covering of $\mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$, there exists a locally finite open covering $\mathscr{W}= (W_{\lambda})_{\lambda\in L}$ (TG, I, p. 70, th. 5) and, for every $\lambda \in L$, functions $u_{1\lambda}, . . . , u_{n\lambda}$, with values in A, defined and infinitely differentiable in $W_{\lambda}$, such that $(z_1-a_1)u_{1\lambda}(\boldsymbol{z}) +\cdots + (z_n-a_n)u_{n\lambda}(\boldsymbol{z}) = 1$ for every $\boldsymbol{z}$ in $W_{\lambda}$. Let $(f_{\lambda})_{\lambda\in L}$ be a partition of unity subordinate to the covering $\mathscr{W}$ formed by infinitely differentiable functions (VAR, R1, p. 40, 5.3.6). Let $i$ be an integer such that $1\leqslant i\leqslant n$. For every $\lambda \in L$, let $u'_{i\lambda}$ be the mapping from $\mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$ into A obtained by extending by 0 the function $f_{\lambda}u_{i\lambda}$ in $(\mathbf{C}^n-$ Sp$^n(\boldsymbol{a}))-W_{\lambda}$. The functions $u'_{i\lambda}$ are infinitely differentiable. The family (Supp($u'_{i\lambda}$))$_{\lambda\in L}$ being locally finite, the function $v_i=\sum_{\lambda\in L}u'_{i\lambda}$ is defined and infinitely differentiable in

$\mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$.

Let $\boldsymbol{z}\in \mathbf{C}^n-$ Sp$^n(\boldsymbol{a})$. Denote by $L'$ the finite set of the $\lambda \in L$ such that $\boldsymbol{z}\in W_{\lambda}$. Then