Lemma 6

Let $\boldsymbol{a}\in A^n$. All sequences adapted to $\boldsymbol{a}$ are linked.

Let $(h, u_1, . . . , u_n)$ and $(h', u'_1, . . . , u'_n)$ be sequences adapted to $\boldsymbol{a}$, and let us denote by $\omega$ and $\omega '$ the associated differential forms.

Define the indefinitely differentiable mappings

so that $w_{ji}=-w_{ij}$, and Supp($s_i$)$\subset$ Supp($h$)$\cup$ Supp($h'$).

Let $u''_i=u'_i-s_i,\boldsymbol{u}= (u_1, . . . , u_n)$ and $\boldsymbol{u}''= (u''_1, . . . , u''_n)$. Let us also denote by $\boldsymbol{v}_{ij}$ the mapping of $\mathbf{C}^n$ into $A^n$ whose $i$-th component is $(z_j-a_j)w_{ij}$, whose $j$-th component is $(z_i-a_i)w_{ji}=-(z_i-a_i)w_{ij}$, and whose other components are zero. Then we have

In fact, for every integer $k$ such that $1\leqslant k\leqslant n$, the $k$-th component of the right-hand side is

By induction, one deduces from Lemma 5, applied to the integers $i$ and $j$ and to the mappings $w_{ij}$, that the sequence $(h, u''_1, . . . , u''_n)$ is adapted to $\boldsymbol{a}$ and is linked to $(h, u_1, . . . , u_n)$. Let $\omega ''$ be the differential form associated with $(h, u''_1, . . . , u''_n)$. Since $u''_i=u'_i-s_i$, we have

which is expressed as a linear combination, with coefficients 1 or $-1$, of differential forms of the form

where $I_1$ (resp. $I_2$) is a nonempty subset (resp. a subset) of $\{1, . . . , n\}$. Each differential form $\xi_{I_1,I_2}$ can also be written in the form

where the support of the differential form $\widetilde{\psi}$ is contained in the support of $s_i$ for every $i\in I_1$. Since $I_1$ is nonempty, this support is contained in Supp($h$)$\cup$ Supp($h'$). Consequently, $(h, u''_1, . . . , u''_n)$ is linked to $(h', u'_1, . . . , u'_n)$, and the lemma results on writing