Let
F be a monoid of countably many functions holomorphic at
y0, and (
Xf)
f∈F be a set of independent variables. We set
F*=
F−{1},
x*=(
xf)
f∈F*. Let (
F1,
F2, ...) be an increasing sequence of finite subsets of
F such that ∪
i≥1 Fi=
F. For
i≥1, let
Ai, denote the ring of all functions of (
xf)
f∈Fi, holomorphic at (
x1, (
xf)
f∈Fi−{1})=(
x10, 0). Define
A=proj lim
Ai. Consider the implicit function
y∈
A defined by
g(
y)=Σ
f∈Fxff(
y) (
y(
x10, 0)=
y0). We have the Taylor expansion of
y at
x*=0:
y=
g−1(
x1)+Σ
α(\frac{
d|α|−1}{
dX1|α|−1} \frac{Π
f∈F*fα(f)(
g−1(
x1))}{
g'(
g−1(
x1))}) \frac{
x*α}{α!},
where the sum runs over all maps α :
F*→{0, 1, 2, ...} such that
|α
| :=Σ
f∈F*α(
f) are positive finite.
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