分からない問題はここに書いてね445

Theorem 7.4.

Let A be open in R^n; let f : A -> R^n; let f(a) = b.
Suppose that g maps a neighborhood of b into R^n, that g(b) = a, and

g(f(x)) = x

for all x in a neighborhood of a. If f is differentiable at a and if g is differentiable at b, then

Dg(b) = [Df(a)]^(-1).

↑は、 James R. Munkres著『Analysis on Manifolds』に書いてある定理です。

なぜ↓のように書かなかったのでしょうか？

Let A be open in R^n.
Let f : A -> R^n.
Let B be open in R^n.
Let g : B -> R^n.
Let a ∈ A.
Let b ∈ B.
Let f(a) = b.
Let f be differentiable at a.
Let g be differentiable at b.
Let g(f(x)) = x for all x in a neighborhood of a.

Then,

Dg(b) = [Df(a)]^(-1).