# The Inverse Function Theorem - proof

I am trying to undresteam Inverse Function Theorem so I googled and found a book

Calculus on Manifolds by Michael Spivak

I was happy because I have proof here but I stucked in first paragraph. Can anyone help me with this equation? I really dont get why it is equal. I tried to do this by definion of frechet's derivative but I got stuck in welter of symbols.

Here is the theorem : And here is part of the proof wich i don't understeand (underlined) Just why $D(\lambda^{-1})(f(a)) = \lambda^{-1}$ ?

For me it is just : $D(\lambda^{-1})(f(a)) = D((Df(a))^{-1})(f(a))$

Because in general, if $L: (E,\| \cdot \|) \to (F,\|\cdot \|)$ is a continuous linear map, then for any $x \in E$, $DL(x) = L$.
$$\frac{1}{\|h\|}\| L(x+h) - L(x) - L(h)\| = 0 \to 0$$