I don't understand the reason for the conclusion, written in boldface at the end, in the following argument (taken from Elon LIMA, Curso De Análise, Vol .2).
If $f:U\longrightarrow V\subset \mathbb{R}^m$ is a diffemorphism which is $C^{k}$, then $g=f^{-1}$ is $C^k$ as well: indeed, by the chain rule, g'(y)=(Inv\circ f'\circ g) (y), where $Inv$ is a $C^{\infty}$ map from $GL(\mathbb{R}^n)$ onto itself. And, since $f$ is $C^k$, it follows, from these facts, that $g$ is $C^k$.
Why?
Thank you.