It says that theorems 12.11 and 12.12 imply Theorem 12.13. But, don't we need some extra conditions? Like existence of $D\_{r,r}f$ and $D\_{k,k}f$? Here $f$ is a function from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $D\_kf$ denotes the partial derivative of $f$ w.r.t. the $k^{th}$ variable.

[![enter image description here](https://i.stack.imgur.com/i4Zxa.png)](https://i.stack.imgur.com/i4Zxa.png)

[![enter image description here](https://i.stack.imgur.com/xb2fT.png)](https://i.stack.imgur.com/xb2fT.png)

[![enter image description here](https://i.stack.imgur.com/ENG8s.png)](https://i.stack.imgur.com/ENG8s.png)

The images are from Tom Apostol's Mathematical Analysis.