Equality of partial derivatives (Schwartz’s theorem) and total differential

I'm reviewing some Physical Chemistry concepts and the two books I'm using show the total differential and right away they present the Schwartz's theorem. I don't get the relationship. In the screenshot below you can see that the author mentions that the mixed partial second derivatives can be obtained from the total differential. What is the relationship between equations like 1.7 and 1.8, in general?


I got the screenshot from this excellent book: <https://peverati.github.io/pchem1/SystemVariables.html#thermodynamic-variables>


Atkins does the same think on Mathematical Background 2, pg 91, ninth edition.


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


