There are grammatical rules to follow but proof writing is just like any other writing and doesn't necessarily follow any format rules. It just needs to consist of logical steps. Most just consist of "We have this. Therefore, that is true. Thus, something else is true." and so on.
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$