| context | $\langle M,T\rangle$ … second-countable Hausdorff space |
| context | $n\in \mathbb N$ |
| definiendum | $A\in$ it |
| inclusion | $A\subseteq$ atlas ($\langle M,T\rangle,n$) |
| forall | $\langle V,\phi\rangle,\langle W,\psi\rangle\in A$ |
| postulate | $\phi\circ\psi^{-1}$ … smooth |
A priori “$\phi\circ\psi^{-1}$” in the postulate doesn't make sense as their domains/codomains will not much. Here, really, one must choose functions with appropriately restricted domain.
For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.
For a smooth atlas there are smooth coordinate changes on $\mathbb R^n$.