The following picture comes from foundations of differential geometry volume I.
According to the definition of complete atlas, would it be more precise to call it maximal atlas?
Secondly, the statement
Every atlas of $M$ compatible with $\Gamma$ is contained in a unique atlas of $M$ compatible with $\Gamma$.
can be proved routinely:
Let $\mathscr A=\{\mbox{all the atlases of }M \mbox{ compatible with }\Gamma\mbox{ that contains }A\}$, I guess it is a set. For each chain $\mathscr A_m$ in $\mathscr A$, let $A_m=\cup \mathscr A_m$, then it is in $\mathscr A$: Suppose $(U_1,\phi_1)\in A_1\in \mathscr A_m, (U_2,\phi_2)\in A_2\in \mathscr A_m$ and $A_1\subset A_2$, then $\phi_2\circ\phi_1^{-1}:\phi_1(U_1\cap U_2)\to\phi_2(U_1\cap U_2) $ is an element of $\Gamma$ whenever $U_1\cap U_2$ is nonempty, since $(U_1,\phi_1)$ and $(U_2,\phi_2)$ are both in $A_2$. Hence $A_m$ is an bound of the chain $\mathscr A_m$. By Zorn's lemma, there is a maximal element in $\mathscr A$.
Can someone verify for me if the proof is correct?