I know that a maximal atlas isnt contained in another atlas. But here is a proposition that if A is a Ck atlas then its contained in a maximal atlas B. and the proof is let B=Union C where C is an atlas Ck related with A . so B is a unique maximal atlas containing A. But i cant recognize this proof . help plz

Here is a rewriting of the proof, using metaphors and visualisation rather than formal language.

For our manifold, there are _many_ $C_k$-atlases. Some of them are $C_k$-related to $A$, and some are not. Take each and every one of those atlases, and if it turns out to be $C_k$-related to $A$, take all the charts in that atlas and throw it into a big pile (removing any duplicate charts we encounter along the way).

Once we've taken _all_ such atlases (including $A$ itself), the pile now contains all possible charts that are $C_k$-related to $A$. This pile is therefore a $C_k$ atlas that is $C_k$-related to $A$, and there can't be any bigger atlas, because we have all possible compatible charts already. So this pile is a _maximal_ $C_k$ atlas that contains $A$, and we call it $B$.