Equivalence of atlases is an equivalence relation. Each atlas on is equivalent to a unique maximal atlas on .
Reflexive: If is a atlas, then is also a atlas.
Symmetry: Let and be two atlases such that is also a atlas. Then certainly is also a atlas.
Transitivity: Let be atlases, such that and are both atlases.
Then is a diffeomorphism since both and are diffeomorphisms due to and being atlases. Also, , implies so is also a atlas.
Let be a atlas on . Define to be the union of all atlases equivalent to . Then . If , then , so that is the unique maximal atlas equivalent to .