Atlas Belongs to Unique Differentiable Structure

Theorem
Let $M$ be a locally Euclidean space of dimension $d$.

Let $\AA$ be an atlas on $M$.

Then there exists a unique differentiable structure $\FF$ on $M$ with $\AA \in \FF$.

Proof
Let $\FF$ be the equivalence class of $\AA$ under the relation of compatibility.

By Compatibility of Atlases is Equivalence Relation, this is indeed an equivalence relation.

By definition we have $\AA \in \FF$.

By Relation Partitions Set iff Equivalence, $\FF$ is an element of the partition of equivalence classes.

By definition, the elements of a partition are pairwise disjoint.

Therefore if $\GG \ne \FF$ is an element of the partition, we must have:
 * $\AA \notin \GG$

Therefore $\AA$ belongs to exactly one differentiable structure.