Two atlases are compatible if and only if their associated maximal atlases are equal

In my class notes a differentiable manifold is defined as an ordered pair $$(M, \mathcal{A}),$$ where $$M$$ is a set and $$\mathcal{A}$$ is a maximal atlas of $$M.$$ There are also the following definitions:

• We say that two atlases $$\mathcal{A}_1$$ and $$\mathcal{A}_2$$ of $$M$$ are compatible if $$\mathcal{A}_1 \cup \mathcal{A}_2$$ is an atlas.
• The maximal atlas $$\mathcal{A}^+$$ associated with an atlas $$\mathcal{A}$$ of $$M$$ is the union of all atlases compatible with $$\mathcal{A}.$$

My question is: Is it true that $$\mathcal{A}_1$$ and $$\mathcal{A}_2$$ are compatible if and only if $$\mathcal{A}_1^+ = \mathcal{A}_2^+?$$

Yes. If $$\mathcal{A}_1$$ and $$\mathcal{A}_2$$ are compatible, then $$\mathcal{A}_2 \subset \mathcal{A}_1^+$$. Hence $$\mathcal{A}_2$$ and $$\mathcal{A}_1^+$$ are compatible and we conclude $$\mathcal{A}_1^+ \subset \mathcal{A}_2^+$$. Similarly $$\mathcal{A}_2^+ \subset \mathcal{A}_1^+$$. Conversely let $$\mathcal{A}_1^+ = \mathcal{A}_2^+ = \mathcal{A}$$. Then $$\mathcal{A}_1, \mathcal{A}_2 \subset \mathcal{A}$$, hence $$\mathcal{A}_1 \cup \mathcal{A}_2 \subset \mathcal{A}$$. Since $$\mathcal{A}$$ is an atlas, also the smaller $$\mathcal{A}_1 \cup \mathcal{A}_2$$ is one.