Equivalence of Definitions of Compatible Atlases

Theorem
Let $M$ be a topological space.

Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$.

1 implies 2
Follows immediately from the definition of $C^k$-atlas.

2 implies 1
Let $(U,\phi)$ and $(V,\psi)$ be charts in $\mathscr F \cup \mathscr G$.

If they are both in $\mathscr F$, they are $C^k$-compatible because $\mathscr F$ is a $C^k$-atlas.

If they are both in $\mathscr G$, they are $C^k$-compatible because $\mathscr G$ is a $C^k$-atlas.

If $(U,\phi) \in \mathscr F$ and $(V,\psi) \in \mathscr G$, they are $C^k$-compatible by assumption.

Thus $\mathscr F \cup \mathscr G$ is a $C^k$-atlas.