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 $M$.

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

Definition $2$ implies Definition $1$
Let $\struct {U, \phi}$ and $\struct {V, \psi}$ 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 $\struct {U, \phi} \in \mathscr F$ and $\struct {V, \psi} \in \mathscr G$, they are $C^k$-compatible.

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