PX262 - C6b - compatibility of operators

if two operators commute, they are called compatible operators
for two compatible operators, $\hat{Q}, \hat{R}$ and a generic wave function: $ψ = \sum_{n} a_{n} ϕ_{n}$
take $ϕ_{n}$ to be the eigenstate of $\hat{Q} :$ $\hat{Q} \hat{R} ϕ_{n} = \hat{R} \hat{Q} ϕ_{n} = \hat{R} q_{n} ϕ_{n} = q_{n} \hat{R} ϕ_{n}$
so, if operators are compatible, then $\hat{R} ϕ_{n}$ is an eigenfunction of $\hat{Q}$