differential geometry - Proof of constructing Action-Angle Coordinates on Hamiltonian System

By Liouville-Arnold Theorem, we know we can construct action-angle coordinates such that the Hamiltonian system, when described in these coordinates, will have a form that is integrable by quadratures. I am looking at a proof of the construction of these coordinates, and I am not certain of a certain part. In page 180 of Mathematical Aspects of Celestial Mechanics it says that the Poisson bracket $\{F_i,\varphi_j\}$ is constant on $M_f$, the Lagrangian tori. I tried to prove it but I am not sure of the proof. Here is my attempt:

Choose Darboux coordinates $\{\bf{p},\bf{q}\}$ and by the coordinate representation of the Poisson bracket, we have

$$\{F_i, \varphi_j\} = \sum\limits_{k=1}^n \dfrac{\partial F_i}{\partial p_k} \dfrac{\partial \varphi_j}{\partial q_k} - \dfrac{\partial F_i}{\partial q_k} \dfrac{\partial \varphi_j}{\partial p_k} = \omega_j\sum\limits_{k=1}^n \dfrac{\partial F_i}{\partial p_k} \bigg(\dfrac{\partial F_1}{\partial p_k}\bigg)^{-1} - \dfrac{\partial F_i}{\partial q_k} \bigg(\dfrac{\partial F_1}{\partial q_k}\bigg)^{-1} = 2\omega_j,$$

where the second equality is because of the Hamiltonian equations (with $H = F_1$) and the fact that $\dfrac{\partial \varphi_j}{\partial t} = \omega_j$ where the $\varphi_j$ are the angle coordinates that are described by linear flow on $M_f$, and the third equality is because of $\dfrac{\partial F_i}{\partial F_1} = \delta_{i1}$. However, I am not so sure of the third equality because it just feels odd to differentiate the $F_i$'s by the $F_1$'s.

Any form of help will be appreciated as I am doing my thesis now and struggling :/