Implicit function theorem proof question

[Syrus]

Homework Statement

I understand the proof of the implicit function theorem up to the point in which I have included a photo. This portion serves to prove the familiar equation for the implicit solution f(x,y) of F(x,y,z)=c. My confusion arises between equations 8.1-4 and 8.1-5 when it is stated to "consider F as a function of x and z only." Why is this at all plausible?

Homework Equations

The Attempt at a Solution

 

[Syrus]

Pardon me, I meant partial derivative f_x of the implicit solution f(x,y) of F(x,y,z).

 

[Ray Vickson]

Homework Statement

I understand the proof of the implicit function theorem up to the point in which I have included a photo. This portion serves to prove the familiar equation for the implicit solution f(x,y) of F(x,y,z)=c. My confusion arises between equations 8.1-4 and 8.1-5 when it is stated to "consider F as a function of x and z only." Why is this at all plausible?

Homework Equations

The Attempt at a Solution

Your image is much too small and is quite fuzzy---basically unreadable. Please type out the relevant line or two that contains your actual question.