Simple proof of $\operatorname{Curl} F = 0$ then F is conservative

Question

Apparently this is just a simple proof with prerequisites for the domain to be a star-like domain.

See picture below for proof:

I don't however understand why

$\operatorname{Curl} F = 0$ means that we could replace:

$(\frac{\partial}{\partial \xi})F_{2}(\xi,\eta,\zeta) $

with

$(\frac{\partial}{\partial \eta})F_{1}(\xi,\eta,\zeta) $

and

$(\frac{\partial}{\partial \xi})F_{3}(\xi,\eta,\zeta) $

with

$(\frac{\partial}{\partial \eta})F_{1}(\xi,\eta,\zeta) $