In page-50 of I.E irodov Basic Laws of Electromagnetism, he derives the relation for the force on surface of a conductor by every other charge distributed across the rest of the surface area.
The proof goes as follows: Let $E_{\sigma}$ be the intensity of field created by the charge on surface area, let $ E_0$ be the field created by rest of the charges.
Inside the conductor, for electrostatic equilibrium:
$$ E_0 - E_{\sigma} = 0 \tag{1}$$
Near the surface:
$$ E = E_0 + E_{\sigma} = 2E_o \tag{2}$$
After some more algebra, we can write:
$$F= \frac{\epsilon_o E^2}{2} \hat{n}$$
My doubt is how we can we say that the fields caused by each surface area portion under consideration for inside portion and outside portion are equal in magnitude?
As in, how do we know:
$$ E_{\sigma inside} = E_{\sigma outside}$$
