Prove that a closed equipotenitial surface with no charge within itself must enclose an equipotential volume.

Explanation

Let's assume contradicting statement that the potential is not same insie the clsoed equipotential surface. Let the potential just inside the surface is different to that of the surface causing is a potential gradient `((dV)/(dr))`. Consequently electric field comes into existence, which is given by as `E=-(dV)/(dr)`.
Consequently field lines pointing inwards or outwards from the surface. these lines cannot be again on the surface, as the surface is equipotential. it is possible only when the other end of the field lines are originated from the charges inside. This contradict the original assumption. hence, the entire volume inside must be equipotential.