Prove that a closed equipotential surface with no charge within itself, must enclose an equipotential value

Let’s assume contradicting statement that the potential is not same inside, the closed equipotential surface. Let the potential.just inside the surface be different to that on the surface having a potential
gradient (dV/dr).Consequently, electric field comes into existence, which is given by, E = -dV/dr
Consequently, field lines point inwards or outwards from the surface. These lines cannot be formed 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 contradicts the original assumption. Hence, the entire volume inside must be equipotential.