Prove this mean value theorem: For charge-free space in the electrostatic limit, the value of the electrostatic potential φ at any point in space is equal to the average of the potential over the surface of any sphere centered on that point. Hint: Use the fact that where there are no charges ∇2φ=0. Then relate the averge φ over the surface of a shpere to the flux of electric field through the surface. Functions which satisfy Laplace's equation are called harmonic functions; harmonic functions obey the above mean value theorem (in other words, you have to prove this mean value theorem, you cannot simply say it is true because harmonic functions obey it!). I need full solution with equations for this problem using guass's law

Prove this mean value theorem: For charge-free space in the electrostatic limit, the value of the electrostatic potential φ at any point in space is equal to the average of the potential over the surface of any sphere centered on that point. Hint: Use the fact that where there are no charges ∇2φ=0. Then relate the averge φ over the surface of a shpere to the flux of electric field through the surface. Functions which satisfy Laplace's equation are called harmonic functions; harmonic functions obey the above mean value theorem (in other words, you have to prove this mean value theorem, you cannot simply say it is true because harmonic functions obey it!). I need full solution with equations for this problem using guass's law