The proof is immediate given some results that we have not yet considered but soon will.
We want to recover the potential from its gradient which we are given; we can construct a potential function at the point r in R by integrating from some reference point r' in R, once we define integration along arbitrary paths t, in three dimensional space.
We will then be able to deduce that the potential so defined is unique up to a constant (independent of the path chosen to define it) within the region R  from  Stokes' Theorem .