It's possible to prove with mathematical terms using a part of the physics knowledge this general assumption? I have several formulas in mind but not a general and big picture for this property of the integrals, and the most important part is that i can't find a rigorous explanation for that.
Sorry, but your question does not make sense. First we'll have to fix the term "integral" we are talking about.
Then we'd have to fix the physical theory we talk about, like classical mechanics. A potential function in classical mechanics is a function from physical space and time to the reals. The (negative of the) gradient of this function is the force, the line integral (Riemann integral) of this function (along a path between two fixed points) is proportional to the energy (difference of a point mass located at the two fixed points) with respect to this force.
In a static situation where the force does not change with time, but does depend on space coordinates only, the energy difference of two points must be independent of the path we choose to integrate along. This is a condition (on the potential function) that is necessary in order for the physical interpretation to make sense. If a function does not satisfy this condition, we cannot accept it as a representation of a potential in our physical theory (classical mechanics in this case).
Am I close to anything you were thinking of?