as far as i can tell, a state function is required to be something whose integral is independent of the path chosen for the integral, this is from searchign with google, and appears to be a thermodynaics thing. if we assume this is the definition of a state function then it is clear why any integral over a loop is zero since we may split the loop into two parts and consider the loop as a path from A to B followed by a path from B to A and the integral over a path from B to A is minus the integral we get if we went along that path backwards so they must add up to zero.
if you're asking why state fucntions are path independent then could i have your definition of state function.
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