Two conditions assuring path independency in any line integral are that the integration domain be a simply connected open set and that the integrand have a symmetric Jacobain matrix in that domain. This paper provides a rigorous account of how a path-dependent line integral can retain path independency in a subset of the original domain. The implicit function theorem, intuitively, proves to be a key avenues toward the establishment of our main result when a positively homogeneous function defines the subset. The Poincare' lemma is used to derive a general proof. An application is made to the consumer surplus analysis in which such a technical concern is commonly ignored.