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This paper gives a proof of a sharpened version of the conjecture in . Let be -dimensional complex vector space and let be the vector of branch admittances of an analog network. A subset of is said to be ample if (i) its complement has Lebesgue measure zero, (ii) it is open, and (iii) it is dense. The sharpened version of the conjecture claims that the -node fault testability condition  is satisfied on an ample subset of values of , if, and only if, for any set of inaccessible nodes, there are at least nodes in (complement of ) each of which is connected with via a branch. This is extremely powerful because the result depends only on the topology of a network and the condition can be checked by inspection. The proof justifies the fault location method developed in .