The present paper quantitatively examines how the process of fast reconnexion depends on the magnitude of the local resistivity enhanced in the vicinity of the magnetic neutral point. It is shown that quasi-steady Petschek-type configurations are set up, one for each of the variously imposed local resistivity enhancements. The fundamental structure of the quasi-steady configuration is largely controlled by the initially indented value of locally enhanced resistivity. It is especially remarked that the width of the diffusion region becomes smaller as the locally enhanced resistivity becomes smaller. We find that each of the quasi-steady configurations presents nothing other than the Petschek-type configuration that corresponds to the allowable maximum reconnexion rate for the relevant magnetic Reynolds number. We also see that the magnitude of fast reconnexion rate has a weak dependence on the local resistivity in the diffusion region. All our numerical results are very consistent with previous theoretical work on the fast reconnexion problem, once the problem is reconsidered from another angle. We hence suggest that the process of fast reconnexion should be viewed as a gross instability, inherent to the current sheet system itself, that can be triggered by some local onset of anomalous resistivity.