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The aim of this paper is to give conditions fer the structural stability of C^∞-proper mappings from 2-Manifolds into 2 Manifolds, by using Mather's Theorem. So, we have obtained these following necessary conditions for stability of f : M² → N² : 1 - f must be an excellent mappinga that is, it has only regular, fold and cusp points. 2 - There is no triple singular value. 3 - A double value cannot be the image of two different cusp points. 4 - If two fold points have the same value, "f is transversal to the image of the general fold", on each of them. Then, by using "jets and singularities", it is proved that "proper" is not necessary in the above case 1. Finally, we have also proved that an excellent mapping has the property of local infinitesimal stability on a neighbourhood of each point. Remark: We were informed later, that more general results have been obtained very recently on the subject in consideration. But we are not able to give more precise information on this matter.