  This paper develops an infinitesimal order of magnitude coupled with overflow
technique that allows nonnumerical proofs of nondegenerate and degenerate
inverse mapping theorems for mappings minimally regular at a point. This
approach is used first to give a transparent proof of the inverse mapping
theorem of Behrens and Nijenhuis and then is deployed to prove an inverse
mapping result for mappings whose linear part vanishes at the differentiable
point. We finish by indicating further possible capacities of this approach.
