If formula_69, formula_70, and formula_71 is a Banach space isomorphism from "Y" onto "Z", then there exist neighbourhoods "U" of "x" and "V" of "y" and a Fréchet differentiable function "g" : "U" → "V" such that "f"("x", "g"("x")) = 0 and "f"("x", "y") = 0 if and only if "y" = "g"("x"), for all formula_72.