  Smooth Equations of the form G[z]=0 are investigated in Banach spaces with
the aim of continuing the basic solution G[0]=0 to a solution curve of G[z]=0
with the implicit function theorem. If the linearization is surjective, then
the transversality condition of the implicit function theorem can be satisfied
in a straightforward way, yielding a regular solution curve, whereas otherwise
the equation G[z]=0 has to be extended appropriately for reaching a surjective
linearization accessible to the implicit function theorem. This extension
process, implying in the first step the standard bifurcation theorem of simple
bifurcation points, is continued arbitrarily, yielding a sequence of
bifurcation results presumably being applicable to bifurcation points with
finite degeneracy.
