  This article is centered around generalizing a previous implicit function
theorem of the author to be applicable for maps f:E sqcap F to F which can be
lifted to Keller C^k_pi maps f_i:E sqcap F_i to F_i with F_i Banach and
F=projlim F_i . We prove theorems about existence and differentiability of
functions g satisfying f(x,g(x))=b constant. In addition to these abstract
theorems, we give several examples of applications to proving smooth dependence
of the solution on initial/boundary values and the nonlinearity in nonlinear
(partial) differential equations.
