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.

AmS-TeX, 36 pages; v2: added some details of proof; v3: '4 Remarks' in Section 2 supplemented and modified, few misprints corrected