  From the point of view of unification of differentiation theory, it is of
interest to note that the general construction principle of Bertram, Gloeckner
and Neeb leading to a C^k differentiability concept from a given C^0 one,
besides subsuming the Keller--Bastiani C_c^k differentiabilities on real
Hausdorff locally convex spaces, also does the same to the "arc-generated"
interpretation of the Lipschitz theory of differentiation by Frolicher and
Kriegl, and likewise to the "compactly generated" theory of Seip's continuous
differentiabilities. In this article, we give the details of the proof for the
assertion concerning Seip's theory. We also give an example indicating that the
premises in Seip's various inverse and implicit function theorems may be too
strong in order for these theorems to have much practical value. Also included
is a presentation of the BGN--setting reformulated so as to be consistent with
the Kelley--Morse--Godel--Bernays--von Neumann type approach to set theory, as
well as a treatment of the function space constructions and development of
their basic properties needed in the proof of the main result.
