Detecting topological groups which are (locally) homeomorphic to LF-spaces

We prove that a topological group G   is (locally) homeomorphic to an LF-space if G=⋃n∈ωGnG=⋃n∈ωGn for some increasing sequence of subgroups (Gn)n∈ω(Gn)n∈ω such that(1)for any neighborhoods Un⊂GnUn⊂Gn, n∈ωn∈ω, of the neutral element e∈Gn⊂Ge∈Gn⊂G, the set ⋃n=1∞U0U1⋯Un is a neighborhood of e in G;(2)each group GnGn is (locally) homeomorphic to a Hilbert space;(3)for every n∈Nn∈N the quotient map Gn→Gn/Gn−1Gn→Gn/Gn−1 is a locally trivial bundle;(4)for infinitely many numbers n∈Nn∈N each Z  -point in the quotient space Gn/Gn−1={xGn−1:x∈Gn}Gn/Gn−1={xGn−1:x∈Gn} is a strong Z-point.