  For a certain family of complete modular lattices, we prove a
Jordan--H\"older--Scheier-like" theorem with no assumptions on cardinality or
well-orderedness. This family includes both lattices which are both join- and
meet-continuous, as well as the lattices of subobjects of any object in an
abelian category satisfying properties related to Grothendieck's axioms (AB5)
and (AB5*). We then give several examples of objects in abelian categories
which satisfy these axioms, including pointwise finite-dimensional persistence
modules, presheaves, and certain Pr\"ufer modules. Moreover, we show that, over
an arbitrary ring, the infinite product of isomorphic simple modules both fails
to satisfy our axioms and admits at least two composition series with distinct
cardinalities. We conclude by giving a lattice-theoretic proof that any object
which is locally finitely generated and satisfies our axioms can be expressed
as a direct sum of indecomposable subobjects. We conjecture that this
decomposition is unique.
