  We find conditions under which a non-orientable closed surface S embedded
into an orientable closed 4-manifold X can be represented by a connected sum of
an embedded closed surface in X and an unknotted projective plane in a
4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem
and the Auckly-Kim-Melvin-Ruberman-Schwartz "one is enough" theorem to the case
of non-orientable surfaces.
