On non-orientable surfaces embedded in 4-manifolds

We find conditions under which a non-orientable closed surface smoothly embedded into an orientable 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.