Local behavior of mappings of metric spaces with branching

The local behavior of mappings with the inverse Poletsky inequality between metric spaces is studied. The case where one of the spaces satisfies the condition of weak sphericalization, is similar to the Riemannian sphere (extended Euclidean space), and is locally linearly connected under a mapping is considered. It is proved that the equicontinuity of the corresponding families of mappings of two domains, one of which is a domain with a weakly flat boundary, and another one is a fixed domain with a compact closure, the corresponding weight in the main inequality being supposed to be integrable.