The edge splittings and connectivity augmentation theorems in directed hypergraphs (dypergraphs) were proved. A dypergraph comprised of a finite collection of hyperedges and hypervertices. The theorem proving process involved extension of the results obtained by Mader and Frank on directed graphs. The equalities in the lemmas were proved by counting the contribution of an edge to the sides.
Berg, A. R., Jackson, B., & Jordán, T. (2003). Edge splitting and connectivity augmentation in directed hypergraphs. Discrete Mathematics, 273(1–3), 71–84. https://doi.org/10.1016/s0012-365x(03)00229-2