The little-known Pappus's area theorem in Euclidean plane geometry can “be thought of as a generalization of the Pythagorean theorem”. However, the Problem 34 in Nicholas Donat Kazarinoff's Geometric Inequalities stated that this Pappus' Theorem may be generalized to three dimensions (which can also be found in the first page of his paper D. K. Kazarinoff's inequality for tetrahedra) as follows:
Let $S$ be a tetrahedron, and construct three triangular prisms which have a common lateral edge and which have for their bases three faces of $S$, and of which all or none lie entirely outside of $S$. Then, if one constructs a fourth triangular prism on the remaining face of $S$ whose lateral edges are translates of the common lateral edge of the first three prisms, the sum of the volumes of the first three prisms is equal to the volume of the fourth prism.
Nevertheless, neither proof nor provenance was provided in the monograph or paper. I attempted to follow the proof idea of the original version; unfortunately, I found it difficult to visualize the corresponding solid figure (in particular, in the second case). So, does there exist some rigorous proof of such a 3D generalization?