For dimension n, the assertion that if f and g are locally flat embeddings of the (n - 1) sphere, S^n-1, in real n space, Rⁿ, with f (S^n-1) in the bounded component of Rⁿ - g(S^n-1), then the closed region in Rⁿ bounded by f (S^n-1) and g(S^n-1) is homeomorphic to the direct product of S^n-1 and the closed interval [0,1]; it is established for n ≠ 4.