A U-shaped prism is a finite gluing of three boxes and is not simple in every coordinate direction

Each $E_i$ is the image of the unit-box construction [L1] under an invertible affine coordinate scaling followed by a translation. Applying the same affine map to its three simple descriptions and six face parametrizations preserves the graph equations, nonzero oriented-area coordinates, projected disjointness, and content-zero parameter boundaries; hence [F2] makes all three pieces elementary solid regions. Their interiors are pairwise disjoint because $E_1$ lies below the plane $y=1$ while $E_2$ and $E_3$ lie above it, and $E_2$ and $E_3$ are separated by the strip $1<x<2$.

The face of $E_1$ in the plane $y=1$ is larger than either matching face of $E_2$ or $E_3$, so it must be subdivided into three rectangles cut at $x=1$ and $x=2$; that refinement preserves the adapted presentation, because it only subdivides one existing graph face into three graph faces with disjoint projections.

For each $y\in (1,3)$ and each $z\in (0,1)$, the section of $E$ in the $x$ direction is $[0,1]\cup [2,3]$, a union of two disjoint intervals. Therefore $E$ is not simple in the $x$ direction, and the gluing clause is genuinely stronger than a single simple description.

Two of the three new rectangles on the face $y=1$ pair with the matching faces of $E_2$ and $E_3$; each pairing is a translation composed with a parameter swap, so its parameter Jacobian determinant is negative and [F1] and [F7] make it orientation-reversing.

Every other face of every piece is declared outer, so the three boxes with this subdivision and pairing data form a finite gluing whose outer presentation is exactly the boundary of $E$.

The divergence of $F$ is the constant $1$ by [F4], so [L2], [L3], [L4], and [L5] give ${\iiint }_E\mathrm{div}F=\mathrm{cont}(E)=3+2+2=7$ and the outward flux through the boundary presentation is the same number.