The closed unit box, with its six faces, is an elementary solid region

Each of the six maps is affine, so its two parameter derivatives are the constant standard basis vectors ${\phi }_{z^{+},u}=e_x$, ${\phi }_{z^{+},v}=e_y$; ${\phi }_{z^{-},u}=e_y$, ${\phi }_{z^{-},v}=e_x$; ${\phi }_{x^{+},u}=e_y$, ${\phi }_{x^{+},v}=e_z$; ${\phi }_{x^{-},u}=e_z$, ${\phi }_{x^{-},v}=e_y$; ${\phi }_{y^{+},u}=e_z$, ${\phi }_{y^{+},v}=e_x$; ${\phi }_{y^{-},u}=e_x$, ${\phi }_{y^{-},v}=e_z$. Computing each cross product from [F1] gives $e_x\times e_y=(0,0,1)=e_z$, $e_y\times e_x=-e_z$, $e_y\times e_z=(1,0,0)=e_x$, $e_z\times e_y=-e_x$, $e_z\times e_x=(0,1,0)=e_y$ and $e_x\times e_z=-e_y$, so the six oriented area vectors are the constants $e_z,-e_z,e_x,-e_x,e_y,-e_y$ as displayed.

The three quadruples $(z,[0,1{]}^2,0,1)$, $(x,[0,1{]}^2,0,1)$ and $(y,[0,1{]}^2,0,1)$, with constant graph functions, are simple descriptions of $B$ in the three directions in the sense of [F4]: the base $[0,1{]}^2$ is compact, Jordan measurable and has nonempty interior; the constants $0<1$ satisfy the weak and the strict inequality; and in each case $\{p:{\pi }_k(p)\in [0,1{]}^2,0\le p_k\le 1\}$ is exactly $[0,1{]}^3$, because ${\pi }_k$ lists the two coordinates other than the $k$th.

Each of the six pairs $(S,\phi )$ is a regular patch in the sense of [F2]: the square $S$ is compact and Jordan measurable, being a rectangle, and is the closure of its nonempty convex, hence connected, interior $(0,1{)}^2$; each $\phi$ is affine and therefore $C^1$ on all of ${\mathbb{R}}^2$; each oriented area vector is a nonzero constant by step 1.1; and each $\phi$ is injective on ${\mathbb{R}}^2$, since its two direction vectors are distinct standard basis vectors and reading the two matching coordinates of the image recovers $(u,v)$, so in particular no interior parameter point shares its image with a distinct point of $S$.

Take ${\mathrm{\Sigma }}_z^{+}=({\phi }_{z^{+}})$, ${\mathrm{\Sigma }}_z^{-}=({\phi }_{z^{-}})$ and ${\mathrm{\Sigma }}_z^0=({\phi }_{x^{+}},{\phi }_{x^{-}},{\phi }_{y^{+}},{\phi }_{y^{-}})$. The image of ${\phi }_{z^{+}}$ is the face $p_z=1$, the graph of ${\gamma }_2=1$, and by step 1.1 the $z$ coordinate of its oriented area vector is $1>0$; the image of ${\phi }_{z^{-}}$ is the graph of ${\gamma }_1=0$ with $z$ coordinate $-1<0$; and the four lateral vectors $\pm e_x,\pm e_y$ have $z$ coordinate $0$, so that coordinate vanishes on the whole parameter square. The projected image ${\pi }_z[{\phi }_{z^{+}}[(0,1{)}^2]]$ is $(0,1{)}^2$, whose complement in $[0,1{]}^2$ is $\mathrm{\partial }([0,1{]}^2)$ of content zero by [F8], and likewise for ${\phi }_{z^{-}}$, where ${\pi }_z(v,u,0)=(v,u)$; each graph sublist is a single patch, so pairwise disjointness is vacuous, and both are nonempty. So the presentation is adapted to the $z$ description of step 1.2, in the sense of [F5].

Take ${\mathrm{\Sigma }}_x^{+}=({\phi }_{x^{+}})$, ${\mathrm{\Sigma }}_x^{-}=({\phi }_{x^{-}})$ and ${\mathrm{\Sigma }}_x^0=({\phi }_{z^{+}},{\phi }_{z^{-}},{\phi }_{y^{+}},{\phi }_{y^{-}})$. By step 1.1 the $x$ coordinates of the oriented area vectors are $1$ for ${\phi }_{x^{+}}$, $-1$ for ${\phi }_{x^{-}}$ and $0$ for the other four. The image of ${\phi }_{x^{+}}$ is the face $p_x=1$ and ${\pi }_x(1,u,v)=(u,v)$, so its projected image is $(0,1{)}^2$; the image of ${\phi }_{x^{-}}$ is $p_x=0$ and ${\pi }_x(0,v,u)=(v,u)$, again with projected image $(0,1{)}^2$. Both complements in the base are $\mathrm{\partial }([0,1{]}^2)$, of content zero. So the same presentation is adapted to the $x$ description.

Take ${\mathrm{\Sigma }}_y^{+}=({\phi }_{y^{+}})$, ${\mathrm{\Sigma }}_y^{-}=({\phi }_{y^{-}})$ and ${\mathrm{\Sigma }}_y^0=({\phi }_{z^{+}},{\phi }_{z^{-}},{\phi }_{x^{+}},{\phi }_{x^{-}})$. By step 1.1 the $y$ coordinates of the oriented area vectors are $1$ for ${\phi }_{y^{+}}$, $-1$ for ${\phi }_{y^{-}}$ and $0$ for the other four. The image of ${\phi }_{y^{+}}$ is the face $p_y=1$ and ${\pi }_y(v,1,u)=(u,v)$, so its projected image is $(0,1{)}^2$; the image of ${\phi }_{y^{-}}$ is $p_y=0$ and ${\pi }_y(u,0,v)=(v,u)$, with projected image $(0,1{)}^2$. So the same presentation is adapted to the $y$ description.

The six images are the six closed faces of $B$, each contained in $\mathrm{\partial }B$, and their union is $\mathrm{\partial }B$ by [F7], since a point of $B$ fails to be interior exactly when one of its coordinates is $0$ or $1$. Two distinct faces meet in a closed edge, a vertex or the empty set; the preimage of such an intersection in either parameter square is contained in $\mathrm{\partial }S$, which has content zero by [F8], so the overlap condition of [F3] holds. Interior parameter points map into the six open faces, which are pairwise disjoint, so no point of an overlap is the image of an interior parameter point of two distinct patches and the normal-agreement condition of [F3] holds with nothing to check. Hence the six patches form a compatible finite patch presentation of $\mathrm{\partial }B$.

Steps 2.1 and 3.1 make the six patches a compatible finite patch presentation of $\mathrm{\partial }B$, step 1.2 supplies the three simple descriptions, and steps 2.2, 2.3 and 2.4 make that one presentation adapted in all three directions. By [F6] the box $B$, with these data, is an elementary solid region.