Internal faces cancel and volume integrals add when elementary solid regions are glued

By [F1] the sum ${\sum }_{i=1}^N{\iint }_{{\mathrm{\Sigma }}_i}⟨G,n⟩$ is the sum of the flux of $G$ over every patch of every ${\mathrm{\Sigma }}_i$, a finite list of real numbers. By [F2] each entry of that list is designated internal or outer.

Each $E_i$ is compact and Jordan measurable by [F4], so $E={\bigcup }_i{E}_i$ is closed and bounded, hence compact by [L6], and each $\mathrm{\partial }{E}_i$ has content zero by [L3]. If $p\in \mathrm{\partial }E$ then $p\in E$, so $p\in E_i$ for some $i$, and $p$ cannot lie in $E_i^{\circ }$, since $E_i\subseteq E$ would then put $p$ in $E^{\circ }$; so $p\in \mathrm{\partial }{E}_i$ and $\mathrm{\partial }E\subseteq {\bigcup }_i\mathrm{\partial }{E}_i$. Concatenating the $N$ finite covers shows that union has content zero, so $E$ is Jordan measurable by [L3] and [F5].

Let $(D,\phi )$ and $(D^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }})$ be a paired internal pair, so by [F2] and [F3] there is a $C^1$ diffeomorphism $h$ between neighbourhoods of $D^{\mathrm{\prime }}$ and $D$ with $h[D^{\mathrm{\prime }}]=D$, ${\phi }^{\mathrm{\prime }}=\phi \circ h$ and $\det Dh<0$; that is an orientation-reversing regular reparametrization. So [L1] gives that the flux of $G$ over $(D^{\mathrm{\prime }},{\phi }^{\mathrm{\prime }})$ is the negative of its flux over $(D,\phi )$, and the two contributions to the sum of step 1.1 add to $0$. The pairing of [F2] is an involution without fixed points, so the internal entries of the list are exhausted by such pairs.

By [L5] the continuous $H$ is bounded on the nonempty compact $E$, and by [L4] it is integrable over $E$ and over each compact Jordan $E_i$. If $i\mathrm{\ne }j$ and $p\in E_i\cap E_j$ then $p$ lies in at most one of the two interiors, so $E_i\cap E_j\subseteq \mathrm{\partial }{E}_i\cup \mathrm{\partial }{E}_j$, which has content zero by step 1.2 and [F5]; and $E\setminus {\bigcup }_i{E}_i$ is empty, hence of content zero. So [L2] applies with $A=E$ and $A_i=E_i$ and gives ${\sum }_i{\int }_{E_i}H={\int }_{E}H$, the integrals being those of [F6].

Deleting the cancelling internal pairs of step 2.1 from the finite sum of step 1.1 leaves exactly the outer entries, whose sum is ${\iint }_{{\mathrm{\Sigma }}^{\mathrm{o}\mathrm{u}\mathrm{t}}}⟨G,n⟩$ by [F1] and [F2]. This is a rearrangement of finitely many reals, so it needs no connectedness of $E$ or of its boundary; for $N=1$ there is no internal patch and the two lists coincide.

Steps 3.1 and 2.2 are the two asserted identities, and step 1.2 is the assertion that $E$ is compact and Jordan measurable.