The Type II boundary identity for the Q dy term

The horizontal endpoint arcs in [L1] have constant $y$, so their contributions to $\int Qdy$ are zero. The right graph contributes ${\int }_c^{d}Q(\rho (y),y)dy$, and [L2] makes the downward left graph contribute $-{\int }_c^{d}Q(\lambda (y),y)dy$.

For each fixed $y$ with $\lambda (y)<\rho (y)$, [L4] in the $x$ variable gives $Q(\rho (y),y)-Q(\lambda (y),y)={\int }_{\lambda (y)}^{\rho (y)}{\mathrm{\partial }}_{x}Q(x,y)dx\mathrm{.}$ Since $\lambda <\rho$ on $(c,d)$, this covers every interior $y$. At $y=c$ and $y=d$ the region definition requires only $\lambda (y)\le \rho (y)$, so both cases occur: where $\lambda (y)<\rho (y)$, as for a rectangle, the same application of [L4] applies verbatim, and where $\lambda (y)=\rho (y)$ both sides are zero. Hence the displayed identity holds for every $y\in [c,d]$.

By [L5] the data satisfy $c<d$ and $\lambda \le \rho$ continuous on $[c,d]$, and $D$ is compact. The coordinate swap $\sigma (x,y)=(y,x)$ is linear and satisfies $\sigma \circ \sigma =\mathrm{i}\mathrm{d}$, and $\sigma (D)=\{(u,v):c\le u\le d,\lambda (u)\le v\le \rho (u)\}$ is the region between the graphs of $\lambda$ and $\rho$ over the first coordinate, so [L6] applies to it with $a=c$, $b=d$, $\alpha =\lambda$, $\beta =\rho$ and makes it compact and Jordan measurable. Since $\sigma (D)$ is therefore a bounded Jordan set, [L7] makes its image $D=\sigma (\sigma (D))$ a bounded Jordan set as well. The hypothesis of [L6], that the region lies between two graphs over an interval of its FIRST coordinate, is verified for $\sigma (D)$ and is never asserted of $D$.

Hence ${\int }_{\mathrm{\partial }D}Qdy={\int }_c^d(Q(\rho (y),y)-Q(\lambda (y),y))dy\mathrm{.}$

$Q$ is $C^1$ on an open neighbourhood of $D$, so ${\mathrm{\partial }}_{x}Q$ is continuous on $D$; with step 1.3 this makes $D$ a compact Jordan measurable set, and [L8] makes ${\mathrm{\partial }}_{x}Q$ integrable over it. For $y\in [c,d]$ the section $\{x:(x,y)\in D\}$ is the compact interval $[\lambda (y),\rho (y)]$, whose boundary is at most two points, so it is Jordan measurable in $\mathbb{R}$; the restriction $x\mapsto {\mathrm{\partial }}_{x}Q(x,y)$ is continuous there, so [L8] makes it integrable over that section. Every section at $y\mathrm{\notin }[c,d]$ is empty. The exceptional set of [L3] may therefore be taken empty.

By steps 1.3 and 2.2, the symmetric-coordinate assertion of [L3] applies to $E=D$ and $g={\mathrm{\partial }}_{x}Q$ with $y$ as the outer coordinate, so ${\iint }_D{\mathrm{\partial }}_{x}QdA={\int }_c^d({\int }_{\lambda (y)}^{\rho (y)}{\mathrm{\partial }}_{x}Q(x,y)dx)dy,$ the outer integrand vanishing off $[c,d]$ because those sections are empty. Substituting step 1.2 into step 2.1 gives the same iterated integral for ${\int }_{\mathrm{\partial }D}Qdy$.

Zero-length endpoint arcs and piecewise-$C^1$ joins contribute nothing beyond subdivision, by [L1] and [L2].