Remap error separates source, target and geometry approximation

Assume Countable Choice. Let $\widehat{\mathrm{\Omega }}$ be bounded, let $\mathrm{\Phi },{\mathrm{\Phi }}_h$ be the geometry maps of Curved-domain geometry approximation errors and assume each is a global orientation-preserving $C^1$ diffeomorphism from $\widehat{\mathrm{\Omega }}$ onto its image, with $J_h\le J^{\ast }$ and $u$ is $L_u$-Lipschitz on the union of their images. On ${\mathrm{\Omega }}_h={\mathrm{\Phi }}_h(\widehat{\mathrm{\Omega }})$ let $\mathcal{S}=(S_i{)}_{i=1}^m$ and $\mathcal{T}=(T_j{)}_{j=1}^n$ be finite families of Lebesgue-measurable sets of positive measure satisfying $\mathrm{\mid }{\mathrm{\Omega }}_h\setminus {\bigcup }_i{S}_i\mathrm{\mid }=\mathrm{\mid }{\mathrm{\Omega }}_h\setminus {\bigcup }_j{T}_j\mathrm{\mid }=0,\mathrm{\mid }{S}_i\cap S_{i^{\mathrm{\prime }}}\mathrm{\mid }=0(i\mathrm{\ne }{i}^{\mathrm{\prime }}),\mathrm{\mid }{T}_j\cap T_{j^{\mathrm{\prime }}}\mathrm{\mid }=0(j\mathrm{\ne }{j}^{\mathrm{\prime }})\mathrm{.}$ Thus each family is a finite measurable partition up to null sets as required by Intersection-volume conservative remapping. Let $v_S$ be source-piecewise-constant data with finite $L^2$ error. Assume the exact target projection error below is finite. Define $P_{T}q$ on each target cell by its exact cell average, ${\eta }_S=\mathrm{\parallel }u-v_S{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_h)}$, and ${\eta }_T=\mathrm{\parallel }u-P_{T}u{\mathrm{\parallel }}_{L^2({\mathrm{\Omega }}_h)}$. If $R$ is the exact overlap remap, then $\mathrm{\parallel }u\circ \mathrm{\Phi }-(R{v}_S)\circ {\mathrm{\Phi }}_h{\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }},J_h)}\le L_u{e}_0\sqrt{J^{\ast }\mathrm{\mid }\widehat{\mathrm{\Omega }}\mathrm{\mid }}+{\eta }_S+{\eta }_T,$ where $e_0=\mathrm{\parallel }\mathrm{\Phi }-{\mathrm{\Phi }}_h{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\parallel }z{\mathrm{\parallel }}_{L^2(\widehat{\mathrm{\Omega }},J_h)}^2={\int }_{\widehat{\mathrm{\Omega }}}\mathrm{\mid }z{\mathrm{\mid }}^2{J}_h$.