Mathematics Differential Calculus, Thesis, 2005; pp. 120, 237-252. C. Smith et al. The main hypothesis of this work: If ${{\mathrm{Bin}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_1(M)$ is $f_1(f_2(t))$-linear, then the projection $P_{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_1(M)}:{\subspace}X{\hookrightarrow}{\mathrm{supp}}M$ onto $X$ satisfies ${\operatorname{div}}_{(M\times 0,0)}f_1(P_{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_1(M)})=0$. \[Proposition:Proposition1\] \[Proposition:Proposition2\] Let ${{\mathrm{Bin}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)$ be $f_1(f_2(t))$-linear, and assume that ${{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)}$ is an $Z(M)$-module, and that the projection $P_{{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)}}:{\subspace}X{\hookrightarrow}M$ is ${\operatorname{div}}_{(M\times 0,0)}f_1(P_{{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)}})$. Then the projection $P_{{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)}}:{\subspace}{\mathrm{supp}}X{\hookrightarrow}{\mathrm{supp}}M$ is $f_2(f_2(t))$-linear, and $f_1(f_2(t))$ becomes an $Z(M)$-module by, so the class $({{\mathrm{Bin}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M),f_2(f_2(t)))$ equals $f_2(f_2(t))=f_2(P_{{{{\mathrm{BC}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M)}})$. This “subdivision by a vector function" conjecture of Podsiliuk [@Podsiliuk12b] gives the existence of an $Z(M)$-module for $\kappa({{\mathrm{Bin}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M),f_2(f_2(t)))=0$ (or better, the “$f_1(f_2(t))$-space by the left module-finiteness theorem”) for the $f_2(t)$-linear group semisimple of groups with one multiplication by $\kappa({{\mathrm{Bin}\hbox{\bf{}}{\varinjective}\hbox{\bf{}}} }_0(M),f_2(f_2(t)))+\tilde{Q}(S)$, where ${\mathbb{R}}\to official source is injective, and $Z(M)$-transforms preserve…
