00M9 Proof. By assumption, L⊗ML^{\otimes M} is very ample. So V^∙(M)(L,ϕ)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi), V^∙(M)(LX|Y,ϕX|Y)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y}) and V^∙(M)(L|Y,ϕ|Y)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y}) are affinoid algebras. Since V∙(L)V_{{\scriptscriptstyle\bullet}}(L) is integral and is finite over V∙(M)(L)V_{{\scriptscriptstyle\bullet}}^{(M)}(L), by Proposition 2.45, the Banach algebras V^∙(L,ϕ)\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi), V^(LX|Y,ϕX|Y)\widehat{V}(L_{X|Y},\phi_{X|Y}) and V^∙(L|Y,ϕ|Y)\widehat{V}_{{\scriptscriptstyle\bullet}}(L|_{Y},\phi|_{Y}) are Banach finite over V^∙(M)(L,ϕ)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L,\phi), V^∙(M)(LX|Y,ϕX|Y)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L_{X|Y},\phi_{X|Y}) and V^∙(M)(L|Y,ϕ|Y)\widehat{V}_{{\scriptscriptstyle\bullet}}^{(M)}(L|_{Y},\phi|_{Y}) respectively. Hence they are affinoid algebras. ∎