By the assumption, the elements of $V_{1}(L)$ induces an embedding

such that $\iota_{1}^{*}\mathscr{O}(1)=L$ with $\mathrm{dim}_{k}V_{1}=d_{1}+1$. Moreover there exists a norm $\lVert\mathord{\cdot}\rVert_{1}$ on $V_{1}(L)$ such that $\phi=\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})$. View $\lVert\mathord{\cdot}\rVert_{1}$ as a norm on $V_{1}(\mathscr{O}(1))$, we get a metric $\psi=\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{1})$ on $\mathscr{O}(1)$. By construction $\psi|_{X}=\phi$.

By Proposition 5.6, the Banach algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(\mathscr{O}(1),\vvvert\mathord{\cdot}\vvvert_{\psi})$ is an affinoid algebra. Hence the quotient Banach algebra $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\vvvert\mathord{\cdot}\vvvert_{\psi,\mathbb{P}^{d_{1}}|X})$ is an affinoid algebra.

By Proposition 3.27, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is the spectral norm of $\vvvert\mathord{\cdot}\vvvert_{\psi,\mathbb{P}^{d_{1}}|X}$. The later is an affinoid norm, hence is equivalent to its spectral norm by Proposition 2.58. So $\vvvert\mathord{\cdot}\vvvert_{\phi}$ is also an affinoid algebra norm. Thus $\widehat{V}_{{\scriptscriptstyle\bullet}}(L,\phi)$ is an affinoid algebra.

Similarly, by Proposition 3.27, on $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$, the algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ is the spectral norm of $\vvvert\mathord{\cdot}\vvvert_{\phi,X|Y}$, hence is itself an affinoid algebra norm. ∎

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

By Corollary 5.8, the Banach algebra norm $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y}}$ is an affinoid algebra norm. By Proposition 2.58, there exists $C(\phi,Y)>0$ such that

Since $\vvvert\mathord{\cdot}\vvvert_{\phi_{X|Y};\mathrm{sp}}=\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}}$ by Corollary 3.27, one gets the bounds. ∎