By Proposition 3.26, $\mathcal{P}(\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}})$ is equal to $\phi|_{Y}$, so it is continuous. Hence for any $\epsilon>0$, by Proposition 3.23, one has

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))=\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}))\subseteq\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon)))).$$

By Proposition 2.88, the topology on $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ coincides the induced topology from $\spec(V_{{\scriptscriptstyle\bullet}}(L|_{Y}))^{\mathrm{an}}$, hence the set $\text{Int}^{\mathrm{top}}_{V_{{\scriptscriptstyle\bullet}}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$ which is open in $\spec(V_{{\scriptscriptstyle\bullet}}(L|_{Y}))^{\mathrm{an}}$ is also open in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$. Therefore this set is contained in $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$. ∎

By Corollary 2.64, every point in $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon)_{X|Y}^{\mathrm{aff}}))$ has a neighbourhood system consisting of affinoid domains. Hence for any $z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$, there exists an affinoid domain neighbourhood $V(z)$. By Lemma 4.1, $z$ has an open neighbourhood $\text{Int}^{\mathrm{top}}_{\mathfrak{M}}(\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon))))$, so we can assume that each $V(z)$ is contained in this open set.

One forms a covering by open sets

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\subseteq\bigcup_{z\in\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))}\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z).$$

Since the left hand side is a compact set by Proposition 2.21, there exist finitely many points $\{z_{1},\dots,z_{m}\}$ such that $\{\text{Int}^{\mathrm{top}}_{\mathfrak{M}}V(z_{i})\}_{i\in\{1,\dots,m\}}$ form a covering of $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$. Let $W_{\epsilon}$ be the union of affinoid domains $\{V(z_{i})\}_{i\in\{1,\dots,m\}}$, then it is a special domain, and satisfies the desired inclusion conditions. ∎

By Proposition 3.5, there are homomorphisms of Banach algebras induced by the identity map on the dense $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$:

$$\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y}^{\mathrm{aff}})\xrightarrow[\sigma(\epsilon/2)|_{Y}]{}\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y})\xrightarrow[\iota(\epsilon/2)]{}\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}).$$

Let $\tau(\epsilon/2)$ denote the composed homomorphism of Banach $k$-algebras. It is a homomorphism from an affinoid algebra to a Banach algebra. It has dense image, so by Proposition 2.27 the induced continuous map

$$\tau(\epsilon/2)^{*}:\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))\rightarrow\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y}^{\mathrm{aff}}))$$

is injective and is closed. As both spaces are compact and Hausdorff, this map is a homeomorphism from its domain to its image. So the homomorphism spectrum $\Sigma_{\tau(\epsilon/2)}$ is homeomorphic to $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y}))$, and is contained in $W_{\epsilon/2}$.

One performs spectral calculus for the homomorphism $\tau(\epsilon/2)$ and the special domain $W_{\epsilon/2}$: by Theorem 2.81, there exist a homomorphism of Banach $k$-algebras

$$\theta(\epsilon/2)_{W_{\epsilon/2}}:\mathcal{W}_{\epsilon/2}\rightarrow\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi_{X|Y})$$

which extends the identity map on the dense sub-$k$-algebra $V_{{\scriptscriptstyle\bullet}}(L_{X|Y})$. ∎

Start from the homomorphism $\theta(\epsilon/2)_{W_{\epsilon/2}}$ constructed in Proposition 4.4. The boundedness (continuity) of this homomorphism of Banach $k$-algebras implies that there exists $C_{\epsilon}>0$ such that

$$\forall\underline{t}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\quad\vvvert\underline{t}\vvvert_{\phi,X|Y}\leq C_{\epsilon}\cdot\vvvert\underline{t}\vvvert_{W_{\epsilon/2}}=C_{\epsilon}\cdot\sup_{z\in W_{\epsilon/2}\cdot}\lvert\underline{t}\rvert_{z}.$$

By Proposition 3.26, one has a canonical homeomorphism

$$\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi(\epsilon/2)_{X|Y}))\simeq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon/2)))$$

from which one deduces

$$W_{\epsilon/2}\subseteq\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon/2))).$$

Remember that since $\vvvert\mathord{\cdot}\vvvert_{\phi|_{Y}(\epsilon/2)}$ is power-mutliplicative, by Theorem 2.30, it is the supremum norm on $\mathfrak{M}(\widehat{V}_{{\scriptscriptstyle\bullet}}(L_{X|Y},\phi|_{Y}(\epsilon/2)))$. Hence by comparing supremum norms on these two closed sets, we get

$$\forall\underline{t}\in V_{{\scriptscriptstyle\bullet}}(L_{X|Y}),\quad\vvvert\underline{t}\vvvert_{W_{\epsilon/2}}\leq\vvvert\underline{t}\vvvert_{\phi|_{Y}(\epsilon/2)},$$

therefore for any $t_{n}\in V_{n}(L_{X|Y})$, one has

$$\begin{split}\lVert t_{n}\rVert_{n\phi,X|Y}&=\vvvert t_{n}\vvvert_{\phi,X|Y}\\ &\leq C_{\epsilon}\cdot\vvvert t_{n}\vvvert_{W_{\epsilon/2}}\\ &\leq C_{\epsilon}\cdot\vvvert t_{n}\vvvert_{\phi|_{Y}(\epsilon/2)}=C_{\epsilon}\cdot\mathrm{e}^{n\epsilon/2}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}.\end{split}$$

Let $n_{Y}$ be the integer $\max\{\lceil\log(C_{\epsilon})/(\epsilon/2)\rceil,M\}$, then for any $n\geq n_{Y}$ and any $t_{n}\in V_{n}(L|_{Y})$, there exists $s_{n}\in V_{n}(L)$ such that

$$\lVert s_{n}\rVert_{n\phi}\leq\mathrm{e}^{n\epsilon}\cdot\lVert t_{n}\rVert_{n\phi|_{Y}}.$$

∎