Proof of (E). We have to prove assumptions (1), (3), (4), (6) of Situation \ref{situation-bootstrap} hold for $A, I, T, M$. Shrinking $T$ to $T'$ weakens assumption (6) and strengthens assumption (4). However, if we have $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ with $\mathfrak p \not \in V(I)$, $\mathfrak r \in V(I) \setminus T'$, $\mathfrak q \in T'$ as in assumption (4) for $A, I, T', M$, then either we can pick $\mathfrak r \in V(I) \setminus T$ and condition (4) for $A, I, T, M$ kicks in or we cannot find such an $\mathfrak r$ in which case we get $\text{depth}(M_\mathfrak p) > s$ by Lemma \ref{lemma-helper-bootstrap}. This proves (4) holds for $A, I, T', M$ as desired. \medskip\noindent Proof of (F). This is straightforward and we omit the details. \medskip\noindent Proof of (G). We have to prove assumptions (1), (3), (4), (6) of Situation \ref{situation-bootstrap} hold for the $I$-adic completions $A', I', T', M'$. Please keep in mind that $\Spec(A') \to \Spec(A)$ induces an isomorphism $V(I') \to V(I)$. \medskip\noindent Assumption (1): The ring $A'$ has a dualizing complex, see Dualizing Complexes, Lemma \ref{dualizing-lemma-ubiquity-dualizing}. \medskip\noindent Assumption (3): Since $I' = IA'$ this follows from Local Cohomology, Lemma \ref{local-cohomology-lemma-cd-change-rings}. \medskip\noindent Assumption (4): If we have primes $\mathfrak p' \subset \mathfrak r' \subset \mathfrak q'$ in $A'$ with $\mathfrak p' \not \in V(I')$, $\mathfrak r' \in V(I') \setminus T'$, $\mathfrak q' \in T'$ then their images $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ in the spectrum of $A$ satisfy $\mathfrak p \not \in V(I)$, $\mathfrak r \in V(I) \setminus T$, $\mathfrak q \in T$. Then we have $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s $$ by assumption (4) for $A, I, T, M$. We have $\text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p)$ and $\text{depth}(M'_{\mathfrak p'}) + \dim((A'/\mathfrak p')_{\mathfrak q'}) = \text{depth}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q)$ by Local Cohomology, Lemma \ref{local-cohomology-lemma-change-completion}. Thus assumption (4) holds for $A', I', T', M'$. \medskip\noindent Assumption (6): Let $\mathfrak q' \in T'$ lying over the prime $\mathfrak q \in T$. Then $A'_{\mathfrak q'}$ and $A_\mathfrak q$ have isomorphic $I$-adic completions and similarly for $M_\mathfrak q$ and $M'_{\mathfrak q'}$. Thus assumption (6) for $A', I', T', M'$ is equivalent to assumption (6) for $A, I, T, M$. \medskip\noindent Proof of (A). We have to check conditions (1), (2), (3), (4), and (6) of Lemmas \ref{lemma-kill-colimit-weak-general} and \ref{lemma-kill-colimit-support-general} for $(A, I, \mathfrak a, M)$. Warning: the set $T$ in the statement of these lemmas is not the same as the set $T$ above. \medskip\noindent Condition (1): This holds because we have assumed $A$ has a dualizing complex in Situation \ref{situation-bootstrap}. \medskip\noindent Condition (2): This is empty. \medskip\noindent Condition (3): Let $\mathfrak p \subset A$ with $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$. Since $I$ is contained in the Jacobson radical of $A$ we see that $V(\mathfrak p) \cap V(I) \not = \emptyset$. Let $\mathfrak q \in V(\mathfrak p) \cap V(I)$ be a generic point. Since $\text{cd}(A_\mathfrak q, I_\mathfrak q) \leq d$ (Local Cohomology, Lemma \ref{local-cohomology-lemma-cd-local}) and since $V(\mathfrak p A_\mathfrak q) \cap V(I_\mathfrak q) = \{\mathfrak q A_\mathfrak q\}$ we get $\dim((A/\mathfrak p)_\mathfrak q) \leq d$ by Local Cohomology, Lemma \ref{local-cohomology-lemma-cd-bound-dim-local} which proves (3). \medskip\noindent Condition (4): Suppose $\mathfrak p \not \in V(I)$ and $\mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a)$. It suffices to show $$ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad\text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim((A/\mathfrak p)_\mathfrak q) > d + s $$ If there exists a prime $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ with $\mathfrak r \in V(I) \setminus T$, then this follows immediately from assumption (4) in Situation \ref{situation-bootstrap}. If not, then $\text{depth}(M_\mathfrak p) > s$ by Lemma \ref{lemma-helper-bootstrap}. \medskip\noindent Condition (6): Let $\mathfrak p \not \in V(I)$ with $V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$. Since $I$ is contained in the Jacobson radical of $A$ we see that $V(\mathfrak p) \cap V(I) \not = \emptyset$. Choose $\mathfrak q \in V(\mathfrak p) \cap V(I) \subset V(\mathfrak a)$. It is clear there does not exist a prime $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ with $\mathfrak r \in V(I) \setminus T$. By Lemma \ref{lemma-helper-bootstrap} we have $\text{depth}(M_\mathfrak p) > s$ which proves (6). \medskip\noindent Proof of (B). We have to check conditions (1), (2), (3), (4) of Lemma \ref{lemma-algebraize-local-cohomology-general}. Warning: the set $T$ in the statement of this lemma is not the same as the set $T$ above. \medskip\noindent Condition (1): This holds because $A$ is complete and has a dualizing complex. \medskip\noindent Condition (2): This is empty. \medskip\noindent Condition (3): This is the same as assumption (3) in Situation \ref{situation-bootstrap}. \medskip\noindent Condition (4): This is the same as assumption (4) in Lemma \ref{lemma-kill-colimit-weak-general} which we proved in (A). \medskip\noindent Proof of (C). This is true because the assumptions in Lemmas \ref{lemma-kill-colimit-weak} and \ref{lemma-kill-colimit-support} are the same as the assumptions in Lemmas \ref{lemma-kill-colimit-weak-general} and \ref{lemma-kill-colimit-support-general} in the local case and we proved these hold in (A). \medskip\noindent Proof of (D). This is true because the assumptions in Lemma \ref{lemma-algebraize-local-cohomology} are the same as the assumptions (1), (2), (3), (4) in Lemma \ref{lemma-algebraize-local-cohomology-general} and we proved these hold in (B).