We can find a finite type $\mathbf{Z}$-subalgebra $A' \subset A$ and a scheme $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$. See Limits, Lemmas \ref{limits-lemma-descend-finite-presentation} and \ref{limits-lemma-descend-separated-finite-presentation}. Let $x' \in X'$ be the image of $x$. If we can prove the lemma for $x' \in X'/A'$, then the lemma follows for $x \in X/A$. Namely, if $U', n', V', Z', z', E'$ provide the solution for $x' \in X'/A'$, then we can let $U \subset X$ be the inverse image of $U'$, let $n = n'$, let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$, let $Z \subset X \times \mathbf{P}^n$ be the scheme theoretic inverse image of $Z'$, let $z \in Z$ be the unique point mapping to $x$, and let $E$ be the derived pullback of $E'$. Observe that $E$ is pseudo-coherent by Cohomology, Lemma \ref{cohomology-lemma-pseudo-coherent-pullback}. It only remains to check (5). To see this set $W = b^{-1}(U) = c^{-1}(V)$ and $W' = (b')^{-1}(U) = (c')^{-1}(V')$  and consider the cartesian square $$ \xymatrix{ W \ar[d]_{(b, c)} \ar[r] & W' \ar[d]^{(b', c')} \\ X \times_A V \ar[r] & X' \times_{A'} V' } $$ By Lemma \ref{lemma-case-of-tor-independence} the schemes $X \times_A V$ and $W'$ are Tor independent over $X' \times_{A'} V'$. Hence the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times_A V$ is $(b, c)_*\mathcal{O}_W$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}. This also uses that $R(b', c')_*\mathcal{O}_{Z'} = (b', c')_*\mathcal{O}_{Z'}$ because $(b', c')$ is a closed immersion and simiarly for $(b, c)_*\mathcal{O}_Z$. Since $E'|_{U' \times_{A'} V'} = (b', c')_*\mathcal{O}_{W'}$ we obtain $E|_{U \times_A V} = (b, c)_*\mathcal{O}_W$ and (5) holds. This reduces us to the situation described in the next paragraph. \medskip\noindent Assume $A$ is of finite type over $\mathbf{Z}$. Choose an affine open neighbourhood $U \subset X$ of $x$. Then $U$ is of finite type over $A$. Choose a closed immersion $U \to \mathbf{A}^n_A$ and denote $j : U \to \mathbf{P}^n_A$ the immersion we get by composing with the open immersion $\mathbf{A}^n_A \to \mathbf{P}^n_A$. Let $Z$ be the scheme theoretic closure of $$ (\text{id}_U, j) : U \longrightarrow X \times_A \mathbf{P}^n_A $$ Since the projection $X \times \mathbf{P}^n \to X$ is separated, we conclude from Morphisms, Lemma \ref{morphisms-lemma-scheme-theoretic-image-of-partial-section} that $b : Z \to X$ is an isomorphism over $U$. Let $z \in Z$ be the unique point lying over $x$. \medskip\noindent Let $Y \subset \mathbf{P}^n_A$ be the scheme theoretic closure of $j$. Then it is clear that $Z \subset X \times_A Y$ is the scheme theoretic closure of $(\text{id}_U, j) : U \to X \times_A Y$. As $X$ is separated, the morphism $X \times_A Y \to Y$ is separated as well. Hence we see that $Z \to Y$ is an isomorphism over the open subscheme $j(U) \subset Y$ by the same lemma we used above. Choose $V \subset \mathbf{P}^n_A$ open with $V \cap Y = j(U)$. Then we see that (3) and (4) hold. \medskip\noindent Because $A$ is Noetherian we see that $X$ and $X \times_A \mathbf{P}^n_A$ are Noetherian schemes. Hence we can take $E = (b, c)_*\mathcal{O}_Z$ in this case, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-identify-pseudo-coherent-noetherian}. This finishes the proof.