We can find a finite type $\mathbf{Z}$-subalgebra $A' \subset A$ and an algebraic space $X'$ separated and of finite presentation over $A'$ whose base change to $A$ is $X$. See Limits of Spaces, Lemmas \ref{spaces-limits-lemma-descend-finite-presentation} and \ref{spaces-limits-lemma-descend-separated-morphism}. Let $x' \in |X'|$ be the image of $x$. If we can prove the lemma for $(X'/A', x')$, then the lemma follows for $(X/A, x)$. Namely, if $n', Z', z', V', E'$ provide the solution for $(X'/A', x')$, then we can let $n = n'$, let $Z \subset X \times \mathbf{P}^n$ be the inverse image of $Z'$, let $z \in Z$ be the unique point mapping to $x$, let $V \subset \mathbf{P}^n_A$ be the inverse image of $V'$, and let $E$ be the derived pullback of $E'$. Observe that $E$ is pseudo-coherent by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pseudo-coherent-pullback}. It only remains to check (5). To see this set $W = c^{-1}(V)$ and $W' = (c')^{-1}(V')$ 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} $X \times_A V$ and $W'$ are tor-independent over $X' \times_{A'} V'$. Thus the derived pullback of $(b', c')_*\mathcal{O}_{W'}$ to $X \times_A V$ is $(b, c)_*\mathcal{O}_W$ by Derived Categories of Spaces, Lemma \ref{spaces-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 \'etale morphism $U \to X$ where $U$ is an affine scheme and a point $u \in U$ mapping to $x$. 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 $$ (\text{id}_U, j) : U \longrightarrow X \times_A \mathbf{P}^n_A $$ Let $z \in Z$ be the image of $u$. 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 Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-scheme-theoretic-image-of-partial-section}. Choose $V \subset \mathbf{P}^n_A$ open with $V \cap Y = j(U)$. Then we see that (2) holds, that $W = (\text{id}_U, j)(U)$, and hence that (3) holds. Part (1) holds because $A$ is Noetherian. \medskip\noindent Because $A$ is Noetherian we see that $X$ and $X \times_A \mathbf{P}^n_A$ are Noetherian algebraic spaces. Hence we can take $E = (b, c)_*\mathcal{O}_Z$ in this case: (4) is clear and for (5) see Derived Categories of Spaces, Lemma \ref{spaces-perfect-lemma-identify-pseudo-coherent-noetherian}. This finishes the proof.