We first make some trivial observations. Note that $V'$ is representable by Lemma \ref{lemma-quasi-affine-local}. It is also quasi-compact (as a quasi-affine scheme over an affine scheme, see Morphisms, Lemma \ref{morphisms-lemma-quasi-affine-separated}). Since $T' \times_T X \to X$ is \'etale (Properties of Spaces, Lemma \ref{spaces-properties-lemma-base-change-etale}) the map $|T' \times_T X| \to |X|$ is open, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-etale-open}. Let $U \subset X$ be the open subspace corresponding to the image of $|V'|$, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-open-subspaces}. As $|V'|$ is quasi-compact we see that $|U|$ is quasi-compact, hence $U$ is a quasi-compact algebraic space, by Properties of Spaces, Lemma \ref{spaces-properties-lemma-quasi-compact-space}. \medskip\noindent By Morphisms, Lemma \ref{morphisms-lemma-locally-quasi-finite-qc-source-universally-bounded} the morphism $T' \to T$ is universally bounded. Hence we can do induction on the integer $n$ bounding the degree of the fibres of $T' \to T$, see Morphisms, Lemma \ref{morphisms-lemma-etale-universally-bounded} for a description of this integer in the case of an \'etale morphism. If $n = 1$, then $T' \to T$ is an open immersion (see \'Etale Morphisms, Theorem \ref{etale-theorem-etale-radicial-open}), and the result is clear. Assume $n > 1$. \medskip\noindent Consider the affine scheme $T'' = T' \times_T T'$. As $T' \to T$ is \'etale we have a decomposition (into open and closed affine subschemes) $T'' = \Delta(T') \amalg T^*$. Namely $\Delta = \Delta_{T'/T}$ is open by Morphisms, Lemma \ref{morphisms-lemma-diagonal-unramified-morphism} and closed because $T' \to T$ is separated as a morphism of affines. As a base change the degrees of the fibres of the second projection $\text{pr}_1 : T' \times_T T' \to T'$ are bounded by $n$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-universally-bounded}. On the other hand, $\text{pr}_1|_{\Delta(T')} : \Delta(T') \to T'$ is an isomorphism and every fibre has exactly one point. Thus, on applying Morphisms, Lemma \ref{morphisms-lemma-etale-universally-bounded} we conclude the degrees of the fibres of the restriction $\text{pr}_1|_{T^*} : T^* \to T'$ are bounded by $n - 1$. Hence the induction hypothesis applied to the diagram $$ \xymatrix{ p_0^{-1}(V') \cap X^* \ar[r] \ar[rd] & X^* \ar[r]_{p_1|_{X^*}} \ar[d] & X' \ar[d] \\ & T^* \ar[r]^{\text{pr}_1|_{T^*}} & T' } $$ gives that $p_1(p_0^{-1}(V') \cap X^*)$ is a quasi-compact scheme. Here we set $X'' = T'' \times_T X$, $X^* = T^* \times_T X$, and $X' = T' \times_T X$, and $p_0, p_1 : X'' \to X'$ are the base changes of $\text{pr}_0, \text{pr}_1$. Most of the hypotheses of the lemma imply by base change the corresponding hypothesis for the diagram above. For example $p_0^{-1}(V') = T'' \times_{T'} V'$ is a scheme quasi-affine over $T''$ as a base change. Some verifications omitted. \medskip\noindent By Properties of Spaces, Lemma \ref{spaces-properties-lemma-subscheme} we conclude that $$ p_1(p_0^{-1}(V')) = V' \cup p_1(p_0^{-1}(V') \cap X^*) $$ is a quasi-compact scheme. Moreover, it is clear that $p_1(p_0^{-1}(V'))$ is the inverse image of the quasi-compact open subspace $U \subset X$ discussed in the first paragraph of the proof. In other words, $T' \times_T U$ is a scheme! Note that $T' \times_T U$ is quasi-compact and separated and locally quasi-finite over $T'$, as $T' \times_T X \to T'$ is locally quasi-finite and separated being a base change of the original morphism $X \to T$ (see Lemmas \ref{lemma-base-change-separated} and \ref{lemma-base-change-quasi-finite}). This implies by More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-finite-separated-quasi-affine} that $T' \times_T U \to T'$ is quasi-affine. \medskip\noindent By Descent, Lemma \ref{descent-lemma-descent-data-sheaves} this gives a descent datum on $T' \times_T U / T'$ relative to the \'etale covering $\{T' \to W\}$, where $W \subset T$ is the image of the morphism $T' \to T$. Because $U'$ is quasi-affine over $T'$ we see from Descent, Lemma \ref{descent-lemma-quasi-affine} that this datum is effective, and by the last part of Descent, Lemma \ref{descent-lemma-descent-data-sheaves} this implies that $U$ is a scheme as desired. Some minor details omitted.