We will prove this lemma in several steps. We will use without further mention that an equivalence relation gives rise to a groupoid scheme and that the restriction of an equivalence relation is an equivalence relation, see Groupoids, Lemmas \ref{groupoids-lemma-restrict-relation}, \ref{groupoids-lemma-equivalence-groupoid}, and \ref{groupoids-lemma-restrict-groupoid-relation}. \medskip\noindent Step 1: We may assume that $s, t : R \to U$ are locally of finite presentation and Cohen-Macaulay morphisms. Namely, as in More on Groupoids, Lemma \ref{more-groupoids-lemma-make-CM} let $g : U' \to U$ be the open subscheme such that $t^{-1}(U') \subset R$ is the maximal open over which $s : R \to U$ is Cohen-Macaulay, and denote $R'$ the restriction of $R$ to $U'$. By the lemma cited above we see that $$ \xymatrix{ t^{-1}(U') \ar@{=}[r] & U' \times_{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^h & R \ar[r]_s & U } $$ is surjective. Since $h$ is flat and locally of finite presentation, we see that $\{h\}$ is a fppf covering. Hence by Groupoids, Lemma \ref{groupoids-lemma-quotient-groupoid-restrict} we see that $U'/R' \to U/R$ is an isomorphism. By the construction of $U'$ we see that $s', t'$ are Cohen-Macaulay and locally of finite presentation. \medskip\noindent Step 2. Assume $s, t$ are Cohen-Macaulay and locally of finite presentation. Let $u \in U$ be a point of finite type. By More on Groupoids, Lemma \ref{more-groupoids-lemma-max-slice-quasi-finite} there exists an affine scheme $U'$ and a morphism $g : U' \to U$ such that \begin{enumerate} \item $g$ is an immersion, \item $u \in U'$, \item $g$ is locally of finite presentation, \item $h$ is flat, locally of finite presentation and locally quasi-finite, and \item the morphisms $s', t' : R' \to U'$ are flat, locally of finite presentation and locally quasi-finite. \end{enumerate} Here we have used the notation introduced in More on Groupoids, Situation \ref{more-groupoids-situation-slice}. \medskip\noindent Step 3. For each point $u \in U$ which is of finite type choose a $g_u : U'_u \to U$ as in Step 2 and denote $R'_u$ the restriction of $R$ to $U'_u$. Denote $h_u = s \circ \text{pr}_1 : U'_u \times_{g_u, U, t} R \to U$. Set $U' = \coprod_{u \in U} U'_u$, and $g = \coprod g_u$. Let $R'$ be the restriction of $R$ to $U$ as above. We claim that the pair $(U', g)$ works\footnote{Here we should check that $U'$ is not too large, i.e., that it is isomorphic to an object of the category $\Sch_{fppf}$, see Section \ref{section-conventions}. This is a purely set theoretical matter; let us use the notion of size of a scheme introduced in Sets, Section \ref{sets-section-categories-schemes}. Note that each $U'_u$ has size at most the size of $U$ and that the cardinality of the index set is at most the cardinality of $|U|$ which is bounded by the size of $U$. Hence $U'$ is isomorphic to an object of $\Sch_{fppf}$ by Sets, Lemma \ref{sets-lemma-what-is-in-it} part (6).}. Note that \begin{align*} R' = & \coprod\nolimits_{u_1, u_2 \in U} (U'_{u_1} \times_{g_{u_1}, U, t} R) \times_R (R \times_{s, U, g_{u_2}} U'_{u_2}) \\ = & \coprod\nolimits_{u_1, u_2 \in U} (U'_{u_1} \times_{g_{u_1}, U, t} R) \times_{h_{u_1}, U, g_{u_2}} U'_{u_2} \end{align*} Hence the projection $s' : R' \to U' = \coprod U'_{u_2}$ is flat, locally of finite presentation and locally quasi-finite as a base change of $\coprod h_{u_1}$. Finally, by construction the morphism $h : U' \times_{g, U, t} R \to U$ is equal to $\coprod h_u$ hence its image contains all points of finite type of $U$. Since each $h_u$ is flat and locally of finite presentation we conclude that $h$ is flat and locally of finite presentation. In particular, the image of $h$ is open (see Morphisms, Lemma \ref{morphisms-lemma-fppf-open}) and since the set of points of finite type is dense (see Morphisms, Lemma \ref{morphisms-lemma-enough-finite-type-points}) we conclude that the image of $h$ is $U$. This implies that $\{h\}$ is an fppf covering. By Groupoids, Lemma \ref{groupoids-lemma-quotient-groupoid-restrict} this means that $U'/R' \to U/R$ is an isomorphism. This finishes the proof of the lemma.