The quotient of an affine etale equivalence relation is an algebraic space

The quotient map is representable. Let $a\text{\hspace{0.17em}⁣}:T\to F$ and let $G=T{\times }_{a,F,c}U$ be the fibre product; by [F4] choose an fppf covering $\{{\phi }_i\text{\hspace{0.17em}⁣}:T_i\to T\}$ with presentations $a_i\text{\hspace{0.17em}⁣}:T_i\to U$ and transition morphisms $r_{i{i}^{\mathrm{\prime }}}$. Then $T_i{\times }_{T}G\cong T_i{\times }_{a_i,U,t}R$, which is a scheme, and the projection $T_i{\times }_{T}G\to T_i$ is the base change of the etale $t$, hence separated and locally quasi-finite. The resulting descent datum for $G$ over $\{T_i\to T\}$ is effective by [F3], so $G$ is representable by a scheme; hence $c$ is representable by schemes.

The quotient map is etale and surjective. With the notation of step 1.1, the morphisms $T_i{\times }_{T}G\to T_i$ are base changes of $t$, hence etale and surjective; since étaleness and surjectivity are fppf-local on the base, the projection $G\to T$ is etale and surjective. As $a$ was arbitrary, $c\text{\hspace{0.17em}⁣}:U\to F$ is representable, etale and surjective.

The diagonal and conclusion. It remains to see that ${\mathrm{\Delta }}_F\text{\hspace{0.17em}⁣}:F\to F{\times }_{S}F$ is representable by schemes. The square with $R\to U{\times }_{S}U$ over $F\to F{\times }_{S}F$ is cartesian: a $T$-point of $U{\times }_{S}U$ whose two components have equal image in $F{\times }_{S}F$ is a pair of $U$-points that are $R$-equivalent, i.e. a $T$-point of $R$. Moreover $U{\times }_{S}U\to F{\times }_{S}F$ is representable, etale and surjective by two applications of step 2.1, so for $a\text{\hspace{0.17em}⁣}:T\to F{\times }_{S}F$ the base change $T^{\mathrm{\prime }}=(U{\times }_{S}U){\times }_{F{\times }_{S}F,a}T\to T$ is an etale covering and $T^{\mathrm{\prime }}{\times }_T(T{\times }_{a,F{\times }_{S}F,{\mathrm{\Delta }}_F}F)=T^{\mathrm{\prime }}{\times }_{U{\times }_{S}U,j}R$ is a scheme whose structure morphism is a base change of $j$, hence separated and locally quasi-finite by [F1]-[F2]. Effective descent by [F3] makes the diagonal representable by schemes. Since $F$ is an fppf sheaf, has representable diagonal and has the representable etale surjective cover $c\text{\hspace{0.17em}⁣}:U\to F$ from the affine scheme $U$, it is an algebraic space and $(U,R,U\to F)$ is a presentation; the Axiom of Choice is inherited from the descent and quotient suppliers [F3]-[F4].