Section Quotient1.
Comments
"We can now define quotient sets, using equivalence relations on sets".
Comments
"A binary relation on a set is a binary relation on its carrier, which is compatible with equality:".
Variable E : Setoid.
Definition rel_compatible (R : relation E) : Prop :=
forall x x' y y' : E,
Equal x x' -> Equal y y' -> app_rel R x y -> app_rel R x' y'.
Record Relation : Type :=
{Rel_fun :> relation E; Rel_compatible_prf : rel_compatible Rel_fun}.

Lemma Rel_comp :
forall (R : Relation) (x x' y y' : E),
Equal x x' -> Equal y y' -> app_rel R x y -> app_rel R x' y'.