Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.Record family : Type := mkfamily
  {ind : R -> Prop;
    f :> R -> R -> Prop;
    cond_fam : forall x:R, (exists y : R, f x y) -> ind x}.Definition family_open_set (f:family) : Prop := forall x:R, open_set (f x).Definition domain_finite (D:R -> Prop) : Prop :=
  exists l : list R, (forall x:R, D x <-> In x l).Definition family_finite (f:family) : Prop := domain_finite (ind f).Definition covering (D:R -> Prop) (f:family) : Prop :=
  forall x:R, D x ->  exists y : R, f y x.Definition covering_open_set (D:R -> Prop) (f:family) : Prop :=
  covering D f /\ family_open_set f.Definition covering_finite (D:R -> Prop) (f:family) : Prop :=
  covering D f /\ family_finite f.

Lemma restriction_family : forall (f:family) (D:R -> Prop) (x:R), (exists y : R, (fun z1 z2:R => f z1 z2 /\ D z1) x y) -> intersection_domain (ind f) D x.