Section Intersection_section1_1.
Variable Z : Ens.
Hypothesis h : IntersectionProp F Z.
Definition Intersection_section1_1_Out : IN Z A := h EmptySet.
End Intersection_section1_1.
Let step1 : forall (Z : Ens) (h : IntersectionProp F Z), IN Z A.
Proof Intersection_section1_1_Out.
Let step2 : Bounded_By A (IntersectionProp F).
Proof step1.
Definition Intersection_section1_Out : Bounded (IntersectionProp F) :=
Bounded_th2 (IntersectionProp F) A step2.
End Intersection_section1.
Definition Intersection_Bounded :
forall F : Ens -> Ens, Bounded (IntersectionProp F) :=
Intersection_section1_Out.
Definition Intersection (F : Ens -> Ens) : Ens :=
REPLACEMENT (IntersectionProp F).
Proof
fun F : Ens -> Ens =>
REPLACEMENT_pr1 (IntersectionProp F) (Intersection_Bounded F).
Proof
fun F : Ens -> Ens =>
REPLACEMENT_pr2 (IntersectionProp F) (Intersection_Bounded F).

Theorem Intersection_th1 :
forall (F : Ens -> Ens) (X : Ens) (h : forall Y : Ens, IN X (F Y)),
IN X (Intersection F).