Proof intersection4_Out.
(******** big Intersection *********)
Definition IntersectionProp (F : Ens -> Ens) (X : Ens) :=
forall Y : Ens, IN X (F Y).
Section Intersection_section1.
Variable F : Ens -> Ens.
Let A := F EmptySet.
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).

Theorem Intersection_pr1 :
forall (F : Ens -> Ens) (X : Ens),
IntersectionProp F X -> IN X (Intersection F).