Definition Comp_INTERp (F G X Y : Ens) := Comp_INTER F G (PAIR X Y).

(* Specifications: *)
Lemma COMPp_Out3 : IN (PAIR X (Comp_INTERp F G X Y)) F.
Lemma COMPp_Out4 : IN (PAIR (Comp_INTERp F G X Y) Y) G.
Theorem COMP_th3p : forall (F G X Y : Ens) (i : IN (PAIR X Y) (COMP F G)), IN (PAIR X (Comp_INTERp F G X Y)) F.
Theorem COMP_th4p : forall (F G X Y : Ens) (i : IN (PAIR X Y) (COMP F G)), IN (PAIR (Comp_INTERp F G X Y) Y) G.
Lemma comp_wd_Out : EQ Y Z.
Theorem composition_well_definedness : forall (F G : Ens) (f : well_definedness F) (g : well_definedness G), well_definedness (COMP F G).
Theorem composition_th1 : forall (F G : Ens) (f : well_definedness F) (g : well_definedness G), is_a_function (COMP F G).
Theorem composition_th2 : forall (F G : Ens) (f : is_a_function F) (g : is_a_function G), is_a_function (COMP F G).
Lemma comp_ev_1_Out : IN W (COMP F G).
Theorem COMP_th5 : forall (F G X Y Z : Ens) (i : IN (PAIR X Y) F) (j : IN (PAIR Y Z) G), IN (PAIR X Z) (COMP F G).
Lemma comp_ev_2_Out : IN (PAIR X Z) (COMP F G).
Theorem composition_th3 : forall (F G X : Ens) (i : IN X (Domain F)) (j : IN (EV F X) (Domain G)), IN (PAIR X (EV G (EV F X))) (COMP F G).
Lemma comp_ev_3_Out : EQ W Z.
Theorem composition_th4 : forall (F G X : Ens) (f : is_a_function F) (g : is_a_function G) (i : IN X (Domain F)) (j : IN (EV F X) (Domain G)), EQ (EV (COMP F G) X) (EV G (EV F X)).
Lemma comp_assoc_1_Out : IN P (COMP (COMP F G) H).
Theorem COMP_assoc_th1 : forall F G H : Ens, SUB (COMP F (COMP G H)) (COMP (COMP F G) H).
Lemma comp_assoc_2_Out : IN P (COMP F (COMP G H)).
Theorem COMP_assoc_th2 : forall F G H : Ens, SUB (COMP (COMP F G) H) (COMP F (COMP G H)).
Theorem COMP_assoc : forall F G H : Ens, EQ (COMP F (COMP G H)) (COMP (COMP F G) H).