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(** * Inclusion and extensionality  *)
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(** The inclusion relation is defined as expected: *)
Definition SUB (X : Typ1) (A : Rel X) (a : X) (Y : Typ1)
(B : Rel Y) (b : Y) : Prop :=
forall (Z : Typ1) (C : Rel Z) (c : Z), ELT Z C c X A a -> ELT Z C c Y B b.
(** Inclusion is clearly reflexive and transitive: *)

(* Specifications: *)
Lemma SUB_refl : forall (X : Typ1) (A : Rel X) (a : X), SUB X A a X A a.
Lemma SUB_trans :
forall (X : Typ1) (A : Rel X) (a : X) (Y : Typ1) (B : Rel Y)
(b : Y) (Z : Typ1) (C : Rel Z) (c : Z),
SUB X A a Y B b -> SUB Y B b Z C c -> SUB X A a Z C c.