Inductive Ens : Type :=
    sup : ∀ A : Type, (A → Ens) → Ens.


Inductive EXType (P : Type) (Q : P → Prop) : Prop :=
    EXTypei : ∀ x : P, Q x → EXType P Q.


Inductive prod_t (A B : Type) : Type :=
    pair_t : A → B → prod_t A B.


Inductive depprod (A : Type) (P : A → Type) : Type :=
    dep_i : ∀ x : A, P x → depprod A P.


Definition EQ : Ens → Ens → Prop.
simple induction 1; intros A f eq1.
simple induction 1; intros B g eq2.
apply and.
exact (∀ x : A, EXType _ (fun y : B ⇒ eq1 x (g y))).
exact (∀ y : B, EXType _ (fun x : A ⇒ eq1 x (g y))).
Defined.


Definition IN (E1 E2 : Ens) : Prop :=
  match E2 with
  | sup A f ⇒ EXType _ (fun y : A ⇒ EQ E1 (f y))
  end.


Definition INC : Ens → Ens → Prop.
intros E1 E2.
exact (∀ E : Ens, IN E E1 → IN E E2).
Defined.


Theorem EQ_refl : ∀ E : Ens, EQ E E.
simple induction E.
intros A f HR.
simpl in |- ×.
split; intros.
∃ x; auto.

∃ y; auto.
Qed.

Theorem EQ_tran : ∀ E1 E2 E3 : Ens, EQ E1 E2 → EQ E2 E3 → EQ E1 E3.
simple induction E1; intros A1 f1 r1; simple induction E2; intros A2 f2 r2;
 simple induction E3; intros A3 f3 r3; simpl in |- *;
 intros e1 e2.
split; (elim e1; intros I1 I2; elim e2; intros I3 I4).
intros a1; elim (I1 a1); intros a2.
elim (I3 a2); intros a3.
∃ a3.
apply r1 with (f2 a2); auto.
intros a3; elim (I4 a3); intros a2; elim (I2 a2); intros a1; ∃ a1.
apply r1 with (f2 a2); auto.
Qed.

Theorem EQ_sym : ∀ E1 E2 : Ens, EQ E1 E2 → EQ E2 E1.
simple induction E1; intros A1 f1 r1; simple induction E2; intros A2 f2 r2;
 simpl in |- *; simple induction 1; intros e1 e2; split.
intros a2; elim (e2 a2); intros a1 H1; ∃ a1; auto.
intros a1; elim (e1 a1); intros a2 H2; ∃ a2; auto.
Qed.

Theorem EQ_INC : ∀ E E' : Ens, EQ E E' → INC E E'.
simple induction E; intros A f r; simple induction E'; intros A' f' r';
 simpl in |- *; simple induction 1; intros e1 e2; unfold INC in |- *;
 simpl in |- ×.
intros C; simple induction 1; intros a ea; elim (e1 a); intros a' ea';
 ∃ a'.
apply EQ_tran with (f a); assumption.
Qed.

Hint Resolve EQ_sym EQ_refl EQ_INC: zfc.


Theorem INC_EQ : ∀ E E' : Ens, INC E E' → INC E' E → EQ E E'.
simple induction E; intros A f r; simple induction E'; intros A' f' r';
 unfold INC in |- *; simpl in |- *; intros I1 I2; split.
intros a; apply I1.
∃ a; auto with zfc.
intros a'; cut (EXType A (fun x : A ⇒ EQ (f' a') (f x))).
simple induction 1; intros a ea; ∃ a; auto with zfc.
apply I2; ∃ a'; auto with zfc.
Qed.

Hint Resolve INC_EQ: zfc.


Theorem IN_sound_left :
 ∀ E E' E'' : Ens, EQ E E' → IN E E'' → IN E' E''.
simple induction E''; intros A'' f'' r'' e; simpl in |- *; simple induction 1;
 intros a'' p; ∃ a''; apply EQ_tran with E; auto with zfc v62.
Qed.

Theorem IN_sound_right :
 ∀ E E' E'' : Ens, EQ E' E'' → IN E E' → IN E E''.
simple induction E'; intros A' f' r'; simple induction E'';
 intros A'' f'' r''; simpl in |- *; simple induction 1;
 intros e1 e2; simple induction 1; intros a' e'; elim (e1 a');
 intros a'' e''; ∃ a''; apply EQ_tran with (f' a');
 assumption.

Qed.


Theorem INC_refl : ∀ E : Ens, INC E E.
unfold INC in |- *; auto with zfc v62.
Qed.

Theorem INC_tran : ∀ E E' E'' : Ens, INC E E' → INC E' E'' → INC E E''.
unfold INC in |- *; auto with zfc v62.
Qed.

Theorem INC_sound_left :
 ∀ E E' E'' : Ens, EQ E E' → INC E E'' → INC E' E''.
simple induction E''; unfold INC in |- *; simpl in |- *;
 intros A f HR e H1 E0 i; apply H1.
apply IN_sound_right with E'; auto with zfc v62.
Qed.

Theorem INC_sound_right :
 ∀ E E' E'' : Ens, EQ E' E'' → INC E E' → INC E E''.
simple induction E'; simple induction E''; unfold INC in |- *; simpl in |- *;
 intros.
elim (H2 E0); try assumption; intros.
elim H1; intros HA HB; elim (HA x); intros.
∃ x0; apply EQ_tran with (e x); auto with zfc v62.
Qed.