+notation "Ω term 90 A \atop ≈" non associative with precedence 70
+for @{ 'ext_powerclass $A }.
+
+interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
+
+ndefinition Full_set: ∀A. 𝛀^A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
+nqed.
+ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
+
+ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
+ #A; @ [ napply (λS,S'. S = S') ] /2/.
+nqed.
+
+ndefinition ext_powerclass_setoid: setoid → setoid1.
+ #A; @ (ext_seteq A).
+nqed.
+
+unification hint 0 ≔ A;
+ R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
+ (* ----------------------------------------------------- *) ⊢
+ carr1 R ≡ ext_powerclass A.
+
+interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+(*
+ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr
+on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
+*)
+
+nlemma mem_ext_powerclass_setoid_is_morph:
+ ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+ [ napply (λx,S. x ∈ S)
+ | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
+ ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
+ ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ A:setoid, x, S;
+ SS ≟ (ext_carr ? S),
+ TT ≟ (mk_binary_morphism1 ???
+ (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S)
+ (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
+ XX ≟ (ext_powerclass_setoid A)
+ (*-------------------------------------*) ⊢
+ fun21 (setoid1_of_setoid A) XX CPROP TT x S
+ ≡ mem A SS x.
+
+nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+ [ napply (λS,S'. S ⊆ S')
+ | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *;/4/]
+nqed.
+
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+ eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+
+nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #A; #S; #S'; @ (S ∩ S');
+ #a; #a'; #Ha; @; *; #H1; #H2; @
+ [##1,2: napply (. Ha^-1‡#); nassumption;
+##|##3,4: napply (. Ha‡#); nassumption]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : setoid, B,C : ext_powerclass A;
+ R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
+
+ (* ------------------------------------------*) ⊢
+ ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
+
+nlemma intersect_is_morph:
+ ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+ #A; @ (λS,S'. S ∩ S');
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔
+ A : Type[0], B,C : Ω^A;
+ R ≟ (mk_binary_morphism1 …
+ (λS,S'.S ∩ S')
+ (prop21 … (intersect_is_morph A)))
+ ⊢
+ fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C
+ ≡ intersect ? B C.
+
+interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+nlemma intersect_is_ext_morph:
+ ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
+#H; #a; #a'; #b; #b'; #H1; #H2; napply H; nassumption;
+nqed.
+
+unification hint 1 ≔
+ A:setoid, B,C : 𝛀^A;
+ R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
+ (λS,S':carr1 (ext_powerclass_setoid A).
+ mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
+ (prop21 … (intersect_is_ext_morph A))) ,
+ BB ≟ (ext_carr ? B),
+ CC ≟ (ext_carr ? C)
+ (* ------------------------------------------------------*) ⊢
+ ext_carr A
+ (fun21
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A)
+ (ext_powerclass_setoid A) R B C) ≡
+ intersect (carr A) BB CC.