+
+definition POW: carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
+ constructor 1;
+ [ apply POW';
+ | intros; constructor 1;
+ [ apply (orelation_of_relation S T);
+ | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
+ | apply orelation_of_relation_preserves_identity;
+ | apply orelation_of_relation_preserves_composition; ]
+qed.
+
+theorem POW_faithful:
+ ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
+ map_arrows2 ?? POW ?? f = map_arrows2 ?? POW ?? g → f=g.
+ intros; unfold POW in e; simplify in e; cases e;
+ unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
+ intros 2; cases (e3 {(x)});
+ split; intro; [ lapply (s y); | lapply (s1 y); ]
+ [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
+ |*: cases Hletin; cases x1; change in f3 with (x =_1 w); apply (. f3‡#); assumption;]
+qed.
+
+
+lemma currify: ∀A,B,C. binary_morphism1 A B C → A → unary_morphism1 B C.
+intros; constructor 1; [ apply (b c); | intros; apply (#‡e);]
+qed.
+
+(*
+alias symbol "singl" = "singleton".
+alias symbol "eq" = "setoid eq".
+lemma in_singleton_to_eq : ∀A:setoid.∀y,x:A.y ∈ {(x)} → (eq1 A) y x.
+intros; apply sym1; apply f;
+qed.
+
+lemma eq_to_in_singleton : ∀A:setoid.∀y,x:A.eq1 A y x → y ∈ {(x)}.
+intros; apply (e^-1);
+qed.
+*)
+
+interpretation "lifting singl" 'singl x =
+ (fun11 ? (objs2 (POW ?)) (singleton ?) x).
+
+theorem POW_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? POW S T g = f).
+ intros; exists;
+ [ constructor 1; constructor 1;
+ [ apply (λx:carr S.λy:carr T. y ∈ f {(x)});
+ | intros; unfold FunClass_1_OF_carr2; lapply (.= e1‡#);
+ [4: apply mem; |6: apply Hletin;|1,2,3,5: skip]
+ lapply (#‡prop11 ?? f ?? (†e)); [6: apply Hletin; |*:skip ]]
+ | whd; split; whd; intro; simplify; unfold map_arrows2; simplify;
+ [ split;
+ [ change with (∀a1.(∀x. a1 ∈ f (singleton S x) → x ∈ a) → a1 ∈ f⎻* a);
+ | change with (∀a1.a1 ∈ f⎻* a → (∀x.a1 ∈ f (singleton S x) → x ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∃y:carr T. y ∈ f (singleton S a1) ∧ y ∈ a) → a1 ∈ f⎻ a);
+ | change with (∀a1.a1 ∈ f⎻ a → (∃y:carr T.y ∈ f (singleton S a1) ∧ y ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a) → a1 ∈ f a);
+ | change with (∀a1.a1 ∈ f a → (∃x:carr S. a1 ∈ f (singleton S x) ∧ x ∈ a)); ]
+ | split;
+ [ change with (∀a1.(∀y. y ∈ f (singleton S a1) → y ∈ a) → a1 ∈ f* a);
+ | change with (∀a1.a1 ∈ f* a → (∀y. y ∈ f (singleton S a1) → y ∈ a)); ]]
+ [ intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)^-1) ? a1);
+ [ intros 2; apply (f1 a2); change in f2 with (a2 ∈ f⎻ (singleton ? a1));
+ lapply (. (or_prop3 ?? f (singleton ? a2) (singleton ? a1)));
+ [ cases Hletin; change in x1 with (eq1 ? a1 w);
+ apply (. x1‡#); assumption;
+ | exists; [apply a2] [change with (a2=a2); apply rule #; | assumption]]
+ | change with (a1 = a1); apply rule #; ]
+ | intros; apply ((. (or_prop2 ?? f (singleton ? a1) a)) ? x);
+ [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f⎻* a); apply (. f3^-1‡#);
+ assumption;
+ | lapply (. (or_prop3 ?? f (singleton ? x) (singleton ? a1))^-1);
+ [ cases Hletin; change in x1 with (eq1 ? x w);
+ change with (x ∈ f⎻ (singleton ? a1)); apply (. x1‡#); assumption;
+ | exists; [apply a1] [assumption | change with (a1=a1); apply rule #; ]]]
+ | intros; cases e; cases x; clear e x;
+ lapply (. (or_prop3 ?? f (singleton ? a1) a)^-1);
+ [ cases Hletin; change in x with (eq1 ? a1 w1); apply (. x‡#); assumption;
+ | exists; [apply w] assumption ]
+ | intros; lapply (. (or_prop3 ?? f (singleton ? a1) a));
+ [ cases Hletin; exists; [apply w] split; assumption;
+ | exists; [apply a1] [change with (a1=a1); apply rule #; | assumption ]]
+ | intros; cases e; cases x; clear e x;
+ apply (f_image_monotone ?? f (singleton ? w) a ? a1);
+ [ intros 2; change in f3 with (eq1 ? w a2); change with (a2 ∈ a);
+ apply (. f3^-1‡#); assumption;
+ | assumption; ]
+ | intros; lapply (. (or_prop3 ?? f a (singleton ? a1))^-1);
+ [ cases Hletin; exists; [apply w] split;
+ [ lapply (. (or_prop3 ?? f (singleton ? w) (singleton ? a1)));
+ [ cases Hletin1; change in x3 with (eq1 ? a1 w1); apply (. x3‡#); assumption;
+ | exists; [apply w] [change with (w=w); apply rule #; | assumption ]]
+ | assumption ]
+ | exists; [apply a1] [ assumption; | change with (a1=a1); apply rule #;]]
+ | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)^-1) ? a1);
+ [ apply f1; | change with (a1=a1); apply rule #; ]
+ | intros; apply ((. (or_prop1 ?? f (singleton ? a1) a)) ? y);
+ [ intros 2; change in f3 with (eq1 ? a1 a2); change with (a2 ∈ f* a);
+ apply (. f3^-1‡#); assumption;
+ | assumption ]]]
+qed.
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