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diff --git a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
index a7041357a..9be9508e0 100644
--- a/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/relations_to_o-algebra.ma
@@ -48,6 +48,9 @@ definition SUBSETS: objs1 SET → OAlgebra.
        assumption]]
 qed.
 
+definition powerset_of_SUBSETS: ∀A.oa_P (SUBSETS A) → Ω \sup A ≝ λA,x.x.
+coercion powerset_of_SUBSETS.
+
 definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
  intros;
  constructor 1;
@@ -169,5 +172,75 @@ definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
     apply (.= (orelation_of_relation_preserves_composition o1 o2 o4 f1 (f3∘f2)));
     apply (#‡(orelation_of_relation_preserves_composition o2 o3 o4 f2 f3)); ]
 qed.
-    
-    
\ No newline at end of file
+
+theorem SUBSETS_faithful:
+ ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
+  map_arrows2 ?? SUBSETS' ?? f = map_arrows2 ?? SUBSETS' ?? g → f=g.
+ intros; unfold SUBSETS' in e; simplify in e; cases e;
+ unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
+ intros 2; lapply (e3 (singleton ? x)); cases Hletin;
+ 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 Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+qed.
+
+theorem SUBSETS_full: ∀S,T.∀f.∃g. map_arrows2 ?? SUBSETS' S T g = f.
+ intros; exists;
+  [ constructor 1; constructor 1;
+     [ apply (λx.λy. y ∈ f (singleton S 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. y ∈ f (singleton S a1) ∧ y ∈ a) → a1 ∈ f⎻ a);
+           | change with (∀a1.a1 ∈ f⎻ a → (∃y.y ∈ f (singleton S a1) ∧ y ∈ a)); ]
+        | 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. 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.