faithful

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 a7041357af3a0cffd16f257a87008e4be21aac7f..7f327c4972130fcb143b5aa1eb45f2eafb6bbc60 100644 (file)
--- 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
@@ -169,5 +169,14 @@ 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.
-    
-    
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+
+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.
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