1 more lemma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
index c2dc0fc60a0e4ddae9ecefd422526218775c138d..c3221a7d99a0068e52b17781c8be2aa01ece8afd 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-concrete_spaces.ma
@@ -85,14 +85,15 @@ intros; constructor 1;
 | intros; apply ((†H)‡(†H1));]
 qed.
 
+interpretation "concrete_space binary ↓" 'fintersects a b = (fun1 _ _ _ (binary_downarrow _) a b).
+
 record convergent_relation_pair (CS1,CS2: concrete_space) : Type ≝
  { rp:> arrows1 ? CS1 CS2;
    respects_converges:
-    ∀b,c. (rp\sub\c)⎻ (Ext⎽CS2 (b ↓ c)) = ?(*  
-     extS ?? rp \sub\c (BPextS CS2 (b ↓ c)) =
-     BPextS CS1 ((extS ?? rp \sub\f b) ↓ (extS ?? rp \sub\f c));
+    ∀b,c. eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (b ↓ c))) (Ext⎽CS1 (rp\sub\f⎻ b ↓ rp\sub\f⎻ c));
    respects_all_covered:
-    extS ?? rp\sub\c (BPextS CS2 (form CS2)) = BPextS CS1 (form CS1)*)
+     eq1 ? (rp\sub\c⎻ (Ext⎽CS2 (oa_one (form CS2))))
+           (Ext⎽CS1 (oa_one (form CS1)))
  }.
 
 definition rp' : ∀CS1,CS2. convergent_relation_pair CS1 CS2 → relation_pair CS1 CS2 ≝
@@ -117,6 +118,13 @@ definition rp'': ∀CS1,CS2.convergent_relation_space_setoid CS1 CS2 → arrows1
 
 coercion rp''.
 
+definition prop_1_SET : 
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
+intros; apply (prop_1 A B w a b H);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop_1_SET _ _ _ _ _ h).
+
 definition convergent_relation_space_composition:
  ∀o1,o2,o3: concrete_space.
   binary_morphism1
@@ -127,28 +135,21 @@ definition convergent_relation_space_composition:
      [ intros; whd in c c1 ⊢ %;
        constructor 1;
         [ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
-        | intros;
-          change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
-          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
-            with (c1 \sub \f ∘ c \sub \f);
-          change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
-            with (c1 \sub \f ∘ c \sub \f);
-          apply (.= (extS_com ??????));
-          apply (.= (†(respects_converges ?????)));
-          apply (.= (respects_converges ?????));
-          apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
+        | intros;  
+          change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (b↓c2))));
+          alias symbol "trans" = "trans1".
+          apply (.= († (respects_converges : ?)));
+          apply (.= (respects_converges : ?));
           apply refl1;
-        | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
-          apply (.= (extS_com ??????));
-          apply (.= (†(respects_all_covered ???)));
-          apply (.= respects_all_covered ???);
+        | change in ⊢ (? ? ? % ?) with (c\sub\c⎻ (c1\sub\c⎻ (Ext⎽o3 (oa_one (form o3)))));
+          apply (.= (†(respects_all_covered :?)));
+          apply (.= (respects_all_covered :?));
           apply refl1]
      | intros;
-       change with (b ∘ a = b' ∘ a');
+       change with (b ∘ a = b' ∘ a'); 
        change in H with (rp'' ?? a = rp'' ?? a');
        change in H1 with (rp'' ?? b = rp ?? b');
-       apply (.= (H‡H1));
-       apply refl1]
+       apply ( (H‡H1));]
 qed.
 
 definition CSPA: category1.