diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index e922543..d078025 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -34,9 +34,9 @@
               "type": "pdf",
               "state": {
                 "file": "Concurrent Systems/slides/class 3.pdf",
-                "page": 13,
+                "page": 14,
                 "left": -26,
-                "top": 182,
+                "top": 41,
                 "zoom": 0.57541567695962
               },
               "icon": "lucide-file-text",
diff --git a/Concurrent Systems/notes/3.md b/Concurrent Systems/notes/3.md
index 1af61e1..ca9ecab 100644
--- a/Concurrent Systems/notes/3.md	
+++ b/Concurrent Systems/notes/3.md	
@@ -226,6 +226,24 @@ unlock(i) :=
 
 #### MUTEX proof
 **Theorem:** if $p_i$ is in the CS, then $p_j$ cannot simultaneously be in the CS.
-*Proof:* by contradiction. a
+*Proof:* by contradiction.
 
-#### proof
\ No newline at end of file
+...
+
+#### Bounded bypass proof
+**Lemma 1:** exactly after n CSs there is a reset of DATE.
+*Proof:*
+- the first CS leads $max_j{DATE[j]}$ to n+1
+- the seconds CS leads ... to n+2
+- ...
+- the n-th read leads ... to n+n = 2n -> RESET
+
+**Lemma 2:** there can be at most one reset of DATE during an invocation of a lock
+*Proof:*
+- let $p_i$ invoke lock, if no reset occurs, ok
+- otherwise, let us consider the moment in which a reset occurs
+	- if pi is the next process that enters the CS, ok
+	- Otherwise let $p_j$ be the process that enters, its next date is $n+1 > DATE[i]$
+		- $p_{j}$ cannot surpass $p_i$ again (before a RESET)
+	- The worst case is then all processes perform lock together and $i = n$ (i am process n)
+		- all $p_{1}\dots p_{n}$ surpass $
\ No newline at end of file