diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index a1fece3..e922543 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -36,7 +36,7 @@
                 "file": "Concurrent Systems/slides/class 3.pdf",
                 "page": 13,
                 "left": -26,
-                "top": 352,
+                "top": 182,
                 "zoom": 0.57541567695962
               },
               "icon": "lucide-file-text",
@@ -216,10 +216,10 @@
       "companion:Toggle completion": false
     }
   },
-  "active": "02dff467d5bcbba5",
+  "active": "6edd4157a160e462",
   "lastOpenFiles": [
-    "Concurrent Systems/notes/3.md",
     "Concurrent Systems/slides/class 3.pdf",
+    "Concurrent Systems/notes/3.md",
     "Concurrent Systems/notes/1 - CS Basics.md",
     "Concurrent Systems/notes/1 - CS Basics2.md",
     "Concurrent Systems/notes/2b - Round Robin algorithm.md",
diff --git a/Concurrent Systems/notes/3.md b/Concurrent Systems/notes/3.md
index dcbf7cf..1af61e1 100644
--- a/Concurrent Systems/notes/3.md	
+++ b/Concurrent Systems/notes/3.md	
@@ -225,5 +225,7 @@ 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
\ No newline at end of file