diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index e1d54a9..eb2d01d 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -231,6 +231,7 @@
   },
   "active": "0d64f519fd7f9420",
   "lastOpenFiles": [
+    "Pasted image 20250325090735.png",
     "Concurrent Systems/slides/class 8.pdf",
     "Concurrent Systems/notes/8 - Enhancing Liveness Properties.md",
     "Concurrent Systems/notes/7- MUTEX-free concurrency.md",
diff --git a/Concurrent Systems/notes/8 - Enhancing Liveness Properties.md b/Concurrent Systems/notes/8 - Enhancing Liveness Properties.md
index 2ddafb1..c3c70e4 100644
--- a/Concurrent Systems/notes/8 - Enhancing Liveness Properties.md	
+++ b/Concurrent Systems/notes/8 - Enhancing Liveness Properties.md	
@@ -51,5 +51,10 @@ It can be proved that there exists no wait-free implementation of $\Omega$ in an
 - need of timing constraints
 
 1. $\exists$ time $\tau_{1}$, time interval $\nabla$ and correct process $p_{L}$ s.t. after $\tau_{1}$ every two consecutive writes to a specific SWMR atomic R/W by $p_{L}$ are at most $\nabla$ time units apart one from the other
+
 2. let t be an upper bound on the number of possible failing processes and f the real number of process failed (hence, $0\leq f\leq t\leq n-1$, with f unknown and t known in advance).
-   Then, there are at least $t-f$ correct processes different from $p_L$ with a timer s.t. $\exists$ time $\tau_{2} \forall$ time interval $\delta$, if their timer is set to $\delta$ after $\tau_{2}$, it expires at least after $\delta$
\ No newline at end of file
+   Then, there are at least $t-f$ correct processes different from $p_L$ with a timer s.t. $\exists$ time $\tau_{2} \forall$ time interval $\delta$, if their timer is set to $\delta$ after $\tau_{2}$, it expires at least after $\delta$.
+
+REMARK: $\tau_{1}, \tau_{2}, \nabla$ and $p_L$ are all unknown.
+
+![[Pasted image 20250325090735.png]]
diff --git a/Pasted image 20250325090735.png b/Pasted image 20250325090735.png
new file mode 100644
index 0000000..1f604b6
Binary files /dev/null and b/Pasted image 20250325090735.png differ