diff --git a/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md b/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md
index 3ad385a..22ebcf7 100644
--- a/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md	
+++ b/Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md	
@@ -93,7 +93,12 @@ lock(i) :=
 
 -  $p_n$ invokes lock alone, completes its CS and its new DATE is n
 
-**Lemma 1:** Suppose we have $n$ processes, then $\not \exists p_{j} : DATE[j]=DATE[i] \forall i \in [0, n]$
+**Lemma 1:** Suppose we have $n$ processes, then $\not \exists p_{j} : DATE[j]=DATE[i] \forall i \in [0, n]$ (non esistono due processi con lo stesso valore per DATE)
+*Proof:*
+- Suppose $p_j$ is the last process to execute unlock, then `DATE[j] = n` and for each i, `DATE[i] < n`, as DATE is either set to $n$ or decreased.
+- by iterating this reasoning, it is clear that until every DATE is > 0, we have no processes with the same value for DATE.
+- What if we reach 0?
+	- Let's suppose that $p_i$ is the first process that reaches `DATE = 0`
 
 **Lemma 2:** DATE's bounds are $[0, n]$
 *Proof:*