diff --git a/Concurrent Systems/notes/6 - Atomicity.md b/Concurrent Systems/notes/6 - Atomicity.md
index 0057b81..0827fae 100644
--- a/Concurrent Systems/notes/6 - Atomicity.md	
+++ b/Concurrent Systems/notes/6 - Atomicity.md	
@@ -56,8 +56,11 @@ We now show that $\to$ is acyclic.
 		- can this be a cycle?
 			- $op1 \to_H op2$ means that $res(op1) <_H inv(op2)$
 			-  $op2 \to_X op3$ entails that $inv(op2) <_H res(op3)$
-				- if not, as is a total order, we would have that $res(op3) <_H inv(op2)$, but we then would have $op3 \to_H op2$ a cycle of lenght 2...
-			-  $op2 \to_H op3$ entails that $inv(op2) <_H res(op3)$
+				- if not, as is a total order, we would have that $res(op3) <_H inv(op2)$, but we then would have $op3 \to_H op2$, forming a cycle of lenght 2 because $op2 \to_X op3$, and we know cycles of lenght 2 are not possible...
+			-  $op3 \to_H op4$ means that $res(op3) <_H inv(op4)$
+
+			- We would then have that $res(op1) <_H inv(op2) <_H res(op3) <_H inv(op4)$
+			- So by transitivity $res(op1) <_H inv(op4)$
 
 
 > [!PDF|red] class 6, p.6>  we would have a cycle of length