diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 99d20e5..77179d9 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -213,9 +213,9 @@
   },
   "active": "cdcc59f1bf6d4ae1",
   "lastOpenFiles": [
-    "Pasted image 20250505090959.png",
     "Concurrent Systems/slides/class 15.pdf",
     "Concurrent Systems/notes/15 - A formal language for LTSs.md",
+    "Pasted image 20250505090959.png",
     "Pasted image 20250505090603.png",
     "Pasted image 20250505085454.png",
     "Pasted image 20250505084741.png",
@@ -234,7 +234,6 @@
     "Pasted image 20250430192526.png",
     "Pasted image 20250430183304.png",
     "Pasted image 20250430175412.png",
-    "Pasted image 20250430171336.png",
     "Concurrent Systems/slides/class 13.pdf",
     "Concurrent Systems/slides/class 14.pdf",
     "Concurrent Systems/slides/class 12.pdf",
diff --git a/Concurrent Systems/notes/15 - A formal language for LTSs.md b/Concurrent Systems/notes/15 - A formal language for LTSs.md
index 56fd666..d9abe4a 100644
--- a/Concurrent Systems/notes/15 - A formal language for LTSs.md	
+++ b/Concurrent Systems/notes/15 - A formal language for LTSs.md	
@@ -45,3 +45,7 @@ To simplify the proof, let us modify the set of formulae by allowing conjunction
 *Inductive step:* Let's assume the thesis for every tree of height at most h. Let h+1 be the height of $\phi$. Let's distinguish on the outmost operator in $\phi$
 ![](../../Pasted%20image%2020250505090603.png)
 ![](../../Pasted%20image%2020250505090959.png)
+
+(<=) We prove that $$R \triangleq \{ (P, Q) : L(P)=L(Q) \}$$ is a simulation; this suffices, since the relation just defined is trivially symmetric (so is a bisimulation too).
+
+Let $(P, Q) \in R$ and $$
\ No newline at end of file