From e2212f7999130414471a5009d9543edf667b09dc Mon Sep 17 00:00:00 2001
From: Michael Zhang <mail@mzhang.io>
Date: Sat, 27 Jul 2024 22:20:41 -0500
Subject: [PATCH] update

---
 html/ProgressHeader.html       |  6 +++++-
 src/HottBook/Chapter2.lagda.md | 21 +++++++++++++++++++--
 2 files changed, 24 insertions(+), 3 deletions(-)

diff --git a/html/ProgressHeader.html b/html/ProgressHeader.html
index d8d931b..5ee429a 100644
--- a/html/ProgressHeader.html
+++ b/html/ProgressHeader.html
@@ -28,4 +28,8 @@
   }
 </style>
 
-<h1>HoTT Book Progress</h1>
\ No newline at end of file
+<h1>HoTT Book Progress</h1>
+
+<p>
+  Last 7 days: <img src="https://wakapi.mzhang.io/api/badge/michael/interval:last_7_days/project:type-theory" />
+</p>
\ No newline at end of file
diff --git a/src/HottBook/Chapter2.lagda.md b/src/HottBook/Chapter2.lagda.md
index db399be..fba3dab 100644
--- a/src/HottBook/Chapter2.lagda.md
+++ b/src/HottBook/Chapter2.lagda.md
@@ -99,10 +99,27 @@ module theorem2∙1∙6 where
   Ω² {l} = Ω[_] {l = l} 2
 
   compose2 : {l : Level} {A : Set l} {a : A} → fst (Ω² (A , a)) → fst (Ω² (A , a)) → fst (Ω² (A , a))
-  compose2 x y = x ∙ y
+  compose2 x y = horiz x y
+    where
+      variable
+        A : Set l
+        a b c : A
+        p q : a ≡ b
+        r s : b ≡ c
+        α : p ≡ q
+        β : r ≡ s
+
+      _∙ᵣ_ : p ≡ q → (r : b ≡ c) → p ∙ r ≡ q ∙ r
+      α ∙ᵣ refl = sym (lemma2∙1∙4.i1 _) ∙ α ∙ lemma2∙1∙4.i1 _
+
+      _∙ₗ_ : (q : a ≡ b) → r ≡ s → q ∙ r ≡ q ∙ s
+      refl ∙ₗ β = sym (lemma2∙1∙4.i2 _) ∙ β ∙ lemma2∙1∙4.i2 _
+
+      horiz : p ≡ q → r ≡ s → p ∙ r ≡ q ∙ s
+      horiz α β = (α ∙ᵣ _) ∙ (_ ∙ₗ β)
 
   compose2-commutative : {l : Level} {A : Set l} {a : A} (x y : fst (Ω² (A , a))) → compose2 x y ≡ compose2 y x
-  compose2-commutative {l} {A} {a} x y = {!  !}
+  compose2-commutative {l} {A} {a} x y = {! x !}
 ```
 
 ### Definition 2.1.7 (pointed type)