diff --git a/FormalBook/Chapter_06.lean b/FormalBook/Chapter_06.lean
index 20acb7c..f33f254 100644
--- a/FormalBook/Chapter_06.lean
+++ b/FormalBook/Chapter_06.lean
@@ -189,8 +189,8 @@ theorem wedderburn (h: Fintype R): IsField R := by
   have finclassa: ∀ (A : ConjClasses Rˣ), Fintype ↑(ConjClasses.carrier A) :=
     fun _ ↦ ConjClasses.instFintypeElemCarrier
 
-  have : ∀ (A :  ConjClasses Rˣ), Fintype ↑(Set.centralizer {Quotient.out' A}) :=
-    fun _ ↦ setFintype (Set.centralizer {Quotient.out' _})
+  have : ∀ (A :  ConjClasses Rˣ), Fintype ↑(Set.centralizer {Quotient.out A}) :=
+    fun _ ↦ setFintype (Set.centralizer {Quotient.out _})
 
   letI fintypea : ∀ (A :  ConjClasses Rˣ), Fintype ↑{A |
       have := finclassa A; Fintype.card ↑(ConjClasses.carrier A) > 1} :=
diff --git a/lake-manifest.json b/lake-manifest.json
index baad312..1b180cf 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -1,145 +1,145 @@
 {"version": "1.1.0",
  "packagesDir": ".lake/packages",
  "packages":
- [{"url": "https://github.com/leanprover-community/batteries",
+ [{"url": "https://github.com/leanprover/doc-gen4",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "485efbc439ee0ebdeae8afb0acd24a5e82e2f771",
-   "name": "batteries",
+   "scope": "",
+   "rev": "82c0223cfb0ba31bcf8bef02519adb92fecf4421",
+   "name": "«doc-gen4»",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
-   "inherited": true,
-   "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/quote4",
+   "inherited": false,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/PatrickMassot/checkdecls.git",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "303b23fbcea94ac4f96e590c1cad6618fd4f5f41",
-   "name": "Qq",
+   "scope": "",
+   "rev": "7edf946a4217aa3aa911290811204096e8464ada",
+   "name": "checkdecls",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
-   "inherited": true,
+   "inputRev": null,
+   "inherited": false,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover-community/aesop",
+  {"url": "https://github.com/leanprover-community/mathlib4.git",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "de91b59101763419997026c35a41432ac8691f15",
-   "name": "aesop",
+   "scope": "",
+   "rev": "b202b867947571f7d316d1f591be7e759dc0165c",
+   "name": "mathlib",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": null,
+   "inherited": false,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/mhuisi/lean4-cli",
+   "type": "git",
+   "subDir": null,
+   "scope": "",
+   "rev": "0c8ea32a15a4f74143e4e1e107ba2c412adb90fd",
+   "name": "Cli",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/ProofWidgets4",
+  {"url": "https://github.com/fgdorais/lean4-unicode-basic",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "1383e72b40dd62a566896a6e348ffe868801b172",
-   "name": "proofwidgets",
+   "scope": "",
+   "rev": "d55279d2ff01759fa75752fcf1a93d1db8db18ff",
+   "name": "UnicodeBasic",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.46",
+   "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover/lean4-cli",
+  {"url": "https://github.com/dupuisf/BibtexQuery",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover",
-   "rev": "726b3c9ad13acca724d4651f14afc4804a7b0e4d",
-   "name": "Cli",
+   "scope": "",
+   "rev": "982700bcf78b93166e5b1af0b4b756b8acfdb54b",
+   "name": "BibtexQuery",
+   "manifestFile": "lake-manifest.json",
+   "inputRev": "master",
+   "inherited": true,
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/acmepjz/md4lean",
+   "type": "git",
+   "subDir": null,
+   "scope": "",
+   "rev": "fe8e6e649ac8251f43c6f6f934f095ebebce7e7c",
+   "name": "MD4Lean",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
-   "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/import-graph",
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/plausible",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "119b022b3ea88ec810a677888528e50f8144a26e",
-   "name": "importGraph",
+   "rev": "8e5cb8d424df462f84997dd68af6f40e347c3e35",
+   "name": "plausible",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "v4.15.0-rc1",
    "inherited": true,
    "configFile": "lakefile.toml"},
   {"url": "https://github.com/leanprover-community/LeanSearchClient",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "86d0d0584f5cd165353e2f8a30c455cd0e168ac2",
+   "rev": "d7caecce0d0f003fd5e9cce9a61f1dd6ba83142b",
    "name": "LeanSearchClient",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/plausible",
+  {"url": "https://github.com/leanprover-community/import-graph",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "42dc02bdbc5d0c2f395718462a76c3d87318f7fa",
-   "name": "plausible",
+   "rev": "ed3b856bd8893ade75cafe13e8544d4c2660f377",
+   "name": "importGraph",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "v4.15.0-rc1",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/mathlib4.git",
-   "type": "git",
-   "subDir": null,
-   "scope": "",
-   "rev": "0e836d6e1a3c5ed008688622e261e19fbef05e0e",
-   "name": "mathlib",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": null,
-   "inherited": false,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/PatrickMassot/checkdecls.git",
-   "type": "git",
-   "subDir": null,
-   "scope": "",
-   "rev": "11fa569b1b52f987dc5dcea97fd80eaff95c2fce",
-   "name": "checkdecls",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": null,
-   "inherited": false,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/acmepjz/md4lean",
+  {"url": "https://github.com/leanprover-community/ProofWidgets4",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "5e95f4776be5e048364f325c7e9d619bb56fb005",
-   "name": "MD4Lean",
+   "scope": "leanprover-community",
+   "rev": "2b000e02d50394af68cfb4770a291113d94801b5",
+   "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "v0.0.48",
    "inherited": true,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/fgdorais/lean4-unicode-basic",
+  {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "2905ab4ec3961d1fd68ddae0ab4083497e579014",
-   "name": "UnicodeBasic",
+   "scope": "leanprover-community",
+   "rev": "43bcb1964528411e47bfa4edd0c87d1face1fce4",
+   "name": "aesop",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "master",
    "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/dupuisf/BibtexQuery",
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/quote4",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "bdc2fc30b1e834b294759a5d391d83020a90058e",
-   "name": "BibtexQuery",
+   "scope": "leanprover-community",
+   "rev": "ad942fdf0b15c38bface6acbb01d63855a2519ac",
+   "name": "Qq",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": "v4.14.0",
    "inherited": true,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover/doc-gen4",
+  {"url": "https://github.com/leanprover-community/batteries",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "7b6a56e8e4fcf54d3834b225b9814a7c9e4d4bda",
-   "name": "«doc-gen4»",
+   "scope": "leanprover-community",
+   "rev": "74dffd1a83cdd2969a31c9892b0517e7c6f50668",
+   "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
-   "inherited": false,
-   "configFile": "lakefile.lean"}],
+   "inherited": true,
+   "configFile": "lakefile.toml"}],
  "name": "FormalBook",
  "lakeDir": ".lake"}
diff --git a/lean-toolchain b/lean-toolchain
index 57a4710..cf25a98 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.14.0-rc2
+leanprover/lean4:v4.15.0-rc1