diff --git a/GlimpseOfLean/Solutions/Topics/GaloisAdjunctions.lean b/GlimpseOfLean/Solutions/Topics/GaloisAdjunctions.lean
index cb78f4a..07a31a3 100644
--- a/GlimpseOfLean/Solutions/Topics/GaloisAdjunctions.lean
+++ b/GlimpseOfLean/Solutions/Topics/GaloisAdjunctions.lean
@@ -232,7 +232,6 @@ lemma isSupUnion {Y : Type} (S : Set (Set Y)) : isSup S (⋃₀ S) := by {
   · intro ht u hu x hx
     apply ht
     use u
-    tauto
   -- sorry
 }
 
diff --git a/lake-manifest.json b/lake-manifest.json
index 7708690..25dcd56 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -1,33 +1,51 @@
-{"version": 4,
+{"version": 5,
  "packagesDir": "lake-packages",
  "packages":
  [{"git":
-   {"url": "https://github.com/EdAyers/ProofWidgets4",
-    "subDir?": null,
-    "rev": "c43db94a8f495dad37829e9d7ad65483d68c86b8",
-    "name": "proofwidgets",
-    "inputRev?": "v0.0.11"}},
-  {"git":
    {"url": "https://github.com/leanprover-community/mathlib4.git",
     "subDir?": null,
-    "rev": "a3b66ecb2f91d93379698626e7ec98a28b893b4d",
+    "rev": "fe2c350a0d0889341aa1c741338f76a73ba03f7e",
+    "opts": {},
     "name": "mathlib",
-    "inputRev?": null}},
+    "inputRev?": null,
+    "inherited": false}},
+  {"git":
+   {"url": "https://github.com/mhuisi/lean4-cli.git",
+    "subDir?": null,
+    "rev": "21dac2e9cc7e3cf7da5800814787b833e680b2fd",
+    "opts": {},
+    "name": "Cli",
+    "inputRev?": "nightly",
+    "inherited": true}},
   {"git":
    {"url": "https://github.com/gebner/quote4",
     "subDir?": null,
-    "rev": "c71f94e34c1cda52eef5c93dc9da409ab2727420",
+    "rev": "e75daed95ad1c92af4e577fea95e234d7a8401c1",
+    "opts": {},
     "name": "Qq",
-    "inputRev?": "master"}},
+    "inputRev?": "master",
+    "inherited": true}},
   {"git":
    {"url": "https://github.com/JLimperg/aesop",
     "subDir?": null,
-    "rev": "ca73109cc40837bc61df8024c9016da4b4f99d4c",
+    "rev": "1a0cded2be292b5496e659b730d2accc742de098",
+    "opts": {},
     "name": "aesop",
-    "inputRev?": "master"}},
+    "inputRev?": "master",
+    "inherited": true}},
   {"git":
    {"url": "https://github.com/leanprover/std4",
     "subDir?": null,
-    "rev": "e68aa8f5fe47aad78987df45f99094afbcb5e936",
+    "rev": "67855403d60daf181775fa1ec63b04e70bcc3921",
+    "opts": {},
     "name": "std",
-    "inputRev?": "main"}}]}
+    "inputRev?": "main",
+    "inherited": true}},
+  {"git":
+   {"url": "https://github.com/EdAyers/ProofWidgets4",
+    "subDir?": null,
+    "rev": "65bba7286e2395f3163fd0277110578f19b8170f",
+    "opts": {},
+    "name": "proofwidgets",
+    "inputRev?": "v0.0.16",
+    "inherited": true}}]}
diff --git a/lean-toolchain b/lean-toolchain
index fd602c4..640f209 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:nightly-2023-06-20
+leanprover/lean4:v4.1.0-rc1