diff --git a/document/core/valid/instructions.rst b/document/core/valid/instructions.rst
index d92dd7d19..6013d4de2 100644
--- a/document/core/valid/instructions.rst
+++ b/document/core/valid/instructions.rst
@@ -235,6 +235,9 @@ $${rule-prose: Instr_ok/br_on_cast}
 
 $${rule: Instr_ok/br_on_cast}
 
+.. note::
+   The existence of a common supertype ${:rt_3} means that ${:rt_1} and ${:rt_2} must be in the same heap subtyping hierarchy.
+
 
 .. _valid-br_on_cast_fail:
 
@@ -245,6 +248,9 @@ $${rule-prose: Instr_ok/br_on_cast_fail}
 
 $${rule: Instr_ok/br_on_cast_fail}
 
+.. note::
+   The existence of a common supertype ${:rt_3} means that ${:rt_1} and ${:rt_2} must be in the same heap subtyping hierarchy.
+
 
 .. _valid-call:
 
diff --git a/specification/wasm-latest/2.3-validation.instructions.spectec b/specification/wasm-latest/2.3-validation.instructions.spectec
index e57e5d90f..aa012b18f 100644
--- a/specification/wasm-latest/2.3-validation.instructions.spectec
+++ b/specification/wasm-latest/2.3-validation.instructions.spectec
@@ -98,7 +98,10 @@ rule Instr_ok/br_on_cast:
   -- if C.LABELS[l] = t* rt
   -- Reftype_ok: C |- rt_1 : OK
   -- Reftype_ok: C |- rt_2 : OK
-  -- Reftype_sub: C |- rt_2 <: rt_1
+  -- Reftype_ok: C |- rt_3 : OK
+  ----
+  -- Reftype_sub: C |- rt_1 <: rt_3
+  -- Reftype_sub: C |- rt_2 <: rt_3
   -- Reftype_sub: C |- rt_2 <: rt
 
 rule Instr_ok/br_on_cast_fail:
@@ -106,7 +109,10 @@ rule Instr_ok/br_on_cast_fail:
   -- if C.LABELS[l] = t* rt
   -- Reftype_ok: C |- rt_1 : OK
   -- Reftype_ok: C |- rt_2 : OK
-  -- Reftype_sub: C |- rt_2 <: rt_1
+  -- Reftype_ok: C |- rt_3 : OK
+  ----
+  -- Reftype_sub: C |- rt_1 <: rt_3
+  -- Reftype_sub: C |- rt_2 <: rt_3
   -- Reftype_sub: C |- $diffrt(rt_1, rt_2) <: rt