diff --git a/Lib/functools.py b/Lib/functools.py
index d450634..1062a45 100644
--- a/Lib/functools.py
+++ b/Lib/functools.py
@@ -68,17 +68,17 @@
 def total_ordering(cls):
     """Class decorator that fills in missing ordering methods"""
     convert = {
-        '__lt__': [('__gt__', lambda self, other: other < self),
-                   ('__le__', lambda self, other: not other < self),
+        '__lt__': [('__gt__', lambda self, other: not (self < other or self == other)),
+                   ('__le__', lambda self, other: self < other or self == other),
                    ('__ge__', lambda self, other: not self < other)],
-        '__le__': [('__ge__', lambda self, other: other <= self),
-                   ('__lt__', lambda self, other: not other <= self),
+        '__le__': [('__ge__', lambda self, other: not self <= other or self == other),
+                   ('__lt__', lambda self, other: self <= other and not self == other),
                    ('__gt__', lambda self, other: not self <= other)],
-        '__gt__': [('__lt__', lambda self, other: other > self),
-                   ('__ge__', lambda self, other: not other > self),
+        '__gt__': [('__lt__', lambda self, other: not (self > other or self == other)),
+                   ('__ge__', lambda self, other: self > other or self == other),
                    ('__le__', lambda self, other: not self > other)],
-        '__ge__': [('__le__', lambda self, other: other >= self),
-                   ('__gt__', lambda self, other: not other >= self),
+        '__ge__': [('__le__', lambda self, other: (not self >= other) or self == other),
+                   ('__gt__', lambda self, other: self >= other and not self == other),
                    ('__lt__', lambda self, other: not self >= other)]
     }
     # Find user-defined comparisons (not those inherited from object).
diff --git a/Lib/test/test_functools.py b/Lib/test/test_functools.py
index f41a144..73a77d6 100644
--- a/Lib/test/test_functools.py
+++ b/Lib/test/test_functools.py
@@ -457,6 +457,8 @@
                 self.value = value
             def __lt__(self, other):
                 return self.value < other.value
+            def __eq__(self, other):
+                return self.value == other.value
         self.assertTrue(A(1) < A(2))
         self.assertTrue(A(2) > A(1))
         self.assertTrue(A(1) <= A(2))
@@ -471,6 +473,8 @@
                 self.value = value
             def __le__(self, other):
                 return self.value <= other.value
+            def __eq__(self, other):
+                return self.value == other.value
         self.assertTrue(A(1) < A(2))
         self.assertTrue(A(2) > A(1))
         self.assertTrue(A(1) <= A(2))
@@ -485,6 +489,8 @@
                 self.value = value
             def __gt__(self, other):
                 return self.value > other.value
+            def __eq__(self, other):
+                return self.value == other.value
         self.assertTrue(A(1) < A(2))
         self.assertTrue(A(2) > A(1))
         self.assertTrue(A(1) <= A(2))
@@ -499,6 +505,8 @@
                 self.value = value
             def __ge__(self, other):
                 return self.value >= other.value
+            def __eq__(self, other):
+                return self.value == other.value
         self.assertTrue(A(1) < A(2))
         self.assertTrue(A(2) > A(1))
         self.assertTrue(A(1) <= A(2))
@@ -524,6 +532,22 @@
             class A:
                 pass
 
+    def test_bug_10042(self):
+        @functools.total_ordering
+        class TestTO:
+            def __init__(self, value):
+                self.value = value
+            def __eq__(self, other):
+                if isinstance(other, TestTO):
+                    return self.value == other.value
+                return False
+            def __lt__(self, other):
+                if isinstance(other, TestTO):
+                    return self.value < other.value
+                raise TypeError
+        with self.assertRaises(TypeError):
+            TestTO(8) <= ()
+
 class TestLRU(unittest.TestCase):
 
     def test_lru(self):