diff --git a/docs/conf.py b/docs/conf.py
index 7d947ac..26b18f1 100644
--- a/docs/conf.py
+++ b/docs/conf.py
@@ -33,9 +33,9 @@
 # built documents.
 #
 # The full version, including alpha/beta/rc tags.
-release = "1.0.0"
+release = "1.1.0"
 # The short X.Y version.
-version = "1.0.0"
+version = "1.1.0"
 
 
 # -- General configuration ---------------------------------------------------
diff --git a/test/test_00_base.py b/test/test_00_base.py
index f946bb8..d7957bf 100644
--- a/test/test_00_base.py
+++ b/test/test_00_base.py
@@ -18,7 +18,7 @@ def test_base_set_name_slots(attr_name: Optional[str]) -> None:
     if attr_name is not None:
         slot_name = attr_name
     else:
-        slot_name = "_x"
+        slot_name = "__x"
     class C:
         x = Attr(int, lambda _, x: x >= 0, attr_name=attr_name)
         __slots__ = (slot_name,)
diff --git a/typed_descriptors/__init__.py b/typed_descriptors/__init__.py
index 237afe8..df9b4f0 100644
--- a/typed_descriptors/__init__.py
+++ b/typed_descriptors/__init__.py
@@ -2,27 +2,13 @@
     Descriptor classes with static and runtime validation features.
 """
 
+# typed-descriptors: Descriptor classes for typed attributes and properties.
 # Copyright (C) 2023 Hashberg Ltd
 
-# This library is free software; you can redistribute it and/or
-# modify it under the terms of the GNU Lesser General Public
-# License as published by the Free Software Foundation; either
-# version 2.1 of the License, or (at your option) any later version.
-
-# This library is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-# Lesser General Public License for more details.
-
-# You should have received a copy of the GNU Lesser General Public
-# License along with this library; if not, write to the Free Software
-# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301
-# USA
-
 from __future__ import annotations
 from .attr import Attr
 from .prop import Prop
 
-__version__ = "1.0.0"
+__version__ = "1.1.0"
 
 __all__ = ("Attr", "Prop")
diff --git a/typed_descriptors/attr.py b/typed_descriptors/attr.py
index d337084..790369f 100644
--- a/typed_descriptors/attr.py
+++ b/typed_descriptors/attr.py
@@ -2,23 +2,9 @@
     Descriptor class for attributes.
 """
 
+# Part of typed-descriptors
 # Copyright (C) 2023 Hashberg Ltd
 
-# This library is free software; you can redistribute it and/or
-# modify it under the terms of the GNU Lesser General Public
-# License as published by the Free Software Foundation; either
-# version 2.1 of the License, or (at your option) any later version.
-
-# This library is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-# Lesser General Public License for more details.
-
-# You should have received a copy of the GNU Lesser General Public
-# License along with this library; if not, write to the Free Software
-# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301
-# USA
-
 from __future__ import annotations
 import sys
 from typing import (
diff --git a/typed_descriptors/base.py b/typed_descriptors/base.py
index c847eb8..f41bc18 100644
--- a/typed_descriptors/base.py
+++ b/typed_descriptors/base.py
@@ -2,23 +2,9 @@
     Abstract base class for descriptors backed by protected/private attributes.
 """
 
+# Part of typed-descriptors
 # Copyright (C) 2023 Hashberg Ltd
 
-# This library is free software; you can redistribute it and/or
-# modify it under the terms of the GNU Lesser General Public
-# License as published by the Free Software Foundation; either
-# version 2.1 of the License, or (at your option) any later version.
-
-# This library is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-# Lesser General Public License for more details.
-
-# You should have received a copy of the GNU Lesser General Public
-# License along with this library; if not, write to the Free Software
-# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301
-# USA
-
 from __future__ import annotations
 from abc import abstractmethod
 import sys
@@ -26,7 +12,6 @@
     Any,
     Generic,
     Literal,
-    MutableSequence,
     Optional,
     Type,
     TypeVar,
@@ -52,8 +37,9 @@ class DescriptorBase(Generic[T]):
 
     - if the optional kwarg ``attr_name`` is specified in the constructor,
       the descriptor uses an attribute with that given name
-    - otherwise, the descriptor uses an attribute obtained from the
-      descriptor name by prepending an underscore
+    - otherwise, the descriptor uses a private attribute obtained from the
+      descriptor name by prepending one or two underscores (depending on
+      whether the descriptor name itself starts with underscore or not, resp.).
 
     If the attribute name starts with two underscores but does not end with
     two underscores, name-mangling is automatically performed.
@@ -186,7 +172,8 @@ def __set_name__(self, owner: Type[Any], name: str) -> None:
         # 2. Get the backing attribute name (not yet name-mangled):
         given_attr_name = self.__given_attr_name
         if given_attr_name is None:
-            given_attr_name = f"_{name}"
+            attr_name_prefix = "_" if name.startswith("_") else "__"
+            given_attr_name = f"{attr_name_prefix}{name}"
         elif given_attr_name == name:
             raise ValueError(
                 f"Name of backing attribute for descriptor {self.name!r} "
diff --git a/typed_descriptors/prop.py b/typed_descriptors/prop.py
index a74ed42..40c61d8 100644
--- a/typed_descriptors/prop.py
+++ b/typed_descriptors/prop.py
@@ -2,23 +2,9 @@
     Descriptor class for cached properties.
 """
 
+# Part of typed-descriptors
 # Copyright (C) 2023 Hashberg Ltd
 
-# This library is free software; you can redistribute it and/or
-# modify it under the terms of the GNU Lesser General Public
-# License as published by the Free Software Foundation; either
-# version 2.1 of the License, or (at your option) any later version.
-
-# This library is distributed in the hope that it will be useful,
-# but WITHOUT ANY WARRANTY; without even the implied warranty of
-# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
-# Lesser General Public License for more details.
-
-# You should have received a copy of the GNU Lesser General Public
-# License along with this library; if not, write to the Free Software
-# Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301
-# USA
-
 from __future__ import annotations
 import sys
 from typing import (