I found the answer to my question, and it's stupid - multiprocessing is already preinstalled in my version of Python (3.5.2) by default.
It will not appear in the package list in Anaconda >> Environments >> root, since it is not a third-party package, but an internal one.
If someone is not sure if this applies to you, just check from multiprocessing import Pool in the Python console.
This is true for all currently supported versions of Python (2.7 and 3.x), and according to Python 2.6, multiprocessing was part of the standard library (including batteries) with Python 2.6. https://bugs.python.org/msg326646
You no longer need to perform pip install multiprocessing using pip install multiprocessing and DO NOT include it in your requirements.txt file if you do not support the Python 2.4 / 2.5 application (please transfer!). In most versions you can just import multiprocessing and everything will be fine.
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