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This question was very interesting for me and I've started investigation. I have never thought to use this approach. I mean to derive one static class from another. So, I found out that .NET Framework doesn't allow to derive one static class from another. The static class must be derived directly from System.Object.
The keyword STATIC defines a singleton class therefore to derive from a static class the base class has to be the same type.
There would be base class function like Public static virtual DoSomething()
This function is overrided in derived class like Public static Override DoSomething()