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I want your advice on writing a small portion of the abstract of a paper. Let me introduce the situation first.

There is a large class of objects, and an excellent way to study these objects is through a property, say "Property P." Now, this large class is subdivided into two sub-classes, say, "object of type X" and "object of type Y." Now, the Property P for objects of type X has been well-studied, nowadays classic stuff. Also, the sub-class of the objects of type Y is further divided into finer sub-classes, say "object of type Y_1" and "object of type Y_2".

Though the Property P for objects of type Y_1 is not written in the literature clearly, it is well-known to the experts in this field and also follows from Property P for objects of type X more or less directly.

Now, the Property P for objects of type Y_2 is neither written in literature nor known to experts very well (but they expect it to be true).

For a moment, assume that I am able to show the Property P does hold for the objects of type Y, i.e., I exhaustively studied the Property P for objects of type Y_1 as well as objects of type Y_2. It is expected that I will write an abstract for a paper as "Property P holds for any object of type Y."

Here is my problem: when one says that "any object of type Y," most of the time, people assume that statement is about "any object of type Y_1", as these type of objects appears more naturally, as I stated earlier. So, as a result, one may ignore the whole paper just reading the phrase "Property P holds for any object of type Y" for the obvious reason: what's new? (as this directly follows from property P for the object of type X)

Question: How do I tackle this? Should I mention this way: "Property P holds for any object of type Y, possibly of the type Y_2". Do you have a better way of writing this?

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If the main and major point of your paper is exactly what you mention, then it deserves an explicit mention that is unambiguous and explores tacit assumptions. All major contributions of your paper acquire space in the abstract corresponding to what you consider to be the importance of that contribution.

We show, for the first time?, that property P holds not just for Y_1 subclass of Y, but also for Y_2. This establihes the previously unclear theorem that P holds for all known Y.

I would advise caution when using "for the first time" if you are not highly certain that it is the case, given your extensive study of the literature. Using a softer tone is always recommended.

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