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?