Proving the Abstract Geometry Property of {S,L} with Intersection

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SUMMARY

The discussion centers on proving that the intersection of two abstract geometries, {S1,L1} and {S2,L2}, results in another abstract geometry, {S,L}. The proof begins by defining S as the intersection of S1 and S2, and L as the intersection of L1 and L2. The author concludes that since both {S1,L1} and {S2,L2} are established as abstract geometries, the intersection {S,L} must also qualify as an abstract geometry. The need for clarity on the term "abstract geometry" and the definitions of S and L is highlighted.

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Homework Statement



Let {S1,L1} and {S2,L2} be abstract geomettries. If S=S1 ^ S2 and L=L1 ^ L2 prove that {S,L} is an abstract geometry ( where ^ = intersection)

Homework Equations





The Attempt at a Solution



Let {S1,L1} and {S2,L2} be abstract geometries. Assume that S=S1 ^ S2. Let x belong to S therefore by definition of an intersection x belongs to S1 and x belongs to S2. Also assume that L=L1 ^ L2. Let y belong to L therefore by definition of an intersection y belongs to L1 and y belongs to L2. Since {S1,L1} and {S2,L2} are abstract geometries {S,L} must also be an abstract geometry.

Now that I have typed all that I am not even sure that what I was trying to prove was possible. Am I even in the right ball park?
 
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I'm not familiar with the term "abstract geometry" -- could you define it?
 
Also, please understand that notation is not always universal. What is the definition of "abstract geometry", and what are S and L here?
 

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