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

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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?