Hey math experts, I need a little help with a proof.

I found a sketch of the proof I want to do on the web(attached) , and I find it difficult to understand.

1. The definition of . (I spouse there is a mistake there and it should be {{ : } : is segment}) what do they mean by "e is a segment"?

2. What is the explanation to the existence of such a " -net" of ?

3. Why does the fact that R is a -net means that each triangle does not intersect more than lines?

Any help will be appreciated.