Let (P,L,I) be an affine plane, and let C be the set of parallelism equiva- lence classes of L. Let ∞ be a element such that ∞ ∈/ L. For the relation I′ given by P ∈ P and m ∈ L and P I m , or PI′m ⇔ P ∈Candm∈Landm∈P,or P ∈ C and m = ∞ the point-line geometry (P ∪ C, L ∪ {∞}, I′) is a projective plane.

Prove the above theorem and explain