The following provides an outline for a proof of Theorem 17.18.
(a) Consider a parallel class of lines given by y = mx + b, where m ∈ F, m ≠ 0. Show that each line in this class intersects each "vertical" line and each "horizontal" line in exactly one point of AP (F). Thus the configuration obtained by labeling the points of AP (F), as in Figs. 17.4, 17.5, and 17.6, is a Latin square.
(b) To show that the Latin squares corresponding to two different classes, other than the classes of slope 0 or infinite slope, are orthogonal, assume that an ordered pair (i, j) appears more than once when one square is superimposed upon the other. How does this lead to a contradiction?