Group of Homography in Real Projective Plane

Using the Mizar system [2], we formalized that homographies of the projective real plane (as defined in [5]), form a group.

Then, we prove that, using the notations of Borsuk and Szmielew in [3]

“Consider in space ℝℙ² points P₁, P₂, P₃, P₄ of which three points are not collinear and points Q₁,Q₂,Q₃,Q₄ each three points of which are also not collinear. There exists one homography h of space ℝℙ² such that h(P_i) = Q_i for i = 1, 2, 3, 4.”

(Existence Statement 52 and Existence Statement 53) [3]. Or, using notations of Richter [11]

“Let [a], [b], [c], [d] in ℝℙ² be four points of which no three are collinear and let [a′],[b′],[c′],[d′] in ℝℙ² be another four points of which no three are collinear, then there exists a 3 × 3 matrix M such that [Ma] = [a′], [Mb] = [b′], [Mc] = [c′], and [Md] = [d′]”

Makarios has formalized the same results in Isabelle/Isar (the collineations form a group, lemma statement52-existence and lemma statement 53-existence) and published it in Archive of Formal Proofs [10], [9].