reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem
  for P being Element of BK_model
  for h being Element of SubGroupK-isometry
  for N being invertible Matrix of 3,F_Real st h = homography(N) holds
  ex u being non zero Element of TOP-REAL 3 st homography(N).P = Dir u &
  u.3 = 1
  proof
    let P be Element of BK_model;
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    assume h = homography(N);
    then reconsider hP = homography(N).P as Element of BK_model by Th31;
    ex u being non zero Element of TOP-REAL 3 st
      Dir u = hP & u.3 = 1 & BK_to_REAL2 hP = |[u.1,u.2]| by BKMODEL2:def 2;
    hence thesis;
  end;
