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 Th19:
  homography(N).P in l implies P in (line_homography(N~)).l
  proof
    assume
A1: homography(N).P in l;
    reconsider P9 = homography(N).P as
      POINT of IncProjSp_of real_projective_plane
      by INCPROJ:3;
    l is LINE of real_projective_plane by INCPROJ:4;
    then
A2: P9 on l by A1,INCPROJ:5;
    (line_homography(N~)).l = {homography(N~).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l} by Def02;
    then homography(N~).P9 in (line_homography(N~)).l by A2;
    hence thesis by ANPROJ_9:15;
  end;
