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;

theorem Th15:
  (line_homography(N)).((line_homography(N~)).l) = l &
  (line_homography(N~)).((line_homography(N)).l) = l
  proof
A1: N~ is_reverse_of N by MATRIX_6:def 4;
    thus (line_homography(N)).((line_homography(N~)).l)
      = (line_homography(N * N~)).l by Th13
     .= (line_homography(1.(F_Real,3))).l by A1,MATRIX_6:def 2
     .= l by Th14;
    thus (line_homography(N~)).((line_homography(N)).l)
      = (line_homography(N~ * N)).l by Th13
     .= (line_homography(1.(F_Real,3))).l by A1,MATRIX_6:def 2
     .= l by Th14;
  end;
