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
  P in (line_homography(N)).l implies homography(N~).P in l
  proof
    assume P in (line_homography(N)).l;
    then P in {homography(N).P where
      P is POINT of IncProjSp_of real_projective_plane : P on l} by Def02;
    then consider P9 be POINT of IncProjSp_of real_projective_plane
    such that
A1: P = homography(N).P9 and
A2: P9 on l;
    P9 is Element of real_projective_plane by INCPROJ:3;
    then
A3: homography(N~).P = P9 by A1,ANPROJ_9:15;
    l is LINE of real_projective_plane by INCPROJ:4;
    hence thesis by A2,A3,INCPROJ:5;
  end;
