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 Th20:
  P in l iff homography(N).P in (line_homography(N)).l
  proof
    hereby
      assume
A1:   P in l;
      reconsider P9 = 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;
      hence homography(N).P in (line_homography(N)).l by A2;
    end;
    assume homography(N).P in (line_homography(N)).l;
    then P in (line_homography(N~)).((line_homography(N)).l) by Th19;
    hence P in l by Th15;
  end;
