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 Th30:
  for P being Point of real_projective_plane
  for h being Element of SubGroupK-isometry,
  N being invertible Matrix of 3,F_Real st h = homography(N) holds
  P is Element of absolute iff homography(N).P is Element of absolute
  proof
    let P be Point of real_projective_plane;
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    assume
A1: h = homography(N);
    h is Element of EnsK-isometry by BKMODEL2:def 8;
    then
A2: homography(N).:absolute = absolute by A1,BKMODEL2:44;
    homography(N~) is Element of SubGroupK-isometry by A1,BKMODEL2:47;
    then homography(N~) is Element of EnsK-isometry by BKMODEL2:def 8;
    then
A3: homography(N~).:absolute = absolute by BKMODEL2:44;
    set hP = homography(N).P;
    hereby
      assume
A4:   P is Element of absolute;
      dom homography(N) = the carrier of ProjectiveSpace TOP-REAL 3
        by FUNCT_2:def 1;
      hence homography(N).P is Element of absolute by A2,A4,FUNCT_1:108;
    end;
    assume
A5: homography(N).P is Element of absolute;
A6: dom homography(N~) = the carrier of ProjectiveSpace TOP-REAL 3
      by FUNCT_2:def 1;
    homography(N~).hP in homography(N~).:absolute by A6,A5,FUNCT_1:108;
    hence P is Element of absolute by A3,ANPROJ_9:15;
  end;
