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 Th31:
  for P being Element of BK_model
  for h being Element of SubGroupK-isometry,
      N being invertible Matrix of 3,F_Real st
  h = homography(N) holds homography(N).P is Element of BK_model
  proof
    let P be Element of BK_model;
    let h be Element of SubGroupK-isometry;
    let N be invertible Matrix of 3,F_Real;
    assume
A1: h = homography(N);
    set hP = homography(N).P;
    assume
A2: not hP is Element of BK_model;
    not hP is Element of absolute
    proof
      assume hP is Element of absolute;
      then P is Element of absolute by A1,Th30;
      hence contradiction by XBOOLE_0:3,BKMODEL2:1;
    end;
    then not hP in BK_model \/ absolute by A2,XBOOLE_0:def 3;
    then consider l such that
A3: hP in l and
A4: l misses absolute by Th29;
    reconsider L = (line_homography(N~)).l as LINE of real_projective_plane
      by INCPROJ:4;
    reconsider L9 = L as LINE of IncProjSp_of real_projective_plane;
    consider P1,P2 be Element of absolute such that
    P1 <> P2 and
A5: P1 in L9 and
    P2 in L9 by A3,Th19,Th12;
A6: homography(N).P1 is Element of absolute by A1,Th30;
    homography(N).P1 in (line_homography(N)).L by A5,Th20;
    then homography(N).P1 in l by Th15;
    hence contradiction by A6,A4,XBOOLE_0:def 4;
  end;
