
theorem
  for P,Q being Element of BK_model holds
  ex h being Element of SubGroupK-isometry,
     N being invertible Matrix of 3,F_Real st
  h = homography(N) & homography(N).P = Q &
  homography(N).Q = P
  proof
    let P,Q be Element of BK_model;
    per cases;
    suppose
A1:   P = Q;
      reconsider N = 1.(F_Real,3) as invertible Matrix of 3,F_Real;
      homography(N) in the set of all homography(N)
        where N is invertible Matrix of 3,F_Real;
      then reconsider h = homography(N) as Element of EnsHomography3
        by ANPROJ_9:def 1;
      h is_K-isometry by Th33;
      then h in EnsK-isometry;
      then reconsider h as Element of SubGroupK-isometry by Def05;
      take h;
      homography(N).P = Q & homography(N).Q = P by A1,ANPROJ_9:14;
      hence thesis;
    end;
    suppose P <> Q;
      then consider N be invertible Matrix of 3,F_Real such that
A2:   (homography(N)).:absolute = absolute and
A3:   (homography(N)). P = Q and
A4:   (homography(N)).Q = P and
      (ex P1,P2 being Element of absolute st P1 <> P2 &
         P,Q,P1 are_collinear & P,Q,P2 are_collinear &
         homography(N).P1 = P2 & homography(N).P2 = P1)
        by Th45;
      homography(N) in the set of all homography(N)
        where N is invertible Matrix of 3,F_Real;
      then reconsider h = homography(N) as Element of EnsHomography3
        by ANPROJ_9:def 1;
      h is_K-isometry by A2;
      then h in EnsK-isometry;
      then reconsider h as Element of SubGroupK-isometry by Def05;
      take h;
      thus thesis by A3,A4;
    end;
  end;
