
theorem
  for P,Q,R,S,T,U being Element of BK_model st
  ex h1,h2 being Element of SubGroupK-isometry,
  N1,N2 being invertible Matrix of 3,F_Real st
  h1 = homography(N1) & h2 = homography(N2) &
  homography(N1).P = R & homography(N1).Q = S &
  homography(N2).R = T & homography(N2).S = U holds
  ex h3 being Element of SubGroupK-isometry,
  N3 being invertible Matrix of 3,F_Real st
  h3 = homography(N3) & homography(N3).P = T &
  homography(N3).Q = U
  proof
    let P,Q,R,S,T,U be Element of BK_model;
    assume ex h1,h2 be Element of SubGroupK-isometry,
       N1,N2 be invertible Matrix of 3,F_Real st
       h1 = homography(N1) & h2 = homography(N2) &
       homography(N1).P = R & homography(N1).Q = S &
       homography(N2).R = T & homography(N2).S = U;
    then consider h1,h2 be Element of SubGroupK-isometry,
                  N1,N2 be invertible Matrix of 3,F_Real such that
A1: h1 = homography(N1) & h2 = homography(N2) &
      homography(N1).P = R & homography(N1).Q = S &
      homography(N2).R = T & homography(N2).S = U;
    reconsider N3 = N2 * N1 as invertible Matrix of 3,F_Real;
    h2 * h1 = homography(N2 * N1) by A1,Th35;
    then reconsider h3 = homography(N3) as Element of SubGroupK-isometry;
    take h3;
    homography(N3).P = T & homography(N3).Q = U by A1,ANPROJ_9:13;
    hence thesis;
  end;
