
theorem Th35:
  for N1,N2 being invertible Matrix of 3,F_Real
  for h1,h2 being Element of SubGroupK-isometry st
  h1 = homography(N1) & h2 = homography(N2) holds
  h1 * h2 is Element of SubGroupK-isometry &
  h1 * h2 = homography(N1 * N2)
  proof
    let N1,N2 being invertible Matrix of 3,F_Real;
    let h1,h2 being Element of SubGroupK-isometry;
    assume that
A1: h1 = homography(N1) and
A2: h2 = homography(N2);
    thus h1 * h2 is Element of SubGroupK-isometry;
    h1 in EnsHomography3 by A1,ANPROJ_9:def 1;
    then reconsider hh1 = h1 as Element of EnsHomography3;
    h2 in EnsHomography3 by A2,ANPROJ_9:def 1;
    then reconsider hh2 = h2 as Element of EnsHomography3;
    set G = GroupHomography3;
    reconsider h1g = hh1, h2g = hh2 as Element of G by ANPROJ_9:def 4;
    h1g * h2g = hh1 (*) hh2 by ANPROJ_9:def 3,def 4
             .= homography(N1 * N2) by A1,A2,ANPROJ_9:18;
    hence h1 * h2 = homography(N1 * N2) by GROUP_2:43;
  end;
