
theorem Th36:
  for N being invertible Matrix of 3,F_Real
  for h being Element of SubGroupK-isometry st
  h = homography(N) holds h" = homography(N~) &
  homography(N~) is Element of SubGroupK-isometry
  proof
    let N being invertible Matrix of 3,F_Real;
    let h being Element of SubGroupK-isometry;
    assume
A1: h = homography(N);
    then h in EnsHomography3 by ANPROJ_9:def 1;
    then reconsider h1 = h as Element of EnsHomography3;
    homography(N~) in EnsHomography3 by ANPROJ_9:def 1;
    then reconsider h2 = homography(N~) as Element of EnsHomography3;
    set G = GroupHomography3;
    reconsider h1g = h1, h2g = h2 as Element of G by ANPROJ_9:def 4;
A2: N is_reverse_of N~ by MATRIX_6:def 4;
A3: h1g * h2g = h1 (*) h2 by ANPROJ_9:def 3,def 4
             .= homography(N * N~) by A1,ANPROJ_9:18
             .= 1_G by A2,MATRIX_6:def 2,Th34;
    h2g * h1g = h2 (*) h1 by ANPROJ_9:def 3,def 4
             .= homography(N~ * N) by A1,ANPROJ_9:18
             .= 1_G by A2,MATRIX_6:def 2,Th34;
    then h2g = h1g" by A3,GROUP_1:5;
    hence h" = homography(N~) by GROUP_2:48;
    hence thesis;
  end;
