
theorem Th34:
  homography(1.(F_Real,3)) = 1_GroupHomography3 &
  homography(1.(F_Real,3)) = 1_SubGroupK-isometry
  proof
    set G = GroupHomography3;
    homography(1.(F_Real,3)) in G by ANPROJ_9:def 1,def 4;
    then reconsider e = homography(1.(F_Real,3)) as Element of G;
    now
      let h being Element of GroupHomography3;
      h in EnsHomography3 by ANPROJ_9:def 4;
      then consider N be invertible Matrix of 3,F_Real such that
A1:   h = homography(N) by ANPROJ_9:def 1;
      h in EnsHomography3 & e in EnsHomography3 by A1,ANPROJ_9:def 1;
      then reconsider h1 = h, h2 = e as Element of EnsHomography3;
      thus h * e = h1 (*) h2 by ANPROJ_9:def 3,def 4
                .= homography(N * 1.(F_Real,3)) by A1,ANPROJ_9:18
                .= h by A1,MATRIX_3:19;
      thus e * h = h2 (*) h1 by ANPROJ_9:def 3,def 4
                .= homography(1.(F_Real,3) * N) by A1,ANPROJ_9:18
                .= h by A1,MATRIX_3:18;
    end;
    hence homography(1.(F_Real,3)) = 1_GroupHomography3 by GROUP_1:4;
    hence thesis by GROUP_2:44;
  end;
