
theorem Th33:
  for h being Element of EnsHomography3 st h = homography(1.(F_Real,3)) holds
  h is_K-isometry
  proof
    let h be Element of EnsHomography3;
    assume
A1: h = homography(1.(F_Real,3));
    reconsider N = 1.(F_Real,3) as invertible Matrix of 3,F_Real;
    h is_K-isometry
    proof
A2:   (homography(N)).:absolute c= absolute
      proof
        let x be object;
        assume x in (homography(N)).:absolute;
        then ex y be object st y in dom homography(N) & y in absolute &
          (homography(N)).y = x by FUNCT_1:def 6;
        hence thesis by ANPROJ_9:14;
      end;
      absolute c= (homography(N)).:absolute
      proof
        let x be object;
        assume
A3:     x in absolute;
        then reconsider y = x as Point of ProjectiveSpace TOP-REAL 3;
A4:     y = homography(N).y by ANPROJ_9:14;
        dom homography(N) = the carrier of ProjectiveSpace TOP-REAL 3
          by FUNCT_2:def 1;
        hence thesis by A4,A3,FUNCT_1:108;
      end;
      then absolute = (homography(N)).:absolute by A2;
      hence thesis by A1;
    end;
    hence thesis;
  end;
