
theorem Th60:
  for a,b being POINT of BK-model-Plane holds a,a equiv b,b
  proof
    let a,b be POINT of BK-model-Plane;
    reconsider A = a, B = b as Element of BK_model;
    reconsider P = Dir |[1,0,1]| as Element of absolute by Th59;
    consider N be invertible Matrix of 3,F_Real such that
A1: homography(N).:absolute = absolute and
A2: (homography(N)).a = b and
    (homography(N)).P = P by BKMODEL2:56;
    homography(N) in the set of all homography(N) where
    N is invertible Matrix of 3,F_Real;
    then reconsider h = homography(N) as Element of EnsHomography3
      by ANPROJ_9:def 1;
    h is_K-isometry by A1,BKMODEL2:def 6;
    then h in EnsK-isometry by BKMODEL2:def 7;
    then reconsider h = homography(N) as Element of SubGroupK-isometry
      by BKMODEL2:def 8;
    ex h being Element of SubGroupK-isometry st
      ex N being invertible Matrix of 3,F_Real st h = homography(N) &
      homography(N).a = b & homography(N).a = b
    proof
      take h;
      take N;
      thus thesis by A2;
    end;
    hence thesis by BKMODEL3:def 8;
  end;
