reserve P for Element of BK_model;
reserve N,N1,N2 for invertible Matrix of 3,F_Real;
reserve l,l1,l2 for Element of the Lines of IncProjSp_of real_projective_plane;
reserve P for Point of ProjectiveSpace TOP-REAL 3,
        l for LINE of IncProjSp_of real_projective_plane;

theorem
  BK-model-Plane is satisfying_CongruenceEquivalenceRelation
  proof
    let P,Q,R,S,T,U be POINT of BK-model-Plane;
    assume that
A1: P,Q equiv R,S and
A2: P,Q equiv T,U;
    consider h1 be Element of SubGroupK-isometry such that
A3:  ex N be invertible Matrix of 3,F_Real st
    h1 = homography(N) & homography(N).P = R & homography(N).Q = S by A1,Def05;
    consider N1 be invertible Matrix of 3,F_Real such that
A4: h1 = homography(N1) & homography(N1).P = R & homography(N1).Q = S by A3;
    reconsider N3 = N1~ as invertible Matrix of 3,F_Real;
    P in BK_model & Q in BK_model;
    then
A5: homography(N3).R = P & homography(N3).S = Q by A4,ANPROJ_9:15;
    reconsider h3 = homography(N3) as Element of SubGroupK-isometry
      by A4,BKMODEL2:47;
    consider h2 be Element of SubGroupK-isometry such that
A6: ex N being invertible Matrix of 3,F_Real st
    h2 = homography(N) & homography(N).P = T & homography(N).Q = U by A2,Def05;
    consider N2 be invertible Matrix of 3,F_Real such that
A7: h2 = homography(N2) & homography(N2).P = T & homography(N2).Q = U by A6;
    reconsider N4 = N2 * N3 as invertible Matrix of 3,F_Real;
    now
      h2 * h3 is Element of SubGroupK-isometry;
      hence homography(N4) is Element of SubGroupK-isometry by A7,BKMODEL2:46;
      thus N4 is invertible Matrix of 3,F_Real;
      R in BK_model & S in BK_model;
      hence homography(N4).R = T & homography(N4).S = U by A5,A7,ANPROJ_9:13;
    end;
    hence R,S equiv T,U by Def05;
  end;
