
theorem
  for N being invertible Matrix of 3,F_Real
  for h being Element of SubGroupK-isometry
  for n11,n12,n13,n21,n22,n23,n31,n32,n33 being Element of F_Real
  for P being Element of BK_model st h = homography(N) &
  N = <* <* n11,n12,n13 *>, <* n21,n22,n23 *>, <* n31,n32,n33 *> *>
  holds homography(N).P = Dir |[
    (n11 * (BK_to_REAL2 P)`1 + n12 * (BK_to_REAL2 P)`2 + n13) /
    (n31 * (BK_to_REAL2 P)`1 + n32 * (BK_to_REAL2 P)`2 + n33),
    (n21 * (BK_to_REAL2 P)`1 + n22 * (BK_to_REAL2 P)`2 + n23) /
    (n31 * (BK_to_REAL2 P)`1 + n32 * (BK_to_REAL2 P)`2 + n33),1 ]|
  proof
    let N be invertible Matrix of 3,F_Real;
    let h be Element of SubGroupK-isometry;
    let n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real;
    let P be Element of BK_model;
    assume
A1: h = homography(N) & N = <* <* n11,n12,n13 *>,
                               <* n21,n22,n23 *>,
                               <* n31,n32,n33 *> *>;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P = Dir u & u.3 = 1 & BK_to_REAL2 P = |[u.1,u.2]| by BKMODEL2:def 2;
    (BK_to_REAL2 P)`1 = u.1 & (BK_to_REAL2 P)`2 = u.2 &
    homography(N).P = Dir |[(n11 * u.1 + n12 * u.2 + n13) /
                            (n31 * u.1 + n32 * u.2 + n33),
                            (n21 * u.1 + n22 * u.2 + n23)
                            / (n31 * u.1 + n32 * u.2 + n33),1 ]|
                              by A1,A2,Th23,EUCLID:52;
    hence thesis;
  end;
