
theorem Th23:
  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
  for u being non zero Element of TOP-REAL 3 st
  h = homography(N) & N = <* <* n11,n12,n13 *>,
                             <* n21,n22,n23 *>,
                             <* n31,n32,n33 *> *> &
  P = Dir u & u.3 = 1 holds
  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 ]|
  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;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: h = homography(N) & N = <* <* n11,n12,n13 *>,
                               <* n21,n22,n23 *>,
                               <* n31,n32,n33 *> *> &
    P = Dir u & u.3 = 1;
    reconsider Q = homography(N).P as Element of BK_model by A1,BKMODEL3:36;
    consider v be non zero Element of TOP-REAL 3 such that
A2: Q = Dir v & v.3 = 1 & BK_to_REAL2 Q = |[v.1,v.2]| by BKMODEL2:def 2;
    n31 * u.1 + n32 * u.2 + n33 <> 0 &
      v.1 = (n11 * u.1 + n12 * u.2 + n13) / (n31 * u.1 + n32 * u.2 + n33) &
      v.2 = (n21 * u.1 + n22 * u.2 + n23) / (n31 * u.1 + n32 * u.2 + n33)
      by A2,A1,Th19;
    then v`1 = (n11 * u.1 + n12 * u.2 + n13) / (n31 * u.1 + n32 * u.2 + n33) &
      v`2 = (n21 * u.1 + n22 * u.2 + n23) / (n31 * u.1 + n32 * u.2 + n33) &
      v`3 = 1 by A2,EUCLID_5:def 1,def 2,def 3;
    hence thesis by A2,EUCLID_5:3;
  end;
