
theorem Th24:
  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 absolute
  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 absolute;
    let u be non zero Element of TOP-REAL 3;
    assume that
A1: h = homography(N) & N = <* <* n11,n12,n13 *>,
                               <* n21,n22,n23 *>,
                               <* n31,n32,n33 *> *> and
A2: P = Dir u & u.3 = 1;
    reconsider Q = homography(N).P as Element of absolute by A1,BKMODEL3:35;
    consider v be non zero Element of TOP-REAL 3 such that
A3: Q = Dir v & v.3 = 1 &
      absolute_to_REAL2(Q) = |[v.1,v.2]| by BKMODEL1:def 8;
    now
      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 A3,A2,A1,Th21;
      hence 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 A3,EUCLID_5:def 1,def 2,def 3;
      thus homography(N).P = Dir |[v`1,v`2,v`3]| by A3,EUCLID_5:3;
    end;
    hence thesis;
  end;
