
theorem Th19:
  for N being invertible Matrix of 3,F_Real
  for n11,n12,n13,n21,n22,n23,n31,n32,n33 being Element of F_Real
  for P being Element of BK_model
  for Q being Point of ProjectiveSpace TOP-REAL 3
  for u,v being non zero Element of TOP-REAL 3 st
  N = <* <* n11,n12,n13 *>,
         <* n21,n22,n23 *>,
         <* n31,n32,n33 *> *> &
  P = Dir u & Q = Dir v & Q = homography(N).P &
  u.3 = 1 & v.3 = 1 holds 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)
  proof
    let N be invertible Matrix of 3,F_Real;
    let n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real;
    let P be Element of BK_model;
    let Q be Point of ProjectiveSpace TOP-REAL 3;
    let u,v be non zero Element of TOP-REAL 3;
    assume that
A1: N = <* <* n11,n12,n13 *>,
              <* n21,n22,n23 *>,
              <* n31,n32,n33 *> *> and
A2: P = Dir u and
A3: Q = Dir v and
A4: Q = homography(N).P and
A5: u.3 = 1 and
A6: v.3 = 1;
    consider a be non zero Real such that
A7: v.1 = a * (n11 * u.1 + n12 * u.2 + n13) and
A8: v.2 = a * (n21 * u.1 + n22 * u.2 + n23) and
A9: v.3 = a * (n31 * u.1 + n32 * u.2 + n33) by A1,A2,A3, A4,A5,Th18;
    thus n31 * u.1 + n32 * u.2 + n33 <> 0 by A6,A9;
    reconsider nn = n31 * u.1 + n32 * u.2 + n33 as non zero Real by A6,A9;
    a = 1 / nn by A6,A9,XCMPLX_1:73;
    hence thesis by A7,A8;
  end;
