
theorem Th29:
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3
  for v being Element of TOP-REAL 3 st u = normalize_proj3 P holds
  |{ dir3a P,dir3b P,v }| = |(u,v)|
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    let v be Element of TOP-REAL 3;
    assume u = normalize_proj3 P;
    then
A1: u.3 = 1 & P = Dir u by Def6;
    then normalize_proj3 P = |[u.1/u.3, u.2/u.3, 1]| by Th17;
    then (normalize_proj3(P))`1 = u.1/u.3 & (normalize_proj3(P))`2 = u.2/u.3;
    then |{ dir3a P,dir3b P,v }| = |{ |[ 1,   0,   -u.1/u.3 ]|,
                                      |[ 0,   1,   -u.2/u.3 ]|,
                                      |[ v`1, v`2, v`3 ]| }|
      .= v`3 - v`1 * (-u.1/u.3) - v`2 * (-u.2/u.3) by Th4
      .= u`1 * v`1 + u`2 * v`2 + u`3 * v`3 by A1
      .= |(u,v)| by EUCLID_5:29;
    hence thesis;
  end;
