
theorem Th21:
  for P being non zero_proj1 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_proj1 P holds
  |{ dir1a P,dir1b P,v }| = |(u,v)|
  proof
    let P be non zero_proj1 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_proj1 P;
    then
A1: u.1 = 1 & P = Dir u by Def2;
    then normalize_proj1 P = |[1, u.2/u.1,u.3/u.1]| by Th11;
    then (normalize_proj1(P))`2 = u.2/u.1 & (normalize_proj1(P))`3 = u.3/u.1;
    then |{ dir1a P,dir1b P,v }| = |{ |[ -u.2/u.1, 1  , 0   ]|,
                                      |[ -u.3/u.1, 0  , 1   ]|,
                                      |[ v`1     , v`2, v`3 ]| }|
      .= v`1 - (-u.2/u.1) * v`2 - v`3 * (-u.3/u.1) by Th2
      .= (1/u.1) * (u`1 * v`1 + u`2 * v`2 + v`3 * u`3) by A1
      .= (1/u.1) * |(u,v)| by EUCLID_5:29;
    hence thesis by A1;
  end;
