
theorem Th25:
  for P being non zero_proj2 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_proj2 P holds
  |{ dir2a P,dir2b P,v }| = - |(u,v)|
  proof
    let P be non zero_proj2 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_proj2 P;
    then
A1: u.2 = 1 & P = Dir u by Def4;
    then normalize_proj2 P = |[u.1/u.2,1,u.3/u.2]| by Th14;
    then (normalize_proj2(P))`1 = u.1/u.2 & (normalize_proj2(P))`3 = u.3/u.2;
    then |{ dir2a P,dir2b P,v }| = |{ |[ 1,   -u.1/u.2, 0   ]|,
                                      |[ 0,   -u.3/u.2, 1   ]|,
                                      |[ v`1, v`2,      v`3 ]| }|
      .= (-u.3/u.2) * v`3 + (-u.1/u.2) * v`1 - v`2 by Th3
      .= -(1/u.2) * (u`1 * v`1 + u`2 * v`2 + u`3 * v`3) by A1
      .= -(1/u.2) * |(u,v)| by EUCLID_5:29;
    hence thesis by A1;
  end;
