
theorem
  for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj2 P holds
  |{ dir2a P,dir2b P,normalize_proj2 P }| = - (u.1 * u.1 + 1 + u.3 * u.3)
  proof
    let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj2 P;
    then
A2: u.2 = 1 by Def4;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir2a P,dir2b P,normalize_proj2 P }| = - |(un,un)| by A1,Th25
      .= - (u.1 * u.1 + u.2 * u.2 + u.3 * u.3) by EUCLID_8:63
      .= - (u.1 * u.1 + 1 + u.3 * u.3) by A2;
  end;
