
theorem
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj3 P holds
  |{ dir3a P,dir3b P,normalize_proj3 P }| = u.1 * u.1 + u.2 * u.2 + 1
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj3 P;
    then
A2: u.3 = 1 by Def6;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir3a P,dir3b P,normalize_proj3 P }| = |(un,un)| by A1,Th29
      .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63
      .= u.1 * u.1 + u.2 * u.2 + 1 by A2;
  end;
