
theorem
  for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st u = normalize_proj1 P holds
  |{ dir1a P,dir1b P,normalize_proj1 P }| = 1 + u.2 * u.2 + u.3 * u.3
  proof
    let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: u = normalize_proj1 P;
    then
A2: u.1 = 1 by Def2;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    thus |{ dir1a P,dir1b P,normalize_proj1 P }| = |(un,un)| by A1,Th21
      .= u.1 * u.1 + u.2 * u.2 + u.3 * u.3 by EUCLID_8:63
      .= 1 + u.2 * u.2 + u.3 * u.3 by A2;
  end;
