theorem Th22:
  for P being Point of ProjectiveSpace TOP-REAL 3
  ex a,b,c being Element of F_Real st P = Dir |[a,b,c]| &
  (a <> 0 or b <> 0 or c <> 0)
  proof
    let P be Point of ProjectiveSpace TOP-REAL 3;
    consider u be Element of TOP-REAL 3 such that
A1: u is non zero and
A2: P = Dir u by ANPROJ_1:26;
A3: u = |[u`1,u`2,u`3]| by EUCLID_5:3
     .= |[u.1,u`2,u`3]| by EUCLID_5:def 1
     .= |[u.1,u.2,u`3]| by EUCLID_5:def 2
     .= |[u.1,u.2,u.3]| by EUCLID_5:def 3;
    reconsider a = u.1,b = u.2,c = u.3 as Element of F_Real by XREAL_0:def 1;
    take a,b,c;
    thus P = Dir |[a,b,c]| by A2,A3;
    thus a <> 0 or b <> 0 or c <> 0 by A1,A3,EUCLID_5:4;
  end;
