
theorem Th13:
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3
  for u being non zero Element of TOP-REAL 3 st P = Dir u holds
  u.2 <> 0
  proof
    let P be non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: P = Dir u;
    consider u9 be non zero Element of TOP-REAL 3 such that
A2: P = Dir u9 and
A3: u9.2 <> 0 by Def3;
    are_Prop u,u9 by A1,A2,ANPROJ_1:22;
    then consider a be Real such that
A4: a <> 0 and
A5: u = a * u9 by ANPROJ_1:1;
    assume u.2 = 0;
    then a * u9.2 = 0 by A5,RVSUM_1:44;
    hence thesis by A3,A4;
  end;
