
theorem Th24:
  for P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3 holds
  Dir dir2a(P) <> Dir dir2b(P)
  proof
    let P being non zero_proj2 Point of ProjectiveSpace TOP-REAL 3;
    assume Dir dir2a(P) = Dir dir2b(P);
    then are_Prop dir2a(P),dir2b(P) by ANPROJ_1:22;
    then consider a be Real such that
A1: a <> 0 and
A2: dir2a(P) = a * dir2b(P) by ANPROJ_1:1;
    0 = (dir2a(P))`3
     .= a * (dir2b(P))`3 by A2,RVSUM_1:44
     .= a * 1;
    hence contradiction by A1;
  end;
