
theorem Th20:
  for P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3 holds
  Dir dir1a(P) <> Dir dir1b(P)
  proof
    let P being non zero_proj1 Point of ProjectiveSpace TOP-REAL 3;
    assume Dir dir1a(P) = Dir dir1b(P);
    then are_Prop dir1a(P),dir1b(P) by ANPROJ_1:22;
    then consider a be Real such that
A1: a <> 0 and
A2: dir1a(P) = a * dir1b(P) by ANPROJ_1:1;
    0 = (dir1a(P))`3
     .= a * (dir1b(P))`3 by A2,RVSUM_1:44
     .= a * 1;
    hence contradiction by A1;
  end;
