
theorem Th43:
  for P being non zero_proj3 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir3a P,Pdir3b P)
  proof
    let P be non zero_proj3 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir3a P,Pdir3b P); then
A1: Pdir3a P,Pdir3b P, P are_collinear by COLLSP:11;
    reconsider u = normalize_proj3 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def6;
    |{ dir3a P,dir3b P,u }| = |( u, u )| by Th29;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;
