
theorem Th41:
  for P being non zero_proj1 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir1a P,Pdir1b P)
  proof
    let P be non zero_proj1 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir1a P,Pdir1b P); then
A1: Pdir1a P,Pdir1b P,P are_collinear by COLLSP:11;
    reconsider u = normalize_proj1 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def2;
    |{ dir1a P,dir1b P,u }| = |( u, u )| by Th21;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;
