
theorem Th42:
  for P being non zero_proj2 Element of ProjectiveSpace TOP-REAL 3 holds
  not P in Line(Pdir2a P,Pdir2b P)
  proof
    let P be non zero_proj2 Element of ProjectiveSpace TOP-REAL 3;
    assume P in Line(Pdir2a P,Pdir2b P); then
A1: Pdir2a P,Pdir2b P, P are_collinear by COLLSP:11;
    reconsider u = normalize_proj2 P as non zero Element of TOP-REAL 3;
A2: P = Dir u by Def4;
    |{ dir2a P,dir2b P,u }| = - |( u, u )| by Th25;
    then |(u, u)| = 0 by A2,A1,BKMODEL1:1;
    hence thesis by Th5;
  end;
