theorem Th64:
  for p,q being Element of ProjectiveSpace TOP-REAL 3 st p <> q & p = Dir u &
  q = Dir v holds u <X> v is non zero
  proof
    let p,q be Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: p <> q & p = Dir u & q = Dir v;
    assume u <X> v is zero;
    then are_Prop u,v by ANPROJ_8:51;
    hence contradiction by A1,ANPROJ_1:22;
  end;
