reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;
reserve P,Q,R for POINT of IncProjSp_of real_projective_plane,
            L for LINE of IncProjSp_of real_projective_plane,
        p,q,r for Point of real_projective_plane;
reserve u,v,w for non zero Element of TOP-REAL 3;

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;
