reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);

theorem Th8:
  u,v,w are_LinDep iff
  (u = v or u = w or v = w or {u,v,w} is linearly-dependent)
  proof
    hereby
      assume u,v,w are_LinDep;
      then ex a,b,c st a*u + b*v + c*w = 0.V & (a<>0 or b<>0 or c <>0)
        by ANPROJ_1:def 2;
      hence u = v or u = w or v = w or {u,v,w} is linearly-dependent
        by RLVECT_4:7;
    end;
    assume u = v or u = w or v = w or {u,v,w} is linearly-dependent;
    then ex a,b,c st a * u + b * v + c * w = 0.V &
      (a <> 0 or b <> 0 or c <> 0) by RLVECT_4:7;
    hence u,v,w are_LinDep by ANPROJ_1:def 2;
  end;
