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 Th9:
  p,q,r are_collinear iff ex u,v,w st p = Dir(u) & q = Dir(v) & r = Dir(w) &
  u is not zero & v is not zero & w is not zero & (u = v or u = w or v = w or
  {u,v,w} is linearly-dependent)
  proof
    hereby
      assume p,q,r are_collinear;
      then consider u,v,w such that
A1:   p = Dir(u) & q = Dir(v) & r =
      Dir(w) & u is not zero & v is not zero & w is not zero &
      u,v,w are_LinDep by ANPROJ_2:23;
      p = Dir(u) & q = Dir(v) & r = Dir(w) & u is not zero & v is not zero &
      w is not zero & (u = v or u = w or v = w or
      {u,v,w} is linearly-dependent) by A1,Th8;
      hence ex u,v,w st p = Dir(u) & q = Dir(v) & r =
      Dir(w) & u is not zero & v is not zero & w is not zero &
      (u = v or u = w or v = w or {u,v,w} is linearly-dependent);
    end;
    given u,v,w such that
A2: p = Dir(u) & q = Dir(v) & r = Dir(w) & u is not zero & v is not zero &
      w is not zero & (u = v or u = w or v = w or
      {u,v,w} is linearly-dependent);
    p = Dir(u) & q = Dir(v) & r = Dir(w) & u is not zero & v is not zero &
      w is not zero & u,v,w are_LinDep by Th8,A2;
    hence p,q,r are_collinear by ANPROJ_2:23;
  end;
