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
  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 & ex a,b,c st
  a*u + b*v + c*w = 0.V & (a<>0 or b<>0 or c <>0)
  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;
      take u,v,w;
      thus p = Dir(u) & q = Dir(v) & r = Dir(w) & u is not zero &
      v is not zero & w is not zero & ex a,b,c st a*u + b*v + c*w = 0.V &
      (a<>0 or b<>0 or c <>0) by A1,ANPROJ_1:def 2;
    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 & ex a,b,c st a*u + b*v + c*w = 0.V &
    (a<>0 or b<>0 or c <>0);
    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 A2,ANPROJ_1:def 2;
    hence p,q,r are_collinear by ANPROJ_2:23;
  end;
