theorem
  for x being object holds x in (0).V iff x = 0.V
  proof let x be object;
  thus x in (0).V implies x = 0.V
  proof
    assume x in (0).V;
    then x in the carrier of (0).V;
    then x in {0.V} by VECTSP_4:def 3;
    hence thesis by TARSKI:def 1;
  end;
  thus thesis by ZMODUL01:33;
end;
