
theorem
  for V being Z_Module, v being divisible Vector of V holds
  - v is divisible
  proof
    let V be Z_Module, v be divisible Vector of V;
    thus for a being Element of INT.Ring st a <> 0.INT.Ring holds
    ex w being Vector of V st - v = a * w
    proof
      let a be Element of INT.Ring such that
      A1: a <> 0.INT.Ring;
      consider u be Vector of V such that
      A2: v = a * u by A1,defDivisibleVector;
      take -u;
      thus -v = a * (-u) by A2,ZMODUL01:6;
    end;
  end;
