
theorem
  for V being Z_Module, v being divisible Vector of V,
      i being Element of INT.Ring holds
    i * v is divisible
  proof
    let V be Z_Module, v be divisible Vector of V, i be Element of INT.Ring;
    thus for a being Element of INT.Ring st a <> 0.INT.Ring holds
    ex w being Vector of V st i * v = a * w
    proof
      let a be Element of INT.Ring such that
      A1: a <> 0.INT.Ring;
      consider v1 be Vector of V such that
      A2: v = a * v1 by A1,defDivisibleVector;
      take i * v1;
      thus i * v = (i * a) * v1 by A2,VECTSP_1:def 16
      .= a * (i * v1) by VECTSP_1:def 16;
    end;
  end;
