
theorem
  for V being Z_Module, v, u being divisible Vector of V holds
  v + u is divisible
  proof
    let V be Z_Module, v, u 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 + u = 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;
      consider u1 be Vector of V such that
      A3: u = a * u1 by A1,defDivisibleVector;
      take v1 + u1;
      thus v + u = a * (v1 + u1) by A2,A3,VECTSP_1:def 14;
    end;
  end;
