
theorem LMLT2:
  for V being divisible Z_Module, W be Submodule of V
  holds VectQuot(V, W) is divisible
  proof
    let V be divisible Z_Module, W be Submodule of V;
    set VW = VectQuot(V, W);
    X0: the carrier of VW = CosetSet(V, W) by VECTSP10:def 6;
    for vw being Vector of VW holds vw is divisible
    proof
      let vw be Vector of VW;
      for a being Element of INT.Ring st a <> 0 holds
      ex u being Vector of VW st a*u = vw
      proof
        let a be Element of INT.Ring;
        assume AS:a <> 0;
        vw in CosetSet(V, W) by X0;
        then consider A be Coset of W such that
        X1: vw = A;
        consider v be Vector of V such that
        X2: A = v + W by VECTSP_4:def 6;
        v is divisible by defDivisibleModule;
        then consider u0 be Vector of V such that
        X3: a*u0 = v by AS;
        u0 + W is Coset of W by VECTSP_4:def 6;
        then u0 + W in CosetSet(V,W);
        then reconsider B = u0 + W as Element of CosetSet(V,W);
        reconsider u = B as Vector of VectQuot(V, W) by VECTSP10:def 6;
        take u;
        thus a*u = lmultCoset(V, W).(a,B) by VECTSP10:def 6
        .= vw by X1,X2,X3,VECTSP10:def 5;
      end;
      hence thesis;
    end;
    hence thesis;
  end;
