reserve x, y, y1, y2 for object;
reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;
reserve u, v for VECTOR of V;
reserve i, j, k, n for Element of NAT;
reserve V,W for finite-rank free Z_Module;
reserve T for linear-transformation of V,W;

theorem LMCLUS1:
  ex V being Z_Module, p being Element of INT.Ring
  st p <> 0.INT.Ring & VectQuot(V, p (*) V) is non trivial
  proof
    reconsider V = ModuleStr(# the carrier of INT.Ring,
    the addF of INT.Ring, the ZeroF of INT.Ring, Int-mult-left(INT.Ring) #)
    as Z_Module by ZMODUL01:164;
    reconsider p = 2 as Element of INT.Ring by INT_1:def 2;
    take V, p;
    thus p <> 0.INT.Ring;
    thus VectQuot(V, p (*) V) is non trivial
    proof
      reconsider i = 1.INT.Ring as Vector of V;
      i + p (*) V is Coset of p (*) V by VECTSP_4:def 6;
      then i + p (*) V in CosetSet(V,p (*) V);
      then reconsider B = i + p (*) V as Element of CosetSet(V, p (*) V);
      reconsider u = B as Vector of VectQuot(V, p (*) V)
      by VECTSP10:def 6;
      u <> 0.VectQuot(V, p (*) V)
      proof
        assume u = 0.VectQuot(V, p (*) V);
        then i + p (*) V = zeroCoset(V, p (*) V) by VECTSP10:def 6
        .= the carrier of p (*) V;
        then i in p (*) V by ZMODUL01:63;
        then consider w be Vector of V such that
        P1: i = p*w;
        reconsider w0 = w as Element of INT.Ring;
        p*w = p*w0 by ZMODUL06:14;
        then 1/2 = w0 by P1;
        hence contradiction by NAT_D:33;
      end;
      hence thesis;
    end;
  end;
