reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem ThQuotBasis5:
  for p being prime Element of INT.Ring,
      V being free Z_Module, I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V,p)
  st Lin(I) = the ModuleStr of V &
  IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I} holds
  Lin(IQ) = the ModuleStr of Z_MQ_VectSp(V,p)
  proof
    let p be prime Element of INT.Ring, V be free Z_Module, I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V,p) such that
    P0: Lin(I) = the ModuleStr of V and
    AS: IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
    for vq being Element of Z_MQ_VectSp(V,p) holds vq in Lin (IQ)
    proof
      let vq be Element of Z_MQ_VectSp(V,p);
      consider v being Vector of V such that
      P3: vq = ZMtoMQV(V,p,v) by ZMODUL03:22;
      consider l be Linear_Combination of I such that
      P4: v = Sum(l) by P0,STRUCT_0:def 5,ZMODUL02:64;
      thus vq in Lin(IQ) by AS,P4,P3,ThQuotBasis5A;
    end;
    hence thesis by VECTSP_4:32;
  end;
