 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem Th35:
  for p being prime Element of INT.Ring,
      V being free Z_Module, I being Basis of V,
  IQ being Subset of Z_MQ_VectSp(V,p)
  st IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I} holds
  IQ is Basis of Z_MQ_VectSp(V,p)
  proof
    let p be prime Element of INT.Ring,
        V be free Z_Module, I be Basis of V,
        IQ be Subset of Z_MQ_VectSp(V,p);
    assume A1: IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
A2: IQ is linearly-independent by Th32,A1;
    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
      A3: vq = ZMtoMQV(V,p,v) by Th22;
      I is base; then
      Lin (I) = the ModuleStr of V;
      then consider l be Linear_Combination of I such that
  A4: v = Sum(l) by STRUCT_0:def 5,ZMODUL02:64;
      thus vq in Lin(IQ) by A1,A4,A3,Th34;
    end; then
    Lin IQ = the ModuleStr of Z_MQ_VectSp(V,p) by VECTSP_4:32; then
    IQ is base by A2;
    hence thesis;
  end;
