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

theorem ThEQRZMV3D:
  for V be free Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V)
  st IQ =(MorphsZQ(V)).:I & I is Basis of V
  holds IQ is Basis of Z_MQ_VectSp(V)
  proof
    let V be free Z_Module,
    I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V);
    assume
    AS: IQ =(MorphsZQ(V)).:(I) & I is Basis of V; then
    I is base; then
    X0: I is linearly-independent & Lin (I) = the ModuleStr of V
    by VECTSP_7:def 3;
    X1: IQ is linearly-independent by AS,ThEQRZMV3C,VECTSP_7:def 3;
    AS0: Z_MQ_VectSp(V) = ModuleStr (# Class EQRZM(V), addCoset(V),
    zeroCoset(V), lmultCoset(V) #) by defZMQVSp;
    for vq being Element of Z_MQ_VectSp(V) holds vq in Lin (IQ)
    proof
      let vq be Element of Z_MQ_VectSp(V);
      consider i be Element of INT.Ring, v be Element of V such that
      P3: i <> 0 & vq=Class(EQRZM(V),[v,i]) by AS0,LMEQRZM1;
      v in Lin (I) by X0; then
      consider l be Linear_Combination of I such that
      P4: v = Sum(l) by ZMODUL02:64;
      thus vq in Lin(IQ) by AS,P4,P3,ThQuotAX;
    end;
    hence thesis by AS0,X1,VECTSP_4:32,VECTSP_7:def 3;
  end;
