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

theorem ThQuotBasis5A:
  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),
  l be Linear_Combination of I
  st IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I}
  holds
  ZMtoMQV(V,p,Sum(l)) in Lin(IQ)
  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),
    l be Linear_Combination of I;
    assume AS: IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
    consider G be FinSequence of V such that
    P1: G is one-to-one & rng G = Carrier(l)
    & Sum(l) = Sum(l (#) G) by VECTSP_6:def 6;
    now let i be Element of NAT;
      assume i in dom (l (#) G);
      then i in Seg (len (l (#) G )) by FINSEQ_1:def 3;
      then i in Seg len G by VECTSP_6:def 5;
      then
      Y3: i in dom G by FINSEQ_1:def 3;
      then G.i in rng G by FUNCT_1:3;
      then reconsider v = G.i as Element of V;
      Y5: (l (#) G ).i = (l.v)*v by Y3,ZMODUL02:13;
      reconsider b = ( (l.v) mod p ) as Element of GF(p) by EC_PF_1:14;
      reconsider a = (l.v) mod p as Element of INT.Ring;
      reconsider k = l.v as Element of INT.Ring;
      reconsider si = (l.v)*v as Element of V;
      reconsider t = ZMtoMQV(V,p,v) as Element of Z_MQ_VectSp(V,p);
      Y7: b*t = ZMtoMQV(V,p, a*v) by ZMODUL03:30
      .= ZMtoMQV(V,p, k*v) by LMX2;
      H1: v in Carrier(l) by Y3,P1,FUNCT_1:3;
      Carrier(l) c= I by VECTSP_6:def 4;
      then t in IQ by AS,H1;
      then b*t in Lin(IQ) by VECTSP_4:21,VECTSP_7:8;
      hence ex si be Vector of V
      st si = (l (#) G ).i & ZMtoMQV(V,p,si) in Lin(IQ) by Y7,Y5;
    end;
    hence ZMtoMQV(V,p,Sum(l)) in Lin(IQ) by AS,P1,ZMODUL03:33;
  end;
