 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 Th34:
  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),
  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 Basis of V,
        IQ be Subset of Z_MQ_VectSp(V,p),
        l be Linear_Combination of I;
    assume A1: IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
    consider G be FinSequence of V such that
    A2: 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
      A3: 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;
      A4: (l (#) G ).i = (l.v)*v by A3,ZMODUL02:13;
      reconsider b = ( (l.v) mod p ) as Element of GF(p) by Lm3;
      reconsider a = ( (l.v) mod p ) as Element of INT.Ring;
      reconsider k = l.v as Element of INT.Ring;
      reconsider t = ZMtoMQV(V,p,v) as Element of Z_MQ_VectSp(V,p);
      A5: b*t = ZMtoMQV(V,p, a*v) by Th30
      .= ZMtoMQV(V,p, k*v) by Lm4;
      A6: v in Carrier(l) by A3,A2,FUNCT_1:3;
      Carrier(l) c= I by VECTSP_6:def 4;
      then t in IQ by A1,A6;
      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 A5,A4;
    end;
    hence ZMtoMQV(V,p,Sum(l)) in Lin(IQ) by A1,A2,Th33;
  end;
