 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 Th22:
  for p being prime Element of INT.Ring, V being Z_Module,
  ZQ being VectSp of GF(p), vq being Vector of ZQ
  st ZQ = Z_MQ_VectSp(V,p)
  holds ex v being Vector of V st vq = ZMtoMQV(V,p,v)
  proof
    let p be prime Element of INT.Ring, V be Z_Module,
    ZQ be VectSp of GF(p), vq be Vector of ZQ such that
    A1: ZQ = Z_MQ_VectSp(V,p);
    vq is Element of CosetSet(V,p(*)V) by A1,VECTSP10:def 6;
    then vq in the set of all A where A is Coset of p(*)V;
    then consider vqq be Coset of p(*)V such that
    A2: vqq = vq;
    consider v be Vector of V such that
    A3: vq = v + p(*)V by A2,VECTSP_4:def 6;
    take v;
    thus thesis by A3;
  end;
