 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 Th24:
  for p being prime Element of INT.Ring, V being free Z_Module,
  I being Basis of V,
  lq being Linear_Combination of Z_MQ_VectSp(V,p)
  holds ex l being Linear_Combination of I
  st for v being Vector of V st v in I holds
  l.v = lq.(ZMtoMQV(V,p,v))
  proof
    let p be prime Element of INT.Ring, V be free Z_Module, I be Basis of V,
    lq be Linear_Combination of Z_MQ_VectSp(V,p);
    consider l being Linear_Combination of I such that
    A1: for v being Vector of V st v in I holds
    ex w be Vector of V st w in I & w in ZMtoMQV(V,p,v)
    & l.w = lq.(ZMtoMQV(V,p,v)) by Th23;
    take l;
    now let v be Vector of V;
      assume A2:v in I;
      then consider w be Vector of V such that
      A3: w in I & w in ZMtoMQV(V,p,v)
      & l.w = lq.(ZMtoMQV(V,p,v)) by A1;
      ZMtoMQV(V,p,w) = ZMtoMQV(V,p,v) by A3,ZMODUL01:67;
      hence l.v = lq.(ZMtoMQV(V,p,v)) by A2,A3,Th21;
    end;
    hence thesis;
  end;
