 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 Th28:
  for p being prime Element of INT.Ring, V being free Z_Module,
  s,t be Element of V holds
  ZMtoMQV(V,p,s)+ZMtoMQV(V,p,t) = ZMtoMQV(V,p,s+t)
  proof
    let p be prime Element of INT.Ring, V be free Z_Module,
      s, t be Element of V;
    set s1 = ZMtoMQV(V,p,s), t1 = ZMtoMQV(V,p,t);
    A1: ZMtoMQV(V,p,s) = s + p(*)V;
    A2: ZMtoMQV(V,p,t) = t + p(*)V;
    A3: s + p(*)V is Element of CosetSet(V,p(*)V) by A1,VECTSP10:def 6;
    A4: t + p(*)V is Element of CosetSet(V,p(*)V) by A2,VECTSP10:def 6;
    thus s1+t1 = addCoset(V,p(*)V).(s + p(*)V,t + p(*)V) by VECTSP10:def 6
    .= ZMtoMQV(V,p,s+t) by A3,A4,VECTSP10:def 3;
  end;
