 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 Th30:
  for p being prime Element of INT.Ring, V being free Z_Module,
  s be Element of V,
  a be Element of INT.Ring,
  b be Element of GF(p) st a = b holds
  b*ZMtoMQV(V,p,s) = ZMtoMQV(V,p,a*s)
  proof
    let p be prime Element of INT.Ring, V be free Z_Module,
    s be Element of V, a be Element of INT.Ring, b be Element of GF(p);
    set t = ZMtoMQV(V,p,s);
    assume A1: a=b;
    A2: ZMtoMQV(V,p,s) = s + p(*)V;
    reconsider t1 = t as Element of VectQuot(V,p(*)V);
    A3: s + p(*)V is Element of CosetSet(V,p(*)V) by A2,VECTSP10:def 6;
    reconsider i = b as Nat;
    A4: b = i mod p by NAT_1:44,NAT_D:24;
    i in INT by INT_1:def 2; then
    reconsider i as Element of INT.Ring;
    thus b*t =(i mod p) * t1 by A4,ZMODUL02:def 11
    .= lmultCoset(V,p(*)V).((i mod p),s + p(*)V) by VECTSP10:def 6
    .= ZMtoMQV(V,p,a*s) by A1,A4,A3,VECTSP10:def 5;
  end;
