reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for p being prime Element of INT.Ring, V being Z_Module,
  s be Element of V, a be Element of INT.Ring, b be Element of GF(p)
  st b = a mod p holds
  b*ZMtoMQV(V,p,s) = ZMtoMQV(V,p,a*s)
  proof
    let p be prime Element of INT.Ring, V be Z_Module, s be Element of V,
    a be Element of INT.Ring, b be Element of GF(p) such that
    A1: b = a mod p;
    A2: ZMtoMQV(V,p,s) = s + p(*)V;
    set t = ZMtoMQV(V,p,s);
    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;
    thus b*t =(a mod p) * t1 by A1,ZMODUL02:def 11
    .= a * t1 by ZMODUL02:2
    .= lmultCoset(V,p(*)V).(a,s + p(*)V) by VECTSP10:def 6
    .= ZMtoMQV(V,p,a*s) by A3,VECTSP10:def 5;
  end;
