reserve x, y, y1, y2 for set;
reserve R for Ring;
reserve V for LeftMod of R;
reserve u, v, w for VECTOR of V;
reserve F, G, H, I for FinSequence of V;
reserve i, j, k, n for Element of NAT;
reserve f, f9, g for sequence of V;

theorem Th1:
  for p be Element of INT.Ring, V be Z_Module, W be Submodule of V,
      x be VECTOR of VectQuot(V, W) st W = p (*) V
  holds p * x = 0.VectQuot(V, W)
  proof
    let p be Element of INT.Ring, V be Z_Module, W be Submodule of V,
        x be VECTOR of VectQuot(V, W) such that
    A1: W = p (*) V;
    A2: x is Element of CosetSet(V,W) by VECTSP10:def 6;
    then x in the set of all A where A is Coset of W;
    then consider xx be Coset of W such that
    A3: xx = x;
    consider v be VECTOR of V such that
    A4: xx = v + W by VECTSP_4:def 6;
    p * v in the carrier of W by A1;
    then A5: p * v in W;
    thus p * x = lmultCoset(V,W).(p, x) by VECTSP10:def 6
    .= (p * v) + W by A2,A3,A4,VECTSP10:def 5
    .= zeroCoset(V,W) by A5,ZMODUL01:63
    .= 0.VectQuot(V, W) by VECTSP10:def 6;
  end;
