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 Th2:
  for p, i be Element of INT.Ring, V be Z_Module, W be Submodule of V,
      x be VECTOR of VectQuot(V, W)
  st p <> 0 & W = p (*) V
  holds i * x = (i mod p) * x
  proof
    let p, i be Element of INT.Ring, V be Z_Module, W be Submodule of V,
    x be VECTOR of VectQuot(V, W) such that
    A1: p <> 0 and
    A2: W = p (*) V;
    consider j be Element of INT.Ring such that
    A3: j = i div p;
    thus i * x = (j*p + (i mod p)) * x by A1,A3,INT_1:59
    .= (j*p) * x + (i mod p) * x by VECTSP_1:def 15
    .= j * (p*x) + (i mod p) * x by VECTSP_1:def 16
    .= 0.VectQuot(V, W) + (i mod p) * x by A2,Th1,ZMODUL01:1
    .= (i mod p) * x  by RLVECT_1:4;
  end;
