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
  for p, q be Element of INT.Ring, V be Z_Module, W be Submodule of V,
  v be VECTOR of V
  st W = p (*) V & p > 1 & q > 1 & p,q are_coprime
  holds q * v = 0.V implies v + W = 0.VectQuot(V, W)
  proof
    let p, q be Element of INT.Ring, V be Z_Module, W be Submodule of V,
    v be VECTOR of V such that
    A1: W = p (*) V and
    A2: p > 1 & q > 1 & p,q are_coprime;
    v + W is Coset of W by VECTSP_4:def 6;
    then v + W in CosetSet(V,W);
    then reconsider x = v + W as VECTOR of VectQuot(V, W) by VECTSP10:def 6;
    p in NAT & q in NAT by A2,INT_1:3;
    then reconsider pp=p, qq=q as Nat;
    consider i, j be Integer such that
    A3: i*pp + j*qq = 1 by A2,EULER_1:10;
    a3: i*pp + j*qq = 1.INT.Ring by A3;
    reconsider i, j as Element of INT.Ring by Lem1;
    assume A4: q * v = 0.V;
    A5: x is Element of CosetSet(V,W) by VECTSP10:def 6;
    A6: q * x = lmultCoset(V,W).(q,x) by VECTSP10:def 6
    .= 0.V + W by A4,A5,VECTSP10:def 5
    .= zeroCoset(V,W) by ZMODUL01:59
    .= 0.VectQuot(V, W) by VECTSP10:def 6;
    (i*p + j*q) * x = (i*p) * x + (j*q) * x by VECTSP_1:def 15
    .= (i*p) * x + j * (q*x) by VECTSP_1:def 16
    .= (i*p) * x + 0.VectQuot(V, W) by A6,ZMODUL01:1
    .= (i*p) * x by RLVECT_1:4
    .= i * (p*x) by VECTSP_1:def 16
    .= 0.VectQuot(V, W) by A1,Th1,ZMODUL01:1;
    hence 0.VectQuot(V, W) = v + W by a3;
  end;
