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

theorem ThQuotAX:
  for V be free Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V),
  l be Linear_Combination of I,
  i be Element of INT.Ring
  st i <> 0.INT.Ring & IQ =(MorphsZQ(V)).:I
  holds Class(EQRZM(V),[Sum(l),i]) in Lin(IQ)
  proof
    let V be free Z_Module,
    I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V),
    l be Linear_Combination of I,
    i be Element of INT.Ring;
    assume AS: i <> 0.INT.Ring & IQ =(MorphsZQ(V)).:I;
    Z0:Z_MQ_VectSp(V) = ModuleStr (# Class EQRZM(V), addCoset(V),
    zeroCoset(V), lmultCoset(V) #) by defZMQVSp;
    consider lq be Linear_Combination of IQ such that
    P1: l = lq * (MorphsZQ(V)) &
    Carrier(lq) = (MorphsZQ(V)).:(Carrier(l)) by ThEQRZMV4,AS;
    P2: (Sum(lq))= (MorphsZQ(V)).(Sum(l)) by AS,P1,XThSum1;
    reconsider a = 1/i as Element of F_Rat by RAT_1:def 2;
    P3: (MorphsZQ(V)).(Sum(l)) = Class(EQRZM(V),[Sum(l),1.INT.Ring])
      by defMorph;
    a*( (MorphsZQ(V)).(Sum(l))) = Class(EQRZM(V),[1.INT.Ring*Sum(l),
      i*1.INT.Ring]) by AS,P3,Z0,DeflmultCoset
    .= Class(EQRZM(V),[Sum(l),i]) by VECTSP_1:def 17;
    hence thesis by P2,VECTSP_4:21,VECTSP_7:7;
  end;
