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

theorem ThEQRZMV3B:
  for V be Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V),
  lq being Linear_Combination of IQ,
  m be Integer, a be Element of F_Rat,
  l be Linear_Combination of I
  st V is Mult-cancelable & IQ =(MorphsZQ(V)).:(I)
  & m <> 0 & m = a
  & l = (a * lq) *(MorphsZQ(V))
  holds a*(Sum(lq)) = (MorphsZQ(V)).(Sum(l))
  proof
    let V be Z_Module,
    I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V),
    lq be Linear_Combination of IQ,
    m be Integer, a be Element of F_Rat,
    l be Linear_Combination of I;
    assume AS: V is Mult-cancelable
    & IQ =(MorphsZQ(V)).:(I)
    & m <> 0 & m = a
    & l = (a * lq) *(MorphsZQ(V));
    reconsider lqa = a * lq as Linear_Combination of IQ by VECTSP_6:31;
    thus a*(Sum(lq)) = Sum(lqa) by VECTSP_6:45
    .= (MorphsZQ(V)).(Sum(l)) by AS,XThSum1;
  end;
