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

theorem ThEQRZMV3C:
  for V be Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V)
  st V is Mult-cancelable & IQ =(MorphsZQ(V)).:(I)
  & I is linearly-independent
  holds IQ is linearly-independent
  proof
    let V be Z_Module,
    I be Subset of V,
    IQ be Subset of Z_MQ_VectSp(V);
    assume
    AS: V is Mult-cancelable & IQ = (MorphsZQ(V)).:I
    & I is linearly-independent;
    assume not IQ is linearly-independent;
    then consider lq being Linear_Combination of IQ such that
    P1: Sum(lq) = 0.(Z_MQ_VectSp(V)) & Carrier(lq) <> {} by VECTSP_7:def 1;
    consider m be Element of INT.Ring, a be Element of F_Rat,
    l be Linear_Combination of I such that
    P2: m <> 0.INT.Ring & m = a & l = (a * lq) *(MorphsZQ(V))
    & (MorphsZQ(V))"Carrier(lq) = Carrier(l) by ThEQRZMV3,AS;
    a*(Sum(lq)) = 0.(Z_MQ_VectSp(V)) by P1,VECTSP_1:15; then
    X2: (MorphsZQ(V)).(Sum(l)) = 0.(Z_MQ_VectSp(V)) by AS,P2,ThEQRZMV3B;
    X3:(MorphsZQ(V)).( 0.V ) = 0.(Z_MQ_VectSp(V)) by AS,defMorph;
    (MorphsZQ(V)) is one-to-one by AS,defMorph; then
    P3: Sum(l) = 0.V by X2,X3,FUNCT_2:19;
    H6: Carrier(lq) c= IQ by VECTSP_6:def 4;
    IQ c= rng (MorphsZQ(V)) by AS,RELAT_1:111; then
    H2: Carrier(lq) = (MorphsZQ(V)).: Carrier(l)
      by H6,P2,FUNCT_1:77,XBOOLE_1:1;
    Carrier(l) <> {} by H2,P1;
    hence contradiction by AS,P3,VECTSP_7:def 1;
  end;
