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

theorem ThEQRZMV3:
  for V be Z_Module,
  I being Subset of V,
  IQ being Subset of Z_MQ_VectSp(V),
  lq being Linear_Combination of IQ
  st V is Mult-cancelable & IQ =(MorphsZQ(V)).:(I) holds
  ex m be Element of INT.Ring, a be Element of F_Rat,
  l be Linear_Combination of I
  st m <> 0.INT.Ring & m = a & l = (a * lq) * (MorphsZQ(V))
  & (MorphsZQ(V))"Carrier(lq) = Carrier(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;
    assume AS: V is Mult-cancelable & IQ = (MorphsZQ(V)).:(I);
    consider m be Integer, a be Element of F_Rat such that
    X3: m <> 0 & m = a & rng (a * lq) c= INT by ThEQRZMV3A;
    reconsider mm = m as Element of INT.Ring by INT_1:def 2;
    P81: rng ((a*lq)*(MorphsZQ(V))) c= rng (a*lq) by RELAT_1:26;
    dom ((a*lq)*(MorphsZQ(V))) = the carrier of V by FUNCT_2:def 1;
    then (a*lq)*(MorphsZQ(V)) is Function of the carrier of V,INT
    by P81,X3,FUNCT_2:2,XBOOLE_1:1;
    then reconsider l = (a*lq)*(MorphsZQ(V))
      as Element of Funcs(the carrier of V, INT) by FUNCT_2:8;
    set T = {v where v is Element of V : l.v <> 0};
    set F = MorphsZQ(V);
    B2: now
      let v be object;
      assume v in T;
      then ex v1 being Element of V st v1 = v & l.v1 <> 0;
      hence v in the carrier of V;
    end;
    R1: T c= F"(Carrier(lq))
    proof
      let x be object;
      assume x in T;then
      consider v be Element of V such that R11: x=v & l.v <> 0;
      RRR: dom F = the carrier of V by FUNCT_2:def 1;
      V1: l.v = (a*lq).(F.v) by FUNCT_1:13,RRR
      .= a*(lq.(F.v)) by VECTSP_6:def 9;
      lq.(F.v) <> 0.F_Rat by V1,R11;
      then F.v in Carrier(lq);
      hence x in F"(Carrier(lq)) by R11,FUNCT_2:38;
    end;
    F is one-to-one by AS,defMorph;
    then F"(Carrier(lq)) = F".:(Carrier(lq)) by FUNCT_1:85;
    then reconsider T as finite Subset of V by B2,R1,TARSKI:def 3;
    for v being Element of V st not v in T holds l.v = 0.INT.Ring;
    then reconsider l as Linear_Combination of V by VECTSP_6:def 1;
    F"(Carrier(lq)) c= T
    proof
      let x be object;
      assume S21: x in F"(Carrier(lq));
      then x in the carrier of V &
      F.x in Carrier(lq) by FUNCT_2:38;
      then consider w be Element of Z_MQ_VectSp(V) such that
      R11: F.x=w & lq.w <> 0.(F_Rat);
      reconsider v = x as Element of V by S21;
      RRR: dom F = the carrier of V by FUNCT_2:def 1;
      RR1: l.v = (a*lq).(F.v) by FUNCT_1:13,RRR
      .= a*(lq.w) by R11,VECTSP_6:def 9;
      reconsider a1 = a, d1 = (lq.w) as Rational;
      l.v <> 0 by RR1,R11,X3;
      hence x in T;
    end; then
    A9: F"(Carrier(lq)) = T by R1;
    A7: T = Carrier(l);
    AA1: F"(Carrier(lq)) c= F" (F.:(I)) by AS,RELAT_1:143,VECTSP_6:def 4;
    dom F = the carrier of V by FUNCT_2:def 1;
    then F is one-to-one & I c= dom F by AS,defMorph;
    then T c= I by A9,AA1,FUNCT_1:94;
    then
    reconsider l1 = l as Linear_Combination of I by A7,VECTSP_6:def 4;
    take mm;
    take a;
    take l1;
    thus mm <> 0.INT.Ring & mm = a & l1 = (a * lq) *(MorphsZQ(V))
    & (MorphsZQ(V))"Carrier(lq) = Carrier(l1) by A9,X3;
  end;
