
theorem ThDivisible3:
  for V being torsion-free Z_Module, r being Element of F_Rat holds
  EMbedding(r, V) is Submodule of DivisibleMod(V)
  proof
    let V be torsion-free Z_Module, r be Element of F_Rat;
    set Z = EMbedding(r, V);
    set D = DivisibleMod(V);
    for x being object st x in the carrier of EMbedding(r, V) holds
    x in the carrier of DivisibleMod(V)
    proof
      let x be object such that
      B1: x in the carrier of EMbedding(r, V);
      x is Vector of Z_MQ_VectSp(V) by B1,rSB01;
      then x is Vector of DivisibleMod(V) by ThDivisibleX1;
      hence thesis;
    end;
    then A1: the carrier of Z c= the carrier of D;
    A2: the addF of Z = (addCoset(V)) || (r * (rng MorphsZQ(V))) by defriV
    .= (addCoset(V)) || the carrier of Z by defriV
    .= (the addF of D) || the carrier of Z by defDivisibleMod;
    A3: 0.Z = zeroCoset(V) by defriV
    .= 0.D by defDivisibleMod;
    A4: [:the carrier of INT.Ring, the carrier of Z:]
     c= [:the carrier of INT.Ring, the carrier of D:] by A1,ZFMISC_1:96;
    (the lmult of D) | [:the carrier of INT.Ring, the carrier of Z:]
    = ((lmultCoset(V)) | [:the carrier of INT.Ring, Class EQRZM(V):])
    |  [:the carrier of INT.Ring, the carrier of Z:]
    by defDivisibleMod
    .= ((lmultCoset(V)) | [:the carrier of INT.Ring, the carrier of D:])
    | [:INT, the carrier of Z:] by defDivisibleMod
    .= (lmultCoset(V)) | [:the carrier of INT.Ring, the carrier of Z:]
    by A4,FUNCT_1:51
    .= (lmultCoset(V)) | [:the carrier of INT.Ring,
    r * (rng MorphsZQ(V)):] by defriV
    .= (the lmult of Z) by defriV;
    hence thesis by A1,A2,A3,VECTSP_4:def 2;
  end;
