
theorem ThDivisible2:
  for V being torsion-free Z_Module holds
  EMbedding(V) is Submodule of DivisibleMod(V)
  proof
    let V be torsion-free Z_Module;
    set EV = EMbedding(V);
    set DV = DivisibleMod(V);
    for x being object st x in the carrier of EV holds x in the carrier of DV
    proof
      let x be object such that
      B1: x in the carrier of EV;
      x in rng MorphsZQ(V) by B1,defEmbedding;
      then consider y be object such that
      B2: y in the carrier of V & (MorphsZQ(V)).y = x by FUNCT_2:11;
      reconsider y as Vector of V by B2;
      B3: Z_MQ_VectSp(V) = ModuleStr(# Class EQRZM(V), addCoset(V),
      zeroCoset(V), lmultCoset(V) #) by ZMODUL04:def 5;
      x in Class EQRZM(V) by B2,B3,FUNCT_2:5;
      hence thesis by defDivisibleMod;
    end;
    then A1: the carrier of EV c= the carrier of DV;
    A2: 0.EV = zeroCoset(V) by defEmbedding
    .= 0.DV by defDivisibleMod;
    A3: the addF of EV = (addCoset(V)) || ( rng MorphsZQ(V) ) by defEmbedding
    .= (the addF of DV) || ( rng MorphsZQ(V) ) by defDivisibleMod
    .= (the addF of DV) || (the carrier of EV) by defEmbedding;
    the lmult of EV = (the lmult of DV) | [:the carrier of INT.Ring,
    rng MorphsZQ(V):]
    proof
      the carrier of EV c= Class EQRZM(V) by A1,defDivisibleMod;
      then B1: rng MorphsZQ(V) c= Class EQRZM(V) by defEmbedding;
      thus the lmult of EV = (lmultCoset(V)) |
      [:the carrier of INT.Ring, rng MorphsZQ(V):]
      by defEmbedding
      .= ((lmultCoset(V)) | [:the carrier of INT.Ring, Class EQRZM(V):])
      | [:the carrier of INT.Ring, rng MorphsZQ(V):]
      by B1,RELAT_1:74,ZFMISC_1:96
      .= (the lmult of DV) | [:the carrier of INT.Ring, rng MorphsZQ(V):]
      by defDivisibleMod;
    end;
    then the lmult of EV = (the lmult of DV) |
    [:the carrier of INT.Ring, (the carrier of EV):]
    by defEmbedding;
    hence thesis by A1,A2,A3,VECTSP_4:def 2;
  end;
