
theorem ThDivisibleL2:
  for L being Z_Lattice, r being Element of F_Rat holds
  EMLat(r, L) is Submodule of DivisibleMod(L)
  proof
    let L be Z_Lattice, r be Element of F_Rat;
    A1: the carrier of EMbedding(r, L)
    = r * (rng MorphsZQ(L)) by ZMODUL08:def 6
    .= the carrier of EMLat(r, L) by defrEMLat;
    A2: the addF of EMLat(r, L) = (addCoset(L)) || (r * (rng MorphsZQ(L)))
    by defrEMLat
    .= the addF of EMbedding(r, L) by ZMODUL08:def 6;
    then reconsider ad = the addF of EMbedding(r, L)
    as BinOp of the carrier of EMLat(r, L);
    A3: 0.EMbedding(r, L) = zeroCoset(L) by ZMODUL08:def 6
    .= 0.EMLat(r, L) by defrEMLat;
    then reconsider ze = 0.EMbedding(r, L) as Vector of EMLat(r, L);
    A4: the lmult of EMLat(r, L)
    = (lmultCoset(L)) | [:the carrier of INT.Ring,r*(rng MorphsZQ(L)):]
    by defrEMLat
    .= the lmult of EMbedding(r, L) by ZMODUL08:def 6;
    then reconsider mu = the lmult of EMbedding(r, L)
    as Function of
    [:the carrier of INT.Ring,the carrier of EMLat(r, L):],
    the carrier of EMLat(r, L);
    reconsider sc = the scalar of EMLat(r, L) as Function of
    [:the carrier of EMbedding(r, L), the carrier of EMbedding(r, L):],
    the carrier of F_Real by A1;
    EMLat(r, L) = GenLat(EMbedding(r, L), sc) by A1,A2,A3,A4;
    then A2: EMLat(r, L) is Submodule of EMbedding(r, L) by ZMODLAT1:2;
    EMbedding(r, L) is Submodule of DivisibleMod(L) by ZMODUL08:32;
    hence thesis by A2,ZMODUL01:42;
  end;
