
theorem ThDivisibleL1:
  for L being Z_Lattice holds EMLat(L) is Submodule of DivisibleMod(L)
  proof
    let L be Z_Lattice;
    A1: the carrier of EMbedding(L) = rng MorphsZQ(L) by ZMODUL08:def 3
    .= the carrier of EMLat(L) by defEMLat;
    A2: the addF of EMLat(L) = (addCoset(L)) || (rng MorphsZQ(L)) by defEMLat
    .= the addF of EMbedding(L) by ZMODUL08:def 3;
    then reconsider ad = the addF of EMbedding(L)
    as BinOp of the carrier of EMLat(L);
    A3: 0.EMbedding(L) = zeroCoset(L) by ZMODUL08:def 3
    .= 0.EMLat(L) by defEMLat;
    then reconsider ze = 0.EMbedding(L) as Vector of EMLat(L);
    A4: the lmult of EMLat(L) = (lmultCoset(L)) |
    [:the carrier of INT.Ring,rng MorphsZQ(L):] by defEMLat
    .= the lmult of EMbedding(L) by ZMODUL08:def 3;
    then reconsider mu = the lmult of EMbedding(L)
    as Function of [:the carrier of INT.Ring,the carrier of EMLat(L):],
      the carrier of EMLat(L);
    reconsider sc = the scalar of EMLat(L) as Function of
    [:the carrier of EMbedding(L), the carrier of EMbedding(L):],
    the carrier of F_Real by A1;
    EMLat(L) = GenLat(EMbedding(L), sc) by A1,A2,A3,A4;
    then A2: EMLat(L) is Submodule of EMbedding(L) by ZMODLAT1:2;
    EMbedding(L) is Submodule of DivisibleMod(L) by ZMODUL08:24;
    hence thesis by A2,ZMODUL01:42;
  end;
