
theorem
  for L being Z_Lattice, A being non empty set, ze being Element of A,
  ad being BinOp of A,
  mu being Function of [:the carrier of INT.Ring, A:],A,
  sc being Function of [:A, A:],the carrier of F_Real
  st A is linearly-closed Subset of DivisibleMod(L) & ze = 0.DivisibleMod(L) &
  ad = (the addF of DivisibleMod(L)) || A &
  mu = (the lmult of DivisibleMod(L)) | [:the carrier of INT.Ring, A:] holds
  LatticeStr (# A, ad, ze,mu, sc #) is Submodule of DivisibleMod(L)
  proof
    let L be Z_Lattice, A be non empty set, ze be Element of A,
    ad be BinOp of A,
    mu be Function of [:the carrier of INT.Ring, A:],A,
    sc be Function of [:A, A:], the carrier of F_Real such that
    A1: A is linearly-closed Subset of DivisibleMod(L) &
    ze = 0.DivisibleMod(L) &
    ad = (the addF of DivisibleMod(L)) || A &
    mu = (the lmult of DivisibleMod(L)) | [:the carrier of INT.Ring, A:];
    set L1 = LatticeStr (# A, ad, ze,mu, sc #);
    set V1 = ModuleStr (# A, ad, ze,mu #);
    A2: V1 is Submodule of DivisibleMod(L) by A1,ZMODUL01:40;
    reconsider V1 as Z_Module by A1,ZMODUL01:40;
    reconsider scc = sc as Function of [:the carrier of V1,
    the carrier of V1 :], the carrier of F_Real;
    L1 = GenLat(V1, scc);
    then L1 is Submodule of V1 by ZMODLAT1:2;
    hence thesis by A2,ZMODUL01:42;
  end;
