
theorem ThSLGM3:
  for L being positive-definite Z_Lattice,
  LX being Z_Lattice
  st LX is Submodule of DivisibleMod(L) &
  the scalar of LX = (ScProductDM(L)) || the carrier of LX
  holds LX is positive-definite
  proof
    let L be positive-definite Z_Lattice, LX be Z_Lattice such that
    A1: LX is Submodule of DivisibleMod(L) &
    the scalar of LX = (ScProductDM(L)) || the carrier of LX;
    for v being Vector of LX st v <> 0.LX holds ||. v .|| > 0
    proof
      let v be Vector of LX such that
      B1: v <> 0.LX;
      reconsider vv = v as Vector of DivisibleMod(L) by A1,ZMODUL01:25;
      B3: ||. v .|| = (ScProductDM(L)).(vv, vv) by A1,FUNCT_1:49;
      consider i be Element of INT.Ring such that
      B4: i <> 0 & i * vv in EMbedding(L) by ZMODUL08:29;
      i * vv in rng MorphsZQ(L) by B4,ZMODUL08:def 3;
      then reconsider iv = i * vv as Vector of EMLat(L) by ZMODLAT2:def 4;
      B5: (i * i) * ||. v .|| = i * (i* (ScProductDM(L)).(vv, vv)) by B3
      .= i * (ScProductDM(L)).(vv, i*vv) by ZMODLAT2:13
      .= (ScProductDM(L)).(i*vv, i*vv) by ZMODLAT2:6
      .= (ScProductEM(L)).(iv, iv) by B4,ZMODLAT2:8
      .= ||. iv .|| by ZMODLAT2:def 4;
      i <> 0.INT.Ring by B4;
      then i*v <> 0.LX by B1,ZMODUL01:def 7;
      then i*vv <> 0.LX by A1,ZMODUL01:29;
      then iv <> 0.DivisibleMod(L) by A1,VECTSP_4:def 2;
      then iv <> zeroCoset(L) by ZMODUL08:def 4;
      then iv <> 0.EMLat(L) by ZMODLAT2:def 4;
      then ||. iv .|| > 0 by ZMODLAT1:def 6;
      hence ||. v .|| > 0 by B5,XREAL_1:63,XREAL_1:131;
    end;
    hence thesis;
  end;
