
theorem ThrEMLat1:
  for L being Z_Lattice, r being non zero Element of F_Rat,
  m, n being Element of INT.Ring, m1,n1 being Element of INT,
  v being Vector of EMLat(r, L)
  st m = m1 & n = n1 & r = m1/n1 & n1 <> 0
  holds ex x being Vector of EMLat(L) st n*v = m*x
  proof
    let L be Z_Lattice, r be non zero Element of F_Rat,
    m, n be Element of INT.Ring,
    m1, n1 be Element of INT,
    v be Vector of EMLat(r, L) such that
    A1: m=m1 & n=n1 & r = m1/n1 & n1 <> 0;
    consider T be linear-transformation of EMbedding(L),EMbedding(r,L)
    such that
    A2: (for u being Element of Z_MQ_VectSp(L) st u in EMbedding(L)
    holds T.u = r*u) & T is bijective by ZMODUL08:27;
    v in the carrier of EMLat(r, L);
    then v in r * (rng MorphsZQ(L)) by defrEMLat;
    then v in the carrier of EMbedding(r, L) by ZMODUL08:def 6;
    then v in rng T by A2,FUNCT_2:def 3;
    then consider ve be object such that
    A3: ve in the carrier of EMbedding(L) & v = T.ve by FUNCT_2:11;
    reconsider vz = ve as Vector of Z_MQ_VectSp(L) by A3,ZMODUL08:19;
    reconsider vd = vz as Vector of DivisibleMod(L) by ZMODUL08:30;
    consider zvd be Vector of DivisibleMod(L) such that
    A4: vd = n * zvd & r * vz = m * zvd by A1,ZMODUL08:31;
    A5: vz in EMbedding(L) by A3;
    vz in rng MorphsZQ(L) by A3,ZMODUL08:def 3;
    then reconsider x = vz as Vector of EMLat(L) by defEMLat;
    B1: EMLat(L) is Submodule of DivisibleMod(L) by ThDivisibleL1;
    B2: EMLat(r, L) is Submodule of DivisibleMod(L) by ThDivisibleL2;
    A7: m * x = m * vd by B1,ZMODUL01:29
    .= (m * n) * zvd by A4,VECTSP_1:def 16
    .= n * (m * zvd) by VECTSP_1:def 16
    .= n * v by A2,A3,A4,A5,B2,ZMODUL01:29;
    take x;
    thus thesis by A7;
  end;
