
theorem ThDivisible4:
  for V being torsion-free Z_Module, i being non zero Integer,
  r1, r2 being non zero Element of F_Rat st r2 = r1/i holds
  EMbedding(r1, V) is Submodule of EMbedding(r2, V)
  proof
    let V be torsion-free Z_Module, i be non zero Integer,
    r1, r2 be non zero Element of F_Rat such that
    A1: r2 = r1/i;
    A2: for x being Vector of DivisibleMod(V) st x in EMbedding(r1, V)
    holds x in EMbedding(r2, V)
    proof
      let x be Vector of DivisibleMod(V) such that
      B1: x in EMbedding(r1, V);
      consider T1 be linear-transformation of EMbedding(V),EMbedding(r1,V)
      such that
      B2: (for v being Element of Z_MQ_VectSp(V) st v in EMbedding(V)
      holds T1.v = r1*v) & T1 is bijective by rSB03A;
      consider T2 be linear-transformation of EMbedding(V),EMbedding(r2,V)
      such that
      B3: (for v being Element of Z_MQ_VectSp(V) st v in EMbedding(V)
      holds T2.v = r2*v) & T2 is bijective by rSB03A;
      reconsider ii = i as Element of INT.Ring by INT_1:def 2;
      reconsider ir = i as Element of F_Rat by INT_1:def 2,NUMBERS:14;
      reconsider iv = 1/i as Element of F_Rat by RAT_1:def 1;
      x in rng T1 by B1,B2,FUNCT_2:def 3;
      then consider v be object such that
      B4: v in the carrier of EMbedding(V) & x = T1.v by FUNCT_2:11;
      reconsider vv = v as Vector of Z_MQ_VectSp(V) by B4,SB01;
      B5: vv in EMbedding(V) by B4;
      then T2.vv = r2*vv by B3;
      then reconsider rv = r2*vv as Vector of EMbedding(r2, V) by B4,FUNCT_2:5;
      B7: ir*iv = i/i
      .= 1.F_Rat by XCMPLX_1:60;
      ii*rv = ir*(r2*vv) by rSB01
      .= (ir*r2)*vv by VECTSP_1:def 16
      .= (r1*(ir*iv))*vv by A1
      .= x by B2,B4,B5,B7;
      hence thesis;
    end;
    EMbedding(r1, V) is Submodule of DivisibleMod(V) &
    EMbedding(r2, V) is Submodule of DivisibleMod(V) by ThDivisible3;
    hence thesis by A2,ZMODUL01:44;
  end;
