
theorem SB03:
  for V being torsion-free Z_Module holds
  ex T being linear-transformation of V, EMbedding(V)
  st T is bijective & T = MorphsZQ(V) &
  (for v being Vector of V holds T.v = Class(EQRZM(V),[v,1]) )
  proof
    let V be torsion-free Z_Module;
    set T = MorphsZQ(V);
    rng T = the carrier of EMbedding(V) by defEmbedding;
    then reconsider T0 = T as Function of V, EMbedding(V) by FUNCT_2:6;
    B0: T0 is additive
    proof
      let x, y be Element of V;
      thus T0.(x+y) = T0.x + T0.y
      proof
        L1: T.(x+y) = T.x + T.y  by ZMODUL04:def 6;
        reconsider v = T0.x, w = T0.y as Vector of EMbedding(V);
        thus T0.(x+y) = T0.x + T0.y by L1,SB01;
      end;
    end;
    for x being Vector of V, i being Element of INT.Ring
    holds T0.(i*x) = i*(T0.x)
    proof
      let x be Vector of V, i be Element of INT.Ring;
      thus T0.(i*x) = i*T0.x
      proof
        reconsider j = i as Element of F_Rat by NUMBERS:14;
        L1: T.(i*x) = j*T.x by ZMODUL04:def 6;
        reconsider v = T0.x as Vector of EMbedding(V);
        thus T0.(i*x) = i*T0.x by L1,SB01;
      end;
    end;
    then T0 is additive homogeneous by B0;
    then reconsider T0 as linear-transformation of V, EMbedding(V);
    take T0;
SS: T0 is one-to-one by ZMODUL04:def 6;
    rng T0 = the carrier of EMbedding(V) by defEmbedding;
    then T0 is onto by FUNCT_2:def 3;
    hence T0 is bijective by SS;
    thus T0 = MorphsZQ(V);
    thus thesis by ZMODUL04:def 6;
  end;
