
theorem rSB03A:
  for V being torsion-free Z_Module, r being Element of F_Rat
  st r <> 0.F_Rat holds
  ex T being linear-transformation of EMbedding(V),EMbedding(r,V)
  st (for v being Element of Z_MQ_VectSp(V) st v in EMbedding(V)
  holds T.v = r*v) &
  T is bijective
  proof
    let V be torsion-free Z_Module,
    r be Element of F_Rat;
    assume AS: r <> 0.F_Rat;
    set EZV = Z_MQ_VectSp(V);
    deffunc F(Vector of EZV) = r*$1;
    consider T be Function of the carrier of EZV, the carrier of EZV
    such that
    P1: for x being Element of the carrier of EZV holds T.x = F(x)
    from FUNCT_2:sch 4;
    set T0 = T | the carrier of EMbedding(V);
    D0: the carrier of EMbedding(V) = rng MorphsZQ(V) by defEmbedding;
    dom T = the carrier of EZV by FUNCT_2:def 1;
    then
    P3: dom T0 = the carrier of EMbedding(V) by D0,RELAT_1:62;
    D1: the carrier of EMbedding(r,V) = r * (rng MorphsZQ(V)) by defriV;
    RX0: for y being object
    holds y in rng T0 iff y in the carrier of EMbedding(r,V)
    proof
      let y be object;
      hereby
        assume y in rng T0;
        then consider x be object such that
        A2: x in dom T0 &  y = T0.x by FUNCT_1:def 3;
        reconsider x as Element of EZV by A2;
        T0.x = T.x by FUNCT_1:49,A2,P3
        .= r*x by P1;
        hence y in the carrier of EMbedding(r,V) by A2,D0,D1,P3;
      end;
      assume y in the carrier of EMbedding(r,V);
      then y in r * (rng MorphsZQ(V)) by defriV;
      then consider x be Vector of EZV such that
      A4: y = r*x & x in rng MorphsZQ(V);
      A5: x in the carrier of EMbedding(V) by A4,defEmbedding;
      T0.x = T.x by FUNCT_1:49,A5
      .= y by A4,P1;
      hence y in rng T0 by A5,P3,FUNCT_1:def 3;
    end;
    then rng T0 = the carrier of EMbedding(r,V) by TARSKI:2;
    then reconsider T0 as Function of EMbedding(V),EMbedding(r,V)
    by P3,FUNCT_2:1;
    B0: T0 is additive
    proof
      let x, y be Element of EMbedding(V);
      F1: x in EMbedding(V) & y in EMbedding(V);
      reconsider x0=x,y0=y as Vector of EZV by SB01;
      F2: x0 + y0 in EMbedding(V) by F1,SB02;
      F3: T.x0 = T0.x by FUNCT_1:49;
      F4: T.y0 = T0.y by FUNCT_1:49;
      thus T0.(x+y) = T0.(x0+y0) by SB01
      .= T.(x0+y0) by FUNCT_1:49,F2
      .= r*(x0+y0) by P1
      .= r*x0 + r*y0 by VECTSP_1:def 14
      .= T.x0 + r*y0 by P1
      .= T.x0 + T.y0 by P1
      .= T0.x + T0.y by rSB01,F3,F4;
    end;
    for x being Element of EMbedding(V), i being Element of INT.Ring
    holds T0.(i*x) = i*(T0.x)
    proof
      let x be Element of EMbedding(V), i be Element of INT.Ring;
      F1: x in EMbedding(V);
      reconsider x0 = x as Vector of EZV by SB01;
      reconsider j = i as Element of F_Rat by NUMBERS:14;
      F2: j*x0 in EMbedding(V) by F1,SB02;
      F3: T.x0 = T0.x by FUNCT_1:49;
      thus T0.(i*x) = T0.(j*x0) by SB01
      .= T.(j*x0) by FUNCT_1:49,F2
      .= r*(j*x0) by P1
      .= (r*j)*x0 by VECTSP_1:def 16
      .= j*(r*x0) by VECTSP_1:def 16
      .= i*T0.x by rSB01,F3,P1;
    end;
    then T0 is additive homogeneous by B0;
    then reconsider T0 as linear-transformation of EMbedding(V),EMbedding(r,V);
    take T0;
    thus
    XX1: for v being Element of Z_MQ_VectSp(V) st v in EMbedding(V)
    holds T0.v = r*v
    proof
      let x be Element of Z_MQ_VectSp(V);
      assume F1: x in EMbedding(V);
      thus T0.x = T.x by FUNCT_1:49,F1
      .= r*x by P1;
    end;
    for x1, x2 being object st
    x1 in the carrier of EMbedding(V)
    & x2 in the carrier of EMbedding(V)
    & T0.x1 = T0.x2 holds x1 = x2
    proof
      let x1, x2 be object;
      assume AS2: x1 in the carrier of EMbedding(V)
      & x2 in the carrier of EMbedding(V) & T0.x1 = T0.x2;
      then reconsider xx1 = x1, xx2 = x2 as Element of EZV by D0;
      Q0: xx1 in EMbedding(V) & xx2 in EMbedding(V) by AS2;
      Q1: T0.x1 = r*xx1 by Q0,XX1;
      Q2: r"*(r*xx1) = r"*(r*xx2) by AS2,Q0,Q1,XX1;
      r"*(r*xx1) = xx1 by AS,VECTSP_1:20;
      hence x1 = x2 by Q2,AS,VECTSP_1:20;
    end; then
T1: T0 is one-to-one by FUNCT_2:19;
    T0 is onto by RX0,FUNCT_2:def 3,TARSKI:2;
    hence thesis by T1;
  end;
