
theorem SB01:
  for V being torsion-free Z_Module holds
  (for x being Vector of EMbedding(V) holds x is Vector of Z_MQ_VectSp(V) ) &
  0.EMbedding(V) = 0.Z_MQ_VectSp(V) &
  (for x, y being Vector of EMbedding(V), v, w being Vector of Z_MQ_VectSp(V)
  st x = v & y = w holds x+y = v+w ) &
  for i being Element of INT.Ring, j being Element of F_Rat,
  x being Vector of EMbedding(V), v being Vector of Z_MQ_VectSp(V)
  st i = j & x = v holds i*x = j*v
  proof
    let V be torsion-free Z_Module;
    set EZV = Z_MQ_VectSp(V);
    D1: EZV = ModuleStr(# Class EQRZM(V), addCoset(V), zeroCoset(V),
    lmultCoset(V) #) by ZMODUL04:def 5;
    set ZS = EMbedding(V);
    AS1: the carrier of ZS = rng MorphsZQ(V) &
    the ZeroF of ZS = zeroCoset(V) &
    the addF of ZS = (addCoset(V)) || ( rng MorphsZQ(V) ) &
    the lmult of ZS = (lmultCoset(V)) |
    [:the carrier of INT.Ring,rng MorphsZQ(V):] by defEmbedding;
    set Cl = the carrier of ZS;
    set Add = (addCoset(V)) ||  Cl;
    set Mlt = (lmultCoset(V)) |
    [:the carrier of INT.Ring,rng MorphsZQ(V):];
    Cl c= the carrier of EZV by AS1;
    hence for x being Vector of ZS holds x is Vector of EZV;
    thus 0.ZS = 0.EZV by D1,defEmbedding;
    thus for x, y being Vector of ZS, v, w being Vector of EZV
    st x = v & y = w holds x+y = v+w
    proof
      let x, y be Vector of ZS,
      v, w be Vector of EZV;
      assume L0: x = v & y = w;
      thus x+y = (addCoset(V)).[x,y] by AS1,FUNCT_1:49
      .= v+ w by D1,L0;
    end;
    thus for i being Element of INT.Ring, j being Element of F_Rat,
    x being Vector of ZS, v being Vector of EZV
    st i = j & x = v holds i*x = j*v
    proof
      let i be Element of INT.Ring,
      j be Element of F_Rat,
      x be Vector of ZS,
      v be Vector of EZV;
      assume L0: i = j & x = v;
      thus i*x = (lmultCoset(V)).[i,x] by AS1,FUNCT_1:49
      .= j*v by D1,L0;
    end;
  end;
