
theorem
  for V being torsion-free Z_Module,
  r being Element of F_Rat holds
  (for v, w being Vector of Z_MQ_VectSp(V)
  st v in EMbedding(r,V) & w in EMbedding(r,V)
  holds v+w in EMbedding(r,V) ) &
  for j being Element of F_Rat,
  v being Vector of Z_MQ_VectSp(V)
  st j in INT & v in EMbedding(r,V)
  holds j*v in EMbedding(r,V)
  proof
    let V be torsion-free Z_Module,
    r be Element of F_Rat;
    set EZV = Z_MQ_VectSp(V);
    set ZS = EMbedding(r,V);
    set Cl = the carrier of ZS;
    set Add =(addCoset(V)) || Cl;
    set Mlt = (lmultCoset(V)) | [:INT,r*(rng MorphsZQ(V)):];
    thus for v, w being Vector of Z_MQ_VectSp(V)
    st v in EMbedding(r,V) & w in EMbedding(r,V)
    holds v+w in EMbedding(r,V)
    proof
      let v, w be Vector of Z_MQ_VectSp(V);
      assume B1: v in EMbedding(r,V) & w in EMbedding(r,V);
      reconsider v1 = v, w1 = w as Vector of EMbedding(r,V) by B1;
      v+w = v1+w1 by rSB01;
      hence thesis;
    end;
    thus for j being Element of F_Rat, v being Vector of Z_MQ_VectSp(V)
    st j in INT & v in EMbedding(r,V)
    holds j*v in  EMbedding(r,V)
    proof
      let j be Element of F_Rat,
      v be Vector of Z_MQ_VectSp(V);
      assume B1: j in INT & v in EMbedding(r,V);
      then reconsider v1 = v as Vector of EMbedding(r,V);
      reconsider i = j as Element of INT.Ring by B1;
      j*v= i*v1 by rSB01;
      hence thesis;
    end;
  end;
