
theorem ThDivisibleX1:
  for V being torsion-free Z_Module holds
  (for v being Vector of Z_MQ_VectSp(V) holds v is Vector of DivisibleMod(V)) &
  (for v being Vector of DivisibleMod(V) holds v is Vector of Z_MQ_VectSp(V)) &
  0.(DivisibleMod(V)) = 0.(Z_MQ_VectSp(V))
  proof
    let V be torsion-free Z_Module;
    A1: Z_MQ_VectSp(V) = ModuleStr(# Class EQRZM(V), addCoset(V),
    zeroCoset(V), lmultCoset(V) #) by ZMODUL04:def 5;
    thus for v being Vector of Z_MQ_VectSp(V) holds
    v is Vector of DivisibleMod(V) by A1,defDivisibleMod;
    thus for v being Vector of DivisibleMod(V) holds
    v is Vector of Z_MQ_VectSp(V) by A1,defDivisibleMod;
    thus 0.(DivisibleMod(V)) = 0.(Z_MQ_VectSp(V)) by A1,defDivisibleMod;
  end;
