reserve V for Z_Module;
reserve W, W1, W2 for Submodule of V;

theorem
  for p being prime Element of INT.Ring, V being free Z_Module
  st Z_MQ_VectSp(V,p) is finite-dimensional
  holds V is finite-rank
  proof
    let p be prime Element of INT.Ring, V be free Z_Module such that
    A1: Z_MQ_VectSp(V,p) is finite-dimensional;
    set I = the Basis of V;
    set IQ = {ZMtoMQV(V,p,u) where u is Vector of V : u in I};
    now let x be object;
      assume x in IQ;
      then consider v be Vector of V such that
  B1: x = ZMtoMQV(V,p,v) & v in I;
      thus x in the carrier of Z_MQ_VectSp(V,p) by B1;
    end;
    then reconsider IQ as Subset of Z_MQ_VectSp(V,p) by TARSKI:def 3;
A3: IQ is Basis of Z_MQ_VectSp(V,p) by ZMODUL03:35;
A2: card IQ = card I by ZMODUL03:26;
    IQ is finite by A1,A3,VECTSP_9:20;
    then I is finite by A2;
    hence thesis by ZMODUL03:def 3;
  end;
