 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem
  for p be prime Element of INT.Ring, V be finite-rank free Z_Module holds
  rank V = dim Z_MQ_VectSp(V,p)
  proof
    let p be prime Element of INT.Ring, V be finite-rank free Z_Module;
    set W = Z_MQ_VectSp(V,p);
    set A = the Basis of V;
    set AQ = {ZMtoMQV(V, p, u) where u is Vector of V : u in A};
    now let x be object;
      assume x in AQ;
      then ex u be Vector of V st x = ZMtoMQV(V, p, u) & u in A;
      hence x in the carrier of Z_MQ_VectSp(V, p);
    end;
    then reconsider AQ as Subset of W by TARSKI:def 3;
    A1: card A = card AQ by Th26;
    AQ is Basis of W by Th35;
    then dim W = card AQ by VECTSP_9:def 1;
    hence thesis by A1,Def5;
  end;
