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

theorem
  for V be finite-rank free Z_Module holds
  rank V = dim Z_MQ_VectSp(V)
  proof
    let V be finite-rank free Z_Module;
    reconsider I = the Basis of V as Subset of V;
    reconsider IQ = (MorphsZQ(V)).:(I) as Subset of Z_MQ_VectSp(V);
    A1: IQ is Basis of Z_MQ_VectSp(V) by ThEQRZMV3D;
    thus rank V = card I by ZMODUL03:def 5
    .= card IQ by ThEQRZMV3E
    .= dim Z_MQ_VectSp(V) by A1,VECTSP_9:def 1;
  end;
